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https://centar.net/tools/example.html | ### 1. Algorithm Derivation
Here LU decomposition is used as an example to show in detail how arrays can be derived directly from mathematical expressions. LU decomposition also serves as a useful benchmark in comparing previous tool and methodology efforts because many of these use it as an example. Here it will be shown how little user intervention is required in the design process and how easy it is for a circuit designer to easily explore a variety of array implementation tradeoffs.
Given a linear system Ax=b, where A is a known NxN matrix , b is a known vector, and x is an unknown vector, a common solution is to decompose A into a product of two matrices, one lower triangular in form (L) and the other upper triangular (U), so that LUx=b. This allows the linear system to be decomposed into two easier to solve systems,
$\small&space;\begin{matrix}&space;Ly=b\\&space;Ux=y&space;\end{matrix}$
where another another unknown vector y has been introduced. Each equation above can then be solved by the process of “back substitution”. For example, Ly=b can be written in the matrix form
$\small&space;\begin{bmatrix}&space;l_{11}&space;&&space;0&&space;0\\&space;l_{21}&space;&&space;l_{22}&&space;0\\&space;l_{31}&&space;l_{32}&space;&&space;l_{33}&space;\end{bmatrix}&space;\begin{bmatrix}&space;y_{1}\\&space;y_{2}\\&space;y_{3}&space;\end{bmatrix}&space;=&space;\begin{bmatrix}&space;b_{1}\\&space;b_{2}\\&space;b_{3}&space;\end{bmatrix}$
whereupon it can be seen that three recursive expressions in y allow the solution
$\small&space;\begin{bmatrix}&space;y_{1}=b_{1}/l_{11}\\&space;y_{2}=b_{2}-l_{21}y_{1}\\&space;y_{3}=b_{3}-l_{31}y_{1}-l_{32}y_{2}&space;\end{bmatrix}$
or in general for L
$\small&space;y_{i}=b_{i}-\sum_{k=1}^{i-1}l_{ik}y_{k}$
A similar expression provides the solution x now that y is known. Thus, using this approach, all that is required to solve the linear system is to decompose A into the product LU. Such a decomposition can be done in a fashion similar to that above, which can be seen directly from the form
$\small&space;\small&space;\begin{bmatrix}&space;l_{11}&space;&&space;0&&space;0\\&space;l_{21}&space;&&space;l_{22}&&space;0\\&space;l_{31}&&space;l_{32}&space;&&space;l_{33}&space;\end{bmatrix}&space;\begin{bmatrix}&space;1&space;&&space;u_{12}&&space;u_{13}\\&space;0&space;&&space;0&&space;u_{23}\\&space;0&space;&&space;0&space;&&space;1&space;\end{bmatrix}&space;=&space;\begin{bmatrix}&space;a_{11}&space;&&space;a_{12}&&space;a_{13}\\&space;a_{21}&space;&&space;a_{22}&space;&&space;a_{23}\\&space;a_{31}&space;&&space;a_{32}&space;&&space;a_{33}&space;\end{bmatrix}$
Hence, it can be determined by induction that
$\small&space;\begin{matrix}&space;l_{11}=a_{11};\;&space;l_{21}=a_{21};\;&space;l_{31}=a_{31};\\&space;u_{12}=a_{12}/l_{11};\;&space;u_{13}=a_{13}/l_{11};\\&space;and\:&space;for\:\:&space;i\geq&space;j,\:&space;j>1\:&space;i\leq&space;3;\\&space;l_{ij}=a_{ij}-\sum_{k=1}^{j-1}l_{i,k}u_{k,j}\\&space;and\:&space;for\:\:&space;j>&space;i,\:&space;i>1\:&space;j\leq&space;3;\\&space;u_{ij}=\left&space;(&space;a_{ij}-\sum_{k=1}^{j-1}l_{i,k}u_{k,j}&space;\right&space;)/l_{ij}&space;\end{matrix}$
From the mathematical expression in above it is possible to go directly to the code form:
for i to N do
for j to N do
if j=1 and i>=1 and i<=N then
l[i,j]:=a[i,j];
elif i=1 and j>1 and j<=N the
u[i,j]:=a[i,j]/l[i,i];
fi;
if i>=j and j>1 and i<=N then
fi;
if j>i and i>1 and j<=N then
fi;
od
od;
Here, the only semantic change has been to use the Maple “add” construct in the place of the mathematical summation sign. Also, N replaces “3″ and is a measure of the “problem size”. In this example it is possible to run this code directly in Maple to verify it’s functionality.
This code is saved as a text file and read into SPADE as a text file at the beginning of processing by the parser. All variables are required to be indexed and of dimension less than three. There is nothing in the space-time mapping methodology that imposes this limitation; rather it was a practical choice based on the observation that this covers most signal processing algorithms of choice.
The FFT design proceeds in the same way with inputs to SPADE derived from first principles. In general there are no steps in the process involving detailed issues related to parallel algorithms or relating to architectural implementation issues. Many other signal processing examples, including the FFT, are provided in the reference links.
### 2. LU Decomposition Design Results
A critical tool function is to provide the circuit designer with a complete range of architectural options so that tradeoffs can be adequately analyzed. This is necessary because available FPGA/ASIC chips, boards and virtual computers are built with different architectural features and any system implementation based on such technologies will have its own unique set of constraints. The examples below show how such considerations can result in very different array (abstract circuit) designs.
### 2.1 Input Constrained to Array Boundary
In this first mapping example the architectural constraints were set to require all input to take place from the systolic array boundary and look for the minimum time latency solution that has the least PE array area. (In this case variable a is the input.) It is a common characteristic of systolic arrays, which are physically tied to sensors to have sensor data stream into the array from one or more array edge boundaries, and also consistent with FPGA based virtual computers that contain a lot of buffer memory at the board inputs.
With the LU decomposition input code above the parser analysis generates 55 search variables that are uniquely associated with the scheduling operation. However, the causality constraints and other high level architectural constraints reduce the number of these search variables to 19. There are also 6 unimodular matrices associated with the reindexing step that in part specifies S, and these matrices must be chosen from a set, typically less than 50. When all high level constraints applied, the search space of possible mappings is reduced to a range that can be explored in a reasonable length of time.
Figure 1. The first of two mappings for LU decomposition with input variable a constrained to appear on the array boundary (N=6). (Variable l is orange, u is grey and a is red.)
The search discovered 303 solutions that satisfied causality. These can be grouped into just 2 time latency categories proportional to 3N and 4N. However, when additive constants are considered (e.g., 4N-3) there were 8 distinct latency values for all solutions. Constraints associated with the boundary I/O and reindexing reduced the total number of feasible solutions to 14 and finally after all constraints were considered there were only six unique mappings with the minimum time latency of 3N-3. These array designs were all triangular in shape. Four of these had less desirable “diagonal” interconnection patterns between PEs. Going through the search process again with the secondary constraint set to find the most regular solutions, yielded just two designs that have the more desired “rectangular” interconnect structure. These two designs are shown in Figures 1 and 2 for the case where N=6.
Here, as with later mappings, the x and y axes represent the spatial coordinates of a 2-D mesh grid with the array of PEs embedded in this grid at intersection points. The grid above is NxN in size and the embedded systolic array is triangular with N(N+1)/2 PEs. The different shadings correspond to regions associated with the different algorithm variables a, l and u and are labeled as such.
Figure 2. The second of two mappings for LU decomposition with input variable a constrained to appear on the array boundary (N=6).
The PE arrays in Figure 1 and 2 are uniform in terms of the interconnection pattern and there are six different flows of data associated with the various variable dependencies, each of which moves along an orthogonal path defined by the arrows in Figure 1 and 2. Thus, some PEs experience six different data streams passing through them. There are three additional dependencies in which data does not move spatially, but rather is updated and reused in the same PE. This corresponds to data “movement” along the time axis in space-time. The picture in Figures 1 and 2 show a superposition of data flow at all times. The actual time variation of data flow is more complex, with the size of uniform sub-regions of PEs growing and shrinking with time. In other words data movement typically begins at one edge of the array and proceeds outwards in a “wavefront-like” fashion.
In space-time all three algorithm variables, a,u and l, are represented by polygons because they are defined with two indices in the algorithm code above. That is, the index points in space-time to which the element values a[i,j], u[i,j] and l[i,j] are mapped lie on a planar surface. When the normal to these surfaces is perpendicular to the time axis, they are projected onto the 2-D mesh array as lines as shown in Figures 1 and 2. In this first mapping example such a constraint was only placed on certain search variables associated with the elements of input variable a. (The boundary alignments of u in Figure 1 and l in Figure 2 occurred only because these lead to the most regular array.)
Figure 3. Position of algorithm variables u, l, and a in space-time for two different orientations.
Projection of these surfaces along the time axis yields the triangular array of Figure 1.
The outline of this projection is shown here at time=0.
SPADE also provides a space-time view of each algorithm array mapping solution, one of which is shown in Figure 3 and corresponds to the spatial-only view in Figure 1. The space-time view is shown from two different perspectives in order to help in its interpretation (in the SPADE environment this view can be easily manipulated along all axes in real-time).
Figure 3 is more complex because it shows additional variables, IM1 and IM2 (IM1[i,j,k]=l[i,k]*u[k,j]; IM2 has the same dependence on l and u, but a different index domain and both are polytopes in the space-time view and are not seen in the original algorithm code above. These two new variables are created automatically in the parser and are there to keep running sums associated with the summations.
This view imparts a good deal more information than the spatial view. For example it shows where and when array activity associated with the different algorithm variables takes place, it provides a visible view of 3-D data flow between algorithm variables, and it imparts a rough estimate of how efficiently PEs are used by what percent of the total space-time volume is occupied by polytopes and polygons.
In Figure 3, it can perhaps been seen more clearly that the normal to the polygons representing variables a and u in space-time is in the plane of the PE array and therefore their projection on this plane is a line.
The design significance of placing a variable entirely along an edge boundary goes beyond its proximity to the rest of the system, which is external to the array. It means that each PE associated with one of the edge points on the grid must have a memory size that’s O(N), where N specifies the size of the problem. This requirement is different from PEs internal to the array for which the memory requirement is small and independent of the problem size.
Given the considerations above, the circuit designer might wonder if there is an architecture that doesn’t place either l or u at the edge of the array as in Figure 1 and 2. This could be potentially motivated by the desire to have all the output data internal to the array for subsequent processing. Such a solution does not appear amongst those that fall into the minimum area category. But by having SPADE generate some of the more sub-optimal area solutions, the desired result can be found and is shown in Figure 4. Cleary the tradeoff in accepting the array shown in Figure 4 is that it is about twice the size of the previous designs. Note that the “gap” between the l and u regions is a result of the fact that the diagonal elements of u are by definition already set to one.
Figure 4. Array design with variables l, u not placed on array edge boundary having a time latency of 3N-3 (N=6).
### 2.2 Input and Output Constrained to Array Boundary
Naturally, it might be of interest as well to inquire about array designs would occur if l, u and a were all constrained to occur along the boundary. Only two solutions were found as shown in Figure 5. However, the latency of the algorithm with this new constraint has increased from the 3N-3 time steps associated with all previous designs to 4N-4 time steps. In addition the area has increased from N(N+1)/2 to (2N-1)N.
It is likely that a circuit designer would prefer working with the array design on the right in Figure 5 because here the actual PE array is square. (Sometimes it is necessary to view the SPADE space-time output in order to ascertain exactly which grid intersections of the 2-D view correspond to PEs.)
### 2.3 Single Divider Implementation
Although the array designs in Figure 1 and 2 look equivalent there is a significant difference between the two in terms of hardware usage. From the code above it can be seen that the line
if j>i and i>1 ...
implies that at each point i>1,j>1, a divider is necessary to compute this statement. Consequently, in Figure 2 a triangular array of dividers corresponds to the internal shaded region labeled u, whereas in Figure 1 only a linear array associated with the linear projection of u is required for divisions.
Figure 5. Two optimal arrays resulting from constraints that all input and output occur on array boundaries (N=6).
Since it is known that systolic arrays that do LU decomposition are possible that use only one divider it is useful to ask how SPADE would achieve such a result. This is best done by introduction of a new variable. This is necessary because it is clear that if a variable positioned in space-time is to project onto the spatial plane at a single point (corresponding to the divider), this variable can’t be represented by a polygon, but must be represented by a line. Hence it must be specified by a single index. Also, it is clear in the code segment
if j>i and i>1 and j<=N then
fi;
that although there are O(N2) divides associated with this statement, only N values of l[i,i] are used and this could be specified by a single index. Therefore it follows that, the number of divisions can be reduced by changing the code segment above to read
if j>i and i>1 and i<=N and j<=N then
fi;
if i>=1 and i<=N then l_inv[i]:=1/l[i,i] fi;
and replacing instances of division by l to multiplication by l_inv elsewhere. If this change is made along with a boundary constraint on l_inv, we get the single solution shown in Figure 6 with an area of N2 (minimum area secondary objective function) and time latency of 3N-3. The corresponding space-time view is shown in Figure 7 where only one PE contains a divider (upper right corner Figure 6.)
The tradeoff above is that the single divider constraint increases the array size by about a factor of two compared to the array design in Figure 1. In addition, the placement of variables l, u and a in this design might or might not be considered favorable. Also there’s data movement in nine vs the six directions for the array in Figure 1. But, while the creation of a new variable often increases the overhead in terms of data movement and computational requirements, this doesn’t happen here because the variable elements l_inv[i] remain in the same PE (no data movement) and the number of total divisions has not increased compared to previous examples.
Figure 6. Optimal array implementation for single divider (upper right hand corner) (N=6).
Once an array design is selected, the actual transformation matrices are saved and available as inputs to other components of the environment; for the particular design shown in Figures 6 and 7 these are summarized in Table 1 and the dependency information (data flow vectors) is provided in Table 2.
Figure 7. Space-time view of single divider array design corresponding to spatial array shown in Figure 6. Variable l_inv is the lightly shaded (yellow) vertical line at x=0, y=0.
Table 1. Transformations for array design of Figures 6 and 7.
Table 2. List of direction vectors ([time,x,y]) of data flow for the Figures 6 and 7. | 2020-06-02 01:36:07 | {"extraction_info": {"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": 6, "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, "math_score": 0.43619394302368164, "perplexity": 779.2839967237626}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347422065.56/warc/CC-MAIN-20200602002343-20200602032343-00554.warc.gz"} |
https://iq.opengenus.org/array-vs-vector-cpp/ | # Arrays vs Vectors in C++
#### Software Engineering C++
Get FREE domain for 1st year and build your brand new site
The differences between array and vectors in C++ are as follows:
• Array can be static or dynamic; Vector is dynamic
• Array can be traversed using indexes, vector uses iterators
• No reallocation in array
• Size of Array is fixed; Size of vector can be changed
• Vector can be copied using assignment statement
• Vector size is not required when we pass a vector to a function
• Vector can be returned from function; Array cannot be returned
• Arrays are deallocated explicitly; Vectors are deallocated automatically
We have explored the points in depth with C++ code examples.
1. An array can be defined as a collection of elements of same data type. Arrays can be implemented statically or dynamically. For static implementation, while initialising the array one needs to specify the number of elements at an earlier stage, but for dynamic implementation, one can use a pointer of the primitive data type and point to the first element and then ask for the contiguous blocks required. New operator is used for dynamic memory allocation. Whereas vector is a template class implemented as a dynamic array. Here, the vector is a sequential container which uses list interface, as one can easily add and remove elements using push_back() and pop_back() repectively.
Example: The below code shows how arrays can be implemented statically and dynamically. It also shows vector implementation.
#include <bits/stdc++.h>
using namespace std;
int main(){
vector<int> myvec; // Vector
int static_arr[50]; // Static Array
int* dy_arr = new int[50]; // Dynamic Array
return 0;
}
1. Elements of an array can be traversed and accessed using numerical indices. A specific element can be accessed using it's index position in the array. In vectors, iterators are used to access any element or just to traverse all the elements present in the vector.
Example:The below code shows how the elements of an array are accessed.
#include<bits/stdc++.h>
using namespace std;
int main () {
// nest is an array containing 6 elements
int nest[]={ 2, 4, 6, 8, 10, 12};
cout << "Element \t Value" << endl;
// accessing each element of the array
for ( int j = 0; j < 6 ; j++ ) {
cout << " " << j+1 <<" \t\t " << nest[ j ] << endl;
}
return 0;
}
Output:
Example:The below code shows how the elements of a vector are accessed.
#include<bits/stdc++.h>
using namespace std;
int main()
{
vector<int> ar ;
ar.push_back(2);
ar.push_back(4);
ar.push_back(6);
ar.push_back(8);
// Declaring iterator to a vector
vector<int>::iterator ptr;
// Displaying vector elements
cout << "The vector elements are : ";
for (ptr = ar.begin(); ptr < ar.end(); ptr++)
cout << *ptr << " ";
return 0;
}
Output:
1. No new elements can be inserted in the array if the array becomes full because no reallocation is done implicitly in array whereas in the case of vectors, memory is reallocated implicitly and new elements are inserted in the vector.
2. Size of an array is fixed whereas size of a vector is modifiable. It means vector can be resized according to the requirement as they are allocated on heap memory.
Example:The below code shows how the vector resizes itself after a few operations are performed on it.
#include <bits/stdc++.h>
using namespace std;
int main(){
vector<int> myVec;
// Inserting Values in Vector
myVec.push_back(1);
myVec.push_back(2);
myVec.push_back(3);
myVec.push_back(4);
myVec.push_back(5);
cout << "Size of vector Before Removal : " << myVec.size() << endl;
// Removing two elements from the vector
myVec.pop_back();
myVec.pop_back();
cout << "Size of vector After Removal : " << myVec.size() << endl;
return 0;
}
Output:
1. In order to create a duplicate array of the original one, we have to follow the iterative method i.e running a loop to copy each element at respective index. Whereas, vectors can be copied or assigned directly using the assignment operator.
Example:The below code shows how vectors can be assigned directly using the assignment operator.
#include <bits/stdc++.h>
using namespace std;
int main()
{
vector<int> vec;
vector<int> dup_vec;
vec.push_back(10);
vec.push_back(100);
vec.push_back(1000);
vec.push_back(10000);
cout << "Contents of vector (vec):\n";
for( int i=0; i<vec.size(); i++){
cout<<vec[i]<<" ";
}
dup_vec = vec; // Copying vec into dup_vec
cout << "\n\nContents of vector (dup_vec):\n";
for( int i=0; i<dup_vec.size(); i++){
cout<<dup_vec[i]<<" ";
}
return 0;
}
Output:
1. Vector size is not required when we pass a vector to a function. This is because vector maintains variables which keeps track of size of container at all times. But when arrays are passed to a function, a separate parameter for size is also passed to the function.
2. C++ does not allows to return the address of a local variable to outside of the function so you would have to define the local variable as static variable. Whereas vectors can be returned from a function.
Example: The code below shows vectors can be returned from a function.
#include <bits/stdc++.h>
using namespace std;
// Function returning vector
vector<int> demo_function() {
vector<int> ans;
int i=5;
while(i>0){
//inserting values in vector
ans.push_back(i);
i--;
}
return ans;
}
int main() {
vector<int> result;
result = demo_function(); // Call function
// Output Values of vector
cout<<"Vector consists of elements : \n";
for (int i=0; i<result.size();i++)
cout << result[i] << " ";
return 0;
}
Output:
1. We can always get the size of the vector in O(1) time using the size() function. But in case of dynamic arrays, the size of array cannot be determined.
2. Dynamic arrays need to be deallocated explicitly whereas vectors are automatically de-allocated from heap memory as soon as variable goes out of scope.
Example:
#include <bits/stdc++.h>
using namespace std;
int main()
{
int* arr_demo = new int[10]; // Dynamic Array
vector<int> v; /* Vector is deallocated implicitly
when the program terminates */
delete[] arr_demo; /* Array needs to be deallocated
explicitly before program terminates */
return 0;
}
## Question
#### WHICH ONE IS MORE MEMORY EFFICIENT, ARRAYS OR VECTORS?
Arrays
Vectors
Vector occupies much more memory in exchange for the ability to manage storage and grow dynamically whereas Arrays are memory efficient data structure.
With this article at OpenGenus, you must have gained a good idea of Arrays V/s vectors in C++. | 2021-04-22 22:49:43 | {"extraction_info": {"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, "math_score": 0.18076394498348236, "perplexity": 3539.751879270749}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618039563095.86/warc/CC-MAIN-20210422221531-20210423011531-00485.warc.gz"} |
https://www.ias.ac.in/listing/bibliography/pram/S._K._Ghosh | • S K Ghosh
Articles written in Pramana – Journal of Physics
• Three-dimensional wake potential in a streaming dusty plasma
The oscillatory wake potential for a slowly moving or static test dust particulate in a finite temperature, collisionless and unmagnetized dusty plasma with a continuous flow of ions and dust particles has been studied. The collective resonant interaction of the moving test particle with the low-frequency and low-phase-velocity dust-acoustic mode is the origin of the periodic attractive force between the like polarity particulates along and perpendicular to the streaming ions and dust grains resulting into dust-Coulomb crystal formation. This wake potential can explain the three-dimensional dust-Coulomb crystal formation in the laboratory conditions.
• Scattering length density profile of Ni film under controlled corrosion: A study in neutron reflectometry
We report the density depth profile of an as-deposited Ni film and density profile for the same film after controlled electrochemical corrosion by chloride ions, measured by unpolarized neutron reflectometry. The neutron reflectometry measurement of the film after corrosion shows density degradation along the thickness of the film. The density profile as a function of depth, maps the growth of pitting and void networks due to corrosion. The profile after corrosion shows an interesting peaking nature.
• Photophysical and laser characteristics of pyrromethene 567 dye: Experimental and theoretical studies
Narrow-band laser performance of alcohol solutions of pyrromethene 567 (PM567) and rhodamine 6G (RH6G) dye was investigated using a home-made GIG- configured dye laser, excited by the second-harmonic radiation (at 532 nm) of a pulsed Nd:YAG laser. Higher laser efficiency was observed with PM567 dye ($\sim 23%$ peak) in comparison to the commonly used RH6G dye (16.5%), in spite of much lower fluorescence quantum efficiency of the PM567 (0.83) vis-à-vis RH6G (0.98) dye solutions in ethanol. First principle-based electronic structure calculations were performed on PM567 dye in the ground ($S_{0}$) and excited states ($S_{1}$) using density functional theory to elucidate the structure and photophysical properties of the dye.
• # Pramana – Journal of Physics
Volume 96, 2022
All articles
Continuous Article Publishing mode
• # Editorial Note on Continuous Article Publication
Posted on July 25, 2019 | 2022-08-12 11:34:06 | {"extraction_info": {"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, "math_score": 0.35165002942085266, "perplexity": 6088.497060421151}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882571692.3/warc/CC-MAIN-20220812105810-20220812135810-00724.warc.gz"} |
http://math.stackexchange.com/questions/254109/is-the-coefficient-of-linear-combination-always-non-zero | # Is the coefficient of linear combination always non-zero?
Suppose that there is some vector. And the vector can be decomposed into linear combination of linearly independent vectors. If we set one of coefficients of these vectors to be zero, and we get the vector, would there be another linear combination with all coefficients non-zero that describes the vector?
Also, are all linear combinations usually set to have non-zero coefficients?
- | 2014-12-22 12:24:55 | {"extraction_info": {"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, "math_score": 0.885057806968689, "perplexity": 256.5191394514552}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-52/segments/1418802775222.147/warc/CC-MAIN-20141217075255-00093-ip-10-231-17-201.ec2.internal.warc.gz"} |
http://mathoverflow.net/revisions/45577/list | 2 edited title; added 174 characters in body
# CocompleteAnycocomplete category (s.t....)withadensesmallfullsubcategory is complete?
Hello,
I with to consider the following statement:
If $C$ is a cocomplete category having a dense small full subcategory $D$, then $C$ is complete.
(a full subcategory $D$ is dense in $C$ if every element of $C$ is canonical colimit of elements of $D$...)
I think I know how to prove it (I give proof below), and I want someone to reassure me that this statement is true exactly as stated, as it seems a little bit surprising.
Proof sketch:
Thank you, Sasha | 2013-05-18 12:13:58 | {"extraction_info": {"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, "math_score": 0.8866720199584961, "perplexity": 262.34579336949434}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368696382396/warc/CC-MAIN-20130516092622-00013-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://www.whsmith.co.uk/products/a-formalization-of-set-theory-without-variables-colloquium-publications-v-41/9780821810415 | # A Formalization of Set Theory without Variables (Colloquium Publications v. 41)
By: Alfred Tarski (author), Steven R. Givant (author)Paperback
1 - 2 weeks availability
£96.95
### Description
Completed in 1983, this work culminates nearly half a century of the late Alfred Tarski's foundational studies in logic, mathematics, and the philosophy of science. Written in collaboration with Steven Givant, the book appeals to a very broad audience, and requires only a familiarity with first-order logic. It is of great interest to logicians and mathematicians interested in the foundations of mathematics, but also to philosophers interested in logic, semantics, algebraic logic, or the methodology of the deductive sciences, and to computer scientists interested in developing very simple computer languages rich enough for mathematical and scientific applications. The authors show that set theory and number theory can be developed within the framework of a new, different, and simple equational formalism, closely related to the formalism of the theory of relation algebras. There are no variables, quantifiers, or sentential connectives.Predicates are constructed from two atomic binary predicates (which denote the relations of identity and set-theoretic membership) by repeated applications of four operators that are analogues of the well-known operations of relative product, conversion, Boolean addition, and complementation. All mathematical statements are expressed as equations between predicates. There are ten logical axiom schemata and just one rule of inference: the one of replacing equals by equals, familiar from high school algebra. Though such a simple formalism may appear limited in its powers of expression and proof, this book proves quite the opposite. The authors show that it provides a framework for the formalization of practically all known systems of set theory, and hence for the development of all classical mathematics.This book contains numerous applications of the main results to diverse areas of foundational research: propositional logic; semantics; first-order logics with finitely many variables; definability and axiomatizability questions in set theory, Peano arithmetic, and real number theory; representation and decision problems in the theory of relation algebras; and, decision problems in equational logic.
### Contents
The formalism $\mathcal L$of predicate logic The formalism $\mathcal L^+$, a definitional extension of $\mathcal L$ The formalism $\mathcal L^+$ without variables and the problem of its equipollence with $\mathcal L$ The relative equipollence of $\mathcal L$ and $\mathcal L^+$, and the formalization of set theory in $\mathcal L^\times$ Some improvements of the equipollence results Implications of the main results for semantic and axiomatic foundations of set theory Extension of results to arbitrary formalisms of predicate logic, and applications to the formalization of the arithmetics of natural and real numbers Applications to relation algebras and to varieties of algebras Bibliography Indices.
### Product Details
• publication date: 15/12/1987
• ISBN13: 9780821810415
• Format: Paperback
• Number Of Pages: 318
• ID: 9780821810415
• weight: 794
• ISBN10: 0821810413
### Delivery Information
• Saver Delivery: Yes
• 1st Class Delivery: Yes
• Courier Delivery: Yes
• Store Delivery: Yes
### Mathematics and SciencesView More
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Close | 2017-02-22 22:48:59 | {"extraction_info": {"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, "math_score": 0.6303356885910034, "perplexity": 1506.3101965562903}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-09/segments/1487501171053.19/warc/CC-MAIN-20170219104611-00277-ip-10-171-10-108.ec2.internal.warc.gz"} |
https://zbmath.org/?q=an:0176.42301&format=complete | ## Fourier $$L_ 2$$-transform of distributions.(English)Zbl 0176.42301
### MSC:
44A10 Laplace transform 46F12 Integral transforms in distribution spaces
### Keywords:
integral equations, integral transforms
Full Text:
### References:
[1] L. Schwartz: Théorie des distributions. vols I, II, Hermann, Paris 1950 and 1951. · Zbl 0037.07301 [2] S. Bochner K. Chandrasekharan: Fourier Transforms. Princeton 1949. · Zbl 0065.34101
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching. | 2022-11-30 01:07:28 | {"extraction_info": {"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, "math_score": 0.5660878419876099, "perplexity": 3696.6176805206082}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710712.51/warc/CC-MAIN-20221129232448-20221130022448-00268.warc.gz"} |
https://solvedlib.com/n/incorrectquestion-10-4-ptslet-d-be-the-following-arbitrary,3448769 | IncorrectQuestion 10 /4 ptsLet D be the following arbitrary region in the cy-plane:Suppose the area of D is 6, then
Question:
Incorrect Question 1 0 /4 pts Let D be the following arbitrary region in the cy-plane: Suppose the area of D is 6, then Js 5dA rounded to 2 decimal places): (enter an integer or a decimal
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How do you simplify (t^2-25)/(t^2+t-20)?... | 2023-03-24 21:19:30 | {"extraction_info": {"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": 2, "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, "math_score": 0.5732893347740173, "perplexity": 6137.273952477319}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296945289.9/warc/CC-MAIN-20230324211121-20230325001121-00523.warc.gz"} |
http://picsdownloadz.com/puzzles/maths-puzzles/can-you-solve-bee-ant-logic-puzzle-with-answers-only-for-genius/ | # Can you solve? ‘Bee & Ant’ Logic Puzzle with Answers- Only For Genius!
Hello Geniuses, Here an interesting and simple math Puzzle for you to solve? Let’s see if can solve it or not?
## Bee and Ant Puzzle – Only For Geniuses
in this bee & ant Puzzle picture, there is Addition and Subtraction is given. But the multiplication is not given. You have to find the Multiplication of Bee & Ant.
The Answer and solution is given below, if you could’t find the solution.
# Bee & Ant Puzzle!Can you solve it?
if you Solve this, Let’s see in the comments…
Share, Comment & Like!!!
Answer
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Facebook Comments | 2017-03-25 23:45:10 | {"extraction_info": {"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, "math_score": 0.8106646537780762, "perplexity": 7311.1658568570465}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-13/segments/1490218189088.29/warc/CC-MAIN-20170322212949-00489-ip-10-233-31-227.ec2.internal.warc.gz"} |
http://jeffrouder.blogspot.com/2015/05/ | Sunday, May 31, 2015
Simulating Bayes Factors and p-Values
I see people critiquing Bayes factors based on simulations these days, and example include recent blog posts by Uri Simonsohn and Dr.-R. These authors assume some truth, say that the true effect size is .4, and then simulate the distribution of Bayes factors are like across many replicate samples. The resulting claim is that Bayes factors are biased, and don't control long run error rates. I think the use of such simulations is not helpful. With tongue-in-cheek, I consider them frequocentrist. Yeah, I just made up that word. Let's pronounce it as "freak-quo-centrists. It refers to using frequentist criteria and standards to evaluate Bayesian arguments.
To show that frequocentric arguments are lacking, I am going to do the reverse here. I am going to evaluate p-values with a Bayescentric simulation.
I created a set of 40,000 replicate experiments of 10 observations each. Half of these sets were from the null model; half were from an alternative model with a true effect size of .4. Let's suppose you picked one of these 40,000 and asked if it were from the null model or from the effect model. If you ignore the observations entirely, then you would rightly think it is a 50-50 proposition. The question is how much do you gain from looking at the data.
Figure 1A shows the histograms of observed effect sizes for each model. The top histogram (salmon) is for the effect model; the bottom, downward going histogram (blue) is for the null model. I drew it downward to reduce clutter.
The arrows highlight the bin between .5 and .6. Suppose we had observed an effect size there. According to the simulation, 2,221 of the 20,000 replicates under the alternative model are in this bin. And 599 of the 20,000 replicates under the null model are in this bin. If we had observed an effect size in this bin, then the proportion of times it comes from the null model is 599/(2,221+599) = .21. So, with this observed effect size, the probability goes from 50-50 to 20-80. Figure 1B shows the proportion of replicates from the null model, and the dark point is for the highlighted bin. As a rule, the proportion of replicates from the null decreases with effect size.
We can see how well p-values match these probabilities. The dark red solid line is the one-tail p-values, and these are miscalibrated. They clearly overstate the evidence against the null and for an effect. Bayes factors, in contrast, get this problem exactly right---it is the problem they are designed to solve. The dashed lines show the probabilities derived from the Bayes factors, and they are spot on. Of course, we didn't need simulations to show this concordance. It falls directly from the law of conditional probability.
Some of you might find this demonstration unhelpful because it misses the point of what a p-value is what it does. I get it. It's exactly how I feel about others' simulations of Bayes factors.
This blog post is based on my recent PBR paper: Optional Stopping: No Problem for Bayesians. It shows that Bayes factors solves the problem they are designed to solve even in the presence of optional stopping.
Monday, May 18, 2015
Ben's Letter: Who Raised This Kid?
UPDATE (5/21): With hurt feelings all around, Ben has been excused for the two days. We are grateful. It is a tough situation because their primary concern is safety, and I get that. Hopefully, the hurt feelings will slowly melt, because the camp folks have done right by us over the years.
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ORIGINAL POST:
"I just wrote camp a letter," said Ben. "Oh God," I thought. "Please, this is a delicate situation," I said to myself. "He is just going to make it worse...."
My 16-year-old son has been going to camp for some six or seven years, and he loves it there. We trusted this camp with our kids and they have delivered year after year. We have a relationship of sorts. He is scheduled to be a first-year counselor, and is excited.
My mother-in-law is 90 and is in failing health. She won't be able to travel to my nephew's wedding or my daughter's Bar Mitzvah in the coming year, assuming she is still alive then.
The conflict is that after much wrangling over dates, my wife's family is honoring my mother-in-law the first weekend of training for Ben's counselor gig. Everyone will be there, and we thought it would be a no-brainer for camp to excuse him for the weekend. He knew everyone; he knew camp; he knew the routines. But they did not. And we are very hurt.
We have been going back and forth with them, expressing our hurt and listening to their reasons, and Ben has been cc'd on the emails.
Dear (I redacted the name, it is not important),
I'm writing this E-mail because I don't think my Mom is going to say this - you are completely missing the point. In your emails you used words such as "Birthday Party" to describe the event that we wish to attend. Not only is this inaccurate, it completely undertones the value of this event. This isn't just some "Birthday Party." It’s meant to be the last time my Grandma will ever be able to see her ENTIRE family alive. It’s about being able to celebrate my Grandma's life while she is still alive, because the scary truth is, if I don't go, the next time I will be in California will probably be her funeral. To our family, this isn't a "Birthday Party," it is like a Bar/Bat Mitzvah, and to others in our family it is even more important and valuable than one. To define this event as a "Birthday Party" is not only severely incorrect, but just reflects a lack of understanding of what this means to my Mom and my Family.
I understand that missing two days is very inconvenient for the camp, but if I do go the plan is to come back on the 14th. If camp starts on the 21st that gives me 7 days to bond with the other counselor's (whom I already know) and to learn the camp rules. I also understand that you have to think about the safety of the campers and I truly do respect that, but to tell me that missing two days of content is going to endanger my campers, and that I won't be able to make up those two days of content, is absurd. If missing two days is truly going to put my campers at risk, then tell us what we’re missing, prove us wrong, because to us it sounds like you're putting bonding time (with people I already know) over seeing my grandma for a possible last time, especially since you said that you would be able to handle it if I missed a few days for a family emergency. If anything I said at all reflects that I don't understand the gravity of missing camp, then tell me exactly how missing camp will put my campers in danger, because I'm trying to understand your situation but I really don't. To us, your situation sounds like an excuse compared to what could be the last time our entire family is united before my grandma dies.
The only point you've made in this argument that I've seen as valid is you mentioning the contract. Yes, I signed the contract saying that I would be there, and yes, technically by not going I would NOT be honoring my contract. But what is more important in life, and dare I say it, in Judaism - honoring a contract or honoring a family? I personally feel like family is a much more important concept in Judaism than honoring a contract, and I feel like that should also be a value a Jewish Camp respects. It shouldn't be camp policy to turn someone down because they want to try to see their grandma one last time with their entire family. My family has already tried to change the date. They tried everything before making me aware of this date, and it just won't work any other day. You mentioned that there would be an exception if there was a family crisis (funeral), but that only further reflected your lack of understanding towards our scenario. In saying this, you send the message that it’s more important to celebrate the life of someone when they are dead, rather than celebrating their life when they are alive. I feel like this contradicts many important values in Judaism. Rules and policies shouldn't restrict the celebration of family, or the values of Judaism.
You could tell me that the fact that I'm putting family over camp is my decision and not the camp's decision, but the point that we are trying to make is that it is camp's policy that is forcing me to have to make a decision, and that is disgraceful. I shouldn't have to choose between my two families, especially if the one forcing the decision is Jewish, but right now I'm being forced to all because in your eyes, two days of bonding is more important than seeing my entire family together one last time. Family is supposed to be an important value in Judaism and should not be a topic you can deescalate by calling our important gathering a mere "Birthday Party". I hope this Email both makes our anger, and disappointment towards your decision clear, but also shows how we view your perspective. If you could help us better understand how your two days of training is more important than seeing a scattered family united one last time before my Grandma dies, then maybe this decision will be easier to make. If the only way you will excuse us is if we have a family emergency, then consider this a family emergency. That is how important this is.
Sincerely,
Ben Rouder
It is a wonderous feeling when your child is more elegant, logical, articulate, and authentic than you could have imagined.
Sunday, May 17, 2015
The Self-Propagated Myth of Bayesian Unity
Substantive psychologists are really uncomfortable with disagreements in the methodological and statistical communities. The reason is clear enough----substantive psychologists by-and-large just want to follow the rules and get on with it.
Although our substantive colleagues would prefer if we had unified and uniform set of rules, we methodologists don't abide. Statistics and methodology are varied fields with important, different points of view that need to be read, understood, and discussed.
Bayesian thought itself is not uniform,. There are critical, deep, and important differences among us, so much so that behind closed doors we have sharp and negative opinions about what others advocate, Yet, at least int he psychological press, we have been fairly tame and reticent to critique each other. We fear our rule-seeking substantive colleagues may use these differences as an excuse to ignore Bayesian methods altogether. That would be a shame.
In what follows, I give the briefest and most coarsest description to the types of Bayesians out there. In the interest of being brief and coarse, I am going to do some points-of-view an injustice. Write me a nice comment if you want to point it out a particular injustice. My hope is simply to do more good than harm.
Also, I am not taking names. You all know who you are:
Strategic vs. Complete Bayesians:
The first and most important dimension of difference is whether one uses Bayes Rule completely or strategically.
Complete Bayesians are those that use Bayes rule always, usually in the form of Bayes factors. They are willing to place probabilities on models themselves and use Bayes rule to update these probabilities in light of data. The outline of the endeavor is that theories naturally predict constraint in data which are captured by models. Model comparison provides a mean of assessing competing theoretical statements of constraint, and the appropriate model comparison is by Bayes factors or posterior odds. In this view, models predict relations among observables and parameters are convenient devices to make conditional statements about these relations. Statements about theories are made based on predictions about data rather than about parameter values. This usage follows immediately and naturally from Bayes rule.
Strategic Bayesians are those that use Bayes rule for updating parameters and related quantities, but not for updating beliefs about models themselves. In this view, parameters and their estimates become the quantities of interest, and the resultants are naturally interpretable in theoretical contexts. These Bayesians stress highest density regions, posterior predictive p-values, and estimation precision. Strategic Bayesians may argue that the level of specification needed for Bayes factors is difficult to justify in practice especially given the attractiveness of estimation.
The Difference: The difference between Complete and Strategic Bayesians may sound small, but it is quite large. At stake are the very premise of why we model, what a model is, how it relates to data, what counts as evidence, and what are the roles of parameters and predictions. Some statisticians, philosophers, and psychologists take these elements very seriously. I am not sure anyone is willing to die on a hill in battle for these positions, but maybe.
I would argue that the difference between Complete and Strategic Bayesians is the most important one in understanding the diversity of Bayesian thought in the social sciences. It is also the most difficult and the most papered over.
Subjectivity vs. Objectivity in Analysis
The nature of subjectivity is debated in the Bayesian community. I have broken out here a few positions that might be helpful.
Subjective Bayesians ask analysts to query their beliefs and represent them as probability statements on parameters and models as part of the process of model specification. For example, if a researcher believes that an effect should be small in size and positive, they may place a normal on effect size centered at .3 with a standard deviation of .2. This prior would then provide constraint for posterior beliefs.
A variant to the subjective approach is to consider the beliefs of a generic, reasonable analyst rather than personal beliefs. For example, I might personally have no faith in a finding (or, in my case, most findings), yet I still may assign probabilities to parameters and hypotheses values that I think capture what a reasonable colleague might feel. This process is familiar and natural---we routinely take the position of others in professional communication.
Objective Bayesians stipulate desirably properties of posteriors and updating factors and choose priors that insure these desired properties hold. A simple example might be that in the large-sample limit, the Bayesian posterior of a parameter should converge to a true value. Such a desirada would necessitate priors that have certain support, say all positive reals for a variance parameter or all values between 0 and 1 for a probability parameter.
There are more subtle examples. Consider a comparison of a null model vs. an alternative model. It may be desirable to place the following constraint on the Bayes factor. As the t-value increases without bound, the Bayes factor should favor without bound the alternative. This constraint is met if a Cauchy prior is placed on effect size, but it is not met if a normal prior is placed on effect size.
There are many other desiderata that have been proposed to place constraints on priors in a variety of situations, and understanding these desiderata and their consequences remains the topic of objective Bayesian development.
The Difference:
My own view is that there is not as much difference between the objective and subjective points of view as there might seem.
1. Almost all objective criteria yield flexibility that still needs to be subjectively nailed down. For example, if one uses a Cauchy prior on effect size, one still needs to specify a scale setting. This specification is subjective.
2. Objective Bayesian statisticians often value substantive information and are eager to incorporate it when available. The call to use desiderata is usually made in the absence of such substantive information.
3. Most subjective Bayesians understand that the desiderata are useful as constraints and most subjective priors adopt some of these properties.
4. My colleagues and I try to merge and balance subjective and objective considerations in our default priors. We think these are broadly though not universally useful. We always recommend they be tuned to reflect reasoned beliefs about phenomena under consideration. People who accuse us as being too objective may be surprised by the degree of subjectivity we recommend; those who accuse us as being too subjective may be surprised by the desiderata we follow.
Take Home
Bayesians do disagree over when and how to apply Bayes rule, and these disagreements are critical. They also disagree about the role of belief and more objectively-defined desiderata, but these disagreements seem more overstated, especially in light of the disagreements over how and when Bayes rule should be used.
Sunday, May 10, 2015
Using Git and GitHub to Archive Data
This blog post is for those of you who have never used Git or GitHub. I use Git and GitHub to archive my behavioral data. These data are uploaded to GitHub, an open web repository where it may be viewed by anyone at any time without any restrictions. This upload occurs nightly, that is, the data are available within 24 hours of their creation. The upload is automatic---no lab personnel is needed to start it or approve it. The upload is comprehensive in that all data files from all experiments are uploaded, even those that correspond to aborted experimental runs or pilot experiments. The data are uploaded with time stamps and with an automatically generated log. The system is versioned so that any changes to data files are logged, and the new and old versions are saved. In summary, if we collect it, it is there, and it is transparent. I call data generated this way as Born Open Data.
Since setting up the born-open-data system, I have gotten a few queries about Git and GitHub, the heart of the system. Git is the versioning software; GitHub is a place on the web (github.com) where the data are stored. They work hand in hand.
In this post, I walk through a few steps of setting up GitHub for archiving. I take the perspective of Kirby, my dog, who wishes to archive the following four photos of himself:
Here are Kirby's steps:
1. The first step is to create a repository on the GitHub server.
1a. Kirby goes to GitHub (github.com) and signs up for a free account (last option). Once the account is set up (with user name KirbyHerby) he is given a screen with a lot of options for exploring GitHub. He ignores these as they are not relevant for his task.
1b. To create his first repository on the server, Kirby presses the green button that says + New repository" on the bottom left.
1c. Kirby now has to make some choices about the repository. He names it data," enters a description of the repository, makes it public,
initializes it with a README and does not specify which files to ignore or a license. He then presses the green Create repository" button on the bottom, and is given his first view of the repository
Kirby's repository is now at github.com/KirbyHerby/data, and he will bark out this URL to anyone interested. The repository contains only the README.md file at this point.
2. The next step is getting a linked copy of this repository on Kirby's local computer.
2a. Kirby downloads the GitHub application for his operating system (mac.github.com} or windows.github.com), and on installation, chooses to install the command-line tools (trust me, you will use these some day).
2b. Kirby enters his GitHub username (KirbyHerby") and password.
2c. He next has to create a local repository and link it to the one on the server. To do so, he chooses to Add repository" and is given a choice to Add," Create," or Clone." Since the repository already exists at GitHub, he presses Clone." A list of his repositories shows up, and in this case, it is a short list of one repository, data." Kirby then selects data" and presses the bottom button Clone repository." The repository now exists on the local computer under the folder data." There are two, separate copies of the same repository: one on the GitHub server and one on Kirby's local machine.
3. Kirby wishes add files to the server repository so others may see them.
3a. Kirby first adds the photo files to the local repository as follows: Kirby copies the photos to the files in the usual way, which for Mac-OSX is by using the Finder. The following screen shot shows Finder window in the foreground and the GitHub client window in the background. As can be seen, Kirby has added three files, and these show up in both applications. Kirby has no more need for the Finder and closes it to get a better view of the local repository in the GitHub client window.
3b. Kirby is now going to save the updated state of the local repository, which is called committing it. Committing a local action, and can be thought of as a snapshot of the repository at this point in time. Kirby turns his attention to the bottom part of the screen. To commit, Kirby must add a log entry, which in this case is, Added three great photos." The log will contain not only this message, but a description of what files were added, when, and by whom. This log message is enforced---one cannot make a commit without it. Finally Kirby presses Commit to master."
3c. Kirby now has to push his changes to the repository to the GitHub server so everyone may see them. He can do so by pressing the sync" button.
That's it. Kirby's additions are now available to everyone at github.com/KirbyHerby/data
Suppose Kirby realizes that he had forgotten his absolutely favorite photo of him hugging his favorite toy, Panda. So he copies the photo over in Finder, commits a new version of the repository with a new message, and syncs up the local with the GitHub server version.
There is a lot more to Git and GitHub than this. Git and GitHub are very powerful, so much so that they are the default for open-source software development world wide. Multiple people may work on multiple parts of the same project. Git and GitHub have support for branches, tagging versions, merging files, and resolving conflicts. More about the system may be learned by studying the wonderful Git Book at git-scm.com/book/en/v2.
Finally, you may wonder why Kirby wanted to post these photos. Well, Kirby doesn't know anything about Bayesian statistics, but he is loyal. He knows I advocate Bayes factors. He also knows that others who advocate ROPEs and credible intervals sell their wares with photos of dogs. Kirby happens to believe that by posting these, he is contributing to my Bayes-factor cause. After all, he is cuter than Kruschke's puppies and perhaps he is more talented. He does know Git and GitHub and has his own repository to prove it. | 2017-06-23 11:54:29 | {"extraction_info": {"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, "math_score": 0.5113956928253174, "perplexity": 2209.791191069341}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-26/segments/1498128320057.96/warc/CC-MAIN-20170623114917-20170623134917-00668.warc.gz"} |
http://math.stackexchange.com/questions/194875/linearly-disjoint-vs-free-field-extensions | # Linearly disjoint vs. free field extensions
Consider two field extensions $K$ and $L$ of a common subfield $k$ and suppose $K$ and $L$ are both subfields of a field $\Omega$, algebraically closed.
Lang defines $K$ and $L$ to be 'linearly disjoint over $k$' if any finite set of elements of $K$ that are linearly independent over $k$ stays linearly independent over $L$ (it is, in fact, a symmetric condition). Similarly, he defines $K$ and $L$ to be 'free over $k$' if any finite set of elements of $K$ that are algebraically independent over $k$ stays algebraically independent over $L$.
He shows right after that if $K$ and $L$ are linearly disjoint over $k$, then they are free over $k$.
Anyway, Wikipedia gives a different definition for linearly disjointness, namely $K$ and $L$ are linearly disjoint over $k$ iff $K \otimes_k L$ is a field, so I was wondering:
do we have a similar description of 'free over $k$' in terms of the tensor product $K \otimes_k L$?
It should be a weaker condition than $K \otimes_k L$ being a field, perhaps it needs to be a integral domain?
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The wikipedia's definition for linear disjointness is different from yours. en.wikipedia.org/wiki/Linearly_disjoint – Makoto Kato Sep 12 '12 at 20:41
You are right, I read this question math.stackexchange.com/questions/57414/… and skimmed through wikipedia and thought that was the same definition. – Niccolò Sep 12 '12 at 21:39
Nonetheless, this definition encyclopediaofmath.org/index.php/Linearly-disjoint_extensions means that when $A$ and $B$ are field extensions and not just $k$-algebras, their tensor products is isomorphic to their compositum, which is a field. And by the way, in the definition you linked, isn't the map $A \otimes_k B \rightarrow AB$, mapping $a \otimes b$ to $ab$, always surjective? – Niccolò Sep 12 '12 at 21:46
The compositum is the sub-$k$-algebra generated by $A\cup B$, it is not a field in general. – user18119 Sep 12 '12 at 21:52
Linear disjointness and freeness are broadly discussed in Zariski-Samuel, Commutative Algebra I, Chapter 3. In particular the point, that QiL mentions in his answer, is emphasized. – Hagen Knaf Sep 13 '12 at 6:55
The condition of being linearly disjoint or free depends much on the "positions" of $K, L$ inside $\Omega$, while the isomorphism class of the $k$-algebra $K\otimes_k L$ doesn't. For instance, consider $\Omega=\mathbb C(X,Y)$, $K=\mathbb C(X)$, $L_1=\mathbb C(Y)$ and $L_2=K$. Then $$K\otimes_\mathbb C L_1\simeq K\otimes_{\mathbb C} L_2$$ as $\mathbb C$-algebras. But $K, L_1$ are linearly disjoint (so free) in $\Omega$, not $K, L_2$. This example shows that in general, the linear disjointness nor the freeness can be determined by intrinsic properties of $K\otimes_k L$.
If $K$ or $L$ is algebraic over $k$, then it is true that linear disjointness is equivalent to $K\otimes_k L$ is a field. But in this situation the freeness is automatic whenever the tensor product is a field or not (can even be non-reduced). | 2016-05-01 18:20:36 | {"extraction_info": {"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, "math_score": 0.972110390663147, "perplexity": 146.87316026340062}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-18/segments/1461860116878.73/warc/CC-MAIN-20160428161516-00013-ip-10-239-7-51.ec2.internal.warc.gz"} |
https://socratic.org/questions/how-do-you-write-an-expression-to-represent-the-perimeter-of-a-triangle-whose-si#336348 | # How do you write an expression to represent the perimeter of a triangle whose side measurements are 5x+3, 2x -7, and 9X?
The perimeter of the triangle is $16 x - 4$.
$P = 5 x + 3 + 2 x - 7 + 9 x$
$P = 16 x - 4$ | 2021-10-24 23:16:58 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 3, "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, "math_score": 0.7741665244102478, "perplexity": 499.710072960298}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323587606.8/warc/CC-MAIN-20211024204628-20211024234628-00686.warc.gz"} |
https://economics.stackexchange.com/questions/9253/derive-the-demand-functions-hotelling-style-model/9256#9256 | # Derive the demand functions: Hotelling-style Model
So I have this economics question that I have been trying for a while now and I can't seem to get the answer correctly. Below is the question and after I will show what I have so far. An explanation would help so much.
Consider the following Hotelling-style model. There are 3 Arms, each offering a single type of milk (always in 1 quart containers) that is horizontally differentiated along a single dimension, the percent of fat.
Firm-1 offers non-fat milk.
Firm-2 offers low fat milk, with 10% fat.
Firm-3 offers high fat milk, with 50% fat.
There are 1,000 potential consumers. Some consumers hate fat and some love it. But the most fat any individual consumer would ideally have in their milk is 50%. In particular, assume that people's tastes for the ideal percent of fat is uniformly distributed from 0% to 50%. That is, the line starts at zero and goes to 50 (instead of 1 as we have done in the past).
The utility that individual i obtains from purchasing a quart of milk from seller j is given by
Uij = 5 - pj - 1/10*|Xi - Xj|
Where Xi € [0,50] is individual i's ideal fat content and Xj € [0,50] is the fat content offered by firm j.
For example, if a consumer is located at Xi = 2 were to purchase from firm 2, and firm 2 happened to set a price P2 = 3, they would obtain utility:
Uij =5 - 3 - 1/10*|2 - 10| = 1.2
1. Derive the demand functions for each of the three firms.
So far I have been able to derive the demand function for firm 1 by first solving for Xi for the indifferent consumer between Firm 1 and Firm 2.
The line looks like this
|--x--|-------------|
with the first line representing 0% fat, second 10%, and third 50%. The x represents the indifferent consumer between firm 1 and firm 2.
The math for the first firm is done by taking the Utility of the Indifferent consumer for firm 1 and setting it equal to the utility of the indifferent consumer for firm 2.
5-P1-1/10*|Xi-0|=5-P2-1/10*|Xi-10|
by doing this, you end up with
Xi=5P2-5P1+5 with the Q1=(Xi-0)*1000/50
Therefore the demand curve for firm 1 is
Q1=100P2-100P1-100
The next step would be to be to solve for the demand curve for firm 3. My problem is when I set U2=U3 the Xis cancel. The math is below.
5-P2-1/10*|Xi-10|=5-P3-1/10*|Xi-50|
This reduces to:
-P2-1+Xi/10 =-P3-5+Xi/10
Finally this reduces to:
-P2+P3+4=0
The "Xi/10" on both sides cancel which does not make sense. Because of this result, I have no idea how to solve for the demand curve of Firm 3.
You are treating the absolute value wrong. You seem to have done it right the first time around, do it like that again. (Make a little drawing, think about which side of 10 and 50 the indifferent $X_i$ will be on, etc.)
Another thing: The demand function you derived for $Q_1$ only holds for certain ranges of $P_1$, $P_2$ and $P_3$, because if the prices of firms 1 and 2 are very high but $P_3$ is low, a customer indiffirent between firms 1 and 2 would go to firm 3. | 2021-10-20 09:45:09 | {"extraction_info": {"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, "math_score": 0.4974551498889923, "perplexity": 891.0119172513375}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585305.53/warc/CC-MAIN-20211020090145-20211020120145-00343.warc.gz"} |
http://people.bath.ac.uk/jdb55/SAMBa_poster/index.html | Numerical weather prediction requires the use of high performance computing. At the core of this we must solve the Navier-Stokes equations, which are a non-linear hyperbolic system of PDES. Simpler forms can be used to demonstrate some of the behaviour of the Navier-Stokes equations, here the advection equation is studied.
Unlike a conformal finite element method, the constraint of continuity between mesh cells is relaxed for a DG method. This is replaced with a numerical flux between cells.
For a higher order DG method, a higher degree polynomial basis is used to approximate the solution, which leads to higher accuracy.
The PDE we solve is the linear advection equation: $$\partial_t u + \beta\cdot\nabla{u} + au = f$$
Where $\partial_t u$ is the rate of change in the solution $u$ in time.
$\beta$ is the advection vector describing which way the solution moves.
$au$ is a reaction term that depends on the value of the solution and the right hand side, $f$, is an external forcing.
We can solve this as a time dependent problem in one dimension, here we look at two different initial conditions.
By setting $\partial_t u = 0$ we can solve a stationary problem.
Removing this restriction we can solve the problem in time.
Using a suitable grid we can also solve the advection problem on the surface of a sphere. Such problems are essential for NWP.
• Kronbichler, Martin and Kormann, Katharina
A generic interface for parallel cell-based finite element operator application
Computers & Fluids, 63:135--147, 2012.
• Vos, Peter EJ and Sherwin, Spencer J and Kirby, Robert M
From h to p efficiently: Implementing finite and spectral/hp element methods to achieve optimal performance for low-and high-order discretisations
Journal of Computational Physics, 229(13):5161--5181, 2010.
• ... | 2019-08-21 07:05:27 | {"extraction_info": {"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, "math_score": 0.7979363799095154, "perplexity": 630.192437778055}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027315811.47/warc/CC-MAIN-20190821065413-20190821091413-00183.warc.gz"} |
https://zbmath.org/?q=an:05532506&format=complete | # zbMATH — the first resource for mathematics
Benford’s law, recurrence relations, and uniformly distributed sequences. II. (Loi de Benford, relations de récurrence et suites équidistribuées. II.) (French) Zbl 1242.11056
This article deals with Benford’s law and uniformly distributed sequences. Let $$b\geq 2$$ be an integer. Each real $$x>0$$ can be written in one way $$x=M_b(x).b^{e_b(x)}$$ where $$e_b(x)\in\mathbb Z$$ and $$M_b(x)$$ is the mantissa of $$x$$ in basis $$b$$. Let $$(a_n)_{n\geq 1}\subset (0,+\infty)$$. The sequence $$(a_n)_{n\geq 1}$$ is said satisfy Bendford’s law in basis $$b$$ if for all $$t\in[1,b)$$ if $\lim_{N\rightarrow\infty} \frac{|\{1\leq n\leq N\;:\;M_b(a_n)<t\}|}{N}=\log_b(t)\text{\;for all\;}t\in [1,b).$ The sequence $$(a_n)_{n\geq 1}$$ is said satisfy strong Benford’s law if it satisfy Bendford’s law in every basis $$b\geq 2$$. The main result of the author generalizing his previous article [Elem. Math. 60, No. 1, 10–18 (2005; Zbl 1084.11005)] is:
Theorem. Let $$\alpha>0, \xi>0$$ and $$\mu$$ be real numbers and $$Q$$ a function defined on $$[1,\infty)$$ satisfying:
1) There exists an integer $$k\geq 1$$ and a real number $$x_0\geq 1$$ such that $$Q$$ be $$k$$-times differentiable on $$(x_0,+\infty)$$;
2) $$\lim_{x\rightarrow\infty} Q^{(k)}$$ exists and is a nonzero rational number.
Let $$(a_n)_{n\geq 1}\in (0,+\infty)$$ be a sequence such that $$\lim_{n\rightarrow\infty}\frac{a_n}{n^\mu\xi^{Q(n)}}=\alpha$$.
Then, for every integer $$b\geq 2$$ such that $$\log_b(\xi)\in \mathbb R\backslash \mathbb Q$$, the sequence $$(a_n)_{n\geq 1}$$ satisfies Benford’s law in basis $$b$$. Moreover, if, for every positive integer $$m$$, $$\xi^m$$ is not integer, then $$(a_n)_{n\geq 1}$$ satisfies strong Benford’s law.
The proof relies on the theory of uniformly distributed sequences. On the historic background of Benford’s law see also [A. Berger, L. A. Bunimovich and T. P. Hill, Trans. Am. Math. Soc. 357, No. 1, 197–219 (2005; Zbl 1123.37006)]; [A. Berger and T. P. Hill, “Newton’s Method obeys Bendford’s law”, Am. Math. Mon. 114, No. 7, 588–601 (2007; Zbl 1136.65048)]; [J. P. Delahaye, “L’étonnante loi de Benford”, Pour la Science 351, 90–95 (2007)]; [N. Hungerbühler, “Benfords Gesetz über führende Ziffern”, EducETH, March (2007), http://www.educ.ethz.ch].
##### MSC:
11K36 Well-distributed sequences and other variations 37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010) 60E99 Distribution theory
##### Keywords:
Benford’s law; uniformly distributed sequences
Full Text: | 2021-08-05 18:28:37 | {"extraction_info": {"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, "math_score": 0.7668253183364868, "perplexity": 892.1915661453212}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046156141.29/warc/CC-MAIN-20210805161906-20210805191906-00032.warc.gz"} |
https://math.stackexchange.com/questions/2552991/find-the-domain-of-a-radical-function | # Find the domain of a radical function
I'm having trouble with this function $f(x)=\sqrt{x\sqrt{x^2-4}}$ for which I have to find the domain.
To find it I first set $x\sqrt{x^2-4}\geq0$ because the argument of the radical must be non negative. Now we have to study the sign of this product, so we study the signs of the single factors.
$x>0$
$\sqrt{x^2-4}>0\rightarrow x^2-4>0\rightarrow x<-2\vee x>2$
And now we have to make the table of signs.
$-2\quad0\quad2$
$-|-|+|+\quad(x>0)$
$+|-|-|+\quad(\sqrt{x^2-4}>0)$
$-|+|-|+\quad(result)$
Since the original inequality is $\geq$ the result is
$-2\le x\le0\vee x\geq2$
but I notice that for $x\in(-2,0]$ the term $\sqrt{x^2-4}$ is not defined (in $\mathbb{R}$), and so the solution should be
$[-2,0]\cup[2,+\infty)\setminus(-2,0]=\{-2\}\cup[2,+\infty)$
so I made a mistake in the previous calculus since from the table of signs I firstly got $-2\le x\le0\vee x\geq2$ which is wrong, but I cannot understand why.
@HagenvonEitzen's answer is very good, you should mark that as the correct answer. However, from your comment, I thought I would point out where you must have made your mistake. You correctly find the domain of $\sqrt{x^2-4}$. Looking at the negative part of that domain, you have $x\le-2$. Then you look at the expression $x\sqrt{x^2-4}$. When $x\le-2$, this expression is negative, which would make $f(x)$ undefined. This is correct. But then it seems that you decided that for $-2<x<0$, $f(x)$ would be defined, apparently because a negative times a negative is a positive. However, we have already determined that $\sqrt{x^2-4}$ is undefined $(-2, 0)$, so $f(x)$ would have to be undefined as well. My guess is you mixed the meanings of negative and undefined when you were reading the table of signs.
• Yes Hagen explanation is very good but it did not answer my question, as you did. Thank you so I understand now where I made the mistake – sound wave Dec 6 '17 at 23:12
$\sqrt{x^2-4}$ is defined (and automatically non-negative) if $x^2\ge 4$, i.e., if $x\le -2$ or $x\ge 2$. We can ignore $(-2,2)$ immediately because $\sqrt{x^2-4}$ is not defined there (even if some smart persons might argue that $x\sqrt{x^2-4}$ ought to be considered defined and zero when $x=0$).
For those points where $\sqrt{x^2-4}=0$, i.e., for $x\in\{-2,2\}$, the value of $x$ does not matter, we will certainly have $x\sqrt{x^2-4}=0\ge 0$ and hence $\sqrt{x\sqrt{x^2-4}}$ defined.
For the other points, i.e., when $\in(-\infty,-2)\cup(2,\infty)$, we need $x\ge 0$. This leaves us only with $(2,\infty)$.
In summary, the domain in question is $\{-2,2\}\cup (2,\infty)=\{-2\}\cup[2,\infty)$.
• Thank you for the answer, I understand but I don't get where my calculus are wrong. I usually solve equations using intersection of solutions (if I have a system of equations) or with the table of signs, as in the case of the product of 2 or more factors. But in this case with the "mechanical" method I get a wrong solution. I know that is better to not use the same method over and over, but it is just a personal curiosity. – sound wave Dec 5 '17 at 22:53
• I have a little question, you said $\sqrt{x^2-4}$ is defined (and automatically non-negative) if $x^2-4\geq0$, but about the non-negative affirmation, I have a doubt because for example $\sqrt{4}=+2$ or $-2$ right? – sound wave Dec 8 '17 at 11:55
• @soundwave No, $\sqrt4=2$ never $-2$ because $\sqrt 4$ is defined as the non-negative number $x$ for which $x^2=4$ this is because we want the square root to be a function which means it must give exactly one value (on it's domain). – kingW3 Dec 8 '17 at 12:57
• @kingW3 ok I understand, the square root has to be a function so every value in the domain has to be linked with only one value in the codomain, right ? so at the start of the exercise I could define a square root for which the codomain contains only negative values (i.e. $\sqrt{4}=-2$) and solve the exercise in this different way ? – sound wave Dec 8 '17 at 13:10
• @soundwave You could define it that way, but that's not the definition of the square root. You would be the only one using that definition, unless explicitly stated otherwise the definition of $\sqrt \cdot$ is understood to return only non-negative values. – kingW3 Dec 8 '17 at 13:21 | 2021-01-27 18:03:15 | {"extraction_info": {"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, "math_score": 0.9358998537063599, "perplexity": 155.15887576738453}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610704828358.86/warc/CC-MAIN-20210127152334-20210127182334-00781.warc.gz"} |
https://thespinery.com/blog/comparing-natural-latex-foams/ | Have any questions? Come visit our Dun Laoghaire showroom or give us a call (01) 284 4093.
# The difference between Natural Latex Foam and Synthetic Latex Foam.
While our mattress or pillow doesn’t contain synthetic latex, we do get a lot of questions about synthetic latex sold by other companies in Ireland. So in this blog, we are going to explain the differences between natural latex, synthetic latex, and blended latex and why those differences are so significant when purchasing a latex mattress or pillow.
## Natural Latex Foam
Natural Latex is a milky white liquid that is extracted from the rubber tree or sometimes known as Hevea-Brasilienis.
Rubber trees are indigenous to South America; however, they were spread around the world in the 19th century due to their high value in a variety of products.
### How Natural Latex Is Made
The way we get our Latex from Rubber trees is very similar to maple trees being tapped for their maple syrup!
A small piece of the bark is cut away, and latex quickly begins flowing from that cut and into containers.
The latex is then collected and shipped to our factory to make our mattresses and pillow.
### How Natural Latex Mattresses Feel
Natural latex foam has a uniquely supportive and springy feel, something we call resiliency.
High quality natural latex foam like the kind we use in our Fitzwilliam mattress is extremely durable and should last longer than any other type of mattress.
While it does have a mild odor ( sometimes describes as vanilla smelling), natural latex foam usually doesn’t have a chemically or abrasive odor than you typically get in synthetic latex.
### Downsides of Natural latex Foam
The only negative of using natural latex foam is that it is more difficult to make aesthetically perfect during the production process as opposed to synthetic latex foam. Small air pockets can be present in the foam.
However, at The Spinery, we maintain strict quality standards to ensure inconsistencies are kept to an absolute minimum.
## Synthetic Latex Foam
Synthetic latex is an artificially produced compound that imitates the properties of natural latex.
However, rather than being harvested from trees, it is made from petrochemicals. There are many types of synthetic latex that exist, but styrene-butadiene latex is a common type of synthetic rubber used in synthetic latex foam.
Synthetic latex foam is described as a more dull, less springy feel than natural latex foam. It is also less durable than natural latex foam and tends to tear easier.
Synthetic latex foam typically has a strong chemically and more abrasive smell. It is rare to find synthetic latex foam that meets industry textile and furniture emissions standards.
## Blended Latex
The last type of latex foam is a mix of both types we’ve already discussed. Retailers mostly use blended latex because it is a cheaper way to produce latex foam that can claim to contain natural latex for many latex mattress retailers and manufacturers. However, often blended latex contains mostly synthetic rather than natural latex. They do this by making claims such as “made with 100% natural latex”, “contains natural latex,” or “100% latex, natural origin”. Others sometimes merely falsely claim their mattress to be “100% natural latex”. Blended latex shares many of the properties of both natural and synthetic latex. However, we find that natural latex outperforms blended latex in terms of comfort, resiliency, health, and eco-friendliness.
## How Can I tell If My Mattress Is Natural Latex or Synthetic Latex?
Buying fake or synthetic latex instead of natural latex can be common.
For example, we get many comments from people that they have a latex mattress that they purchased from a well-known big box store. But it’s not the real thing, and that’s why they are not getting the results that should expect.
Or, people looking to get a latex mattress or pillow primarily rely on a website to represent their product honestly. Unfortunately, as we mentioned earlier, some mattress retailers use misleading language to convince you that their synthetic latex mattresses are natural.
### Here are some tips so you can check yourself at your local showroom:
To recognize synthetic and blended latex, look at the latex sample provided in the store as a demo.
### Signs the ‘latex’ you’re looking at might be Styrene Butadiene (SBR):
• Turns yellow as it ages.
• Crumbles as it ages.
• The foam easily tears and pulls apart, especially on the sides of the pillow or mattress.
• Keeps a chemical smell for months after purchase.
#### How to tell if the “latex” mattress is really only a layer of Latex laminated to polyurethane foam:
Time needed: 5 minutes.
How to tell if a mattress is made from natural latex
1. Try Bending the mattress
An all-latex mattress will fold in half much easier than a typical spring or polyurethane mattress.
2. Look on the law label for the contents by percentage
Unfortunately, the latex listed will not specify whether it is natural or synthetic however it will help you identifier if only a layer of latex is being used.
3. Smell the mattress
You should notice a chemical smell if polyurethane is used. Natural Latex has a vanilla-like smell.
4. Lift the mattress
Natural Latex is heavier than most synthetics. You should be able to notice a significant difference. That’s why you can’t ship natural latex in a box – it’s just too heavy.
5. Feel the mattress through the sides
You may be able to feel the transition points for the layers. Plus, polyfoam feels noticeably different than latex – it doesn’t have that rebound, or what we call resiliency. And remember, resiliency is the ‘holy grail’ for mattress designs.
Hopefully, some of these tips will help in ensuring that you get the real thing: a real natural latex mattress that will give you better sleep, stay durable and supportive, and feel genuinely comfortable.
We hope this video helped explain the differences between natural and artificial. If you’re curious about what else makes our Fitzwilliam natural latex mattress special, click here. | 2021-01-22 23:17:30 | {"extraction_info": {"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, "math_score": 0.8756574392318726, "perplexity": 4175.042473420467}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703531429.49/warc/CC-MAIN-20210122210653-20210123000653-00501.warc.gz"} |
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# Math: Number Theory
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Intern
Joined: 12 Sep 2017
Posts: 5
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22 Sep 2017, 04:11
Thank you Bunuel and everyone for their contribution here
Intern
Joined: 01 Apr 2017
Posts: 6
GMAT 1: 720 Q48 V40
### Show Tags
14 Oct 2017, 03:51
Nothing is better than this
Intern
Status: hope for the best, prepare for the worst
Joined: 13 Dec 2015
Posts: 6
Location: Pakistan
GPA: 3.19
### Show Tags
26 Dec 2017, 14:39
How many powers of 900 are in 50!? the statement written in following question wa
We need all the prime {2,3,5} to be represented twice in 900, 5 can provide us with only 6 pairs, thus there is 900 in the power of 6 in 50!
what does it mean?
Math Expert
Joined: 02 Sep 2009
Posts: 43896
### Show Tags
26 Dec 2017, 20:11
abbas57 wrote:
How many powers of 900 are in 50!? the statement written in following question wa
We need all the prime {2,3,5} to be represented twice in 900, 5 can provide us with only 6 pairs, thus there is 900 in the power of 6 in 50!
what does it mean?
I tried to elaborate this in the posts below:
https://gmatclub.com/forum/math-number- ... ml#p678298
https://gmatclub.com/forum/math-number- ... ml#p710819
_________________
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Joined: 14 Jul 2015
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### Show Tags
28 Dec 2017, 12:13
Finding the power of non-prime in n!:
How many powers of 900 are in 50!
Make the prime factorization of the number: 900=22∗32∗52900=22∗32∗52, then find the powers of these prime numbers in the n!.
Find the power of 2:
502+504+508+5016+5032502+504+508+5016+5032=25+12+6+3+1=47=25+12+6+3+1=47
= 247247
Find the power of 3:
503+509+5027=16+5+1=22503+509+5027=16+5+1=22
=322322
Find the power of 5:
505+5025=10+2=12505+5025=10+2=12
=512512
We need all the prime {2,3,5} to be represented twice in 900, 5 can provide us with only 6 pairs, thus there is 900 in the power of 6 in 50!.
Hi Bunuel,
I read the explanation you gave to defoue but couldn't apply it to the above example of 900. Could you please elaborate?
Thanks.
Intern
Joined: 14 Jul 2015
Posts: 4
### Show Tags
28 Dec 2017, 12:18
I just read the remaining explanations...got it! Sorry
Intern
Joined: 05 Jan 2018
Posts: 1
### Show Tags
05 Jan 2018, 05:18
A die is rolled 14 times.
a) What is the probability that we obtain exactly one 6?
b) What is the probability that we obtain 4 times 4, 2 times 6 and 8 times 3?
Intern
Joined: 18 May 2016
Posts: 12
### Show Tags
22 Feb 2018, 20:16
Bunuel
Excellent Post!
Edit suggestions:
1. If p is a prime number and p is a factor of ab, then p is a factor of a or p is a factor of b.
If p = a = b = 2, the above point fails
as p will be a factor of ab and a and b
2. Take the last digit, double it, and subtract it from the rest of the number, if the answer is divisible by 7 (including 0), then the number is divisible by 7.
I think you should change the underlined "number" to digits for better understanding
and then provide an example of the working
eg. 203/7
Last digit -> 3
Double it -> 3*2 = 6
Rest of the digits = 20 -> 20-6 = 14
Answer (14) -> Divisible by 7 -> Yes
3. Verifying the primality (checking whether the number is a prime) of a given number n can be done by trial division ...
Wouldn't it be easier & faster to just use another method that you've mentioned earlier in the post?
all prime numbers above 3 are of the form 6n−1 or 6n+1
-> Divide a number by 6 and +1/-1 to check if it is a prime
Thanks
P
Math Expert
Joined: 02 Sep 2009
Posts: 43896
### Show Tags
22 Feb 2018, 21:30
1
KUDOS
Expert's post
ppnimkar wrote:
Bunuel
Excellent Post!
Edit suggestions:
1. If p is a prime number and p is a factor of ab, then p is a factor of a or p is a factor of b.
If p = a = b = 2, the above point fails
as p will be a factor of ab and a and b
2. Take the last digit, double it, and subtract it from the rest of the number, if the answer is divisible by 7 (including 0), then the number is divisible by 7.
I think you should change the underlined "number" to digits for better understanding
and then provide an example of the working
eg. 203/7
Last digit -> 3
Double it -> 3*2 = 6
Rest of the digits = 20 -> 20-6 = 14
Answer (14) -> Divisible by 7 -> Yes
3. Verifying the primality (checking whether the number is a prime) of a given number n can be done by trial division ...
Wouldn't it be easier & faster to just use another method that you've mentioned earlier in the post?
all prime numbers above 3 are of the form 6n−1 or 6n+1
-> Divide a number by 6 and +1/-1 to check if it is a prime
Thanks
P
1. x or y means x or y or both. So, everything is correct there.
2. I don't think it would be better.
3. Any prime number p, which is greater than 3, could be expressed as $$p=6n+1$$ or $$p=6n+5$$ or $$p=6n-1$$, where n is an integer greater than 1.
Any prime number p, which is greater than 3, when divided by 6 can only give the remainder of 1 or 5 (remainder cannot be 2 or 4 as in this case p would be even and the remainder cannot be 3 as in this case p would be divisible by 3).
So, any prime number p, which is greater than 3, could be expressed as $$p=6n+1$$ or $$p=6n+5$$ or $$p=6n-1$$, where n is an integer greater than 1.
But:
Not all number which yield a remainder of 1 or 5 upon division by 6 are primes, so vise-versa of the above property is not true. For example 25 yields the remainder of 1 upon division be 6 and it's not a prime number.
_________________
Re: Math: Number Theory [#permalink] 22 Feb 2018, 21:30
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Display posts from previous: Sort by | 2018-02-24 19:30:07 | {"extraction_info": {"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, "math_score": 0.7014660239219666, "perplexity": 1265.9075822027812}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891815934.81/warc/CC-MAIN-20180224191934-20180224211934-00352.warc.gz"} |
https://computergraphics.stackexchange.com/questions/126/how-can-i-raytrace-a-scene-that-does-not-fit-into-memory/9882#9882 | # How can I raytrace a scene that does not fit into memory?
If the scene to be raytraced cannot be stored in memory, then without adding more RAM to the machine it seems unrealistic to render it in a practical time span, due to the need to load different parts of the scene from disk potentially several times per pixel.
Is there any way around this? I'm trying to think of some way of performing a large number of the calculations involving a particular subset of the scene all at once, to reduce the number of times it needs to be loaded into memory. Is there any other way of improving the speed in such a case?
If the scene does not entirely fit into memory, you are entering the field of out-of-core rendering. There are essentially two approaches here: a) Generate your scene on-demand b) Load your scene on-demand
The former approach aligns well with most animation workflows, where models are heavily subdivided using e.g. Catmull-Clark and can become very memory-intensive, but the base meshes themselves easily fit into memory. Pixar have a few papers about this (e.g. Ray Differentials and Multiresolution Geometry Caching for Distribution Ray Tracing in Complex Scenes), but the gist of it is that models are only subdivided when they are hit by a ray, and only subdivided as much as is reasonable for such a ray (e.g. diffuse interreflection need less accuracy than mirror reflections). The rest is handled by a geometry cache, which keeps the subdivided models in memory and hopefully makes the process efficient by a good eviction strategy.
As long as all your base meshes comfortably fit into memory, you can easily go out-of-core and render meshes at subdivision levels that would never fit into memory. The geometry cache also scales nicely with the amount of memory you have, allowing you to weigh RAM vs. render times. This was also used in Cars I believe.
The second approach is more general and does not rely on heavy use of subdivision. Instead, it relies on the fact that your scene was most likely made by an artist and already comes partitioned into reasonably small objects that fit into memory individually. The idea is then to keep two hierarchies (kD-tree or bounding volume hierarchy): A top-level hierarchy that only stores bounding boxes of the objects in your scene, and a low-level hierarchy that stores the actual geometry. There is one such low-level hierarchy for each object.
In this approach, you ideally already store a bounding box along with each object on disk. As the scene is loaded, you only build the top-level hierarchy initially, meaning you only have to look at the bounding boxes and not the geometry. You then start tracing rays and traverse them through the hierarchy. Whenever a ray hits a leaf node in the top-level hierarchy (i.e. it hits the bounding box of an object), that object is loaded into memory and its low-level hierarchy is built. The ray then continues down into tracing that object. Combined with an object cache that keeps as much of the low-level hierarchy in memory as possible, this can perform reasonably well.
The first benefit of such an approach is that objects that are never hit are never loaded, meaning that it automatically adapts to the visibility in your scene. The second benefit is that if you are tracing many rays, you don't have to load an object immediately as it is hit by a ray; instead, you can hold that ray and wait until enough rays have hit that object, amortizing the load over multiple ray hits.
You can also combine this approach with a ray sorting algorithm such as Sorted Deferred Shading for Production Path Tracing to avoid thrashing due to incoherent rays. The mentioned paper describes the architecture of Disney's Hyperion renderer, used for Big Hero 6 I believe, so it most likely can handle scenes at production scale.
• This is super interesting! So is the Disney paper you linked. Aug 9 '15 at 23:41
• +1 So many answers to things I've always wondered about! Oct 22 '15 at 20:53
If you organize your scene in a spatial structure (the usual way being a Bounding Volume Hierarchy), you can use a sort of virtual scene (I am making up this term, in reference to virtual textures).
A memory manager would keep only a limited number of bounding boxes loaded at a time, and abstract the operation consisting in retrieving one.
This way, a box would be loaded only as needed: when a ray hits a bounding box, the box get loaded to resolve the collision. Later on when another box needs to be loaded, the unused one is deleted to make room for the new one.
With all these boxes getting loaded and deleted, the ray coherency would be a major factor in speed. I suppose a further improvement could be to defer loading, by reordering rays to treat first the boxes that are already loaded.
• Yes something like this. Aug 9 '15 at 11:40
What you do is you load triangles into memory from disk based on what has been hit previously. You can begin with triangles in close proximity first. The reasoning is that in one area the rays are likely to hit same triangles repeatedly. And eventually you will be somewhat efficient. (For this reason it is a good idea to cache last hit triangle in occlusion tracing that dont care of order)
Second you store the triangles in a spatial tree that allows you to do quick searching from the disk, to renew what portions you are having in memory by proximity. So load only branches that will be in the way of the ray. If its some kind of voxel tree, like a octree you can even sort secondary rays and solve them by coherence. A BSP tree is also somewhat good at pruning areas.
There are cases where this fails but its reasonably efficient in most scene buckets if your not rendering noise...
The approaches mentioned by the 3 answers up to now (May 2020) mention:
• subdivision surfaces
• BVH per object tracing/eviction + ray cache + memory management
There are other orthogonal techniques that can be added or used by themselves. That is decimation based on something else than OpenSubdiv. Like:
A different angle of thought: if your scene doesn't fit in memory, it has an entropy problem. You only have 4k res (~8 Million pixels?), therefore in the sense of Shannon "more than memory" cannot reasonably map to less data than can be projected to these 8Mp. Which means a scene that uses "more than memory" is most likely a super overkill. Thus again, the simplification approach will probably not only speed up things, but also could help aliasing.
• I strongly disagree with your interpretation of information theory. It is true, that in the end you will need #pixels * #channels * #bits_per_channel but you don't want to guess that data, you want to compute them by evaluating the rendering equation. Note that you have many free variables like camera and light parameters, etc. You don't want to pre-bake an optimized scene for each possible camera position or light setup! For example that would mean to precompute all textures for all possible perspectives. May 25 '20 at 9:37
• Another wrong assumption of yours is perhaps that only the visible objects contribute to the final image. However, there may be shadow casters all around, layers of transparent objects that accumulate together, specular highlights and caustics from behind the camera reflecting towards object in front of the camera and possibly back again. In order to compute all the light bounces correctly, you will easily end up with scenes much more information content (orders of magnitude more) than the resulting image. May 25 '20 at 9:41
• The last bit of additional information you need to take into account is that if you use a spectral renderer, you will try to have more than just RGB data for your materials, but the rendering output will be in most cases just RGB. May 25 '20 at 9:50
• @Isolin Your first remark I would argue, yes we do. That's what we do with mipmaps. When we prepare any sort of map, especially impostors, we do pre-bake them for any sort of camera distance or angle. Any sort of LOD of meshes is exactly that, it's a pre-baked optimization for each possible camera position. Now your second remark is interesting. It made me ponder about the universe. We can say that the faint GI caused by galaxies contribute to the final image. Though I doubt that it's chaotic; meaning they can be reduced to few parameters. But caustics are. good catch May 25 '20 at 10:44 | 2021-10-18 00:20:02 | {"extraction_info": {"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, "math_score": 0.3400142788887024, "perplexity": 1062.978951473902}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585186.33/warc/CC-MAIN-20211018000838-20211018030838-00338.warc.gz"} |
https://math.stackexchange.com/questions/3549593/how-are-these-two-banach-spaces-related-weighted-l2-type-space-involving-a | # How are these two Banach spaces related ? (weighted $L2$ type space involving a logarithm and Besov type space)
First the standard $$L2$$ space : $$L^2(\mathbb{R}) = \Big \{ f : \| f \|_2 = \left( \int_{\mathbb{R}} | f(x) |^2 dx \right)^{1/2} < \infty \Big \}.$$
Let $$s \geqslant 1/2$$. Define a weighted $$L2$$ space as follows :
$$L^2_{s} := \{ f \in L^2(\mathbb{R}) : \| (2+|x|)^s f(x) \|_2 < \infty \}.$$
There is also another Banach space $$B$$ (in the literature that I have seen it either doesn't have a special name, or is just called a "Besov" space) :
$$B := \{ f \in L^2(\mathbb{R}) : \sum_{n=0} ^{\infty} \sqrt{2^n} \| \mathbb{1}_{\Omega_n} (x) f(x) \|_2 <\infty \}.$$
Here $$\mathbb{1}_{\Omega_n}(x)$$ is the indicator function onto the sets $$\Omega_n := \{ x \in \mathbb{R} : 2^{n-1} \leqslant |x| < 2^n \}$$, $$n \geqslant 1$$, and $$\Omega_0 := \{x \in \mathbb{R} : |x| < 1 \}$$. Note $$\{ \Omega_n \}$$ is a partition of $$\mathbb{R}$$.
Then one can show that the following inclusions hold (I can include a proof if requested) :
$$L^2_s \subsetneq B \subsetneq L^2_{1/2}, \quad \forall s >1/2.$$
Now let us define another type of weighted $$L2$$ space involving logarithms. For $$s,p \geqslant 1/2$$,
$$L^2_{s,p} := \{ f \in L^2(\mathbb{R}) : \| (2+|x|)^s \left(\log(2+|x|)\right)^p f(x) \|_2 < \infty \}.$$
Then one can also show that the following inclusions hold :
$$L^2_s \subsetneq L^2 _{1/2,p} \subsetneq B \subsetneq L^2_{1/2}, \quad \forall s,p >1/2.$$
We also have trivially
$$L^2_s \subsetneq L^2 _{1/2,p} \subsetneq L^2_{1/2,1/2} \subsetneq L^2_{1/2}, \quad \forall s,p >1/2.$$
My question is : what is the relationship between $$L^2_{1/2,1/2}$$ and $$B$$ ? Is one included in the other ? Thanks for any tips or references
• Is there a reason you take indicator functions instead of a smooth partition of unity in the definition of $B$? – MaoWao Feb 17 '20 at 12:43
• @MaoWao Since we're talking about $L^2$ and not continuous functions, smoothness of the partition of unity is irrelevant. – David C. Ullrich Feb 17 '20 at 13:15
• @MaoWao That's the way it seems to me, yes. – David C. Ullrich Feb 17 '20 at 13:32
• @DavidC.Ullrich Sorry, I deleted my previous comment. The space $B$ is of course not $F^{-1}(B^s_{2,2})$, but $F^{-1}(B^s_{2,1})$, so the claim $B\subsetneq L^2_{1/2}$ makes sense. – MaoWao Feb 17 '20 at 13:34
• @MaoWao "Sorry, I deleted my previous comment.": heh, sorry I agreed with it then. Wasn't paying attention to the details, sorry. – David C. Ullrich Feb 17 '20 at 13:37
Let $$a_n:= \sqrt{\int_{\Omega_n}f^2}$$. A function $$f$$ belongs to $$B$$ if and only if $$\sum_{n\geqslant 0}2^{n/2}a_n$$ is finite, and to $$L^2_{1/2,1/2}$$ if and only if $$\sum_{n\geqslant 0} 2^nna_n^2$$ is finite.
• Let $$a_n=2^{-n/2}n^{-1}(\log(n+2))^{-3/4}$$: then $$f$$ belongs to $$L^2_{1/2,1/2}$$ but not to $$B$$.
• If $$a_{2^N}=N^{-4}2^{-2^N/2}$$ and $$a_n=0$$ if $$N$$ is not of the form $$2^N$$ for some $$N$$, then $$f$$ belongs to $$B$$ but not to $$L^2_{1/2,1/2}$$.
• thanks a lot Davide! (maybe just add a - in $2^2^N$) – Marc_Adrien Feb 18 '20 at 0:08 | 2021-01-24 03:33:16 | {"extraction_info": {"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": 39, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9377675652503967, "perplexity": 191.84378402398096}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703544403.51/warc/CC-MAIN-20210124013637-20210124043637-00738.warc.gz"} |
https://datascience.stackexchange.com/questions/89300/how-do-you-use-ks-test-in-a-data-science-report/89304 | # How do you use KS-test in a data science report?
I'm writing a data science report, I want to find an exist distribution to fit the sample. I got a good looking result , but when I use KS-test to test the model, I got a low p-value,1.2e-4, definitely I should reject the model.
I mean, whatever what distribution/model you use to fit the sample, you cannot expect to have a perfect result, especially working with huge amount of data. So what does KS-test do in a data science report? Does it means only if we got high p-value in KS-test then the model is correct?
• this must be a stupid question, in real world, do we always be able to find a fitting curve with high enough p-value? And what if we can't, then we have to disprove our assumption? – Carl Feb 12 at 20:27
• In my sole opinion: I do use to measure the discriminatory power of the model, In other words if my model does distinguish between events and non-events(looking at first 4 deciles). In this way KS I do use for model selection aka what model does perform better for my problem/task I try to solve (not only logit models). And I do use on the large amount of data (hundreds of millions of obs) ... on what u do show is to test if your data comes from same distribution and I do not see usage on validating model performance ... if u agree With my approach I will post detailed answer. – n1tk Feb 13 at 16:17
• There is no contradiction. Your chosen distribution is a pretty good fit. The low p-value (loosely speaking) says that it is not a perfect fit. If you have a fairly large sample size, then you have the sensitivity to detect even small deviations from the fitted distribution. You might be interested in this discussion over at the statistics Stack: stats.stackexchange.com/q/2492/247274. – Dave Apr 14 at 10:13
In your case, the null hypothesis $$H_0$$ is that your sample follows your the distribution that your model has learned. The alternative hypothesis $$H_1$$ is that it follows some other distribution. Assuming you have fixed your significance level $$\alpha$$ to be $$0.05$$ (the most common choice for $$\alpha$$, but up to you if you want to go lower), getting a p-value lower than that means you should reject the null hypothesis.
The p-value can be interpreted as the probability of a type I error, in other words a false positive: the probability that you reject the null hypothesis when it is in fact true. In your case, rejecting the hypothesis means stating that there is statistically significant evidence that the distribution your model has learned is not the underlying distribution of the sample. So yes, you would like as large a p-value as possible.
You are using a Kolmogorov-Smirnov test to compare your sample to a reference distribution, in this case, so it's a one-sample KS test. The way I would put it is that getting a high p-value means that: "it is highly unlikely that your model has learned a wrong distribution". In other words, it is highly likely it has learned a pretty good approximation of the underlying distribution. However, nothing is certain when doing statistical hypothesis testing!
I'm not sure what you're showing on your plots though, since there doesn't seem to be an empirical cumulative distribution function on them (lines look smooth).
• Yeah, I think I understand the meaning of p-value. To clarify, I used two Pareto distribution to form an empirical distribution and find the parameters by maximum likelihood, and the blue line in first plot is the sample's CDF, and the red line is the fitting CDF. What confused me is, even if I used log scale when plotting, this empirical distribution seem fit the sample great, but I still got a pretty low p-value. So in this case, I should change my assumption to find other distribution to fit the sample? – Carl Feb 12 at 19:40
• The test statistic of the KS test is the Kolmogorov Smirnov statistic, which is the greatest distance between your empirical and predicted distributions. The test statistic of a test is itself assumed to follow a distribution "in the wild", here the Kolmogorov distribution. The p-value is defined (here, for a one-sided right-tail test) as the probability of something more extreme (i.e. distance between the two distributions even greater). One weakness of the KS test is few & far between data points, which leads to large KS statistic which leads to low p-value. How many data points do you have? – David Cian Feb 13 at 0:09
• When you're saying "empirical distribution" do you mean you sampled the two Pareto distributions to give a discrete set of samples or just used them as is? – David Cian Feb 13 at 0:10
The p-value is interpreted as the probability of a type I error. In other words a false positive: the probability that you reject the null hypothesis when it is in fact true.
You are invoking a Kolmogorov-Smirnov test.
" when I use KS-test to test the model, I got a low p-value,1.2e-4, definitely I should reject the model." Answer - Your low p-value does not indicate that the observed distribution does not fit expected distribution. p value simply indicates the chance for comiting type - 1 error which is quite low in your case. The low value of p i.e. alpha implies that your model predicts very well. The null hypothesis of no difference between two distributions (observed and predicted) is accepted. In nutshell, the test confirms validity of your model.
• -1 The p-value, loosely speaking, is the probability of getting the observations you got if the null hypothesis is true. Thus, the low p-value is evidence against the null hypothesis. – Dave Apr 14 at 10:08 | 2021-05-12 14:05:29 | {"extraction_info": {"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": 5, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.732630729675293, "perplexity": 451.8096761960495}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243990929.24/warc/CC-MAIN-20210512131604-20210512161604-00142.warc.gz"} |
https://www.aimsciences.org/article/doi/10.3934/dcdsb.2013.18.1217 | # American Institute of Mathematical Sciences
July 2013, 18(5): 1217-1251. doi: 10.3934/dcdsb.2013.18.1217
## The existence of weak solutions to immiscible compressible two-phase flow in porous media: The case of fields with different rock-types
1 Laboratoire de Mathématiques et de leurs Applications, CNRS-UMR 5142, Université de Pau, Av. de l’Université, 64000 Pau 2 Department of Mathematics, B.Verkin Institute for Low Temperature Physics and Engineering, 47, av. Lenin, 61103, Kharkov 3 Narvik University College, Postbox 385, Narvik, 8505
Received September 2011 Revised December 2012 Published March 2013
We study a model describing immiscible, compressible two-phase flow, such as water-gas, through heterogeneous porous media taking into account capillary and gravity effects. We will consider a domain made up of several zones with different characteristics: porosity, absolute permeability, relative permeabilities and capillary pressure curves. This process can be formulated as a coupled system of partial differential equations which includes a nonlinear parabolic pressure equation and a nonlinear degenerate diffusion-convection saturation equation. Moreover the transmission conditions are nonlinear and the saturation is discontinuous at interfaces separating different media. There are two kinds of degeneracy in the studied system: the first one is the degeneracy of the capillary diffusion term in the saturation equation, and the second one appears in the evolution term of the pressure equation. Under some realistic assumptions on the data, we show the existence of weak solutions with the help of an appropriate regularization and a time discretization. We use suitable test functions to obtain a priori estimates. We prove a new compactness result in order to pass to the limit in nonlinear terms. This passage to the limit is nontrivial due to the degeneracy of the system.
Citation: Brahim Amaziane, Leonid Pankratov, Andrey Piatnitski. The existence of weak solutions to immiscible compressible two-phase flow in porous media: The case of fields with different rock-types. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1217-1251. doi: 10.3934/dcdsb.2013.18.1217
##### References:
[1] H. W. Alt and E. Di Benedetto, Nonsteady flow of water and oil through inhomogeneous porous media,, Ann. Scu. Norm. Sup. Pisa Cl. Sci., 12 (1985), 335. Google Scholar [2] H. W. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations,, Math. Z., 3 (1983), 311. doi: 10.1007/BF01176474. Google Scholar [3] B. Amaziane, S. Antontsev, L. Pankratov and A. Piatnitski, Homogenization of immiscible compressible two-phase flow in porous media: Application to gas migration in a nuclear waste repository,, SIAM J. Multiscale Model. Simul., 8 (2010), 2023. doi: 10.1137/100790215. Google Scholar [4] B. Amaziane and M. Jurak, A new formulation of immiscible compressible two-phase flow in porous media,, C. R. Mécanique, 336 (2008), 600. Google Scholar [5] B. Amaziane, M. Jurak and A. Žgaljić-Keko, Modeling and numerical simulations of immiscible compressible two-phase flow in porous media by the concept of global pressure,, Transport in Porous Media, 84 (2010), 133. doi: 10.1007/s11242-009-9489-8. Google Scholar [6] B. Amaziane, M. Jurak and A. Žgaljić-Keko, An existence result for a coupled system modeling a fully equivalent global pressure formulation for immiscible compressible two-phase flow in porous media,, J. Differential Equations, 250 (2011), 1685. doi: 10.1016/j.jde.2010.09.008. Google Scholar [7] Y. Amirat, K. Hamdache and A. Ziani, Mathematical analysis for compressible miscible displacement models in porous media,, Math. Models Methods Appl. Sci., 6 (1996), 729. doi: 10.1142/S0218202596000316. Google Scholar [8] Y. Amirat and M. Moussaoui, Analysis of a one-dimensional model for compressible miscible displacement in porous media,, SIAM J. Math. Anal., 26 (1995), 659. doi: 10.1137/S003614109223297X. Google Scholar [9] Y. Amirat and V. Shelukhin, Global weak solutions to equations of compressible miscible flows in porous media,, SIAM J. Math. Anal., 38 (2007), 1825. doi: 10.1137/050640321. Google Scholar [10] ANDRA, Couplex-Gas Benchmark, 2006., Available online at: , (). Google Scholar [11] S. N. Antontsev, A. V. Kazhikhov and V. N. Monakhov, "Boundary Value Problems in Mechanics of Nonhomogeneous Fluids,", North-Holland, (1990). Google Scholar [12] T. Arbogast, The existence of weak solutions to single-porosity and simple dual-porosity models of two-phase incompressible flow,, Nonlinear Anal., 19 (1992), 1009. doi: 10.1016/0362-546X(92)90121-T. Google Scholar [13] J. Bear and Y. Bachmat, "Introduction to Modeling of Transport Phenomena in Porous Media,", Kluwer Academic Publishers, (1991). Google Scholar [14] A. Bourgeat and A. Hidani, A result of existence for a model of two-phase flow in a porous medium made of different rock types,, Appl. Anal., 56 (1995), 381. doi: 10.1080/00036819508840332. Google Scholar [15] G. Chavent and J. Jaffré, "Mathematical Models and Finite Elements for Reservoir Simulation,", North-Holland, (1986). Google Scholar [16] Z. Chen, Degenerate two-phase incompressible flow I: Existence, uniqueness and regularity of a weak solution,, J. Differential Equations, 171 (2001), 203. doi: 10.1006/jdeq.2000.3848. Google Scholar [17] Z. Chen and G. Huan and Y. Ma, "Computational Methods for Multiphase Flows in Porous Media,", SIAM, (2006). doi: 10.1137/1.9780898718942. Google Scholar [18] C. Choquet, On a fully coupled nonlinear parabolic problem modelling miscible compressible displacement in porous media,, J. Math. Anal. Appl., 339 (2008), 1112. doi: 10.1016/j.jmaa.2007.07.037. Google Scholar [19] F. Z. Daïm, R. Eymard and D. Hilhorst, Existence of a solution for two phase flow in porous media: The case that the porosity depends on the pressure,, J. Math. Anal. Appl., 326 (2007), 332. doi: 10.1016/j.jmaa.2006.02.082. Google Scholar [20] B. K. Fadimba, On existence and uniqueness for a coupled system modeling immiscible flow through a porous medium,, J. Math. Anal. Appl., 328 (2007), 1034. doi: 10.1016/j.jmaa.2006.06.012. Google Scholar [21] G. Gagneux and M. Madaune-Tort, "Analyse Mathématique de Modèles Non Linéaires de l'Ingénierie Pétrolière,", Springer-Verlag, (1996). Google Scholar [22] X. Feng, Strong solutions to a nonlinear parabolic system modeling compressible miscible displacement in porous media,, Nonlinear Anal., 23 (1994), 1515. doi: 10.1016/0362-546X(94)90202-X. Google Scholar [23] C. Galusinski and M. Saad, On a degenerate parabolic system for compressible, immiscible, two-phase flows in porous media,, Adv. Differential Equations, 9 (2004), 1235. Google Scholar [24] C. Galusinski and M. Saad, Water-gas flow in porous media,, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 281. Google Scholar [25] C. Galusinski and M. Saad, Two compressible immiscible fluids in porous media,, J. Differential Equations, 244 (2008), 1741. doi: 10.1016/j.jde.2008.01.013. Google Scholar [26] C. Galusinski and M. Saad, Weak solutions for immiscible compressible multifluid flows in porous media,, C. R. Acad. Sci. Paris, 347 (2009), 249. doi: 10.1016/j.crma.2009.01.023. Google Scholar [27] D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Springer-Verlag, (1983). Google Scholar [28] R. Helmig, "Multiphase Flow and Transport Trocesses in the Subsurface,", Springer, (1997). Google Scholar [29] Z. Khalil and M. Saad, Solutions to a model for compressible immiscible two phase flow in porous media,, Electronic Journal of Differential Equations, 122 (2010), 1. Google Scholar [30] Z. Khalil and M. Saad, On a fully nonlinear degenerate parabolic system modeling immiscible gas-water displacement in porous media,, Nonlinear Analysis: Real World Applications, 12 (2011), 1591. doi: 10.1016/j.nonrwa.2010.10.015. Google Scholar [31] D. Kröner and S. Luckhaus, Flow of oil and water in a porous medium,, J. Differential Equations, 55 (1984), 276. doi: 10.1016/0022-0396(84)90084-6. Google Scholar [32] A. Mikelić, An existence result for the equations describing a gas-liquid two-phase flow,, C. R. Mécanique, 337 (2009), 226. Google Scholar [33] OECD/NEA, Safety of geological disposal of high-level and long-lived radioactive waste in France,, An International Peer Review of the, (2006). Google Scholar [34] J. Simon, Compact sets in the space $L^p(0,t; B)$,, Ann. Mat. Pura Appl., 146 (1987), 65. doi: 10.1007/BF01762360. Google Scholar [35] F. Smaï, A model of multiphase flow and transport in porous media applied to gas migration in underground nuclear waste repository,, C. R. Math. Acad. Sci. Paris, 347 (2009), 527. doi: 10.1016/j.crma.2009.03.011. Google Scholar [36] F. Smaï, Existence of solutions for a model of multiphase flow in porous media applied to gas migration in underground nuclear waste repository,, Appl. Anal., 88 (2009), 1609. doi: 10.1080/00036810902942226. Google Scholar [37] L. M. Yeh, On two-phase flow in fractured media,, Math. Models Methods Appl. Sci., 12 (2002), 1075. doi: 10.1142/S0218202502002045. Google Scholar [38] L. M. Yeh, Hölder continuity for two-phase flows in porous media,, Math. Methods Appl. Sci., 29 (2006), 1261. doi: 10.1002/mma.724. Google Scholar
show all references
##### References:
[1] H. W. Alt and E. Di Benedetto, Nonsteady flow of water and oil through inhomogeneous porous media,, Ann. Scu. Norm. Sup. Pisa Cl. Sci., 12 (1985), 335. Google Scholar [2] H. W. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations,, Math. Z., 3 (1983), 311. doi: 10.1007/BF01176474. Google Scholar [3] B. Amaziane, S. Antontsev, L. Pankratov and A. Piatnitski, Homogenization of immiscible compressible two-phase flow in porous media: Application to gas migration in a nuclear waste repository,, SIAM J. Multiscale Model. Simul., 8 (2010), 2023. doi: 10.1137/100790215. Google Scholar [4] B. Amaziane and M. Jurak, A new formulation of immiscible compressible two-phase flow in porous media,, C. R. Mécanique, 336 (2008), 600. Google Scholar [5] B. Amaziane, M. Jurak and A. Žgaljić-Keko, Modeling and numerical simulations of immiscible compressible two-phase flow in porous media by the concept of global pressure,, Transport in Porous Media, 84 (2010), 133. doi: 10.1007/s11242-009-9489-8. Google Scholar [6] B. Amaziane, M. Jurak and A. Žgaljić-Keko, An existence result for a coupled system modeling a fully equivalent global pressure formulation for immiscible compressible two-phase flow in porous media,, J. Differential Equations, 250 (2011), 1685. doi: 10.1016/j.jde.2010.09.008. Google Scholar [7] Y. Amirat, K. Hamdache and A. Ziani, Mathematical analysis for compressible miscible displacement models in porous media,, Math. Models Methods Appl. Sci., 6 (1996), 729. doi: 10.1142/S0218202596000316. Google Scholar [8] Y. Amirat and M. Moussaoui, Analysis of a one-dimensional model for compressible miscible displacement in porous media,, SIAM J. Math. Anal., 26 (1995), 659. doi: 10.1137/S003614109223297X. Google Scholar [9] Y. Amirat and V. Shelukhin, Global weak solutions to equations of compressible miscible flows in porous media,, SIAM J. Math. Anal., 38 (2007), 1825. doi: 10.1137/050640321. Google Scholar [10] ANDRA, Couplex-Gas Benchmark, 2006., Available online at: , (). Google Scholar [11] S. N. Antontsev, A. V. Kazhikhov and V. N. Monakhov, "Boundary Value Problems in Mechanics of Nonhomogeneous Fluids,", North-Holland, (1990). Google Scholar [12] T. Arbogast, The existence of weak solutions to single-porosity and simple dual-porosity models of two-phase incompressible flow,, Nonlinear Anal., 19 (1992), 1009. doi: 10.1016/0362-546X(92)90121-T. Google Scholar [13] J. Bear and Y. Bachmat, "Introduction to Modeling of Transport Phenomena in Porous Media,", Kluwer Academic Publishers, (1991). Google Scholar [14] A. Bourgeat and A. Hidani, A result of existence for a model of two-phase flow in a porous medium made of different rock types,, Appl. Anal., 56 (1995), 381. doi: 10.1080/00036819508840332. Google Scholar [15] G. Chavent and J. Jaffré, "Mathematical Models and Finite Elements for Reservoir Simulation,", North-Holland, (1986). Google Scholar [16] Z. Chen, Degenerate two-phase incompressible flow I: Existence, uniqueness and regularity of a weak solution,, J. Differential Equations, 171 (2001), 203. doi: 10.1006/jdeq.2000.3848. Google Scholar [17] Z. Chen and G. Huan and Y. Ma, "Computational Methods for Multiphase Flows in Porous Media,", SIAM, (2006). doi: 10.1137/1.9780898718942. Google Scholar [18] C. Choquet, On a fully coupled nonlinear parabolic problem modelling miscible compressible displacement in porous media,, J. Math. Anal. Appl., 339 (2008), 1112. doi: 10.1016/j.jmaa.2007.07.037. Google Scholar [19] F. Z. Daïm, R. Eymard and D. Hilhorst, Existence of a solution for two phase flow in porous media: The case that the porosity depends on the pressure,, J. Math. Anal. Appl., 326 (2007), 332. doi: 10.1016/j.jmaa.2006.02.082. Google Scholar [20] B. K. Fadimba, On existence and uniqueness for a coupled system modeling immiscible flow through a porous medium,, J. Math. Anal. Appl., 328 (2007), 1034. doi: 10.1016/j.jmaa.2006.06.012. Google Scholar [21] G. Gagneux and M. Madaune-Tort, "Analyse Mathématique de Modèles Non Linéaires de l'Ingénierie Pétrolière,", Springer-Verlag, (1996). Google Scholar [22] X. Feng, Strong solutions to a nonlinear parabolic system modeling compressible miscible displacement in porous media,, Nonlinear Anal., 23 (1994), 1515. doi: 10.1016/0362-546X(94)90202-X. Google Scholar [23] C. Galusinski and M. Saad, On a degenerate parabolic system for compressible, immiscible, two-phase flows in porous media,, Adv. Differential Equations, 9 (2004), 1235. Google Scholar [24] C. Galusinski and M. Saad, Water-gas flow in porous media,, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 281. Google Scholar [25] C. Galusinski and M. Saad, Two compressible immiscible fluids in porous media,, J. Differential Equations, 244 (2008), 1741. doi: 10.1016/j.jde.2008.01.013. Google Scholar [26] C. Galusinski and M. Saad, Weak solutions for immiscible compressible multifluid flows in porous media,, C. R. Acad. Sci. Paris, 347 (2009), 249. doi: 10.1016/j.crma.2009.01.023. Google Scholar [27] D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Springer-Verlag, (1983). Google Scholar [28] R. Helmig, "Multiphase Flow and Transport Trocesses in the Subsurface,", Springer, (1997). Google Scholar [29] Z. Khalil and M. Saad, Solutions to a model for compressible immiscible two phase flow in porous media,, Electronic Journal of Differential Equations, 122 (2010), 1. Google Scholar [30] Z. Khalil and M. Saad, On a fully nonlinear degenerate parabolic system modeling immiscible gas-water displacement in porous media,, Nonlinear Analysis: Real World Applications, 12 (2011), 1591. doi: 10.1016/j.nonrwa.2010.10.015. Google Scholar [31] D. Kröner and S. Luckhaus, Flow of oil and water in a porous medium,, J. Differential Equations, 55 (1984), 276. doi: 10.1016/0022-0396(84)90084-6. Google Scholar [32] A. Mikelić, An existence result for the equations describing a gas-liquid two-phase flow,, C. R. Mécanique, 337 (2009), 226. Google Scholar [33] OECD/NEA, Safety of geological disposal of high-level and long-lived radioactive waste in France,, An International Peer Review of the, (2006). Google Scholar [34] J. Simon, Compact sets in the space $L^p(0,t; B)$,, Ann. Mat. Pura Appl., 146 (1987), 65. doi: 10.1007/BF01762360. Google Scholar [35] F. Smaï, A model of multiphase flow and transport in porous media applied to gas migration in underground nuclear waste repository,, C. R. Math. Acad. Sci. Paris, 347 (2009), 527. doi: 10.1016/j.crma.2009.03.011. Google Scholar [36] F. Smaï, Existence of solutions for a model of multiphase flow in porous media applied to gas migration in underground nuclear waste repository,, Appl. Anal., 88 (2009), 1609. doi: 10.1080/00036810902942226. Google Scholar [37] L. M. Yeh, On two-phase flow in fractured media,, Math. Models Methods Appl. Sci., 12 (2002), 1075. doi: 10.1142/S0218202502002045. Google Scholar [38] L. M. Yeh, Hölder continuity for two-phase flows in porous media,, Math. Methods Appl. Sci., 29 (2006), 1261. doi: 10.1002/mma.724. Google Scholar
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2019 Impact Factor: 1.27 | 2021-01-16 10:02:41 | {"extraction_info": {"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, "math_score": 0.6490615606307983, "perplexity": 8542.679049854913}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703505861.1/warc/CC-MAIN-20210116074510-20210116104510-00361.warc.gz"} |
https://math.stackexchange.com/questions/3369338/discuss-the-existence-and-uniquness-of-ivp-y-gx-dfracdydx-y0-1 | # Discuss the existence and uniquness of IVP $y=g(x)\dfrac{dy}{dx}, y(0)=1$
Discuss the existence and uniquness of IVP $$y=g(x)\dfrac{dy}{dx}, y(0)=1$$ where
$$g(x)= \dfrac{\sin(x)}{x}; x\ne 0$$
and
$$g(x)=1; x=0$$
I calculated $$dy/dx=f(x)$$ first:
$$f(x)= \dfrac{xy}{\sin(x)}; x \ne 0$$ and $$f(x)=y; x=0$$
For existence, $$f(x)$$ must be continuous, since the limit $$\dfrac{xy}{\sin(x)}$$ for $$(0,1)$$ tends to $$1$$, the function $$f(x)$$ is continuous. Hence, a solution to the IVP exists.
Now, for uniqueness $$\dfrac{\partial f(x)}{\partial y}$$ must be continuous.
$$\dfrac{\partial f(x)}{\partial y} = \dfrac{x}{\sin x}; x \ne 0$$ and $$\dfrac{\partial f(x)}{\partial y} = 1; x=0$$ Hence, $$\dfrac{\partial f(x)}{\partial y}$$ is continuous.
Therefore, the solution to the given IVP exists and is unique.
My question is, are there any errors in what I did? Or, if I write this solution in my math exam, due to what missing conditions/calculations/writing style may I lose credits? | 2022-08-08 22:08:45 | {"extraction_info": {"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": 15, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9512470960617065, "perplexity": 347.8149671517305}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882570879.1/warc/CC-MAIN-20220808213349-20220809003349-00388.warc.gz"} |
https://math.stackexchange.com/questions/3110079/prove-a-figure-is-a-cyclic-quadrilateral | # Prove a figure is a cyclic quadrilateral
In the figure below, $$O$$ is the center of the circle. If angle CPB is $$90^\circ$$, then prove that $$AOEF$$ is a cyclic quadrilateral.
Connect $$OB$$. $$\angle AOP=\angle POB$$ and $$\angle AOP+\angle POB=2\angle AFE$$ implies $$\angle AOP=\angle AFE$$ and therefore $$AOEF$$ cocyclic. | 2019-06-18 20:37:53 | {"extraction_info": {"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": 8, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9473589658737183, "perplexity": 207.79208461400628}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627998817.58/warc/CC-MAIN-20190618203528-20190618225528-00443.warc.gz"} |
https://greekgodofstats.com/2019/11/26/what-is-tpa/ | # What is TPA?
Chances are if you’re into sports you’ve seen the famed charts from @NBA_Math that feature overlapping pictures of NBA players in a conventional Cartesian plot with a line plotted on it (y=-x). Anything above the line is good, anything below is bad, and average values will tend to walk the line. The graph is a visual attempt to quantify some mystery statistic known as TPA.
What is TPA?
This is something of a loaded question; the acronym TPA stands for “Total Points Added”. The basic idea behind it is that a player adds points on offense and defense. You then total these subsections to get TPA.
Unfortuantely, the definition above is rather incomplete. We now need to understand Offensive Points Added (OPA) and Defensive Points Saved (DPS), the components which make up TPA. The two subcategories are much more complex than TPA alone.
To get OPA and DPS, we need to use “Box Plus/Minus,” an all-in-one statistic created by Daniel Myers and hosted at basketball-reference.com. I promise we are almost at the bottom of the well here in terms of stat definitions. Box Plus/Minus is a relativistic stat that gauges a player’s impact on team performance when s/he is on the court. S/he again will have an impact on both defense and offense, so accordingly Box Plus/Minus can break down into two stats: Offensive Box Plus/Minus (OPBM) and Defensive Box Plus/Minus (DPBM). We can already see that TPA has the same structure as Box Plus Minus; both purport to measure a player’s impact on both ends of the floor. What is the difference between the two?
According to NBA Math, Offensive Points Added and Defensive Points Saved are the Box Plus/Minus (BPM) stats scaled by the number of possessions played by a player. Looking at their numbers, however, I am not totally convinced this is true. Maybe my math is off, but it appears that TPA is a function of the minutes played by each player, and not exactly the possessions. However, if we assume all NBA teams play the same number of possessions, this works out.
Is that a fair assumption? Does pace matter?
### Impact of Pace
If we look at league Pace statistics (Figure 1, below), it is a reasonably safe assumption that teams play nearly the same amount of possessions (although I personally am not fond of this generalization). Pace is the amount of possessions played by each team per game or per 48 minutes. The average pace of the NBA is 101.74 possessions per game, so assuming that pace is consistent across the board is not that bad considering the relative standard deviation is only 2.50%.
Is Box Plus/Minus the best metric for individual performance? In my mind it is not the best metric, but it is a decent one. I will defer to others to produce a better model. For the purposes of the present study, however, we can move forward with a basic understanding of BPM as the grounding of Total Points Added.
Please, do not be fooled here, Box Plus/Minus is not agnostic to team performance. This stat is a function of the individual and the team. The idea is that the individual’s performance is a much larger contributor, and with the varied lineups a player will play in, it all should average out. It is absolutely necessary to acknowledge that assumption and comprehend its implications before moving forward.
### Calculating TPA
Having these definitions ready to hand, we can now break down Offensive Points Added and Defensive Points Saved into terms of quantifiable quantities OBPM and DBPM. (To have complete transparency here, OBPM and DBPM also break down further into a host of other measurable stats as well (on which, cf. https://www.sports-reference.com/blog/2014/10/introducing-box-plusminus-bpm-2/).
We have a little math to do now:
$\boldsymbol{\mathbf{OPA}}&space;\alpha&space;(&space;OBPM*&space;Pace*Minutes&space;Played*Fudge&space;Factor)$ $\boldsymbol{\mathbf{DPS}}&space;\alpha&space;(&space;DBPM*&space;Pace*Minutes&space;Played*Fudge&space;Factor)$
Equation 2(a and b). OPA and DPS functions
Do not get scared away here, the hardest math here will be algebra, so do not fret! The payoff is coming. I have suffered with the math and made a trend-line to break down the function for OPA to be the following:
$OPA=&space;OBPM*(Pace*Minutes&space;Played*Fudge&space;Factor_1+Fudge&space;Factor_2&space;)$
Equation 3. The fudge factors. These numbers, fudge factors as I like to call them, help massage the correlation between OPA and OBPM. This will help us ascertain the relationship a little better and allow us to get numbers similar to NBA Math’s numbers.
For those interested in the math, I plotted OPA/OPBM vs. Minutes played (for the Houston Rockets to keep Pace the same) and generated a best fit line to ascertain an equation for the standard line, where m= Fudge Factor1 and b= Fudge Factor2 . If you do not like math, or think there is a better way to generate your correlation skip forward and don’t give this any mind.
DPS is calculated with the same method AND it uses the same fudge factors! How convenient! This discovery exposes a flaw in the method, since the method assumes that teams play the same proportion of minutes on offensive possessions and defensive possessions. This is not always true. If we had more granularity we could break down time spent on offense and time spent on defense, then multiply by their respective box score statistics. I do not know if the impact is significant, but again, it is an assumption worth noting nonetheless.
### Checking our Solution
A general rule of thumb in science to check if an equation is correct is to check if the units balance out (I told you the units would be important later on). If I have an equation where meters = seconds, I might have done something wrong.
The units are as follows:
$Contribution=&space;\frac{Contributions}{Possessions}*\frac{Possessions}{Minutes}*Minutes$
So now let’s do some algebra:
$Contribution=&space;\frac{Contributions}{{\color{Red}&space;Possessions}}*\frac{{\color{Red}&space;Possessions}}{{\color{Red}&space;Minutes}}*{\color{Red}&space;Minutes&space;}$
Since all the units marked in red cancel out, we are left with the following:
$Contribution=&space;Contributions$
Cool. This is intuitive and it makes sense. Now we know how Total Points Added is derived. In the following posts I will outline how you too can make those NBA Math TPA charts in Excel, Python, and then R. You will notice the trend of automatibility, difficulty, and overall versatility follows the order listed above.
About the author: My name is Alan Moghaddam and I am a chemist. I got my BS in Biochemistry and my Ph.D. in Chemistry. I’m not a true stats guy, even though I love math and numbers. My approach and my goal is to give anyone the tools they need to be able to understand what is being fed to them by sports writers and stat heads. So, if anything, my posts will be a journey for both of us, and I hope you enjoy it. | 2020-09-19 10:03:46 | {"extraction_info": {"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": 12, "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, "math_score": 0.5191843509674072, "perplexity": 1277.799467230949}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400191160.14/warc/CC-MAIN-20200919075646-20200919105646-00041.warc.gz"} |
https://electronics.stackexchange.com/questions/444231/controlling-mosfet-with-attiny85-and-ir-reflective-sensor/444468 | # Controlling Mosfet with ATTiny85 and IR Reflective Sensor
My plan is to start a motor, then stop the motor when the gear teeth gets to 15 (out of 16) I will be able to tell at what position the gear is by placing an IR reflective sensor under the teeth. The sensor is a VCNT2020. I am controlling the motor with three MOSFETs in the schematic here (Leaving out M1):
And using the VCNT2020 Sensor in the schematic here:
And finally using an ATtiny85 as the microcontroller. I was originally going to use an arduino, but there is not that much IO so I thought it wouldn't be worth it.
My question is, will the motor be able to stop in time, should I begin to stop it on an earlier gear or will I need to do some kind of braking where I use a fourth FET to short the motor terminals. Also, I would like you to take a look at my unfinished code and ask if you could suggest anything. The reading of the sensor pin and the digital writing to the Mosfet need to be very fast otherwise the motor could do another few rotations and the gear go another tooth, going past the 15th one. I don't mind if it is exactly 15, but it needs to be 16 or below, and 13 or above.
Here is the motor, I couldn't find the datasheet but it runs on a maximum of 30 amps and 12.6v 3s lipo. The maximum rpm is 34,000
Code:
int gearcounter = 0;
int noiseAllowance = 10 // Ill change this when I 'calibrate' the design on a breadboard.
void setup() {
Serial.begin(9600);
pinMode(A1, INPUT); // Ir sensor pin
pinMode(5, OUTPUT); // Mosfet gate pin
}
void loop() {
delayMicroseconds(10)
if ((gearNow < gear10us - noiseAllowance) || (gear10us + noiseAllowance < gearNow)) {
gearcounter += 1;
}
}
if (gearcounter == 16) {//Maximum gear is 16, I just want to stop it one before the maximum
gearcounter = 1;
}
while(gearcounter < 15){
digitalWrite(5, HIGH);
}
}
Thanks
EDIT: My current schematic -
• Anybody want to 'tap' Olin for a 'cartoon' statement? – Voltage Spike Jun 18 at 18:28
• @laptop2d It's breadboard wiring diagrams that are cartoons. Other than that we've already have Olin's statement here. – Nick Alexeev Jun 20 at 1:36
• @NickAlexeev Thanks, forgot about that – Voltage Spike Jun 20 at 3:07
• My schematic is terrible, I didn’t use labels, the mosfets are different symbols which really annoyes me and the 5v regulator is attached to the mosfet circuits for some reason. I would fix this but I haven’t had that much time recently and I just plugged everything together in kicad. – Vosem Media Jun 20 at 6:28
Also, I would like you to take a look at my unfinished code and ask if you could suggest anything.
### The not-equal comparison will not work well with a real analog signal because of noise
gear10us = analogRead(A1);
delayMicroseconds(10)
// ...
}
Suppose you stop your motor, and your sensor is staring at the same thing. The signal from the IR sensor will be constant. But there also is going to be some noise riding on the signal. If you make two ADC readings you will get two integer numbers. These numbers will almost always be not-equal because of noise.
So, gear10us != analogRead(A1) could be always true, depending on the ADC resolution and the amount of noise.
gear10us = analogRead(A1);
delayMicroseconds(10)
if ((gearNow < gear10us - noiseAllowance) || (gear10us + noiseAllowance < gearNow)) {
// ...
}
### The while loop will never exit; it will loop infinitely
while(gearCounter < 15) {
You monitor the counter in the while loop. The counter would have to change while the loop is looping. But there's no way for that counter to change during the while loop, so the program will be stuck in the while loop.
Check the counter one time during the pass through the loop() function. Check for the dear (one time), then check the counter (one time without a while loop). | 2019-10-14 19:01:15 | {"extraction_info": {"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, "math_score": 0.21654045581817627, "perplexity": 2123.4179972257757}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986654086.1/warc/CC-MAIN-20191014173924-20191014201424-00455.warc.gz"} |
https://countrymusicstop.com/how-big-is-500-cubic-feet-update-new/ | Home » How Big Is 500 Cubic Feet? Update New
# How Big Is 500 Cubic Feet? Update New
Let’s discuss the question: how big is 500 cubic feet. We summarize all relevant answers in section Q&A of website Countrymusicstop in category: Technology. See more related questions in the comments below.
## How many square feet is 500 cubic feet?
125 cubic feet So for our example: 0.25 foot depth times 500 square feet equals 125 cubic feet.
## How many cubic feet are in a 10×10 room?
800 cubic feet Storage Unit Conversion Chart Storage Unit Size Storage Unit Space Equivalent ABF Trailer Space 5′ x 10′ Storage 400 cubic feet 6 linear feet 5′ x 15′ Storage 600 cubic feet 9 linear feet 10′ x 10′ Storage 800 cubic feet 12 linear feet 10′ x 15′ Storage 1200 cubic feet 17 linear feet 1 hàng khác
## How do you visualize a cubic foot?
A cubic foot is a space that measures 1 foot by 1 foot by 1 foot. To determine how many cubic feet a certain piece will be multiply the length x width x height of the piece. For example, if a dresser measures 4 feet long x 2 feet wide x 5 feet high it is 40 cubic feet.
## What size room is cubic feet?
Volume of a room: For the volume in cubic feet of our example room from above, simply multiply the width by the length by the height: (10 feet x 12 feet x 8 feet = 960 cubic feet). 27 thg 3, 2015
## What is the cubic feet of a 5 gallon bucket?
The volume of a five-gallon bucket can be expressed in cubic inches or cubic feet. One U.S. liquid gallon is equal to 0.134 cubic feet, which means a five-gallon bucket is equal to 0.670 cubic feet. A five-gallon bucket is equal to 1,155 cubic inches. 21 thg 7, 2017
## How many cubic feet is 24 square feet?
Cubic feet to Square feet Calculator 1 cubic feet = 1 ft2 1 ft2 = 23 cubic feet = 8.0876 ft2 23 ft2 = 24 cubic feet = 8.3203 ft2 24 ft2 = 25 cubic feet = 8.5499 ft2 25 ft2 = 26 cubic feet = 8.7764 ft2 26 ft2 = 25 hàng khác
How to Find Volume in Cubic Feet
How to Find Volume in Cubic Feet
## How many sq ft is 12×12?
144 square feet How many square feet is a 12×12 room? The square footage of a room 12 feet wide by 12 feet long is 144 square feet. Find the square footage by multiplying the width (12 ft) by the length (12 ft).
## Is cubic feet bigger than square feet?
Square feet are units of area and cubic feet are units of volume. Here is an example. 16 + 4 16 = 80 square feet on liner. The volume of the pool is 4 4 4 = 64 cubic feet so I need 64 cubic feet of water.
## How big is a cubic foot in feet?
Definition: A cubic measurement is the three-dimensional derivative of a linear measure, so a cubic foot is defined as the volume of a cube with sides 1 ft in length. In metric terms a cubic foot is a cube with sides 0.3048 metres in length. 22 thg 7, 2018
## Is it true that there are 12 cubic inches in a cubic foot?
There are 12 (vertical) layers, each one inch high. These layers are made up of 12 rows of cubes, each containing 12 cubes. So each layer has 12 12 unit cubes and, since there are 12 layers, there are 12 \times 12) total cubic inches in the cubic foot.
## What’s a cubic yard look like?
A cubic yard is the volume of a cube with the length, width and height of one yard (3 feet or 36 inches). One cubic yard is equal to 27 cubic feet. To help you picture this, the volume of two washing machines is just over a cubic yard.
## How large is a cubic?
A cubic foot is the volume of a cube with sides 1 ft in length, and it is a three-dimensional derivative of a linear measure. A 20 ft x 8 ft x 8 ft 6 in shipping container has a volume of over a thousand square feet.
## Is cubic feet the same as square feet?
The volume in cubic feet is equal to the area in square feet times the height in feet.
## How many cubic feet is my house?
Floor area (3000 sq feet) times height equals volume, so if you have 8 ft rooms, then you have 24000 cubic feet. If your rooms average 7 ft, you have 21000 cubic feet and so on. Divide the volume of your house by 6000 and you get the number of foggers you will need.
## How do you calculate cubic volume of a room?
Calculate the volume by multiplying the measured length and width of the space together, then multiply the result by the height of the room.
## How many gallons is 600 square feet?
Multiply the cubic feet by the amount of water per cubic foot, which is 7.48 gallons. In this example, you would multiply 7.48 times 600 to get 4,488 gallons of water.
## What is the standard size of a bucket?
What is the size of a standard bucket? The 5 gallon bucket is the most common size of bucket that is in use today.
## How many gallons go in a cubic foot?
7.48 gallons There are 7.48 gallons of water in one cubic foot of water.
## How do you convert square feet to cubic feet?
Square feet = cubic feet ÷ depth.
## How do you calculate square feet to cubic feet?
Square feet: You can find the area or square footage of a space by multiplying the length and width. Then multiply this figure by the height to find the cubic space, or cubic footage, within the object. 24 thg 2, 2022
## How do you convert square feet into cubic feet?
To convert a measurement in square feet to cubic feet, multiply by the measurement of the missing dimension (either length, width or height). Measure the object’s height in feet. (You have already measured the object’s length and width to get its measurement in square feet.) 6 thg 11, 2020
## How big is a room that is 600 square feet?
If you can’t quite visualize how big 600 square feet is yet, no worries. Six hundred square feet is about the size of a standard three-car garage that has enough space to park the cars and walk around comfortably. 14 thg 9, 2021
## How many square feet is an 8×10 room?
80 square feet Example: If the room is 8 feet long and 10 feet wide, the square footage of the room is 80 square feet (8 x 10 = 80). 19 thg 10, 2021
## How many square feet is 12×24?
288 sq ft At 12 feet by 24 feet, you’re looking at 12/3= 4 yards and 24/3 = 8 yards, giving you 32 square yards (288 sq ft).
## How tall is 7 cubic feet refrigerator?
LG Smart Inverter Refrigerator (7 cubic feet capacity / 5 feet tall)
## How much does a cubic foot cover?
Coverage Chart- Bagged Material (2 Cubic Foot Bags) Depth One 2 Cubic Foot Bag Covers 1” 24 sq feet 2” 12 sq feet 3” 8 sq feet 4” 6 sq feet 9 hàng khác • 29 thg 1, 2010
## How tall is 4 cubic feet?
ENDMEMO 1 cubic feet = 1 feet 1 cubic feet 3 cubic feet = 1.4422 feet 27 cubic feet 4 cubic feet = 1.5874 feet 64 cubic feet 5 cubic feet = 1.71 feet 125 cubic feet 6 cubic feet = 1.8171 feet 216 cubic feet 25 hàng khác
## How many cubic feet is a 50 lb bag of concrete?
One 50-pound bag of Quikrete Fast Setting Concrete Mix yields about . 375 cubic feet.
## How many cubic feet is an 80 pound bag of concrete?
80 lb bag of concrete cubic feet 80 lb bag of concrete yields 0.60 cubic feet which is approximately around 0.0222 cubic yards that will cover around 3.6 sq ft area up to standard depth of 2 inch excavation or backfills, and for 1 cubic feet of landfill you will need 1.66 bags of 80 lb concrete.
## What is the difference between an inch and a cubic inch?
A square inch is the area of a square with 1 inch sides. So when you have 15 inch sides, you simply multiply them together and get 225 square inches. Cubic inches are the same concept, except in 3 dimensions. A cubic inch is a 1x1x1 inch box.
## How many cubic yards is a pickup truck?
A regular size pick-up will hold three cubic yards of mulch (a full load). Two cubic yards is about body level full. When picking up soils, sands and gravels, one cubic yard is all that is recommended on a pick-up truck.
## How much cubic yards do I need?
Length in feet x Width in feet x Depth in feet (inches divided by 12). Take the total and divide by 27 (the amount of cubic feet in a yard). The final figure will be the estimated amount of cubic yards required.
## How many wheelbarrows are in a yard of dirt?
Depending upon your wheelbarrow size (i.e. 2 or 3 cubic feet per wheelbarrow load), it will take 9 to 14 full loads to equal 1 cubic yard.
## What is a cubic inch in inches?
One cubic inch is approximately 16.387 mL. One cubic foot is equal to exactly 1,728 cubic inches because 123 = 1,728. One U.S. gallon is equal to exactly 231 cubic inches. … Cubic inch 1 in3 in … … is equal to … SI derived units ≈ 16.387 mL US customary 1⁄231 US Gallon nonstandard 1⁄1728 ft3 7 hàng khác
## How many cubic are in a yard?
27 cubic feet A yard is 3 feet or 36 inches, and therefore, a cubic yard is 3 x 3 x 3, or 27 cubic feet(ft3).
## How many inches is 1 cubic?
ENDMEMO 1 cubic inch = 1 inch 1 cubic inch 2 cubic inch = 1.2599 inch 8 cubic inch 3 cubic inch = 1.4422 inch 27 cubic inch 4 cubic inch = 1.5874 inch 64 cubic inch 5 cubic inch = 1.71 inch 125 cubic inch 25 hàng khác
## How many square feet is in a cubic yard?
27 cubic feet One yard is equal to 3 feet. Hence to determine the number of cubic feet in a cubic yard, take the cube of both sides, and you will get 1 cubic yard = 27 cubic feet . 2 thg 2, 2022
## What is a cubic foot of soil?
bags of topsoil are in a cubic yard. There are 25.71404638 Dry Quarts in a Cubic Foot, so a 25 quart bag of potting soil would equal approximately 1 Cubic Foot. 2 thg 10, 2017
## How do you calculate cubic feet for landscaping?
Estimating Cubic Yards Measure square foot area (width x length) Choose the desired depth of material (convert inches to feet) 1″” = .08 ft. 2″” = .16 ft. 3″” = .25 ft. 4″” = .33 ft. 5″” = .42 ft. 6″” = .50 ft. … Square Feet Area x Depth (in feet) = Cubic Feet (width x length x depth) Divide Cubic Feet by 27 to get Cubic Yards.
## How many squares is my house?
Measure the length and width, in feet, of each room. Then, multiply the length by the width to calculate that room’s square footage. 14 thg 2, 2022
## What is the average volume of a house?
The average single family house in the United States has overall increased in size since 2000. It reached its peak of 2,467 square feet in 2015 before falling to 2,261 square feet by 2020. 28 thg 3, 2022
See also Внешние диски G-drive серии R и обычные для фотографа g technology ssd
## What is the rule of volume?
Volume = l × w × h , where l is length, w is width and h is height.
## How is the volume of a rectangular room measured?
Answer: To find the volume of a rectangular prism, multiply its 3 dimensions: length x width x height. The volume is expressed in cubic units. 21 thg 6, 2021
## How many square feet is a 10 gallon tank?
Gallon to Square feet Calculator 1 gallon = 0.2614 ft2 7.4805 gallon 10 gallon = 1.2135 ft2 236.5548 gallon 11 gallon = 1.2931 ft2 272.9108 gallon 12 gallon = 1.3704 ft2 310.9594 gallon 13 gallon = 1.4455 ft2 350.6282 gallon 25 hàng khác
## How many gallons is a 450 square foot pool?
13,500 gallons VOLUME OF POOL A pool that is 15 feet by 30 feet with a depth of 5 feet in the deep end and 3 feet in the shallow end would have a surface area of 450 square feet and an average depth of 4 feet and a volume of 13,500 gallons.
## How do you calculate cubic feet of water?
Multiply the length, width and height to get the volume in cubic feet. The formula is volume = length_width_height. The result is the volume of one half gallon in cubic feet, or in other words, the number of cubic feet in a half of a gallon. 25 thg 4, 2017
## How many gallons is a Home Depot bucket?
5 Gal The Home Depot 5 Gal.
## How tall is a 10 gallon bucket?
17 1/8 in CHOOSE OPTIONS U.S. Metric Height: 17 1/8 in 43.5 cm Volume Capacity [Nom]: 10 gal 37.9 L Volume Capacity [Max]: Volume Capacity [Min]: 10 hàng khác
## What sizes do plastic buckets come in?
Shop Buckets By Size 1 Gallon Buckets & Lids. 2 Gallon Buckets & Lids. 3 Gallon Buckets & Lids. 3.5 Gallon Buckets & Lids. 4 Gallon Buckets & Lids. 4.25 Gallon Buckets & Lids. 5 Gallon Buckets & Lids. 5.25 Gallon Buckets & Lids. Mục khác…
## What is the cubic feet of 1 gallon?
Gallon (US) to Cubic Foot Conversion Table Gallon (US) [gal (US)] Cubic Foot [ft^3] 1 gal (US) 0.1336805556 ft^3 2 gal (US) 0.2673611111 ft^3 3 gal (US) 0.4010416667 ft^3 5 gal (US) 0.6684027778 ft^3 7 hàng khác
## What is the cubic feet of a 5 gallon bucket?
The volume of a five-gallon bucket can be expressed in cubic inches or cubic feet. One U.S. liquid gallon is equal to 0.134 cubic feet, which means a five-gallon bucket is equal to 0.670 cubic feet. A five-gallon bucket is equal to 1,155 cubic inches. 21 thg 7, 2017
## What is a cubic foot of water?
7.48 gallons One cubic foot of water is equivalent to 7.48 gallons. One hundred cubic feet would equal 748 gallons.
## How many cubic feet is 500 square feet?
125 cubic feet So for our example: 0.25 foot depth times 500 square feet equals 125 cubic feet.
## How many cubic feet are in a 10×10 area?
800 cubic feet Storage Unit Conversion Chart Storage Unit Size Storage Unit Space Equivalent ABF ReloCubes 5′ x 10′ Storage 400 cubic feet 1-2 5′ x 15′ Storage 600 cubic feet 2 10′ x 10′ Storage 800 cubic feet 3 10′ x 15′ Storage 1200 cubic feet 3-4 1 hàng khác
## How wide is 9 cubic feet?
ENDMEMO 1 cubic feet = 12 inch 1 inch = 8 cubic feet = 24 inch 8 inch = 9 cubic feet = 24.961 inch 9 inch = 10 cubic feet = 25.8532 inch 10 inch = 11 cubic feet = 26.6878 inch 11 inch = 25 hàng khác
## How do you measure cubic feet of soil?
How to Calculate Cubic Feet for Soil Use a measuring tape to determine the length of the location where you will be placing the soil. … Measure the width in feet. Determine the necessary depth of the soil in feet. … Multiply the width by the length by the depth to find the number of cubic feet of soil you need.
## Is a 500 square-foot apartment small?
If you’re not interested in a studio but are looking for a one-bedroom, 500 square feet is definitely small. The standard square footage for one-beds is between 550 and 1,000 square feet. This extra square footage accounts for the fact that the bedroom is a separate space from the living room. 2 thg 3, 2022
## Is 550 square feet small?
If we’re being driven by cross-country data, I’d say a “small apartment” is somewhere between the average of the lower limit—around 250 square feet—and the upper limit—about 850 square feet. So safe to say a small apartment is one around 550 square feet or less. 4 thg 2, 2019
## Is 600 square feet small for an apartment?
How big is a 600-square-foot apartment? If you can imagine four Volkswagen vans or a three-car garage, that’s about 600 square feet. It’s nothing to gawk at, but it’s a good amount of space to work with, especially if you’re living alone. 12 thg 8, 2020
## What is the square footage of an 8×12 room?
Multiply 12 x 8 = 96 square feet for each wall, then multiply 96 x 4 (since there are four walls with 96 square feet each)= 384 total square feet for the room.
## How many square feet is a 5×10 room?
5×10 Storage Unit Size A 5×10 self storage unit is a 5 feet wide and 10 feet long small storage solution, totaling 50 square feet. For comparison, a 5×10 space is a small storage unit that’s about the size of an average walk-in closet. Many units have an 8-foot ceiling, giving you up to 400 cubic feet of storage space.
## How many sq ft is 12×12?
144 square feet How many square feet is a 12×12 room? The square footage of a room 12 feet wide by 12 feet long is 144 square feet. Find the square footage by multiplying the width (12 ft) by the length (12 ft).
## How many square feet is 16×40?
1193 Sq Ft 16×40 House 1193 Sq Ft PDF Floor Plan Instant | Etsy.
## How many 12×12 tiles do I need for 100 square feet?
Thus, approximately 115 nos or 10 box of 12×12 tiles you needed to cover 100 square feet area of room or patio.
## How big is a room that is 800 square feet?
What does 800 square feet look like? To visualize it better, 800 square feet is about the size of five parking spaces or a little smaller than three school buses combined. Typically, in an 800-square-foot apartment, you’ll find either a one- or two-bedroom apartment. 17 thg 9, 2021
## How big is a 5 cubic foot freezer?
A 5 cubic feet chest freezer has 27 inches or 69 centimeters in width, a height of 33 inches or 84 centimeters, and a depth of 22 inches or 56 centimeters. In contrast, an upright freezer has 47 inches or 119 centimeters in height, a width of 21 inches or 53 centimeters, and a depth of 22 inches or 55 centimeters.
## How tall is 7.4 cubic feet refrigerator?
55.5 in. Depth: 22.5 in. Height: 55.5 in. Width: 21.75 in.
## How tall is a 4.5 cubic foot refrigerator?
A larger 4.5 cubic foot model with a separate freezer measures around 19 inches wide, 20 inches deep and 43 inches tall.
## How big is a cubic foot in feet?
Definition: A cubic measurement is the three-dimensional derivative of a linear measure, so a cubic foot is defined as the volume of a cube with sides 1 ft in length. In metric terms a cubic foot is a cube with sides 0.3048 metres in length. 22 thg 7, 2018
## How big is a cubic foot?
1 foot by 1 foot by 1 foot A cubic foot is a space that measures 1 foot by 1 foot by 1 foot. To determine how many cubic feet a certain piece will be multiply the length x width x height of the piece. For example, if a dresser measures 4 feet long x 2 feet wide x 5 feet high it is 40 cubic feet.
## What is the difference between cubic feet and square feet?
The square foot is a unit of area (two dimensions) and is commonly calculated by multiplying width and length together. The cubic foot is a unit of volume (three dimensions) – commonly calculated by multiplying width, length and height measurements together.
## How many cubic feet is 24 square feet?
Cubic feet to Square feet Calculator 1 cubic feet = 1 ft2 1 ft2 = 23 cubic feet = 8.0876 ft2 23 ft2 = 24 cubic feet = 8.3203 ft2 24 ft2 = 25 cubic feet = 8.5499 ft2 25 ft2 = 26 cubic feet = 8.7764 ft2 26 ft2 = 25 hàng khác
## How tall is 7 cubic feet refrigerator?
LG Smart Inverter Refrigerator (7 cubic feet capacity / 5 feet tall)
## Is cubic feet the same as feet?
Feet is a measurement of length and cubic feet is a measurement of volume. You can convert feet into inches or yards since all three measure length. You can convert cubic feet into gallons since they both measure volume. But you can’t convert cubic feet into feet.
## How many cubic feet Does a 50 lb bag of quikrete?
One 50-pound bag of Quikrete Fast Setting Concrete Mix yields about . 375 cubic feet.
## How many bags of concrete do I need for a 10×10 slab?
The thickness of your 10′ x 10′ slab will determine just how many bags you need. The average thickness of a 10′ x 10′ concrete slab for a shed is 4 inches thick. At 4″ thick, your 10 x 10 slab will take 56 bags of concrete that weigh 80 lbs.
## How many yards of concrete do I need for a 24×24 slab?
7.11 yards For example, for a concrete slab that is 24′ X 24′ X 4”, simply enter 4 in the Thickness/Depth field, 24 in the Width field, and 24 in the Length field. Click “Calculate”. Your answer should be 7.11 yards. Note: The Concrete Volume Calculator can also be used to determine yardage for aggregate products.
## How many cubic feet is a bag of quikrete?
.60 cu ft. One 80lbs bag of Quikrete Concrete Mix will yield approximately . 60 cu ft. So it will take 45 bag to equal one cubic yard of concrete. If you have a project over a 1/2 a cubic yard you should consider our Mix On-site Concrete especially if you are hand mixing.
## Is it true that there are 12 cubic inches in a cubic foot?
There are 12 (vertical) layers, each one inch high. These layers are made up of 12 rows of cubes, each containing 12 cubes. So each layer has 12 12 unit cubes and, since there are 12 layers, there are 12 \times 12) total cubic inches in the cubic foot.
## How many cubic inches is my box?
To calculate cubic inches, start by measuring the length, width, and depth of the box you’re measuring in inches. Then, multiply the length by the width. Finally, multiply the product of the length and the width by the depth of the box to find the volume in cubic inches.
## Will a cubic yard fit in a pickup?
A regular size pick-up will hold three cubic yards of mulch (a full load). Two cubic yards is about body level full. When picking up soils, sands and gravels, one cubic yard is all that is recommended on a pick-up truck.
## How many cubic yards can a f150 hold?
The bed of a full size standard pickup has dimensions of: 8′ long X 5.33′ wide X 1.5′ high. When loaded level full, a truck of this size will hold 2.5 cubic yards of material.
## What’s a cubic yard look like?
A cubic yard is the volume of a cube with the length, width and height of one yard (3 feet or 36 inches). One cubic yard is equal to 27 cubic feet. To help you picture this, the volume of two washing machines is just over a cubic yard.
## How many cubic are in a yard?
27 cubic feet A yard is 3 feet or 36 inches, and therefore, a cubic yard is 3 x 3 x 3, or 27 cubic feet(ft3).
## How many yards is a bobcat scoop?
A scoop is measured out by our Bobcat machine. One scoop is the equivalent of 2/3 of a cubic yard.
## How many cubic yards of dirt will fit in a pickup truck?
2 cubic yards Full-size Pickup Trucks: Can usually handle 2 cubic yards of soil, 2-3 cubic yards of mulch, and 1 cubic yard of stone or gravel. Small Pickups and Trailers: Can usually handle 1 cubic yard of soil to maybe 1½ of mulch.
## How do you calculate cubic feet from inches?
Calculating cubic feet from inches Multiply your length, width and height figures together, giving you a total in cubic inches (in3) Divide the total by 1728 (as there are 1728 cubic inches in a cubic foot).
## How big is a cubic?
A cubic foot is the volume of a cube with sides 1 ft in length, and it is a three-dimensional derivative of a linear measure. A 20 ft x 8 ft x 8 ft 6 in shipping container has a volume of over a thousand square feet.
## How much cubic yards do I need?
Length in feet x Width in feet x Depth in feet (inches divided by 12). Take the total and divide by 27 (the amount of cubic feet in a yard). The final figure will be the estimated amount of cubic yards required.
## Is 1 yard the same as 1 cubic yard?
Under the United States’ Customary System, 1 yard is equal to 3 feet or 36 inches. And a cubic yard is the volume of material that fits in a space that is 1 yard wide by 1 yard deep by 1 yard high. This is important because quite a few common materials are measured in cubic yards — here are some of them: Concrete. 17 thg 3, 2020
## What is one cubic foot box?
1 Cubic Foot may sound small, but is actually quite a large box. As a cube, a cubic foot is 12″ x 12″ x12″. Most E-Commerce boxes aren’t a cube, and any size box whose sides, when multiplied together equal 1 cubic foot, or 1728 cubic inches, or less, is fine. Here is a box you may be more familiar with, 18″ x 16″ x 6″. 19 thg 4, 2019
## How do you convert square feet to cubic feet?
Square feet: You can find the area or square footage of a space by multiplying the length and width. Then multiply this figure by the height to find the cubic space, or cubic footage, within the object. 24 thg 2, 2022
## How many cubic yards is 100 square feet?
100 square feet ≈ 0.309 cubic yards You can convert 100 square feet to cubic yards with any height (depth) of inches below.
## What is a yard of soil?
A cubic yard is a measurement that is 3 feet by 3 feet by 3 feet. A cubic yard measures volume where a ton measures weight. A yard of topsoil usually weighs about 1,800 pounds and a yard of gravel usually weighs about 2,200 pounds.
## How many bags of soil do I need for 12 cubic feet?
If you are buying soil that is sold by the cubic foot just divide the amount you need by the amount that is in the bag to get the number of bags you need. In this example, the reader needs 12 cubic feet. If she is buying 1 cubic foot bags then she will need 12 bags (12/1 = 12). 11 thg 9, 2013
## How many bags of soil do I need for a 4×8 raised bed?
For a 4×8–foot raised bed with a 10” height, about 1 cubic yard of soil is needed. For a 4×8-foot raised bed with a 6” height, using Mel’s Mix: about 5 cubic feet each of compost, peat moss, and vermiculite is needed.
## How many cubic feet are in a 40 lb bag of soil?
For reference: One cubic yard equals 27 cubic feet. A 40 pound bag of topsoil usually contains about . 75 Cubic Feet of soil. 2 thg 10, 2017
## How much will 10 cubic feet cover?
2.4 sq feet Coverage Chart- Bagged Material (2 Cubic Foot Bags) Depth One 2 Cubic Foot Bag Covers 9” 2.7 sq feet 10” 2.4 sq feet 11” 2.2 sq feet 12” 2 sq feet 9 hàng khác • 29 thg 1, 2010
## How many cubic feet of stone do I need?
Formula for Crushed Stone for a Project Multiply the length (L), in feet, by the width (W), in feet, by the height (H), in feet, and divide by 27. This number is how many cubic yards of crushed stone you need. 25 thg 2, 2022
## How many sq ft are in a cubic yard?
27 cubic feet One yard is equal to 3 feet. Hence to determine the number of cubic feet in a cubic yard, take the cube of both sides, and you will get 1 cubic yard = 27 cubic feet . 2 thg 2, 2022
## Do closets count as square footage?
Stairways and closets will usually be included in the square footage length. When it comes to porches and other outdoor spaces, they’re usually only included if they use the same heating system as the rest of the house. 14 thg 3, 2020
## What is considered a big house?
There is not one specific size for a McMansion type of home. Generally speaking, these homes are larger than the median size of a newly built single-family home, which according to the U.S. Census Bureau for 2020, was 2,261 square feet. 16 Most McMansions between 3,000 or 5,000 square feet or larger.
SnapSafe, Dehumidifier, Rechargable, Effective in Spaces Up To 500 Cubic Feet 4-8-22
SnapSafe, Dehumidifier, Rechargable, Effective in Spaces Up To 500 Cubic Feet 4-8-22
## Do stairs count as square footage?
Stairs: Runs/treads and landings both count in square footage totals. They are measured as a part of the floor “from which they descend,” so are generally counted twice in a typical two-story home with a basement. 27 thg 10, 2016
## How many cubic feet is my house?
Floor area (3000 sq feet) times height equals volume, so if you have 8 ft rooms, then you have 24000 cubic feet. If your rooms average 7 ft, you have 21000 cubic feet and so on. Divide the volume of your house by 6000 and you get the number of foggers you will need.
## How do you measure cubic feet in a house?
Cubic feet represents the total usable space within the building footprint (length by width by height). This is found by multiplying your usable square feet by the height of the building.
## How do you calculate the volume of your house?
Calculate the volume by multiplying the measured length and width of the space together, then multiply the result by the height of the room. From the example, 10 * 25 feet = 250 square feet, and 5 * 10 feet = 50 square feet.
## How do you learn volume?
Units of Measure Volume = length x width x height. You only need to know one side to figure out the volume of a cube. The units of measure for volume are cubic units. Volume is in three-dimensions. You can multiply the sides in any order. Which side you call length, width, or height doesn’t matter.
## How do you explain volume to a child?
Volume refers to the amount of space the object takes up. In other words, volume is a measure of the size of an object, just like height and width are ways to describe size. If the object is hollow (in other words, empty), volume is the amount of water it can hold. 6 thg 12, 2021
## What is cube formula?
Length = Breadth = Height = a. Thus, the measure of each edge of the cube = a. Therefore, the volume of cube formula is a × a × a = a3. It is to be noted that the number obtained using cube formula is the perfect cube number.
## How do you calculate cubic volume?
To find the volume of a box, simply multiply length, width, and height — and you’re good to go! For example, if a box is 5×7×2 cm, then the volume of a box is 70 cubic centimeters.
## What is the formula of rectangular room?
That is, A = l x w where l is the length and w is the width of the rectangle. For example, the area of a rectangle of length 35 m and width 25 m is 35 times 25 or 875 square meters. A rectangular room has a length of 16 feet and a width of 18 feet.
## How do you calculate volume of a box?
You can calculate the volume of a box by multiplying length x width x height.
## How many gallons is 600 square feet?
Multiply the cubic feet by the amount of water per cubic foot, which is 7.48 gallons. In this example, you would multiply 7.48 times 600 to get 4,488 gallons of water.
## How many gallons is a 450 square foot pool?
13,500 gallons VOLUME OF POOL A pool that is 15 feet by 30 feet with a depth of 5 feet in the deep end and 3 feet in the shallow end would have a surface area of 450 square feet and an average depth of 4 feet and a volume of 13,500 gallons.
## How many gallons is 1000 square feet?
On average you would use about 2 gallons of water per 1000 sq/ft.
## How many gallons is a 15×30 pool?
Standard In-Ground Pool Volumes in Gallons by Size Pool Size (in feet) 3.5 ft Avg Depth 4.5 ft Avg Depth 12×24 7,600 9,700 14×28 10,300 13,200 15×30 11,800 15,200 8 hàng khác • 9 thg 9, 2021
## How many gallons are in a 18×36 pool?
How Many Gallons Of Water Are In My Pool ? POOL SIZE 4′ AVG. DEPTH 6′ AVG. DEPTH 14′ X 28′ 11,760 17,640 15′ X 30′ 13,500 20,250 16′ X 32′ 15,400 23,040 18′ X 36′ 19,400 29,160 5 hàng khác • 3 thg 4, 2019
## What is the square footage of a pool?
Measure the length of the rectangular or square pool. Measure the width. Multiply the length by the width of the pool to find the square footage. For instance, 15 feet (length) times 12 feet (width) equals 180 square feet.
## How big is a cubic foot in feet?
Definition: A cubic measurement is the three-dimensional derivative of a linear measure, so a cubic foot is defined as the volume of a cube with sides 1 ft in length. In metric terms a cubic foot is a cube with sides 0.3048 metres in length. 22 thg 7, 2018
## How big is a cubic foot?
1 foot by 1 foot by 1 foot A cubic foot is a space that measures 1 foot by 1 foot by 1 foot. To determine how many cubic feet a certain piece will be multiply the length x width x height of the piece. For example, if a dresser measures 4 feet long x 2 feet wide x 5 feet high it is 40 cubic feet.
## How many cubic feet is 24 square feet?
Cubic feet to Square feet Calculator 1 cubic feet = 1 ft2 1 ft2 = 23 cubic feet = 8.0876 ft2 23 ft2 = 24 cubic feet = 8.3203 ft2 24 ft2 = 25 cubic feet = 8.5499 ft2 25 ft2 = 26 cubic feet = 8.7764 ft2 26 ft2 = 25 hàng khác
## Where is the 5 gallon mark on a bucket?
Lights! Total volume on standard 5 gallon buckets is usually to the first tab line down on the side of the bucket. The most efficient way to tell would be to put RODI water in the bucket tared on a scale and fill it to 41.7 lbs. 1 thg 9, 2017
## How tall is a 5 gallon bucket with lid?
between 12-15 inches tall 5 gallon bucket dimensions vary depending on the manufacturer. 5 gallon bucket height is usually between 12-15 inches tall, with a 10-12 inch diameter. 8 thg 12, 2021
## Are orange Home Depot buckets food Safe?
Probably the most commonly misused bucket is the Home Depot “Homer” bucket in its signature bright orange color. These are not rated safe for handling food. 13 thg 2, 2017
## How tall is a 7 gallon bucket?
17 5/16″” Protect Both Food and Children with This Natural 7 Gallon Plastic Bucket Capacity 7 Gallon Weight 3.74 lbs Diameter 12 1/2″” Height 17 5/16″” Thickness 100 Mil 3 hàng khác
## How tall is a 8 gallon bucket?
12 3/4″” Keeping Food Fresh and Tamper Proof Has Never Been So EZ Capacity 8 Gallon Length 16 1/4″” Width 13 1/4″” Height 12 3/4″” Standards FDA Compliant 5 hàng khác
## How tall is a 6 gallon bucket?
17.5 6 Gallon Round White Plastic Pail w/ Metal Handle – P062 Capacity 6 Gallon Material HDPE Length 12.4 Width 12.4 Height 17.5 3 hàng khác
## How big are standard buckets?
5 gallon What is the size of a standard bucket? The 5 gallon bucket is the most common size of bucket that is in use today.
## How tall is a 10 gallon bucket?
17 1/8 in CHOOSE OPTIONS U.S. Metric Height: 17 1/8 in 43.5 cm Volume Capacity [Nom]: 10 gal 37.9 L Volume Capacity [Max]: Volume Capacity [Min]: 10 hàng khác
## How many gallons is a Home Depot bucket?
5 Gal The Home Depot 5 Gal.
## What is the cubic feet of a 5 gallon bucket?
The volume of a five-gallon bucket can be expressed in cubic inches or cubic feet. One U.S. liquid gallon is equal to 0.134 cubic feet, which means a five-gallon bucket is equal to 0.670 cubic feet. A five-gallon bucket is equal to 1,155 cubic inches. 21 thg 7, 2017
See also C Array - Part 3 | C Language Tutorial naresh i technology c language
## How many cubic feet are in a 5 gallon bucket of dirt?
On average, a 5-gallon bucket holds about 16 quarts of dry soil and dirt and around . 7 cubic feet of soil. A 40-pound bag of soil usually does the work as it can easily fill up a 5-gallon bucket.
## How many gallons is one cubic?
There are 7.48 gallons of water in one cubic foot of water. To convert gallons to cubic feet, divide the total gallons by 7.48. To convert cubic feet to gallons, multiply the cubic feet by 7.48.
## How many 5 gallon buckets are in a cubic yard of soil?
There are 202 gallons in a cubic yard. So if you completely fill a 5 gallon bucket up, it would take approximately 40 of those buckets to make up a yard.
## How much can I fit in a 5 gallon bucket?
Based on my math, a typical 5 gallon bucket holds 0.71 cubic feet of anything you want. Filled with water, it will hold 42 pounds of the stuff, not counting the weight of the bucket (typically 2 pounds.)
## What is the square footage of a 5 gallon bucket?
150 square feet About 30 square feet per gallon, so 150 square feet per 5 Gallon pail.
## What is the cubic feet of 1 gallon?
Gallon (US) to Cubic Foot Conversion Table Gallon (US) [gal (US)] Cubic Foot [ft^3] 1 gal (US) 0.1336805556 ft^3 2 gal (US) 0.2673611111 ft^3 3 gal (US) 0.4010416667 ft^3 5 gal (US) 0.6684027778 ft^3 7 hàng khác
## How much does 1 cubic foot of dirt weigh?
How much does 1 cubic foot of dirt weigh:- on average, one cubic foot of dirt mixed of topsoil can weigh 80 pounds or 0.04 short tons, in general it can be range between 74 – 110 pounds per cubic foot, dry loose dirt can weight around 76 lbs per cubic foot and while moist loose dirt can weigh around 78 lbs per cubic …
## How many dry gallons are in a cubic foot?
6.43 gallons As per the unit of conversion 1 cubic foot equals 6.43 gallons. 23 thg 2, 2022
## How many cubic feet are in a 10×10 area?
800 cubic feet Storage Unit Conversion Chart Storage Unit Size Storage Unit Space Equivalent ABF ReloCubes 5′ x 10′ Storage 400 cubic feet 1-2 5′ x 15′ Storage 600 cubic feet 2 10′ x 10′ Storage 800 cubic feet 3 10′ x 15′ Storage 1200 cubic feet 3-4 1 hàng khác
## How many cubic are in a yard?
27 cubic feet A yard is 3 feet or 36 inches, and therefore, a cubic yard is 3 x 3 x 3, or 27 cubic feet(ft3).
## What are the measurements for a cubic yard?
One cubic yard is equivalent to a box measuring 3 feet wide by 3 feet long by 3 feet deep for a total of 27 cubic feet. 2. As a rule of thumb, 1 cubic yard of mulch covers approximately 100 square feet to a depth of 3 inches.
## How many cubic feet is 500 square feet?
125 cubic feet So for our example: 0.25 foot depth times 500 square feet equals 125 cubic feet.
## Is cubic feet bigger than square feet?
Square feet are units of area and cubic feet are units of volume. Here is an example. 16 + 4 16 = 80 square feet on liner. The volume of the pool is 4 4 4 = 64 cubic feet so I need 64 cubic feet of water.
## How do I convert cubic feet to feet?
Calculating cubic feet from measurements in feet If you have measured the width, length and height of your item in feet then the calculation process is straight forward: Simply multiply the 3 figures together.
## How big is a 5 cubic foot freezer?
A 5 cubic feet chest freezer has 27 inches or 69 centimeters in width, a height of 33 inches or 84 centimeters, and a depth of 22 inches or 56 centimeters. In contrast, an upright freezer has 47 inches or 119 centimeters in height, a width of 21 inches or 53 centimeters, and a depth of 22 inches or 55 centimeters.
## How tall is a 7 cu ft refrigerator?
LG Smart Inverter Refrigerator (7 cubic feet capacity / 5 feet tall)
## What is 7 cubic feet in feet?
ENDMEMO 1 cubic feet = 1 feet 1 cubic feet 7 cubic feet = 1.9129 feet 343 cubic feet 8 cubic feet = 2 feet 512 cubic feet 9 cubic feet = 2.0801 feet 729 cubic feet 10 cubic feet = 2.1544 feet 1000 cubic feet 25 hàng khác
## How do I know how much soil I need?
To estimate soil volume for any area, all you need is a tape measure. “The basic formula is simple: Length x Width x Height = Volume,” says Michael Dean, co-founder of Pool Research. Then divide the number of cubic feet by 27. So one cubic yard = 27 cubic feet = 1,728 cubic inches. 26 thg 10, 2021
## How many cubic feet do I need for my garden?
Multiply the length by the width by the depth. All the units of measurements must be in feet. For example, a raised garden bed that is 4 feet by 3 feet by 1.5 feet equals 18 cubic feet.
## How do I calculate how much soil I need?
How do I calculate soil volume in raised beds? Measure the width, length, and depth of the raised beds. Multiply the width, length, and depth together. The number you get will be the volume of soil you need to fill your raised beds. 5 ngày trước
## How big is a 500 square-foot home?
Measure each room’s length and width and add up all the square footage for a total count of the entire space. In an apartment, that may mean measuring the bedroom, bathroom, kitchen and living area all separately. However, the basic dimensions of a 500-square-foot space are 25 feet by 20 feet. 2 thg 3, 2022
## How do I build a 500 sq ft apartment?
5 ways to decorate a 500-square-foot apartment Vertical storage. When you’ve run out of room on the floor, start moving up! … Focus on lighting. Keep your space looking bright with great lighting. … Simplify furniture. In a 500-square-foot apartment, size matters. … Add some drama. … Use symmetry. 11 thg 8, 2020
Yick Wo – \”500 cubic feet\”
Yick Wo – \”500 cubic feet\”
## Is 500 sqft small?
500 Square Feet Five hundred square feet is already pretty small for an American home, although it might seem about par for the course for our European neighbors. In Lauren’s home, 500 square feet means a separate kitchen and small dining room, spacious living room/bedroom, and a pretty generous closet. 19 thg 6, 2019
## What does 600 sq feet look like?
So what does 600 square feet actually look like? If you can’t quite visualize how big 600 square feet is yet, no worries. Six hundred square feet is about the size of a standard three-car garage that has enough space to park the cars and walk around comfortably. 14 thg 9, 2021
## How do you measure 550 square feet?
Multiply length and width to calculate square feet. To measure the approximate square footage of a space, you measure the length and width of a space.
## Is 550 square feet small?
If we’re being driven by cross-country data, I’d say a “small apartment” is somewhere between the average of the lower limit—around 250 square feet—and the upper limit—about 850 square feet. So safe to say a small apartment is one around 550 square feet or less. 4 thg 2, 2019
## Is 400 square feet small?
How big is a 400-square-foot apartment? Four-hundred square feet is about the size of a two-car garage. It’s not a huge apartment, but it’ll get the job done. After all, two cars take up a lot of space and you won’t actually have two cars parked in your house — the furniture you have will be much smaller! 10 thg 8, 2020
## How big is a 700 sq ft apartment?
How big is a 700-square-foot apartment? Seven hundred square feet covers one-fourth of a standard tennis court. If it were a single room, it would measure about 26 by 27 feet. 13 thg 8, 2020
## How many square feet is a 10×8 shed?
80 sq. ft 10×8 Sheds. Square footage: 80 sq. ft.
## How many sq ft is a 10×12 room?
120 square feet Area of the floor or ceiling: Multiply the length by the width (10 feet x 12 feet = 120 square feet of area). 27 thg 3, 2015
## How many sq ft is 12×12?
144 square feet How many square feet is a 12×12 room? The square footage of a room 12 feet wide by 12 feet long is 144 square feet. Find the square footage by multiplying the width (12 ft) by the length (12 ft).
## How many feet is a 10×10 room?
100 square feet A room that is 10×10 is 10 feet by 10 feet in size, or 100 square feet in total.
## How many feet is 7 square feet?
Square feet to Feet Calculator 1 ft2 = 1 feet 1 feet = 5 ft2 = 2.2361 feet 5 feet = 6 ft2 = 2.4495 feet 6 feet = 7 ft2 = 2.6458 feet 7 feet = 8 ft2 = 2.8284 feet 8 feet = 25 hàng khác
## How do you visualize square feet?
In any given space, you can multiply the length by the width to get the total square footage. For example, a closet that’s 4 feet long and 3 feet wide is 12 square feet. 23 thg 7, 2020
## How many square feet is 12×24?
288 sq ft At 12 feet by 24 feet, you’re looking at 12/3= 4 yards and 24/3 = 8 yards, giving you 32 square yards (288 sq ft).
## How many sq ft is 30×40?
1,200 square feet A building of this size (30′ x 40′ = 1,200 square feet) is a good choice for a 4 car garage or 3 car garage with shop & storage space.
## How many sq ft is 20×20?
Here, it is said that 20×20 means 20ftx20ft. Therefore 20 ft multiplying by 20 ft equal to 400 square feet. 19 thg 7, 2018
## What is footage of an acre?
acre, unit of land measurement in the British Imperial and United States Customary systems, equal to 43,560 square feet, or 4,840 square yards.
## How many 12X12 tiles are in a box price?
Production Capacity: 10000 BOXES. Delivery Time: 8 DAYS. Packaging Details: Size :30X30 CM (12X12 Inch)Packing :1 Box = 10 PcsThickness :8-9 mmWeight :1 Box = 12 KgWork Area :1 Box = 9.69 Sq. Ft.
## How do I calculate how much tile to buy?
Multiply the tile length by the width to figure the area that one tile will cover in square inches. Divide the result by 144 to convert it to square feet. Then, divide the area you’re tiling by the square footage of one tile to determine how many tiles you need. 17 thg 2, 2021
## How big is a 700 square-foot home?
A 700-square-foot apartment is equal to about a quarter the size of a full-length tennis court. They are usually available as studios or have one small bedroom.
## Is 800 sq ft big for an apartment?
To visualize it better, 800 square feet is about the size of five parking spaces or a little smaller than three school buses combined. Typically, in an 800-square-foot apartment, you’ll find either a one- or two-bedroom apartment. The bedrooms are small but they’re definitely livable. 17 thg 9, 2021
## What is a good size for a bedroom?
The average bedroom size is about 132 square feet while there are bedrooms much larger than even 144 square feet. The basic guideline to bear in mind while allocating space for a standard bedroom is that it should measure at least 120 square feet to comfortably accommodate a full-size bed. 14 thg 6, 2021
## What size freezer do I need for a family of 4?
6 cubic foot The rule of thumb to plan for is 1.5 cubic feet of freezer space per person in your family. Thus a family of 4 should get at least an 6 cubic foot freezer. Choose a freezer that’s not too big and not too small but just right. For the best energy efficiency your freezer should always be 2/3s full or more.
## What size freezer do I need for 400 pounds of beef?
You’ll be filling at least four deep freezers when you order a whole beef. With approximately 320 – 440 pounds of freezer beef, you’ll need at least 40 cubic feet of freezer space. 16 thg 1, 2017
## How much meat can a 5 cubic foot freezer hold?
A 5 cubic foot small deep freezer should be sufficient for a two-person household. It may accommodate a family of four (but it’ll likely be a tight squeeze!). A freezer of this size holds approximately 175 pounds (79 kilograms) of food. If you’re wondering how much meat fits in, it’s roughly a quarter beef. 11 thg 10, 2021
## How tall is a 7.5 cu ft refrigerator?
55.31″” height The adjustable shelves in the interior of the Thomson 7.5 cu. ft. Top-Freezer Refrigerator (55.31″” height x 22.83″” width x 22.36″” depth) provide flexibility in food storage. Its easily-accessed temperature control keeps food at the proper temperature for food safety.
## How many cubic feet is an apartment size refrigerator?
A 7.3 or 7.4 cubic foot refrigerator is large enough for most apartment dwellers, but you may prefer a model with more than ten cubic feet of storage space.
## How tall is a Magic Chef mini fridge?
Magic Chef 1.7 Cubic’ Mini Fridge, White, One Size Color White Form Factor Compact Pattern Solid Configuration Compact Freezerless Item Dimensions LxWxH 17.7 x 18.6 x 19.4 inches 5 hàng khác
## How tall is a 4.6 cubic foot refrigerator?
about 48 inches high A: AnswerThis fridge is about 48 inches high it fits perfect in a dorm room.
## How tall is a 4.4 cubic foot refrigerator?
SPT RF-444SS 4.4 cu. ft. Compact Refrigerator in Stainless Steel – Energy Star Color Stainless Steel Form Factor Stand Alone Pattern Solid Configuration Compact Internal Freezer Item Dimensions LxWxH 21.13 x 19.5 x 33.69 inches 5 hàng khác
## What size is a dorm size refrigerator?
Standard-size dorm refrigerators measure about 2 cubic feet — a small, square model — or approximately 4.5 cubic feet, which measures desk height. You may find dorm fridges slightly smaller than 2 cubic feet or slightly larger than 4.5 cubic feet.
## How many cubic feet is 24 square feet?
Cubic feet to Square feet Calculator 1 cubic feet = 1 ft2 1 ft2 = 23 cubic feet = 8.0876 ft2 23 ft2 = 24 cubic feet = 8.3203 ft2 24 ft2 = 25 cubic feet = 8.5499 ft2 25 ft2 = 26 cubic feet = 8.7764 ft2 26 ft2 = 25 hàng khác
## What are the dimensions of 200 cubic feet?
Each liftvan measures 200 cubic feet and they are generally 7 foot long by 4 foot wide x 7 foot tall. This means if a crate is used you will be charged for 200 cubic feet, even if your shipment does not entirely fill the crate.
## What is a cubic foot in square feet?
Cubic feet = square feet × depth. So: 20 × 0.25 = 5.
## What is a cubic foot of soil?
bags of topsoil are in a cubic yard. There are 25.71404638 Dry Quarts in a Cubic Foot, so a 25 quart bag of potting soil would equal approximately 1 Cubic Foot. 2 thg 10, 2017
## What does CU mean in dryers?
While there’s no official definition of large capacity or extra-large capacity in a dryer, you can consider anything over 7.0 cubic feet (cu.
## How many cubic feet is a 6 foot freezer?
Compact – 5 cubic feet. Small – 6 to 9 cubic feet. Medium – 12 to 18 cubic feet. Large – 18+ cubic feet. 2 thg 4, 2020
## How much does a cubic foot cover?
Coverage Chart- Bagged Material (2 Cubic Foot Bags) Depth One 2 Cubic Foot Bag Covers 1” 24 sq feet 2” 12 sq feet 3” 8 sq feet 4” 6 sq feet 9 hàng khác • 29 thg 1, 2010
## How many cubic feet is 3000 square feet?
How high is your ceiling? Floor area (3000 sq feet) times height equals volume, so if you have 8 ft rooms, then you have 24000 cubic feet. If your rooms average 7 ft, you have 21000 cubic feet and so on.
## How do I calculate cubic feet?
If you prefer to or have to calculate cubic footage by hand, you can find cubic feet by multiplying three linear measurements—length, width, and height—in feet. For instance, to find the volume of a cube, you would calculate the following: length x width x height. 24 thg 2, 2022
## How many cubic feet is 500 square feet?
125 cubic feet So for our example: 0.25 foot depth times 500 square feet equals 125 cubic feet.
## Is cubic feet the same as square feet?
The volume in cubic feet is equal to the area in square feet times the height in feet.
## What is the difference between cubic feet and square feet?
Square feet are units of area and cubic feet are units of volume. Here is an example. 16 + 4 16 = 80 square feet on liner. The volume of the pool is 4 4 4 = 64 cubic feet so I need 64 cubic feet of water.
## How tall is 7.4 cubic feet refrigerator?
55.5 in. Depth: 22.5 in. Height: 55.5 in. Width: 21.75 in.
## How tall is a 4.5 cubic foot refrigerator?
A larger 4.5 cubic foot model with a separate freezer measures around 19 inches wide, 20 inches deep and 43 inches tall.
## How tall is a 7 cubic foot freezer?
7 Cubic Feet Freezer Dimensions A 7-cubic feet chest freezer has 36 inches or 91 centimeters in width, a height of 34 inches or 87 centimeters, and a depth of 25 inches or 63 centimeters.
## Is cubic feet bigger than feet?
Feet is a measurement of length and cubic feet is a measurement of volume. You can convert feet into inches or yards since all three measure length. You can convert cubic feet into gallons since they both measure volume. But you can’t convert cubic feet into feet.
## How many cubic feet is 1ft?
ENDMEMO 1 cubic feet = 1 feet 1 cubic feet 2 cubic feet = 1.2599 feet 8 cubic feet 3 cubic feet = 1.4422 feet 27 cubic feet 4 cubic feet = 1.5874 feet 64 cubic feet 5 cubic feet = 1.71 feet 125 cubic feet 25 hàng khác
## Is a cubic foot bigger than a foot?
The cubic foot (symbol ft3 or cu ft) is an imperial and US customary (non-metric) unit of volume, used in the United States and the United Kingdom. It is defined as the volume of a cube with sides of one foot (0.3048 m) in length. … cubic foot US Customary 1728 in3 1⁄27 yd3 SI units 0.02831685 m3 6 hàng khác
## How many bags of concrete do I need for a 4×4 slab?
How many bags of concrete do I need for a 4×4 slab?”, at 4 inches thick, generally you will need approximately either 12 bags of 60lb or 9 bags of 80lb of premixed concrete for a 4×4 slab, at 5 inches thick slab, either 15 bags of 60lb or 11 bags of 80lb premixed concrete are required, while at 6 inches thick slab, …
## How much concrete does a 50lb bag make?
One 50-pound bag of Quikrete Fast Setting Concrete Mix yields about . 375 cubic feet.
## How many bags of concrete do I need for a 10×10 slab?
The thickness of your 10′ x 10′ slab will determine just how many bags you need. The average thickness of a 10′ x 10′ concrete slab for a shed is 4 inches thick. At 4″ thick, your 10 x 10 slab will take 56 bags of concrete that weigh 80 lbs.
## How many 80 bags of concrete make a yard?
If you’re using 80lb bags of concrete, you’ll need 45 bags to make a yard.
## How many bags of concrete do I need for a 10×8 slab?
Regarding this, “how many bags of concrete do I need for a 10×8 slab?”, at 4 inches thick, generally you will need approximately either 58 bags of 60lb or 45 bags of 80lb of premixed concrete for a 10×8 slab, at 5 inches thick slab, either 73 bags of 60lb or 56 bags of 80lb premixed concrete are required, while at 6 …
## How much concrete is needed for a 12×12 slab?
Regarding this, “how much concrete do I need for a 12×12 slab?”, at 4 inches thick, generally you will need approximately 1.76 cubic yards or 47.52 cubic feet or 1.35 m3 (either 104 bags of 60lb or 80 bags of 80lb) of premixed concrete for a 12×12 slab, at 5 inches thick slab, 2.22 cubic yards or 59.90 cubic feet or …
## How many yards of concrete do I need for a 10×10 slab?
1.3 cubic yards The amount you will need for a 10 x 10 slab is 1.3 cubic yards, we always add an extra 10% to allow for any slab depth variations or spills that may occur. 25 thg 9, 2020
## How thick do I need my concrete?
Standard concrete floor slab thickness in residential construction is 4 inches. Five to six inches is recommended if the concrete will receive occasional heavy loads, such as motor homes or garbage trucks. To prepare the base, cut the ground level to the proper depth to allow for the slab thickness.
## How many square feet is a yard of concrete 4 inches thick?
81 square feet One Cubic Yard of Concrete: 4-inches thick – covers 81 square feet.
## How many bags of concrete do I need for 1 cubic foot?
In order to have enough concrete for one cubic foot, you’ll require 2.2 bags of 60-lb concrete. Since bags are only sold in whole (not half, or quarter) quantities, you’ll need to purchase three (3) 60-lb. bags for one cubic foot.
## How many bags of concrete do I need for 100 square feet?
Ans. :- 7.6 no of cement bags (381.30 kgs) are required for 100 sq ft rcc roof slab 4 inch thick m20 grade concrete. Ans. :- 7.6 bags (381.30 kgs), 9.5 bags (476.10 kgs) & 11.4 bags ( 570.8 kgs) cement are required for 100 sq ft rcc roof slab 4 inch, 5inch & 6 inch thick respectively in m20 grade of concrete.
## How many cubic feet Does a 50 lb bag of concrete cover?
Volume of a 50 lb bag of concrete:- a 50 lb bag of concrete yields 0.375 cubic feet volume which is approximately equal as 0.0138 cubic yards or 2.8 gallons or 10.62 liters volume of concrete.
## What’s a cubic yard look like?
A cubic yard is the volume of a cube with the length, width and height of one yard (3 feet or 36 inches). One cubic yard is equal to 27 cubic feet. To help you picture this, the volume of two washing machines is just over a cubic yard.
## How many cubic feet is an 80 pound bag of concrete?
80 lb bag of concrete cubic feet 80 lb bag of concrete yields 0.60 cubic feet which is approximately around 0.0222 cubic yards that will cover around 3.6 sq ft area up to standard depth of 2 inch excavation or backfills, and for 1 cubic feet of landfill you will need 1.66 bags of 80 lb concrete.
## How many standard inches are in a cubic foot?
1728 cubic inches According to an online calculator, there are 1728 cubic inches in a cubic foot.
## What is a cubic inch in inches?
One cubic inch is approximately 16.387 mL. One cubic foot is equal to exactly 1,728 cubic inches because 123 = 1,728. One U.S. gallon is equal to exactly 231 cubic inches. … Cubic inch 1 in3 in … … is equal to … SI derived units ≈ 16.387 mL US customary 1⁄231 US Gallon nonstandard 1⁄1728 ft3 7 hàng khác
## What is a cubic inch compared to an inch?
Cubic inches are the same concept, except in 3 dimensions. A cubic inch is a 1x1x1 inch box.
## How do you calculate cubic inches of a package?
Calculate the cubic size of your package in inches by multiplying the height (number 1 in the diagram), by the length (number 2 in the diagram), by the width (number 3 in the diagram). Round each measurement to the nearest whole inch. The resulting total is the cubic size of your package.
## How many cubic feet is a truckload?
Full Truckload Shipping: Are You Maximizing Your Trailer Space? Specifications Dry Van Floor Space: 389.81 sq. ft. 433.13 sq. ft. Usable Capacity: 3,000 cu. ft. 3,800 cu. ft. Tare Weight (empty): 10,850 lbs. 15,000 lbs. Max Payload: 45,000 lbs. 29,000 lbs. 5 hàng khác • 29 thg 11, 2018
## What does a cubic foot of dirt weigh?
Weight of dirt/ soil per cubic foot:- on average, weight of dirt mixed of top soil can weigh 80 pounds per cubic foot which is approximately equal as 0.04 short tons, in general it can be range between 74 – 110 pounds per cubic foot, dry loose dirt can weight around 76 lbs per cubic foot and while moist loose dirt can …
## How many cubic yards fit in a dump truck?
Dump Truck Cubic Yardage – The Basics While there is room for variance, most full-size dump trucks have a capacity of between 10 and 16 cubic yards. These types of vehicles often have their capacity listed in the owner’s manual. 4 thg 2, 2021
## Will a cubic yard fit in a pickup?
A regular size pick-up will hold three cubic yards of mulch (a full load). Two cubic yards is about body level full. When picking up soils, sands and gravels, one cubic yard is all that is recommended on a pick-up truck.
## How much does a yard of soul weigh?
What is this? By using the standard soil density which is 100 lbs/ft3 , you can calculate that a cubic yard of topsoil is roughly 2,700 lbs. This can definitely be less or more depending on how much moisture and other materials are in the soil. 28 thg 10, 2021
## Can my truck handle a yard of dirt?
Full-size Pickup Trucks: Can usually handle 2 cubic yards of soil, 2-3 cubic yards of mulch, and 1 cubic yard of stone or gravel. Small Pickups and Trailers: Can usually handle 1 cubic yard of soil to maybe 1½ of mulch.
## How much cubic yards do I need?
Length in feet x Width in feet x Depth in feet (inches divided by 12). Take the total and divide by 27 (the amount of cubic feet in a yard). The final figure will be the estimated amount of cubic yards required.
## How many wheelbarrows are in a yard of dirt?
Depending upon your wheelbarrow size (i.e. 2 or 3 cubic feet per wheelbarrow load), it will take 9 to 14 full loads to equal 1 cubic yard.
## What is the cubic feet of a 5 gallon bucket?
The volume of a five-gallon bucket can be expressed in cubic inches or cubic feet. One U.S. liquid gallon is equal to 0.134 cubic feet, which means a five-gallon bucket is equal to 0.670 cubic feet. A five-gallon bucket is equal to 1,155 cubic inches. 21 thg 7, 2017
## Is 1 yard the same as 1 cubic yard?
Under the United States’ Customary System, 1 yard is equal to 3 feet or 36 inches. And a cubic yard is the volume of material that fits in a space that is 1 yard wide by 1 yard deep by 1 yard high. This is important because quite a few common materials are measured in cubic yards — here are some of them: Concrete. 17 thg 3, 2020
## How many yards of concrete do I need for a 24×24 slab?
7.11 yards For example, for a concrete slab that is 24′ X 24′ X 4”, simply enter 4 in the Thickness/Depth field, 24 in the Width field, and 24 in the Length field. Click “Calculate”. Your answer should be 7.11 yards. Note: The Concrete Volume Calculator can also be used to determine yardage for aggregate products.
## How many cubic feet is a tractor bucket?
Let’s see, according to the chart Kennyd linked to… The 49″ bucket holds 6.0 cubic feet of material struck and 7.3 cubic feet heaped. It’s easy enough to do the math to convert cubic feet to cubic yards… 28 thg 9, 2016
## What size is a 1 yard bucket?
Available in 58″ (1/2 yard) and 62” (1 yard) widths. These buckets are meant for landscaping material facilities that require precision measurement of products being sold in a bucket that allows for dumping into the bed of a pickup truck if need be!
## How many cubic yards is a 66 inch skid steer bucket?
skid steer loader buckets Model Width(IN) Rated Bucket Capacity(YD3) 1372mm 54 0.39 1524mm 60 0.44 1676mm 66 0.48 1829mm 72 0.53 1 hàng khác
## How many cubic yards can a f150 hold?
The bed of a full size standard pickup has dimensions of: 8′ long X 5.33′ wide X 1.5′ high. When loaded level full, a truck of this size will hold 2.5 cubic yards of material.
## Can a 1/2 ton truck carry a yard of dirt?
How Many Yards Of Soil Are In A Half-ton Truck? Usually, a half-ton truck has the capacity to hold between 1-2 cubic yards of bulk soil or garden mix. A smaller pickup truck can hold about 1 cubic yard.
## How many 40 pound bags of topsoil are in a cubic yard?
There are 36 bags of 40lb of topsoil in a cubic yard. A 40lb bag of topsoil yields 0.75 cubic feet, and 1 cubic yard is equal as 27 cubic feet, so number of bags of 40lb of topsoil in a yard = 27/0.75 = 36 bags, so, there are 36 bags of 40 lb of topsoil in a cubic yard.
## How many cubic feet is a 30 inch stove?
Size. The standard width of a residential range is 30 inches, but higher-end ranges can extend to 36 inches and beyond. The oven compartment on a typical range is 5 cubic feet, compared to just over 3 cubic feet in a typical wall oven.
## How big is a cubic foot in feet?
Definition: A cubic measurement is the three-dimensional derivative of a linear measure, so a cubic foot is defined as the volume of a cube with sides 1 ft in length. In metric terms a cubic foot is a cube with sides 0.3048 metres in length. 22 thg 7, 2018
## How tall is a 7 cu ft refrigerator?
LG Smart Inverter Refrigerator (7 cubic feet capacity / 5 feet tall)
## What is a cubic foot of soil?
bags of topsoil are in a cubic yard. There are 25.71404638 Dry Quarts in a Cubic Foot, so a 25 quart bag of potting soil would equal approximately 1 Cubic Foot. 2 thg 10, 2017
## What is a cubic foot in square feet?
Cubic feet to Square feet Calculator 1 cubic feet = 1 ft2 1 cubic feet 2 cubic feet = 1.5874 ft2 2.8284 cubic feet 3 cubic feet = 2.0801 ft2 5.1962 cubic feet 4 cubic feet = 2.5198 ft2 8 cubic feet 5 cubic feet = 2.924 ft2 11.1803 cubic feet 25 hàng khác
## How many cubic are in a yard?
27 cubic feet A yard is 3 feet or 36 inches, and therefore, a cubic yard is 3 x 3 x 3, or 27 cubic feet(ft3).
## How many sq ft are in a cubic yard?
27 cubic feet One yard is equal to 3 feet. Hence to determine the number of cubic feet in a cubic yard, take the cube of both sides, and you will get 1 cubic yard = 27 cubic feet . 2 thg 2, 2022
## How do I calculate how much dirt I need?
To estimate soil volume for any area, all you need is a tape measure. “The basic formula is simple: Length x Width x Height = Volume,” says Michael Dean, co-founder of Pool Research. Then divide the number of cubic feet by 27. So one cubic yard = 27 cubic feet = 1,728 cubic inches. 31 thg 3, 2022
## How do you convert cubic feet to cubic yards?
To convert a cubic foot measurement to a cubic yard measurement, divide the volume by the conversion ratio. The volume in cubic yards is equal to the cubic feet divided by 27.
## What is a yard of soil?
A cubic yard is a measurement that is 3 feet by 3 feet by 3 feet. A cubic yard measures volume where a ton measures weight. A yard of topsoil usually weighs about 1,800 pounds and a yard of gravel usually weighs about 2,200 pounds.
## How many cubic yards are in 4 yards?
ENDMEMO 1 cubic yard = 1 yards 1 yards = 3 cubic yard = 1.4422 yards 3 yards = 4 cubic yard = 1.5874 yards 4 yards = 5 cubic yard = 1.71 yards 5 yards = 6 cubic yard = 1.8171 yards 6 yards = 25 hàng khác
## How tall is a cubic foot?
A cubic foot is a space that measures 1 foot by 1 foot by 1 foot. To determine how many cubic feet a certain piece will be multiply the length x width x height of the piece.
## How do you measure one cubic foot?
If you prefer to or have to calculate cubic footage by hand, you can find cubic feet by multiplying three linear measurements—length, width, and height—in feet. For instance, to find the volume of a cube, you would calculate the following: length x width x height. 24 thg 2, 2022
## How do you calculate cubic feet?
Measure the length, width and height of the box in inches. 2. Multiply the length, width and height and divide the resulting number by 1,728. This is the container’s volume in cubic feet.
## What does CU mean in dryers?
While there’s no official definition of large capacity or extra-large capacity in a dryer, you can consider anything over 7.0 cubic feet (cu.
## How many cubic feet is 500 square feet?
125 cubic feet So for our example: 0.25 foot depth times 500 square feet equals 125 cubic feet.
## How many cubic feet are in a 10×10 area?
800 cubic feet Storage Unit Conversion Chart Storage Unit Size Storage Unit Space Equivalent ABF ReloCubes 5′ x 10′ Storage 400 cubic feet 1-2 5′ x 15′ Storage 600 cubic feet 2 10′ x 10′ Storage 800 cubic feet 3 10′ x 15′ Storage 1200 cubic feet 3-4 1 hàng khác
## How many cubic feet is 3000 square feet?
How high is your ceiling? Floor area (3000 sq feet) times height equals volume, so if you have 8 ft rooms, then you have 24000 cubic feet. If your rooms average 7 ft, you have 21000 cubic feet and so on.
## How many cubic yards is 5000 square feet?
5000 square feet ≈ 15.432 cubic yards As you can see, answers are rounded if necessary. You can convert 5000 square feet to cubic yards with any height (depth) of inches below.
## How many sq ft does a yard of concrete cover?
How much concrete is in a yard? Concrete volume is measured in cubic yards. When poured on a perfectly level surface, one cubic yard will cover 27 square feet at one foot thick.
## What’s a cubic yard of concrete?
One Cubic Yard of Concrete: 4-inches thick – covers 81 square feet. 5-inches thick – covers 65 square feet. 6-inches thick – covers 54 square feet.
## How many inches of topsoil do I need for grass?
You need at least 4 to 6 inches of nutrient-rich topsoil to grow good grass.
## How much top soil do I need for sod?
New Construction Sites : It’s advisable to order soil based on a minimum 4 – 5 inch depth of new top soil. Once spread, leveled and packed with a roller it will result in a 3.5 – 4 inch depth of top soil for your new sod.
## How much topsoil do I need for 1000 square feet?
Prepared topsoil can be spread over existing soil and immediately used as the medium for any type of planting, including bushes, trees, sod, seed, flowers, etc. Rate: 3 CY per 1000 sq ft (equal to 1″ of soil) • Example: The area for top dress or fill is 10,000 sq ft.
## How many 25l bags of soil do I need?
All of our soils and mulches are available to purchase by volume, weight or bag. There are 25 bags (40 litre) of mulch or bark per 1 cubic metre and 40 bags (25 litre) of soil per 1 cubic metre.
## How much soil do I need for a 4×4 raised bed?
How much soil will that take? Answer: The cost to fill a raised bed with bagged fertile soil (planting mix) adds up quickly. The volume of soil you need is 12 feet times 4 feet times 1.5 feet (length times width times depth equals volume), which comes to 72 cubic feet. 12 thg 4, 2017
## How many bags of soil do I need for a 4×8 raised bed?
For a 4×8–foot raised bed with a 10” height, about 1 cubic yard of soil is needed. For a 4×8-foot raised bed with a 6” height, using Mel’s Mix: about 5 cubic feet each of compost, peat moss, and vermiculite is needed.
## How do you fill a raised bed cheaply?
Put down a few layers of cardboard to kill any weeds or grass. Then, fill the core of your raised bed. The best option for this is to use straw bales, but you can also use leaves, grass clippings, or old twigs. You can mix together a few of those options if you choose, too. 9 thg 3, 2021
## How deep does a raised bed need to be for corn?
Soil Depth Requirements for Common Garden Vegetables Shallow Rooting 12″ – 18″ Medium Rooting 18″ – 24″ Deep Rooting 24″ – 36″+ Corn Cucumber Squash, winter Endive Eggplant Sweet potatoes Garlic Kale Tomatoes Kohlrabi, Bok Choy Peas Watermelon 13 hàng khác
## How deep does a raised bed need to be?
A raised bed does not have to be very deep to be effective. Eight to 12 inches is usually adequate. If drainage is a problem, or if the plants you are growing prefer drier soil, the bed could be taller and filled with a porous growing medium. Vegetable beds should be 12 to 18 inches deep.
## How many bags of soil do I need for 12 cubic feet?
If you are buying soil that is sold by the cubic foot just divide the amount you need by the amount that is in the bag to get the number of bags you need. In this example, the reader needs 12 cubic feet. If she is buying 1 cubic foot bags then she will need 12 bags (12/1 = 12). 11 thg 9, 2013
## How many pounds is 1 cubic feet?
Cubic feet to pounds table Cubic feet Pounds 1 ft³ 62.43 lb 2 ft³ 124.86 lb 3 ft³ 187.29 lb 4 ft³ 249.72 lb 96 hàng khác
## What is the weight of a cubic foot of soil?
How much does a cubic foot of dirt weight:- on average, a cubic foot of dirt mixed of top soil can weigh about 80 pounds or 0.04 short tons, in general it can be range between 74 – 110 pounds per cubic foot, dry loose dirt can weight around 76 lbs per cubic foot and while moist loose dirt can weight around 78 lbs per …
## Is cubic feet the same as square feet?
The volume in cubic feet is equal to the area in square feet times the height in feet.
## What’s a cubic yard look like?
A cubic yard is the volume of a cube with the length, width and height of one yard (3 feet or 36 inches). One cubic yard is equal to 27 cubic feet. To help you picture this, the volume of two washing machines is just over a cubic yard.
## How do you calculate cubic feet for landscaping?
Estimating Cubic Yards Measure square foot area (width x length) Choose the desired depth of material (convert inches to feet) 1″” = .08 ft. 2″” = .16 ft. 3″” = .25 ft. 4″” = .33 ft. 5″” = .42 ft. 6″” = .50 ft. … Square Feet Area x Depth (in feet) = Cubic Feet (width x length x depth) Divide Cubic Feet by 27 to get Cubic Yards.
## How much will 10 cubic feet cover?
2.4 sq feet Coverage Chart- Bagged Material (2 Cubic Foot Bags) Depth One 2 Cubic Foot Bag Covers 9” 2.7 sq feet 10” 2.4 sq feet 11” 2.2 sq feet 12” 2 sq feet 9 hàng khác • 29 thg 1, 2010
## How do I calculate how much stone I need for my driveway?
To measure the driveway in cubic feet, multiply the length by width by depth. For cubic yards, divide the total cubic feet by 27. Because one cubic yard of gravel is equal to 1.13 tons, you can multiply your total cubic yards by 1.13 to convert this measurement to tons.
## How do I calculate how much rock I need?
Length in feet x Width in feet x Depth in feet (inches divided by 12). Take the total and divide by 21.6 (the amount of cubic feet in a ton). The final figure will be the estimated amount of tons required.
## How many cubic yards is 100 square feet?
100 square feet ≈ 0.309 cubic yards You can convert 100 square feet to cubic yards with any height (depth) of inches below.
## Is a laundry room considered living space?
Well, we have some good news. If your laundry room is a finished room under the main roof of your home, which means it has floors, ceilings, finished walls, and heating and air-conditioning, then it can be counted. In addition, if you have a laundry closet, it will also be counted if it meets the above requirements. 3 thg 4, 2021
## Does garage count in square footage?
The garage does not count towards the square footage of a house, as that is considered an unfinished space. A garage will only count towards the square footage of a home is if it has been legally converted into a living space. 21 thg 2, 2022
## Is a walk in closet considered living space?
In general, areas like staircases and closets count as finished square footage. Spaces like garages, three-season porches and unfinished basements or attics are not included in the square footage of a house. 7 thg 1, 2020
## What is the ideal square footage for a family of 4?
around 2400 square feet How Much Space Does A Family Need? The average house size for a family of four to live comfortably is around 2400 square feet. It is widely believed that each person in a home requires 200-400 square feet of living space. The average cost to build a home of that size will range between $147,000 to$436,000. 17 thg 6, 2019
## Is a 3500 sq ft house big?
Houses of 3000 to 3500 square feet are large enough to create a luxury home that can suit almost any family. Features such as high ceilings, an expansive master suite, home office, media room, or separate guest space can easily fit in an upper-mid-size home plan. 30 thg 3, 2021
## Is a 3800 square-foot house big?
A 3700 to 3800 square foot home may be perfect for a larger family that needs plenty of extra space or an average-sized family or couple that enjoys having a home with individual spaces dedicated to specific needs.
## How many square feet is 13 stairs?
You’ll need between 80 and 110 square feet for 13 stairs. 20 thg 1, 2021
## How does an appraiser determine the square footage of a house?
Measure the length and width, in feet, of each room. Then, multiply the length by the width to calculate that room’s square footage. 14 thg 2, 2022
## How many square feet is 12 stairs?
54 square feet For a typical set of 12 box stairs, you’ll multiply the 36” width by the 18” tread and riser, then multiply by 12 stairs. The total is 7776 square inches of carpet, which equates to 54 square feet.
## What does cubic feet mean in a fridge?
The capacity of refrigerators and freezers is expressed in terms of cubic feet. This is a measurement of the volume of the inside portion of the appliance. The capacity shows how much food can be stored inside the refrigerator or freezer.
## How do you measure cubic feet of dirt?
How to Calculate Cubic Feet for Soil Use a measuring tape to determine the length of the location where you will be placing the soil. … Measure the width in feet. Determine the necessary depth of the soil in feet. … Multiply the width by the length by the depth to find the number of cubic feet of soil you need.
## How many cubic feet is 24 square feet?
Cubic feet to Square feet Calculator 1 cubic feet = 1 ft2 1 ft2 = 23 cubic feet = 8.0876 ft2 23 ft2 = 24 cubic feet = 8.3203 ft2 24 ft2 = 25 cubic feet = 8.5499 ft2 25 ft2 = 26 cubic feet = 8.7764 ft2 26 ft2 = 25 hàng khác
## How much is 1 cubic feet in square feet?
Square feet = cubic feet ÷ depth. So: 5 ÷ 0.25 = 20. We have our total: 20 square feet.
## How big is a cubic foot in feet?
Definition: A cubic measurement is the three-dimensional derivative of a linear measure, so a cubic foot is defined as the volume of a cube with sides 1 ft in length. In metric terms a cubic foot is a cube with sides 0.3048 metres in length. 22 thg 7, 2018
## What is the average volume of a house?
The average single family house in the United States has overall increased in size since 2000. It reached its peak of 2,467 square feet in 2015 before falling to 2,261 square feet by 2020. 28 thg 3, 2022
## What is the volume of a bedroom?
So, how do we calculate room volume which is the length, width, height? Good simple formula. Take the length of your room, the width of your room and the height of your room and multiply the three. That will give you your cubic volume, cubic feet of volume. 10 thg 12, 2019
## How do you get volume from density?
Divide the mass by the density of the substance to determine the volume (mass/density = volume). Remember to keep the units of measure consistent. For example, if the density is given in grams per cubic centimeter, then measure the mass in grams and give the volume in cubic centimeters. 24 thg 4, 2017
## What does cm3 mean in maths?
A cubic centimetre (or cubic centimeter in US English) (SI unit symbol: cm3; non-SI abbreviations: cc and ccm) is a commonly used unit of volume that corresponds to the volume of a cube that measures 1 cm × 1 cm × 1 cm. One cubic centimetre corresponds to a volume of one millilitre.
## How do you find volume 5th grade?
V = l×w×h is a formula. A 3-dimensional solid shape that has 6 faces. A measure of how long an object is. A measure of how wide an object is. 31 thg 7, 2021
## Does volume mean in math?
In math, volume is the amount of space in a certain 3D object. For instance, a fish tank has 3 feet in length, 1 foot in width and two feet in height. To find the volume, you multiply length times width times height, which is 3x1x2, which equals six. So the volume of the fish tank is 6 cubic feet.
## How do I teach my kindergarten capacity?
Capacity Activities For Kindergarten & Preprimary select the right tool to measure the object being measured. select the same unit and not different units when measuring an object. measure without gaps or overlaps. start and end the measuring in the right spots. measure it straight. be precise.
## How many sides cuboid have?
A cuboid is any shape with flat faces that are quadrilateral (have four sides).
## What is a cuboid in math?
A cuboid is a 3D shape that has six faces, twelve edges and eight vertices. Each of its faces is a rectangle. A cuboid is also a prism, as it has the same cross-section all the way through. It’s known as a rectangular prism.
## What is cuboid perimeter?
Perimeter of a cuboid = 4 (l + b + h)
## How tall is 4 cubic feet?
ENDMEMO 1 cubic feet = 1 feet 1 cubic feet 3 cubic feet = 1.4422 feet 27 cubic feet 4 cubic feet = 1.5874 feet 64 cubic feet 5 cubic feet = 1.71 feet 125 cubic feet 6 cubic feet = 1.8171 feet 216 cubic feet 25 hàng khác
## How wide is 9 cubic feet?
ENDMEMO 1 cubic feet = 12 inch 1 inch = 8 cubic feet = 24 inch 8 inch = 9 cubic feet = 24.961 inch 9 inch = 10 cubic feet = 25.8532 inch 10 inch = 11 cubic feet = 26.6878 inch 11 inch = 25 hàng khác
Genç Çoban Hüseyin ve Keçi Sürüsü Aşkı | Belgesel ▫️4K▫️
Genç Çoban Hüseyin ve Keçi Sürüsü Aşkı | Belgesel ▫️4K▫️
## Is the ceiling a wall?
A part of a building which encloses and is exposed overhead in a room, protected shaft or circulation space. (The soffit of a rooflight is included as part of the surface of the ceiling, but not the frame. An upstand below a rooflight would be considered as a wall.) 11 thg 6, 2021
## What is the area of four walls of cuboid?
FORMULA USED: Area of four walls = 2 H (L + B)
## How is rectangular room volume?
To find the volume of a rectangular prism, multiply its 3 dimensions: length x width x height. The volume is expressed in cubic units.
## How do I find cubic feet?
If you prefer to or have to calculate cubic footage by hand, you can find cubic feet by multiplying three linear measurements—length, width, and height—in feet. For instance, to find the volume of a cube, you would calculate the following: length x width x height. 24 thg 2, 2022
## How do you find the volume of a box in cubic feet?
Measure the length, width and height of the box in inches. 2. Multiply the length, width and height and divide the resulting number by 1,728. This is the container’s volume in cubic feet.
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You have just come across an article on the topic how big is 500 cubic feet. If you found this article useful, please share it. Thank you very much. | 2022-06-28 16:30:44 | {"extraction_info": {"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, "math_score": 0.50533527135849, "perplexity": 2361.9175637597946}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103556871.29/warc/CC-MAIN-20220628142305-20220628172305-00685.warc.gz"} |
https://academy.vertabelo.com/course/introduction-to-r/vectors/what-are-vectors/combining-vectors | Spring Deals - hours only!Up to 70% off on all courses and bundles.-Close
What are vectors?
7. Combining vectors
Vector operations
Indexing and filtering
Simple analysis
Summary
## Instruction
Great! You can also combine vectors of the same data type to produce a bigger vector.
Suppose the four vectors that follow represent the salaries of employees in Versico's four departments:
salaries_dep_1 <- c(3400, 2800, 5600, 4200)
salaries_dep_2 <- c(3100, 3800, 2700)
salaries_dep_3 <- c(6100, 6200)
salaries_dep_4 <- c(2400, 1800, 2600)
You can easily combine these vectors with the c() function. Pass the individual vector names, separated by commas, into the function as the arguments, like so:
salaries <- c(salaries_dep_1, salaries_dep_2, salaries_dep_3, salaries_dep_4)
The vector salaries will now contain all elements from all four vectors.
## Exercise
We've created vectors names_dep1, names_dep2, names_dep3, and names_dep4 that store the names of employees in each department.
Combine these vectors into a single vector called names. | 2019-04-25 18:25:27 | {"extraction_info": {"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, "math_score": 0.22020354866981506, "perplexity": 8200.353713462398}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-18/segments/1555578732961.78/warc/CC-MAIN-20190425173951-20190425195951-00197.warc.gz"} |
https://stats.stackexchange.com/questions/143156/useful-representation-of-continuous-and-nominal-variables | # Useful Representation of Continuous and Nominal variables
I want to develop a prediction model (e.g. using SVM, Neural Networks...etc) to predict the relationship between a protein and its DNA target. Each proteins is represented using ~100 continuous [-infinity,+infinity] numerical variables + one categorical (nominal) variable. However, its DNA target is a sequence of A,C,G and T letter and will be represented in also a categorical variable.
One feature vector should combine features (variables) from both of the protein and its target DNA sequence. So, I have to represent mixture of continuous and categorical (nominal) variables.
The categorical (nominal variables) are two types:
1) One type is to represent DNA Sequence (e.g. AACTT) [Note: we have four possibilities for DNA letters: A,C,G or T]
2) Another type is the category of the protein (I have 69 classes).
So, my questions are:
1) I am wondering what is the best representation for both types of categorical variables? (e.g. I saw people represent A,C,G and T as 0001,0010,0100 and 1000, respectively, while two binary digits were sufficient). What about the 69 classes variable?
2) Can I combine the continuous and categorical variables in one feature vector?
I have looked into similar questions in this group, but could not find relevant answer to what I have.
• To clarify, the DNA may be represented as {ATCGGATCAAGCTT....(20 such characters)} and protein as {1,38,-705,50,986,-5,7,-890,...(100 such numbers)}+{1 of 69 categories} ? And you want to combine and make a another entity which has one DNA and one protein? So the new entity will have 3 components?
– rnso
Mar 24, 2015 at 14:47
• correct @rnso!. (The DNA is of fixed length e.g. I selected it to be of length 20 letters) Mar 24, 2015 at 14:49
• Can you not have a smaller range for protein numbers rather than [-infinity,+infinity] ? What does it actually represent?
– rnso
Mar 24, 2015 at 14:52
• Initially you write that you want to "predict the relationship between a protein and its DNA target". Then you write that you want to combine them to a single vector. These 2 can be 2 different questions.
– rnso
Mar 24, 2015 at 14:54
• For your first question: actually some variables range between 0 and 1; some between -100, +200, some are more varient... (but none of the 100 variables have value <-10000 or >10000). These variables represent the chemical properties of the protein (e.g. its hydrophobic score..etc) Mar 24, 2015 at 14:56
To clarify, the DNA may be represented as {ATCGGATCAAGCTT....(20 such characters)} and protein as {1,38,-705,50,986,-5,7,-890,...(100 such numbers)}+{1 of 69 categories}. And you want to combine and make a another entity which has one DNA and one protein. So the new entity will have 3 components.
It seems that for protein part you want to show 101 features: 100 as numbers that represent the chemical properties of the protein (e.g. its hydrophobic score..etc) and last 1 as category (1 of 69 categories). It is like a table showing different features of different persons:
person_name height weight waist age gender family_name ...
So your can create a table with following columns:
DNA_sequence
Protein_name
Chemical_feature1
Chemical_feature2
Chemical_feature3
..
Protein_category_1_to_69
Then you can try to find which chemical feature or category is associated with which DNA sequence.
I believe DNA sequences can also be broken down to 'triplets'. That may also be helpful in finding associations.
I think you should break down your information into separate variables for better analysis. Each row of above table will be your combined entity.
Edit: To have analysis as described in the comment below, one can simply create a text string from 100 chemical features of proteins, eg:
"100,-52,-1,0.5,259,365,...."
This will then become one categorical variable (column) from 100 different numeric variables (columns). One can use 'paste' or 'paste0' function of R for this.
• Thanks for the description of the questions.However, I do not think this solution can work to solve the problem. Simply because what you said "Then you can try to find which chemical feature or category is associated with which DNA sequence" is not applicable here. We want the model to learn that when a protein with a specific values for the set of chemical properties (100 values) + one categorical value then it binds to this dna traget. It learns the same for each sample.So, consider the training data for the model as a matrix(rows=samples of proteins with their dna,columns=100+1+20 features) Mar 24, 2015 at 15:20 | 2022-07-01 23:44:28 | {"extraction_info": {"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, "math_score": 0.4860146641731262, "perplexity": 1421.3182746926202}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103947269.55/warc/CC-MAIN-20220701220150-20220702010150-00324.warc.gz"} |
https://www.mathlearnit.com/decimal-numbers.html | # Decimal Numbers
Decimal numbers are similar to fractions in basic principle.
They are often used as a way of writing a number that isn’t whole, a number that will contain a decimal point.
Take the mixed number 4 $\fn_jvn&space;\tfrac{1}{4}$, there is a whole number, and a fraction.
The whole number is 4, and the fraction, which is part of a whole number, is $\fn_jvn&space;\tfrac{1}{4}$.
Likewise, with the decimal 4.25.
The whole number is 4, and the decimal part is 0.25.
One can think in terms of money, cents are parts of a dollar, 100 cents make 1 full dollar.
If you have \$3 and 55c, you have 3 full dollars, and 55 parts of another dollar.
Which looks like \$3.55 as a decimal.
You don’t have a full \$4, but you’re some of the way there, having more than \$3.
Above is a basic introduction to the idea of numbers in decimal form, the pages listed below delve into the topic Mathematically in more detail.
## Decimal Numbers Pages
- Decimal place value, infinite, repeating
A section covering decimal place value, along with explanations of infinite and repeating decimals.
- Rounding Decimals
Decimals can be rounded a certain place to make the number smaller or simpler.
Like whole numbers, numbers in decimal form can be added and subtracted.
- Multiplication of Decimals
Multiplying decimals is quite similar to multiplying whole numbers.
- Division of Decimals
Section featuring a variety of examples of division involving decimals.
- Change Decimals to Fractions
Examples of how to approach changing various decimals to fractions.
› Decimals | 2018-09-26 07:15:17 | {"extraction_info": {"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": 2, "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, "math_score": 0.6108853220939636, "perplexity": 1765.6954963169355}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267163704.93/warc/CC-MAIN-20180926061824-20180926082224-00471.warc.gz"} |
https://learn.careers360.com/ncert/question-in-how-many-ways-can-a-team-of-3-boys-and-3-girls-be-selected-from-5-boys-and-4-girls/ | # Q.4. In how many ways can a team of 3 boys and 3 girls be selected from 5 boys and 4 girls?
S seema garhwal
A team of 3 boys and 3 girls be selected from 5 boys and 4 girls.
3 boys can be selected from 5 boys in $^5C_3$ ways.
3 girls can be selected from 4 boys in $^4C_3$ ways.
Therefore, by the multiplication principle, the number of ways in which a team of 3 boys and 3 girls can be selected $=^5C_3\times ^4C_3$
$=\frac{5!}{2!3!}\times \frac{4!}{1!3!}$
$=10\times 4=40$
Exams
Articles
Questions | 2020-03-31 22:30:57 | {"extraction_info": {"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": 5, "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, "math_score": 0.2354070246219635, "perplexity": 122.0880820927611}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370504930.16/warc/CC-MAIN-20200331212647-20200401002647-00237.warc.gz"} |
http://physics.stackexchange.com/questions/14652/fluid-mechanics-from-a-variational-principle | # Fluid Mechanics from a variational principle
It is possible to define a good variational principle to describe Fluid Mechanics? if so, what is the correct treatment of the issue. I guess something like:
$$I=\int d^4x \left(\frac{1}{2}\rho v^2-P-\rho g x\right)$$
-
You should probably be more specific about what equation you want to reproduce. The full Navier-Stokes equations? Incompressible? Euler equations? Any of the above? Also, I know its just a first attempt, but the equation you wrote down has many flaws. – BebopButUnsteady Sep 14 '11 at 16:00
I'm not an expert in the field, but I found a reference that claims this is possible. prl.aps.org/abstract/PRL/v58/i14/p1419_1 (behind a PRL paywall). – Gerben Sep 14 '11 at 17:29
Thanks very much you both. BebopButUnsteady You are rigth, i meant Euler's equation. – Adolfo_Toloza Sep 15 '11 at 14:06
A nice lagrangian treatment is available for incompressible potential flows with hydrostatic pressure, or for flows allowing one component of vorticity. Also, for a general Hamiltonian description one may resort to Clebsch variables, and then define the resultant lagrangian from these. One can check out some of Zakharov's reviews on the subject. – user8260 Apr 14 '12 at 22:27
This is the reason for the Lagrangian coordinates in fluid mechanics.
The velocity field is a momentum, so that the Lagrangian variational description needs the correpsonding coordinate. The corresponding coordinate is the map which tells you where each fluid particle ends up if you follow the flow up to time t. This is a diffeomorphism, and the Hamiltonian formulation is on a phase space of all diffeomorphisms and its tangent space, which is the velocity vector fields.
The kinetic energy is just the integral of the square of the velocity, and there is a pressure which is best put in by enforcing the constraint that the fluid is incompressible by Lagrange multipliers (if it is incompressible). The Lagrangian formulation is covered in many places. It is not particularly computationally convenient because the diffeomorphism generated by a flow is completely impossible to determine, and irrelevant because the diffeomorphisms are a homogenous group.
V.A. Arnold has a treatment of this point of view in his book "Topological Methods in Hydrodynamics", which is very good, and emphasizes the geometry.
-
Thank you very much. – Adolfo_Toloza Sep 15 '11 at 14:06 | 2015-03-05 20:32:11 | {"extraction_info": {"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, "math_score": 0.8754177689552307, "perplexity": 471.136509192487}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-11/segments/1424936464840.47/warc/CC-MAIN-20150226074104-00131-ip-10-28-5-156.ec2.internal.warc.gz"} |
http://bootmath.com/show-that-hom_rrnm-cong-mn-for-r-modules.html | # Show that $Hom_R(R^n,M) \cong M^n$ for R-modules
We want to show that $Hom_R(R^n,M) \cong M^n$ for $n\in\Bbb Z_{\ge0}$
I have already shown that $Hom_R(R,M) \cong M$ by letting $f:Hom_R(R,M)\rightarrow M$ given by $f(\phi) = \phi(1)$.
I showed that $f$ is bijective and is a group homomorphism, thus $Hom_R(R,M) \cong M$.
It seems too easy to define the same function for $Hom_R(R^n,M) \cong M^n$, I must be doing something wrong. Injectivity and surjectivity were both straightforward, and showing its a homomorphism seemed to go okay… Can you just use the same $f$? How can you show these are isomorphic?
#### Solutions Collecting From Web of "Show that $Hom_R(R^n,M) \cong M^n$ for R-modules"
In general, following theorem holds:
$$\operatorname{Hom}_R\left(\bigoplus_{i\in I}A_i, B \right)\cong \prod_{i\in I}\operatorname{Hom}_R(A_i,B)$$
Let define a map $\phi$ from the former to the latter defined as
$$\phi(f) := \langle f\circ \iota_i\rangle_{i\in I}$$
(where $\iota_i:A_i\to \bigoplus_{i\in I} A_i$ be the canonical insertion) and define map $\psi$ from the latter to the former as
$$\psi(\langle g_i\rangle_{i\in I})(a) = \sum_{i\in I}g_i(a_i).$$
Above sum ranges for all $i\in I$. Each $g_i$ is a morphism from $A_i$ to $B$ and $a = \langle a_i\rangle_{i\in I} \in \bigoplus_{i \in I} A_i$. $a_i=0$ but finitely many so $\psi(\langle g_i\rangle_{i\in I})$ is well-defined. You can check that $\phi$ and $\psi$ are homomorphism and inverses each other.
You can find the finite version of above theorem, and its proof is essentially same. | 2018-07-16 20:07:19 | {"extraction_info": {"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, "math_score": 0.9879046082496643, "perplexity": 150.20230855913107}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676589455.35/warc/CC-MAIN-20180716193516-20180716213516-00035.warc.gz"} |
https://www.esaral.com/q/avinash-can-run-with-a-speed-of-13791/ | Avinash can run with a speed of
Question:
Avinash can run with a speed of $8 \mathrm{~m} \mathrm{~s}^{-1}$ against the frictional force of $10 \mathrm{~N}$, and Kapil can move with a speed of $3 \mathrm{~ms}^{-1}$ against the frictional force of $25 \mathrm{~N}$. Who is more powerful and why?
Solution:
$P=F u$
Power of Avinash $=10 \times 8=80 \mathrm{~W}$
Power of Kapil $=25 \times 3=75 \mathrm{~W}$
So, Avinash is more powerful. | 2022-05-17 11:47:42 | {"extraction_info": {"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, "math_score": 0.5250863432884216, "perplexity": 833.2204331045672}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662517245.1/warc/CC-MAIN-20220517095022-20220517125022-00184.warc.gz"} |
https://semiautomaticclassificationmanual.readthedocs.io/en/latest/importSignaturesTab.html | # 4. Import signatures¶
The tab Import signatures allows for importing spectral signatures from various sources.
## 4.1. Import library file¶
Import library file
This tool allows for importing spectral signatures from various sources: a previously saved Training input (.scp file); a USGS Spectral Library (.asc file); a previously exported CSV file. In case of USGS Spectral Library, the library is automatically sampled according to the image band wavelengths defined in the Band set, and added to the ROI & Signature list;
## 4.2. Import vector¶
Import vector
This tool allows for importing a vector (shapefile or geopackage), selecting the corresponding fields of the Training input.
• Select a vector : open a vector;
• MC ID field : select the vector field corresponding to MC ID;
• MC Name field : select the vector field corresponding to MC Name;
• C ID field : select the vector field corresponding to C ID;
• C Name field : select the vector field corresponding to C Name;
• Calculate sig.: if checked, the spectral signature is calculated while the ROI is saved to Training input;
• Import vector : import all the vector polygons as ROIs in the Training input;
TIP: Spectral libraries downloaded from the USGS Spectral Library can be used with Minimum Distance or Spectral Angle Mapping algorithms, but not Maximum Likelihood because this algorithm needs the covariance matrix that is not included in the spectral libraries. | 2022-07-05 13:22:44 | {"extraction_info": {"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, "math_score": 0.4550475776195526, "perplexity": 6193.862115195595}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104576719.83/warc/CC-MAIN-20220705113756-20220705143756-00610.warc.gz"} |
http://www.maths.ox.ac.uk/node/10154 | # Optimization meets Statistics: Fast global convergence for high-dimensional statistical recovery
21 February 2013
14:00
Professor Martin Wainwright
Abstract
Many methods for solving high-dimensional statistical inverse problems are based on convex optimization problems formed by the weighted sum of a loss function with a norm-based regularizer. \\ Particular examples include $\ell_1$-based methods for sparse vectors and matrices, nuclear norm for low-rank matrices, and various combinations thereof for matrix decomposition and robust PCA. In this talk, we describe an interesting connection between computational and statistical efficiency, in particular showing that the same conditions that guarantee that an estimator has good statistical error can also be used to certify fast convergence of first-order optimization methods up to statistical precision. \\ \\ Joint work with Alekh Agarwahl and Sahand Negahban Pre-print (to appear in Annals of Statistics) \\ http://www.eecs.berkeley.edu/~wainwrig/Papers/AgaNegWai12b_SparseOptFull.pdf
• Computational Mathematics and Applications Seminar | 2017-10-20 23:46:30 | {"extraction_info": {"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, "math_score": 0.9995656609535217, "perplexity": 919.1920982264384}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187824471.6/warc/CC-MAIN-20171020230225-20171021010225-00083.warc.gz"} |
http://math.stackexchange.com/questions/65923/how-does-one-compute-the-sign-of-a-permutation | # How does one compute the sign of a permutation?
The sign of a permutation $\sigma\in \mathfrak{S}_n$, written ${\rm sgn}(\sigma)$, is defined to be +1 if the permutation is even and -1 if it is odd, and is given by the formula
$${\rm sgn}(\sigma) = (-1)^m$$
where $m$ is the number of transpositions in the permutation when written as a product of transpositions.
Alternatively the sign is -1 if, when we express $\sigma$ as a product of disjoint cycles, the result contains an odd number of even-length cycles. Otherwise the sign is +1.
My permutations are expressed as tuples $(\sigma_1,\dots,\sigma_n)$, so neither the expression of the tuple as a product of disjoint cycles nor as a product of transpositions are immediately available to me.
This suggests two high-algorithms to compute the sign of a permutation:
1. Express the permutation as a product of transpositions and count the number of transpositions.
2. Express the permutation as a product of disjoint cycles and count the number of even-length cycles.
What are typical algorithms for accomplishing these tasks, and how do they vary in running type depending on $n$? Are there more efficient algorithms for calculating the sign of a permutation?
The motivation is to quickly decide whether instances of the fifteen puzzle are solvable. I want to generate a large number of solvable instances of the fifteen puzzle for testing some search algorithms. At the moment I generate a random instance, and test whether it is solvable by trying to solve it with depth-first search, which is fairly slow going, and won't generalize well to larger puzzles (24 puzzle, 35 puzzle...) due to time and memory limitations. Since solvable instances of the fifteen puzzle are in 1-1 correspondence with even elements of $\mathfrak{S}_{16}$, I figure that there must be a faster way of generating solvable instances.
It has just occurred to me that a better way of generating solvable instances of the puzzle might be to generate an even number of transpositions and multiply them together to generate an even permutation. I'd prefer an algorithm that was guaranteed to return an even distribution over $\mathfrak{S}_n$ though, and in fact I'm now sufficiently interested in the answer to this question in its own right.
-
The parity of a permutation is also the number of inversions in the permutation modulo $2$. Since the number of inversions can be computed in $O(n \log n)$ time, parity can certainly be. It's an interesting question if even faster algorithms are possible. Off the top of my head, I can't think of any reason why $O(n)$ algorithm is impossible. (This is the best reference I could pull off right now for finding inversions: stackoverflow.com/questions/6523712/… .) – Srivatsan Sep 19 '11 at 23:58
It seems the situation is more complicated. PengOne's answer in the other question says that there is a long-standing $O(n \frac{\log n}{\log \log n})$ algorithm and a recent $O(n \sqrt{\log n})$ algorithm for counting inversions. Actually, I think this question would be a good fit for cstheory.SE as well. – Srivatsan Sep 20 '11 at 0:02
I think you can find this in Knuth. (No, I don't know where. It just feels like it's in there somewhere.) – Michael Lugo Sep 20 '11 at 0:16
You should probably specify what encoding of a permutation you're using (in cycle notation, shouldn't you be able to compute the parity in $O(n)$ time?). – Qiaochu Yuan Sep 20 '11 at 0:27
– lhf Sep 20 '11 at 0:52
So you have a permutation $f: X \to X$ for which you can efficiently compute $f(x)$ from $x$.
I think a good way to do any of the things you mentioned is to make a checklist for the elements of $X$. Then you can start with the first unchecked element and follow the chain $x, f(x), f(f(x))$, etc. checking off each element as you find it until you reach an element that is already checked. You have now traversed one cycle. Then you can pick the next unchecked element and traverse that cycle, and so on until all elements are checked.
While you traverse a cycle you can easily
1. Count the cycle length
2. Record the cycle, or
3. Record transpositions
All this works in roughly linear time. Obviously just counting the cycle lengths is going to be the fastest.
-
Nice :). I totally missed that the cycle representation can indeed be obtained in linear time... – Srivatsan Sep 20 '11 at 1:05
Thanks for your answer Niels. – Chris Taylor Sep 23 '11 at 14:11
If $c_e(n)$ is the number of even-length cycles in a permutation $p$ of length $n$, then one of the formulas for the sign of a permutation $p$ is $\text{sgn}(p) = (-1)^{c_e(n)}$.
Here is an $O(n)$ Matlab function that computes the sign of a permutation vector $p(1:n)$ by traversing each cycle of $p$ and (implicitly) counting the number of even-length cycles. The number of cycles in a random permutation of length $n$ is $O(H_n)$, where $H_n$ is the $n$-th Harmonic Number.
function sgn = SignPerm(p);
% ----------------------------------------------------------
% Calculates the sign of a permutation p.
% p is a row vector p(1,n), which represents the permutation.
% sgn(p) = (-1)^(No. of even-length cycles)
% Complexity : O(n + ncyc) ~ O(n + Hn) ~~ O(n+log(n)) steps.
%
% Derek O'Connor 20 March 2011.
% ----------------------------------------------------------
n = length(p);
visited(1:n) = false; % Logical vector which marks all p(k)
% not visited.
sgn = 1;
for k = 1:n
if ~visited(k) % k not visited, start of new cycle
next = k;
L = 0;
while ~visited(next) % Traverse the current cycle k
L = L+1; % and find its length L
visited(next) = true;
next = p(next);
end
if rem(L,2) == 0 % If L is even, change sign.
sgn = -sgn;
end
end % if ~visited(k)
end % for k
-
Every cycle of length $n$ can be factored into a product of $n-1$ transpositions (on the same elements). For example, $(abc\cdots)=(ab)(ac)\cdots$. This explains the $c_{\epsilon}(n)$ formula, since disjoint cycles factor into disjoint transpositions. – bgins Dec 15 '11 at 21:34
The inverse permutation can be constructed as a sequence of $n-1$ transpositions via Gaussian Elimination with partial pivoting, $P A = L U$, where $A$ is the original permutation matrix, $P=A^{-1}$, and $L=U=I$. Since the signature of the inverse permutation is the same as that of the original permutation, this procedure yields the sign of the permutation.
Thankfully, this algorithm can be run in linear time by maintaining the permutation and its inverse throughout a short-circuited Gaussian Elimination procedure since it can easily be seen that the Schur complement updates will always be zero.
-
It's worth mentioning the quadratic time algorithm, since it can be faster for small permutations:
$$\textrm{sgn}(\sigma) = (-1)^{\sum_{0 \le i<j<n} (\sigma_i>\sigma_j)}$$
I.e., the sign is negative iff there are an odd number of misordered pairs of indices. This algorithm also works if we're interested in the permutation defined by an unsorted list of integers where the cycle structure can't be determined in linear time.
-
Wouldn't you need a sum over all $1 \leq i < j \leq n$? – Lynn Oct 13 '15 at 15:02
Mauris: Fixed. I had left off the $0 \le$. – Geoffrey Irving Oct 13 '15 at 23:02
$sgn(\sigma)=\prod_{i<j}\frac{\sigma(i)-\sigma(j)}{i-j}$ is quadratic, but straightforward to code.
- | 2016-02-09 12:18:49 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8790392875671387, "perplexity": 297.29774528384803}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-07/segments/1454701157075.54/warc/CC-MAIN-20160205193917-00102-ip-10-236-182-209.ec2.internal.warc.gz"} |
https://stats.stackexchange.com/questions/302464/wilcoxon-signed-rank-vs-kruskal-wallis | # Wilcoxon-signed rank vs Kruskal-Wallis?
I am rather stressed out about my methodology. For my PhD I ran a year long literacy intervention (small scale longitudinal study) that was supposed to test the rate of change in individual students' literacy skills over the year. I chose an individual rate of change per student because I cannot compare students with different socio-economic backgrounds and from different schools etc.
My sample was roughly 90 (3 classess = small-scale). The students had to write multiple writing drafts throughout the year-long intervention so that I could track their changing literacy scores. I numbered the literacy assignments from 1 to 10 and then at the end when I did the analysis I looked at the literacy scores pre-intervention and post-intervention (assignments 1 and 10) for an overview of whether students developed better literacy skills. But the strength of my study (or so I thought, and my supervisors) was in the process/based analysis. I compared the literacy scores for assignment 10 & 9; 9 & 8; 8 & 7 etc to get a detailed analysis of what was happening during the intervention. I wanted to know if there was a better literacy score between assignment 10 & 9; a better score between 9 & 8 etc to see if progress was being made, each one building on the one before. I wanted more than just a pre and post score because often things happen during the year that impact on teaching and learning and we wanted to see if they impacted on development, where in the year to get an idea of what aspect of the syllabus was not working. So I used a Wilcoxon signed rank test and now that I am trying to publish I am being told it was the wrong test to use. Out of 4 reviewers, 3 are happy with the Wilcoxon but one said it should have been Kruskal-Wallis. I am being told that because I looked at pairwise comparisons over a year, and not just a once off pre and post, I have inflated the error? I am confused and would love some help if possible.
• I'm not quite sure what you did. Did you do one Wilcoxon test for assignments 10 vs 9, then another for 9 vs 8 etc, or did you somehow combine all the comparisons into one test? Also, what do you mean by "3 classes"? How do the classes enter the analysis? – Gordon Smyth Sep 11 '17 at 0:01
• I just gave context about the sample with the 3 classes so the 90 students came from 3 separate classes. I taught a section, students responded with a writing task, I taught some more, they responded again and this process carried on for the year. Each writing response was given a task number hence 1 - 10. All I wanted to check was whether the students showed an increase in their literacy scores as we progressed so I was told to use the wilcoxon to test matched pairs - was task 10's score greater than task 9. Was task 9's score greater than task 8 etc all the way down to task 1. – Tracey Bunn Sep 11 '17 at 0:20
• Sorry, forgot to add. I think we ran a separate wilcoxon signed rank test for each pairwise comparison for all 90 odd students. – Tracey Bunn Sep 11 '17 at 0:22
• You need to be sure! Did you run the analysis yourself? – Gordon Smyth Sep 11 '17 at 0:35
• Kruskal Wallis doesn't make sense because it ignores the student effect. – Glen_b Sep 11 '17 at 2:35
In my opinion, the Wilcoxon signed rank tests are fine. It's not the only way you could analyse this data, but it's ok. You need to present nine test p-values in your manuscript, one for each increment (2 vs 1, 3 vs 2, ..., 10 vs 9.).
Your data is not in any sense a one-way anova so the Kruskal-Wallis test is inappropriate. A Kruska-Wallis test would assume that all observations are independent, whereas repeat observations on the same student are related.
The Wilcoxon signed rank test correctly accounts for the fact that observations are paired by student by making a pairwise comparisons.
• Thank you Gordon. I did have separate p values for all pairwise samples which enabled me to check if each of the paired scores (the 'changes') were statistically significant. I was very concerned when a journal reviewer came back and said I should not have used a wilcoxon but rather the Kruskal-Wallis (despite using the same student's pairwise comparisons). – Tracey Bunn Sep 11 '17 at 0:57
• As you publish more papers, you'll find that reviewers do make mistakes. BTW, if you think I've covered everything that was concerning you, you could mark my answer as "accepted". Or just wait for other answers if you want more opinions. – Gordon Smyth Sep 11 '17 at 6:19
Wilcoxon signed rank tests seem appropriate since they reflects the fact that measures are taken repeatedly on the same subjects (which increases the tests power to detect real effects) but are modest enough to recognize that grades on writing assignments are only ordinal, but not interval variables (i.e. the difference between a B+ and an A could be much smaller than the difference between an A and an A+ etc.).
Doings the Wilcoxon signed rank tests assumes, that even if the steps between grades are not uniformly high, you as a grader would know for any two evolutions you compare among the students which step is bigger of the two. If you cannot do that, you cannot rank the changes and without a ranking there will not be a rank test. You would be limited to a sign test: basically just counting how many students improved, how many stayed put and how many deteriorated. If there are many more improvements than deteriorations, your test will be significant. Such a test is obviously less powerful since it has no notion of how large any of the improvements were. I do not think you need to use this one. If only you can establish a ranking of improvements, you don't.
If on the other hand your literacy score is much more objectively countable like for example counting the number of mistakes per 100 words (I'm no expert in the field, but you see what I mean with objective I believe), then you can even use paired t-tests. They will have higher statistical power to detect real effects.
When used in the right conditions as described above, the power of the tests to detect existing changes compares as follows:
$$\text{t-test} \geq \text{wilcoxon signed rank test} \geq \text{sign test}$$
In any of the three options, use two sided tests since the possibility that a training session might have deteriorated performance is real. That is what everybody does. Doctors also hope their medication works better than a placebo, but they use two sided tests because it might be even worse than doing nothing. (One sided tests would just make your $\alpha$ level less stringent and are frowned upon.)
Just to be sure, these 3 school classes only exist so that you can get a big enough sample size? You have not chosen the three classes to purposefully represent for example one posh private school, one average school and one underprivileged school? If yes, you will need a more complex statistical methodology to include that information in your analysis as well.
Now to the most important part: The preceding caveats and options notwithstanding, you still need to control your p-value cutoffs for multiple testing. It is very important not to confuse two concepts here:
• your tests are paired on the student level since you observe the same student multiple times (as opposed to observing a different class of 90 students each time after one of your training intervals)
• your tests are pairwise since you compare multiple intermediary situations (as opposed to only comparing the before-after states)
Being paired is taken care of by signed rank tests (or paired t-tests or sign tests), being pairwise requires the following additional precautions:
When the null hypothesis is true and there is no real effect, you still have the possibility of a false discovery proportional to the $\alpha$ cutoff that you compare your p-value to. That is true for every test, so if you do enough tests, you are bound to find some significant results that are false. You need to correct for this inflated chance of false discovery. The easiest way is the Bonferroni correction, just divide your $\alpha$ level by the number of tests. For example, you would be comparing all your p-values against a cutoff of $\alpha/9=0.55\%$ because you perform 9 tests instead of the usual $5\%$ cutoff for a single test. You can see that this is quite a stringent restriction. Holm's method will be a little bit less stringent while still not inflating the chance of false discoveries, it is preferable for that reason.
Practically speaking, are you sure your results are actionable? If you find out for example that the first two training intervals didn't help, but the third and fourth did help, then the fifth and sixth did active harm, the seventh was neutral again and the last two helped, can you translate such mixed results into actionable recommendations? Recommending to skip the intervals 5 and 6 and to put more emphasis on intervals 3 and 4 is only actionable if each interval did something different with the students. If you can explain based on some theory (not only the statistical data) why it could be that some training sessions helped but others didn't, that's insightful. If all the intervals were supposed to do more of the same but didn't, this will be hard to make sense of.
Also, even if you do the 9 intermediary comparisons, you can still also do a before-after comparison. You just need to adjust your cutoffs for one more test (10 instead of 9)
• (+1) But you'd perform the Wilcoxon signed-rank text by calculating the change in each student's grade from one assignment to the next before ranking the magnitude of those changes. So is going from a B+ to an A a bigger or smaller improvement than going from an A to an A+, or is it the same? For that matter is it the same magnitude of change as going from an A to a B+? If you really wanted to be completely non-committal about all that you'd use the sign test. – Scortchi Sep 11 '17 at 9:41
• I assumed that the grader would know which grade improvement has which value even if the steps are not uniformly high. I will clarify. – David Ernst Sep 11 '17 at 13:04 | 2019-10-14 08:31:38 | {"extraction_info": {"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, "math_score": 0.6528363227844238, "perplexity": 726.920850230301}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986649841.6/warc/CC-MAIN-20191014074313-20191014101313-00420.warc.gz"} |
https://economics.stackexchange.com/questions/37786/what-is-the-meaning-of-labor-input-in-the-context-of-incentive-theories | What is the meaning of “labor input” in the context of incentive theories?
In the article "Multitask Principal-Agent Analyses: Incentive Contracts, Asset Ownership, and Job Design" (Holmström and Milgrom, 1991) it is said that two identical agents ($$i=1,2$$) devote attention $$t_i(k)$$ to a task $$k$$, i.e., they allocate $$t_i(k)$$ across a continuum of tasks indexed by $$k\in[0,1]$$. Later it is said that total labor input $$\overline t_i$$ is equal to $$\int t_i(k)dk$$.
With this in mind, my questions are the next: what is the meaning of labor input in this context? How can I graph attention $$t_i(k)$$ and labor input $$\overline t_i$$?
I have an intuitive idea of the general meaning of labor input, but I can't figure out its meaning in this context. For me, there is no difference between labor input and attention. The problem that I have is that I can't clearly understand the meaning of labor input beyond its mathematical definition as the summation of attention.
In particular, I'm more interested in a verbal and familiar (easy to understand for a freshman) interpretation of "labor input". For example, if attention is the derivative of labor input, does this mean that attention is a measure of the productivity of labor input? (this doesn't make much sense to me) How would you define "labor input" with familiar words?
I will appreciate any help.
Since the authors state that the total labor input is:
$$\int t_i(k)dk$$
the meaning of the total labor input in this case would be that it is the sum of all attention $$t_i$$ allocated over those tasks $$k$$.
For example if we would assume that $$t_i (k) = k$$ then the labor supply across continuum given by $$[0,1]$$ would be equal to $$\frac{1}{2}$$ because $$\int k dk = \frac{1}{2} k^2 + c$$ and when you evaluate it between bounds $$[0,1]$$ the area would sum to $$\frac{1}{2}$$ (note that geometric interpretation of an integral is an area under the curve that you are integrating).
You can graph it by making assumption about the function, for example following my assumption of $$t_i(k)=k$$ the labor supply can be plotted (using tikz in LaTex) as:
Where the total labor supply is the area under the curve bounded by $$[0,1]$$. Of course you might want to impose some less simplistic assumption on the function $$t_i (k)$$.
• Thanks so much. I thought the same, although you went a little bit farther than me. The problem that I have is that I can't clearly understand the meaning of labor input beyond its mathematical definition as the summation of attention. In particular, I'm more interested in a verbal and familiar (easy to understand for a freshman) interpretation of "labor input". For example, if attention is the derivative of labor input, does this mean that attention is a measure of the productivity of labor input? (this doesn't make much sense to me) How would you define "labor input" with familiar words? – David Fernando Jiménez Jul 16 at 22:06
• @DavidFernandoJiménez if you care only about purely intuitive explanation then I would say that in this case the labor supply depends on the intensity of effort a person puts into the task. Any student should be familiar with the fact that for example studying is not all or nothing - you can study with more or less concentration put more or less effort. In this case the labor supply is simply the sum of all that effort over tasks – 1muflon1 Jul 16 at 22:09
• I really appreciate your comments. They help me a lot. There are left some gaps, but I think I can fill them up based on your ideas. – David Fernando Jiménez Jul 17 at 21:22 | 2020-10-24 18:30:27 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 21, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9340170621871948, "perplexity": 462.40010389142566}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107884322.44/warc/CC-MAIN-20201024164841-20201024194841-00456.warc.gz"} |
https://www.rdocumentation.org/packages/ciftiTools/versions/0.1.6.0/topics/read_cifti_separate | 0th
Percentile
##### Read a CIFTI file with optional resampling
Read a CIFTI file by writing each component into a GIFTI and NIFTI file (separate_cifti), optionally resampling the GIFTIs, (resample_gifti), and then reading each separated component into R (make_xifti). Surfaces can also be provided; they will be resampled along with the CIFTI for viewing.
Keywords
internal
##### Usage
read_cifti_separate(
cifti_fname,
surfL_fname = NULL,
surfR_fname = NULL,
brainstructures = c("left", "right"),
resamp_res = NULL,
sep_keep = FALSE,
sep_fnames = NULL,
resamp_keep = FALSE,
resamp_fnames = NULL,
write_dir = NULL,
mwall_values = c(NA, NaN),
wb_path = NULL,
verbose = TRUE
)
##### Arguments
cifti_fname
File path of CIFTI-format data (ending in ".d*.nii").
surfL_fname
(Optional) File path of GIFTI surface geometry file representing the left cortex.
surfR_fname
(Optional) File path of GIFTI surface geometry file representing the right cortex.
brainstructures
Character vector indicating which brain structure(s) to obtain: "left" (left cortical surface), "right" (right cortical surface) and/or "subcortical" (subcortical and cerebellar gray matter). Can also be "all" (obtain all three brain structures). Default: c("left","right") (cortical surface only).
If a brain structure is indicated but does not exist, a warning will be raised and that brain structure will be skipped.
resamp_res
(Optional) Target resolution for resampling (number of cortical surface vertices per hemisphere). If NULL (default) or FALSE, do not perform resampling.
sep_keep
If separated files are created, should they be kept or deleted at the end of this function call? Default: FALSE (delete). Keeping the separated files may help speed up certain tasks, for example when repeatedly iterating over subjects--the CIFTI will only be separated once instead of at each iteration.
sep_fnames
(Optional) Where to write the separated files (override their default file names). This is a named list where each entry's name is a file type label, and each entry's value is a file name indicating where to write the corresponding separated file. The recognized file type labels are: "cortexL", "cortexR", "ROIcortexL", "ROIcortexR", "subcortVol", and "subcortLabs".
Entry values can be NULL, in which case a default file name will be used: see cifti_component_suffix. Default file names will also be used for files that need to be separated/written but without a corresponding entry in sep_fnames.
Entries in sep_fnames will be ignored if they are not needed based on [ROI_]brainstructures. For example, if brainstructures="left", then sep_fnames$cortexR will be ignored if specified. The write_dir argument can be used to place each separated file in the same directory. resamp_keep If resampled files are created, will they be kept or deleted at the end of this function call? Default: FALSE (delete). Keeping the resampled files may help speed up certain tasks, for example when repeatedly iterating over CIFTI files--resampling will only be done once instead of every new iteration. resamp_fnames Where to write the resampled files. This is a named list where each entry's name is a file type label, and each entry's value is a file name indicating where to write the corresponding resampled file. The recognized file type labels are: "cortexL", "cortexR", "ROIcortexL", "ROIcortexR", "validROIcortexL", and "validROIcortexR". Entry values can be NULL, in which case a default file name will be used: see resample_cifti_default_fname. Default file names will also be used for files that need to be resampled/written but without a corresponding entry in resamp_fnames. Entries in resamp_fnames will be ignored if they are not needed based on [ROI_]brainstructures. For example, if brainstructures="left", then resamp_fnames$cortexR will be ignored if specified.
The write_dir argument can be used to place each resampled file in the same directory.
write_dir
Where should any output files be written? NULL (default) will write them to the current working directory.
Files flagged for deletion will be written to a temporary directory, and thus are not affected by this argument. So if sep_keep is TRUE, the separated files will be written to write_dir, but if sep_keep is FALSE, they will be written to tempdir() and later deleted. resamp_keep works similarly.
For read_cifti_separate, the surface files (surfL or surfR) are deleted if resamp_keep is FALSE, so in this case they will be written to tempdir(). But for resample_cifti, the surface files are kept even if resamp_keep is FALSE, so they will always be written to write_dir.
Different subfolders for the separated, resampled, and final output files cannot be specified by write_dir. Instead, modify the individual file names in sep_fnames and resamp_fnames.
write_dir must already exist, or an error will occur.
mwall_values
If the medial wall locations are not indicated in the CIFTI, use these values to infer the medial wall mask. Default: c(NA, NaN). If NULL, do not attempt to infer the medial wall.
wb_path
(Optional) Path to Connectome Workbench folder or executable. If not provided, should be set with ciftiTools.setOption("wb_path", "path/to/workbench").
verbose
Should occasional updates be printed? Default: TRUE.
##### Details
The subcortical component (NIFTI) is not resampled.
##### Value
A "xifti" object. See is.xifti.
##### Connectome Workbench Requirement
This function uses a system wrapper for the 'wb_command' executable. The user must first download and install the Connectome Workbench, available from https://www.humanconnectome.org/software/get-connectome-workbench . The wb_path argument is the full file path to the Connectome Workbench folder. (The full file path to the 'wb_cmd' executable also works.) | 2020-12-04 14:43:58 | {"extraction_info": {"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, "math_score": 0.42801377177238464, "perplexity": 8874.333527325913}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141737946.86/warc/CC-MAIN-20201204131750-20201204161750-00336.warc.gz"} |
http://rreece.github.io/outline-of-philosophy/naturalism.html | # Naturalism
Naturalism is a unifying view of philosophy and science. It is an attitude about how philosophy should be done (a meta-philosophy) and about how philosophy relates to science. Regardless of how one views the realism debate (discussed in the previous outline), naturalism shows a provocative, unifying way of viewing the projects of philosophy and science.
Here we also discuss additional meta-philosophical issues, other worldviews in constrast to naturalism, and the culture wars between worldviews.
This could have been called the outline of “Meta-philosophy.”
## What is naturalism?
### First pass
• Natural philosophy
Sellars:
The aim of philosophy, abstractly formulated, is to understand how things in the broadest possible sense of the term hang together in the broadest possible sense of the term. Under ‘things in the broadest possible sense’ I include such radically different items as not only ‘cabbages and kings,’ but numbers and duties, possibilities and finger snaps, aesthetic experience and death. To achieve success in philosophy would be, to use a contemporary turn of phrase, to ‘know one’s way around’ with respect to all these things…1
TODO: Manifest image and scientific image - Sellars
• Sellars, W. (1963). “Philosophy and the scientific image of man” in Sellars (1963)
These days, as more and more philosophers count themselves as naturalists, the term has come to mark little more than a vague science-friendliness. To qualify as unnaturalistic, a contemporary thinker has to insist, for example, that epistemology is an a priori discipline with nothing to learn from empirical psychology or that metaphysical intuitions show quantum mechanics to be false.2
So our inquirer will continue her investigation of the world in her familiar ways, despite her encounter with Descartes and his meditator. She will ask traditionally philosophical questions about what there is and how we know it, just as they do, but she will take perception as a mostly reliable guide to the existence of medium-sized physical objects, she will consult her astronomical observations and theories to weigh the existence of black holes, and she will treat questions of knowledge as involving the relations between the world—as she understands it in her physics, chemistry, optics, geology, and so on—and human beings—as she understands them in her physiology, cognitive science, neuroscience, linguistics, and so on. While Descartes’s meditator begins by rejecting science and common sense in the hope of founding them more firmly by philosophical means, our inquirer proceeds scientifically and attempts to answer even philosophical questions by appeal to its resources. For Descartes’s meditator, philosophy comes first; for our inquirer, it comes second—hence ‘Second Philosophy’ as opposed to ‘First.’ Our Character now has a name: she is the Second Philosopher.3
Jacobs:
Naturalism is an approach to philosophical problems that interprets them as tractable through the methods of the empirical sciences or at least, without a distinctively a priori project of theorizing.4
Naturalism could be defined as a support for philosophy that is filtered for plausibility given the information from science.5 See the Outline on the science method.
Maudlin gives clear naturalist attitude at the start of The Metaphysics within Physics:
[M]etaphysics, insofar as it is concerned with the natural world, can do no better than to reflect on physics. Physical theories provide us with the best handle we have on what there is, and the philosopher’s proper task is the interpretation and elucidation of those theories. In particular, when choosing the fundamental posits of one’s ontology, one must look to scientific practice rather than to philosophical prejudice.6
Papineau:
The term “naturalism” has no very precise meaning in contemporary philosophy. Its current usage derives from debates in America in the first half of the last century. The self-proclaimed “naturalists” from that period included John Dewey, Ernest Nagel, Sidney Hook and Roy Wood Sellars. These philosophers aimed to ally philosophy more closely with science. They urged that reality is exhausted by nature, containing nothing “supernatural,” and that the scientific method should be used to investigate all areas of reality, including the “human spirit” (Krikorian 1944, Kim 2003).
So understood, “naturalism” is not a particularly informative term as applied to contemporary philosophers. The great majority of contemporary philosophers would happily accept naturalism as just characterized—that is, they would both reject “supernatural” entities, and allow that science is a possible route (if not necessarily the only one) to important truths about the “human spirit.”7
### Second pass
A very diverse set of thinkers are often characterized as naturalists or aligned with naturalism, at the expense of much clarity in the term, but naturalism generally consists of varying degrees of either or both:
1. epistemological/methodological naturalism - an epistemic respect for science and empiricism; a methodological commitment to the scientific method of justifying empirical claims as a route to knowledge, if not the chief or perhaps (with a sufficiently broad definition) the only route to knowledge. Science should be guiding in what we claim to know.
2. metaphysical/ontological naturalism - has a variety of claims and interpretations, but they center on the premise that if we have any claim to what there is, it better be informed by and consistent with science. It often involves a skepticism of a priori metaphysics (statements about what there is that come prior to empirical information), and sometimes has further qualified ontological commitments to the products of science, which concerns the debate of scientific realism. Another claim associated with ontological naturalism and closely related to epistemological naturalism is a rejection of ontologies to which we do not have demonstrable, access i.e. supernatural entities, which can be seen as a claim to a type of monism as opposed to dualism about ontology. There is one (natural) world. At the least, it is a claim that science should be guiding in what we claim there is.
Also:
• Methodological vs ontological naturalism
• Jenkins, C. (2016). Epistemological naturalisms.8
• Sean Carroll: “There is one (natural) world.”
• Prasetya, Y. (2021). Methodological naturalism and scientific success.9
### History
• Discussion by McEvilley10
• Archilochus
• Thales
• Anaximander
• Xenophanes
Riepe on the characteristics of naturalism in Indian thought:
1. The naturalist accepts sense experience as the most important avenue of knowledge.
2. The naturalist believes that knowledge is not esoteric, innate, or intuitive (mystical).
3. The naturalist believes that the external world, of which man is an integral part, is objective and hence not “his idea” but an existent apart from his, your, or anyone’s consciousness.
4. The naturalist believes that the world minifests order and regularity and that, contrary to some opinion, this does not exclude human responsibility. This order cannot be changed merely by thought, magic, sacrifice, or prayer, but requires an actual manipulation of the external world in some physical way.
5. The naturalist rejects supernatural teleology. The direction of the world is caused by the world itself.
6. The naturalist is humanistic. Man is not simply a mirror of deity or the absolute but a biological existent whose goal it is to do what is proper to man. What is proper to man is discovered in a naturalistic context by the moral philosopher.11
• New York in the 1940s
• John Dewey (1859-1952) - Columbia University
• Sidney Hook (1902-1989) - New York University
• Herbert W. Schneider (1892-1984) - Columbia University
• Ernest Nagel (1901-1985) - Columbia University
• Their manifesto: Krikorian, Y.H. (1944). Naturalism and the Human Spirit.12
• Quine, W.V.O. (1908-2000) - Harvard University
• Naturalized epistemology
• Anti-foundationalism
## Unity of philosophy and science
### Continuity
• Philosophy and science are continuous in concerns.
• Philosophy is the nursury of budding programs that become specialized sciences.
• Costa, C.F. (2012). Philosophy as a protoscience.13
Descartes (often seen as anti-naturalist, explain, but he) wrote in 1644,
Philosophy as a whole is like a tree; of which the roots are Metaphysics, the trunk is Physics, and the branches emerging from this trunk are all the other branches of knowledge. These branches can be reduced to three principal ones, namely, Medicine, Mechanics, and Ethics.14
Reichenbach:
What unites the members of this group is not the maintenance of a philosophical “system,” but a community of working methods—an agreement to treat philosophical problems as scientific problems whose answers are capable of soliciting universal assent. Philosophical problems, in other words, do not differ in principle from problems of the positive sciences. The strength of this group lies in its common working program and not in a common doctrine—a program which distinguishes it from philosophical sects, and makes possible progress in research.15
Russell:
In the welter of conflicting fanaticisms, one of the few unifying forces is scientific truthfulness, by which I mean the habit of basing our beliefs upon observations and inferences as impersonal, and as much divested of local and temperamental bias, as is possible for human beings. To have insisted upon the introduction of this virtue into philosophy, and to have invented a powerful method by which it can be rendered fruitful, are the chief merits of the philosophical school of which I am a member. The habit of careful veracity acquired in the practice of this philosophical method can be extended to the whole sphere of human activity, producing, wherever it exists, a lessening of fanaticism with an increasing capacity of sympathy and mutual understanding. In abandoning a part of its dogmatic pretensions, philosophy does not cease to suggest and inspire a way of life.16
• Lugg, A. (2006). Russell as a precursor of Quine.17
• In the boat with Neurath.
• International Encyclopedia of Unified Science
Sellars:
Philosophy in an important sense has no special subject-matter which stands to it as other subject-matters stand to other special disciplines. If philosophers did have such a special subject-matter, they could turn it over to a new group of specialists as they have turned other special subject-matters to non-philosophers over the past 2500 years, first with mathematics, more recently psychology and sociology, and, currently, certain aspects of theoretical linguistics. What is characteristic of philosophy is not a special subject-matter, but the aim of knowing one’s way around with respect to the subject-matters of all the special disciplines.18
### Consilience
• William Whewell (1794-1866)
• Philosophy of the Inductive Sciences (1840)
• Wilson, E.O. (1998). Consilience: The Unity of Knowledge.19
• Convergence of evidence
• Unity of knowledge
• The Canberra Plan
Viena Circle manifesto:
The scientific world conception is characterised not so much by theses of its own, but rather by its basic attitude, its points of view and direction of research. The goal ahead is unified science. The endeavour is to link and harmonise the achievements of individual investigators in their various fields of science. From this aim follows the emphasis on collective efforts, and also the emphasis on what can be grasped intersubjectively; from this springs the search for a neutral system of formulae, for a symbolism freed from the slag of historical languages; and also the search for a total system of concepts.20
Bunge:
I believe that a philosophy is spineless without ontology, confused without semantics, acephalous without epistemology, deaf without ethics, paralytic without social philosophy, and obsolete without scientific support—and no philosophy at all with neither.21
### Progress
• Does philosophy make any progress?
• Does science make any progress?
• Philosophy as a nursury for budding sciences.
• Bunge, M. (2012). Evaluating Philosophies.22
• Callard, A. (2018). How philosophy makes progress.
• Priest, G. (2020). Philosophy and its history: An essay in the philosophy of philosophy.23
Carnap:
If we allot to the individual in philosophical work as in the special sciences only a partial task, then we can look with more confidence into the future: in slow careful construction insight after insight will be won. Each collaborator contributes only what he can endorse and justify before the whole body of his co-workers. Thus stone will be carefully added to stone and a safe building will be erected at which each following generation can continue to work.24
Bunge:
Good philosophy is worth doing because it is a vantage point for the study of anything, whether concrete things or abstract ideas. Indeed, although it may not see the world, good philosophy helps looking at it—just as bad philosophy blocks the view of ideas and things, in denying that there are any, or in claiming that they can be understood without the help of either reason or experience.
If good philosophy is both valuable and currently in short supply, then it should be reconstructed. Which materials and tools should be used to rebuild philosophy? I suggest that the materials—the substance—are provided by science and technology, as well as by the history of philosophy; and the tools—the form—by logic and mathematics. This is, at least, the kind of philosophy I care for: one capable of tackling interesting philosophical questions in the light of the best available factual knowledge, and with the help of precision tools forged by formal science.25
## Rejection of a priori metaphysics
### Early modern
Kant in his Prolegomena to Any Future Metaphysics:
My object is to persuade all those who think Metaphysics worth studying, that it is absolutely necessary to pause a moment, and, neglecting all that has been done, to propose first the preliminary question, ‘Whether such a thing as metaphysics be at all possible?’
If it be a science, how comes it that it cannot, like other sciences, obtain universal and permanent recognition? If not, how can it maintain its pretensions, and keep the human mind in suspense with hopes, never ceasing, yet never fulfilled? Whether then we demonstrate our knowledge or our ignorance in this field, we must come once for all to a definite conclusion respecting the nature of this so-called science, which cannot possibly remain on its present footing. It seems almost ridiculous, while every other science is continually advancing, that in this, which pretends to be Wisdom incarnate, for whose oracle every one inquires, we should constantly move round the same spot, without gaining a single step. And so its followers having melted away, we do not find men confident of their ability to shine in other sciences venturing their reputation here, where everybody, however ignorant in other matters, may deliver a final verdict, as in this domain there is as yet no standard weight and measure to distinguish sound knowledge from shallow talk.26
### Analytic/continental divide
#### Introduction
• Counter-Enlightenment
• Johann Georg Hamann (1730-1788)
• G.W.F. Hegel (1770-1831)
• Franz Brentano (1838-1917)
• Edmund Husserl (1859-1938)
• Martin Heidegger (1889-1976)
• Criticism of Kant
• Berlin, I. (1973). The Counter-Enlightenment.
• Parekh, B. (2006). Rethinking Multiculturalism: Cultural Diversity and Political Theory.27
• Analytic philosophy starts with the rejection of idealism
### Criticism
• Putnam, H. (1997). A half century of philosophy, viewed from within.33
### Discussion
TODO: While promoting the careful gathering of empirical information, naturalism is not against (largely non-empirical) rational pursuits like mathematics. Indeed some naturalist may even see logic and mathematics as scientific pursuits of a priori truths.
TODO: Naturalism shares much in common with schools of thought surrounding positivism, through its respect for science and skepticism of a priori metaphysics, but naturalism is a more general distinction, being a more much longer thread throughout the history of philosophy and science.
Quine in “Natural kinds”:
At this point let me say that I shall not be impressed by protests that I am using inductive generalizations, Darwin’s and others, to justify induction, and thus reasoning in a circle. The reason I shall not be impressed by this is that my position is a naturalistic one; I see philosophy not as an a priori propaedeutic or groundwork for science, but as continuous with science. I see philosophy and science as in the same boat—a boat which, to revert to Neurath’s figure as I so often do, we can rebuild only at sea while staying afloat in it. There is no external vantage point, no first philosophy. All scientific findings, all scientific conjectures that are at present plausible, are therefore in my view as welcome for use in philosophy as elsewhere.34
More Quine:
I also expressed, at the beginning, my unswerving belief in external things—people, nerve endings, sticks, stones. This I reaffirm. I believe also, if less affirmly, in atoms and electrons, and in classes. Now how is all this robust realism to be reconciled with the barren scene that I have just been depicting? The answer is naturalism: the recognition that it is within science itself, and not some prior philosophy, that reality is properly to be identified and described.35
Opening lines to ETMG:
This is a polemical book. One of its main contentions is that contemporary analytic metaphysics, a professional activity engaged in by some extremely intelligent and morally serious people, fails to qualify as part of the enlightened pursuit of objective truth, and should be discontinued.36
## Reductionism
### Introduction
Leibniz:
If controversies were to arise, there would be no more need of disputation between two philosophers than between two accountants. For it would suffice to take their pencils in their hands, and say to each other: Calculemus—Let us calculate.37
Wittgenstein:
Every statement about complexes can be analysed into a statement about their constituent parts, and into those propositions which completely describe the complexes.38
• Nagel, E. (1961). The Structure of Science: Problems in the Logic of Scientific Explanation.40
• Nagel, E. (2008). Issues in the logic of reductive explanations.41
• Bunge, M. (1991). The power and limits of reduction.42
• Reductive and non-reductive physicalism
• Morris, K. (2019). Physicalism Deconstructed: Levels of reality and the mind-body problem.43
• Fundamentality
• Physicalism and Hempel’s dilemma
• Robert Sapolsky (2010) lecture: Chaos and Reductionism
• Rosaler, J. (2019). Reduction as an a posteriori relation.46
### Criticism
• Fodor, J.A. (1974). Special sciences (Or: The disunity of science as a working hypothesis).47
• Schweber, S.S. (1993). The ebb and flow of reductionism.48
• Laughlin, R.B. & Pines, D. (2000). The theory of everything.49
Wittgenstein:
There are, indeed, things that cannot be put into words. They make themselves manifest. They are what is mystical.50
## Natural kinds
### Introduction
Part of metaphysical naturalism.
According to the account we will give, science tells us many surprising things, but it does not impugn the everyday status of objects like tables and baseballs. These are, we will argue, aspects of the world with sufficient cohesion at our scale that a group of cognitive systems with practically motivated interest in tracking them would sort them into types for book-keeping purposes.55
### Criticism
• Azzouni, J. (2000). Knowledge and Reference in Empirical Science.60
• Hennig, C. (2015). What are the true clusters?61
• Haslanger, S. (2012). Resisting Reality: Social Construction and Social Critique.62
## Physicalism
### Introduction
• Physicalism and monism
• Parmenides
• Materialism, physicalism, functionalism
• Alyssa Ney on dualism, physicalism, neutral monism
• Ruja, H. (1957). Are naturalists materialists?63
### Completeness thesis
• “the domain of the physical is causally closed.”
• Spurrett, D.J. (1999). The Completeness of Physics.64
• Spurrett, D.J. & Papineau, D. (1999). A note on the completeness of ‘physics.’65
• Ney, A. (2021). The fundamentality of physics: Completeness or maximality?66
• Principle of Naturalistic Closure (PNC)67
### Uniformity of nature
• Part of methodological naturalism, but also a result of empirical confirmation.
• TODO: find ref for Cox’s art, that stars have standard spectra.
• Limits on deviations in $$\alpha_\mathrm{EM}$$
• TODO
## Rejection of the supernatural
### Introduction
• How to define magic and what it could be.
• Aleister Crowley: Magick “is the Science and Art of causing change to occur in conformity with Will.”
• To the degree you can control the physical world, naturalists would call this “technology.”
• Part of methodological naturalism.
• Not an a priori assumption but an empirical conclusion
• Dennett: Naturalism is a method of explaining a magic trick without appealing to magic.
• Rejection of supernatural, magic, and paranormal.
• Blackmore, S.J. (1996). Near-death experiences.68
• Blackmore, S.J. (1997). Probability misjudgment and belief in the paranormal: A newspaper survey.69
• Carrier, R. (2020). Naturalism is not an axiom of the sciences but a conclusion of them.
• Paul Kurtz (1925-2012)
• “father of secular humanism”
Jenkins says naturalism is
the view that a broadly scientific world-view is correct, and there exists nothing supernatural or otherwise spooky.70
### Miracles
• Hume - “Of Miracles”
• The Lewis-Anscombe debate
• Scott Aaronson. (2001). Letter to James Randi.72
• The Low Kolmogorov Complexity (LKC) Principle: Under any reasonable encoding, the universe has low Kolmogorov complexity.
### Criticism
• Koons, R.C. (2000). The incompatibility of naturalism and scientific realism.73
• Paul, L.A. (2012). Metaphysics as modeling: the handmaiden’s tale.74
## Scientism
### As a pejorative
Schopenhauer:
Mere experience is no more a substitute for thinking than reading is. Pure empiricism is related to thinking as eating is to digestion and assimilation. When empiricism boasts that it alone has, through its discoveries, advanced human knowledge, it is as if the mouth should boast that it alone keeps the body alive.75
• Sorell76
• Science Unlimited? The Challenges of Scientism77
• TODO: find examples
### Honorific reinterpretation
• Mario Bunge
• Exact Philosophy
• Rosenberg, A. (2011). The Atheist’s Guide to Reality.78
• Glymour, C. (2011). Manifesto on positivism.
• Nelson Goodman: “There are two kinds of people in the world: the logical positivists and the god-damned English professors.”
• “Carnap was the grandfather of artificial intelligence: his students, Walter Pitts and Herbert Simon, were among the fathers.”
• Lawhead, J. (2016). My Scientism.79
### Naturalized metaphysics
• Methodological naturalism
• Is there any unique project left for metaphysics?
• Ney, A. (2019). Are the questions of metaphysics more fundamental than those of science?80
• Bennett, K. (2015). There is no special problem with metaphysics.81
• What are the limits of science?
• Bunge, M. (1971). Is scientific metaphysics possible?82
• Ross, D., Ladyman, J., & Kincaid, H. (2013). Scientific Metaphysics.83
### Other ways of knowing
Boghossian:
Especially within the academy, but also and inevitably to some extent outside of it, the idea that there are “many equally valid ways of knowing the world,” with science being just one of them, has taken very deep root. In vast stretches of the humanities and social sciences, this sort of “postmodernist relativism” about knowledge has achieved the status of orthodoxy. I shall call it (as neutrally as possible) the doctrine of
Equal Validity:
There are many radically different, yet “equally valid” ways of knowing the world, with science being just one of them.86
Scruton:
In our present circumstances the impression arises that, outside the hard sciences, just about anything goes, and that the humanities have neither a method nor a received body of knowledge, it being up to the professor to decide what to teach in his class.87
Art critics have a discipline, and it is one that involves reasoning and judgment. It is not a science, and what it describes forms no part of the physical world, which does not contain Olympia or anything else that you see in Manet’s painting. Yet someone who thought that art criticism is therefore deficient, and ought to be replaced by the study of pigments, is surely missing the point. There are forms of human understanding that can be neither reduced to science nor enhanced by it.88
## Moral naturalism
TODO.
See the Outline on ethics.
## Criticisms of naturalism
### Counter rebutals
Dang & Bright:
Fortunately, these norms of assertion don’t constrain science. We rightly tolerate the fact that it is, essentially, what you could politely call guesswork. Researchers must constantly be open to nature surprising them, and spread out over conceptual space, exploring whatever may be found. The trouble is that the general social esteem in which people hold science makes it natural for them to make an unhelpful assumption. That if scientific claims differ from the sort of claims each of us make every day, it is because the scientific ones have a better standing—better checked, have more evidence behind them, carry greater weight than our everyday assertions.115
## Other worldviews
• Worldview/Weltanschauung
• Other meta-philosophies
• Dualism, religion, mysticism
• Theology, appologetics
• Fideism
• New Age Movement
• Lots can be learned from other worldviews without 100% agreement.
• Presocratics
• Stoicism
• Buddhism
• Continental philosophy
• Existentialism
• Native American philosophy
• Etc.
## My thoughts
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## Annotated bibliography
• TODO
• TODO
• TODO
• TODO
• TODO
• TODO
• TODO
• TODO
• TODO
• TODO
### Sellars, W. (1963). Empiricism and Philosophy of Mind.
• Sellars (1963)
#### My thoughts
• TODO.
• Putnam (2016).
## References
Aaronson, S. (2001). Letter to James Randi. https://www.scottaaronson.com/writings/randi.html
Azzouni, J. (2000). Knowledge and Reference in Empirical Science. Routledge.
Bennett, K. (2015). There is no special problem with metaphysics. Philosophical Studies, 173, 21–37.
Berenstain, N. (2014). Necessary laws and chemical kinds. Australasian Journal of Philosophy, 92, 631–647. https://philpapers.org/rec/BERNLA
Bird, A. (2018). The metaphysics of natural kinds. Synthese, 195, 1397–1426. https://d1wqtxts1xzle7.cloudfront.net/38555439/Metaphysics_Natural_Kinds_final-libre.pdf
Bird, A. & Tobin, E. (2015). Natural kinds. Stanford Encyclopedia of Philosophy. http://plato.stanford.edu/entries/natural-kinds/
Blackmore, S. J. (1996). Near-death experiences. Journal of the Royal Society of Medicine, 89, 73–76.
———. (1997). Probability misjudgment and belief in the paranormal: A newspaper survey. British Journal of Psychology, 88, 683–689.
Boghossian, P. (2006). Fear of Knowledge. Oxford University Press.
Boudry, M. & Pigliucci, M. (2018). Science Unlimited? The Challenges of Scientism. University of Chicago Press.
Bunge, M. (1971). Is scientific metaphysics possible? The Journal of Philosophy, 68, 507–520.
———. (1991). The power and limits of reduction. In E. Agazzi (Ed.), The Problem of Reductionism in Science (pp. 31–49). Dordrecht: Springer.
———. (2001). Philosophy in Crisis: The Need for Reconstruction. Prometheus Books.
———. (2010). Mind and Matter: A Philosophical Inquiry. Dordrecht: Springer.
———. (2012). Evaluating Philosophies. Springer.
Cadwalladr, C. (2017). Daniel Dennett: I begrudge every hour I have to spend worrying about politics. The Guardian. February 12, 2017. https://www.theguardian.com/science/2017/feb/12/daniel-dennett-politics-bacteria-bach-back-dawkins-trump-interview
Cañizares-Esguerra, J. (2022). Rethinking the "Western" revolution in science. Science, 376, 467. https://www.science.org/doi/full/10.1126/science.abo5229
Carnap, R. (1934). On the character of philophical problems. Philosophy of Science, 1, 5–19.
———. (1959). The elimination of metaphysics through logical analysis of language. In A. J. Ayer (Ed.), Logical Positivism (pp. 60–81). New York: The Free Press. (Translated by Arthur Pap; originally published in 1932 as “Überwindung der Metaphysik durch logische Analyse der Sprache“ in Erkenntnis, 2, 219–241.). https://philpapers.org/archive/TEO.pdf
———. (2003). The Logical Structure of the World. (R. A. George, Trans.). Chicago: Open Court. (Originally published in 1928 as Der logische Aufbau der Welt).
Costa, C. F. (2012). Philosophy as a protoscience. Disputatio, 4, 591–608. https://sciendo.com/pdf/10.2478/disp-2012-0022
Dang, H. & Bright, L. K. (2021). How to make sense of contradictory science papers. Nautilus. June 2, 2021. https://nautil.us/issue/100/outsiders/how-to-make-sense-of-contradictory-science-papers
Descartes, R. (1982). Principles of Philosophy. (V. R. Miller & R. P. Miller, Trans.). Kluwer Academic Publishers. (Originally published in 1644 as Principia Philosophiæ).
Feigl, H. (1958). Critique of intuition according to scientific empiricism. Philosophy East and West, 8, 1.
Fodor, J. A. (1974). Special sciences (Or: The disunity of science as a working hypothesis). Synthese, 28, 97–115.
Friedman, M. (2000). A Parting of the Ways: Carnap, Cassirer, and Heidegger. Open Court Publishing.
———. (2002). Carnap, Cassirer, and Heidegger: The Davos disputation and twentieth century philosophy. European Journal of Philosophy, 10, 263–274.
Gabriel, G. (2003). Carnap’s "Elimination of metaphysics through logical analysis of language". In Logical Empiricism: Historical and contemporary perspectives (pp. 30–42). University of Pittsburgh Press.
Garfield, J. L. & Van Norden, B. W. (2016). If philosophy won’t diversify, let’s call it what it really is. New York Times Opinion. May 11, 2016. https://www.nytimes.com/2016/05/11/opinion/if-philosophy-wont-diversify-lets-call-it-what-it-really-is.html
Hahn, H., Neurath, O., & Carnap, R. (1973). The scientific conception of the world: The Vienna Circle. In M. Neurath & R. S. Cohen (Eds.), Empiricism and Sociology (pp. 298–318). Dordrecht: Reidel. (Originally published in German in 1929 as “Wissenschaftliche Weltauffassung: Der Wiener Kreis“). http://rreece.github.io/philosophy-reading-list/docs/the-scientific-conception-of-the-world-the-vienna-circle.pdf
Haslanger, S. (2012). Resisting Reality: Social Construction and Social Critique. Oxford University Press.
Heer, J. (2017). America’s first postmodern president. New Republic. July 8, 2017. https://newrepublic.com/article/143730/americas-first-postmodern-president
Heidegger, M. (1929). What is metaphysics? Lecture at the U. of Freiburg. https://antilogicalism.com/wp-content/uploads/2017/07/what-is-metaphysics.pdf
Hennig, C. (2015). What are the true clusters? Pattern Recognition Letters, 64, 53–62. https://arxiv.org/abs/1502.02555
Hicks, S. (2011). Explaining Postmodernism: Skepticism and socialism from Rousseau to Foucault. Ockham’s Razor Publishing. (Originally published in 2004).
Jacobs, J. (2002). Naturalism. Internet Encyclopedia of Philosophy. http://www.iep.utm.edu/naturali/
Jenkins, C. (2014). Naturalism and norms of inference. In A. Fairweather & O. Flanagan (Eds.), Naturalizing Epistemic Virtue (pp. 53–69). Cambridge University Press.
———. (2016). Epistemological naturalisms. In K. Clark (Ed.), The Blackwell Companion to Naturalism. Oxford: Blackwell.
Kant, I. (1912). Prolegomena to Any Future Metaphysics. (P. Carus, Trans.). Chicago: Open Court. (Originally published in 1783 as Prolegomena zu einer jeden künftigen Metaphysik).
Konnikova, M. (2012). Humanities aren’t a science. Stop treating them like one. Scientific American Blogs. August 10, 2012. https://blogs.scientificamerican.com/literally-psyched/humanities-arent-a-science-stop-treating-them-like-one/
Koons, R. C. (2000). The incompatibility of naturalism and scientific realism. In W. L. Craig & J. P. Moreland (Eds.), Naturalism: A critical analysis. Routledge.
Krikorian, Y. H. (1944). Naturalism and the Human Spirit. Columbia University Press.
Ladyman, J., Ross, D., Spurrett, D., & Collier, J. (2007). Every Thing Must Go: Metaphysics Naturalised. Oxford University Press.
Laughlin, R. B. & Pines, D. (2000). The theory of everything. Proceedings of the National Academy of Sciences, 97, 28–31.
Leah, R. (2018). “How do you separate fact and opinion?” Rudy Giuliani mused. Then, Stephen Colbert offered an answer. Salon. May 8, 2018. https://www.salon.com/2018/05/08/how-do-you-separate-fact-and-opinion-rudy-giuliani-mused-then-stephen-colbert-offered-an-answer
Leibniz, G. (1951). The Art of Discovery. In P. Wiener (Ed.), Leibniz: Selections. Scribner. (Originally published in 1685).
———. (1989). Dissertatio de Arte Combinatoria. In L. E. Loemker (Ed.), Philosophical Papers and Letters: The New Synthese Historical Library (Texts and Studies in the History of Philosophy), vol 2 (pp. 73–84). Springer. (Originally published in 1666).
Lewis, C. S. (1947). Miracles: A Preliminary Study. Geoffrey Bles.
Lugg, A. (2006). Russell as a precursor of Quine. Bertrand Russell Society Quarterly, 128, 9–21. https://bertrandrussellsocietyorg.files.wordpress.com/2020/04/brsq_0128-0129.pdf
Maddy, P. (2007). Second Philosophy. Oxford University Press.
Maudlin, T. (2007). The Metaphysics Within Physics. Oxford University Press.
McCrea, A. (2019). The magical thinking of guys who love logic. The Outline. February 15, 2019. https://theoutline.com/post/7083/the-magical-thinking-of-guys-who-love-logic
McEvilley, T. (2002). The Shape of Ancient Thought: Comparative studies in Greek and Indian philosophies. Allworth Press.
McManus, M. (2020). Myth and mayhem: A leftist critique of Jordan Peterson. https://areomagazine.com/2020/04/19/myth-and-mayhem-a-leftist-critique-of-jordan-peterson/
Morris, K. (2019). Physicalism Deconstructed: Levels of reality and the mind-body problem. Cambridge University Press.
Nagel, E. (1961). The Structure of Science: Problems in the Logic of Scientific Explanation. Harcourt.
———. (2008). Issues in the logic of reductive explanations. In M. A. Bedau & P. Humphreys (Eds.), Emergence: Contemporary Readings in Philosophy and Science (pp. 359–373). MIT Press. https://perpus.univpancasila.ac.id/repository/EBUPT180357.pdf
Ney, A. (2018). The Politics of Fundamentality. https://fqxi.org/community/forum/topic/3095
———. (2019). Are the questions of metaphysics more fundamental than those of science?. Philosophy and Phenomenological Research. (forthcoming).
———. (2021). The fundamentality of physics: Completeness or maximality. Oxford Studies in Metaphysics, 12, 203. https://philpapers.org/rec/NEYTFO
Oppenheim, P. & Putnam, H. (1958). Unity of science as a working hypothesis. In H. F. et al. (Ed.), Minnesota Studies in the Philosophy of Science, Vol. II. Minnesota University Press.
Papineau, D. (2007). Naturalism. Stanford Encyclopedia of Philosophy. https://plato.stanford.edu/entries/naturalism/
Parekh, B. (2006). Rethinking Multiculturalism: Cultural Diversity and Political Theory (2nd ed.). Palgrave MacMillan.
Paul, L. A. (2012). Metaphysics as modeling: the handmaiden’s tale. Philosophical Studies, 160, 1–29.
Perrin, A. J. (2017). Stop blaming postmodernism for post-truth Politics. The Chronicle of Higher Education. August 4, 2017. http://www.chronicle.com/article/Stop-Blaming-Postmodernism-for/240845
Pluckrose, H. (2016). Why I no longer identify as a feminist. Areo. December 26, 2016. https://areomagazine.com/2016/12/29/why-i-no-longer-identify-as-a-feminist/
———. (2017). How French “intellectuals” ruined the west: Postmodernism and its impact, explained. Areo. March 27, 2017. https://areomagazine.com/2017/03/27/how-french-intellectuals-ruined-the-west-postmodernism-and-its-impact-explained/
Pomerantsev, P. (2016). Why we’re post-fact. Granta. July 20, 2016. https://granta.com/why-were-post-fact/
Poskett, J. (2022). Horizons: The Global Origins of Modern Science. Mariner Books.
Prasetya, Y. (2021). Methodological naturalism and scientific success: Lessons from the scientific realism debate. European Journal for Philosophy of Religion. (forthcoming). https://philpapers.org/rec/PRAMNA
Prescod-Weinstein, C. (2017). Scientists must challenge what makes studies scientific. American Scientist. August 15, 2017. https://www.americanscientist.org/blog/macroscope/scientists-must-challenge-what-makes-studies-scientific
Priest, G. (2020). Philosophy and its history: An essay in the philosophy of philosophy. Analytic Philosophy, 61, 297–303.
Putnam, H. (1997). A half century of philosophy, viewed from within. Daedalus, 126, 175–208.
———. (2016). Naturalism, Realism, and Normativity. Harvard University Press.
Quine, W. V. O. (1969). Ontological Relativity and Other Essays. Columbia University Press.
———. (1981a). Reply to Stroud. Midwest Studies in Philosophy, 6, 473–476.
———. (1981b). Theories and Things. Harvard University Press.
Reichenbach, H. (1936). Logistic empiricism in Germany and the present state of its problems. The Journal of Philosophy, 33, 141–160.
Riepe, D. (1961). The Naturalistic Traddition in Indian Thought. University of Washington Press.
Rosaler, J. (2019). Reduction as an a posteriori relation. The British Journal for the Philosophy of Science, 70, 269–299. https://www.journals.uchicago.edu/doi/10.1093/bjps/axx026
Rosenberg, A. (2011a). The Atheist’s Guide to Reality. Norton.
———. (2011b). Why I am a naturalist. New York Times. September 17, 2011. https://opinionator.blogs.nytimes.com/2011/09/17/why-i-am-a-naturalist/
Ross, D., Ladyman, J., & Kincaid, H. (2013). Scientific Metaphysics. Oxford University Press.
Ruja, H. (1957). Are naturalists materialists? Philosophy and Phenomenological Research, 17, 555–557.
Russell, B. (2003). History of Western Philosophy. Routledge. (Originally published in 1945).
Sartwell, C. (2015). Philosophy returns to the real world. New York Times. April 13, 2015. http://opinionator.blogs.nytimes.com/2015/04/13/philosophy-returns-to-the-real-world/
Schopenhauer, A. (2014). Essays and Aphorisms. (R. J. Hollingdale, Trans.). Penguin.
Schweber, S. S. (1993). The ebb and flow of of reductionism. In L. M. Brown (Ed.), Renormalization: From Lorentz to Landau (and Beyond) (pp. 155–166). Springer-Verlag.
Scruton, R. (2015). Scientism and the humanities. In R. N. Williams & D. N. Robinson (Eds.), Scientism: The New Orthodoxy (pp. 131–146). Bloomsbury.
Sellars, W. (1963). Science, Perception, and Reality. Atascadero, CA: Ridgeview Publishing Co.
Siegel, E. (2016). No, science is not faith-based. Medium.com. March 15, 2016. https://medium.com/starts-with-a-bang/no-science-is-not-faith-based-ddc9be25efba
Silvers, S. (1997). Nonreductive naturalism. Theoria, 12, 163–184. https://addi.ehu.es/bitstream/handle/10810/40244/Theoria%2028%20163-184.pdf
Snow, C. P. (1959). The Two Cultures and the Scientific Revolution. Oxford University Press.
Sokal, A. D. (1996a). A physicist experiments with cultural studies. http://www.physics.nyu.edu/sokal/lingua_franca_v4.pdf
———. (1996b). Transgressing the Boundaries: Towards a Transformative Hermeneutics of Quantum Gravity. http://www.physics.nyu.edu/sokal/transgress_v2_noafterword.pdf
Sokal, A. D. & Bricmont, J. (1998). Fashionable Nonsense: Postmodern Intellectuals’ Abuse of Science. Picador.
Sorell, T. (1991). Scientism: Philosophy and the Infatuation with Science. Routledge.
Spurrett, D. J. (1999). The Completeness of Physics. Durban: The University of Natal. (Ph.D. thesis). https://philpapers.org/rec/SPUTCO
Spurrett, D. J. & Papineau, D. (1999). A note on the completeness of ’physics’. Analysis, 59, 25–29. https://www.davidpapineau.co.uk/uploads/1/8/5/5/18551740/analysis-1999-spurrett-25-9.pdf
Tahko, T. E. (2022). Natural kinds, mind-independence, and unification principles. Synthese, 200, 144. https://link.springer.com/article/10.1007/s11229-022-03661-7
Thomas, B. (2015). A disease of scienceyness. Medium.com. March 4, 2015. https://medium.com/%40writingben/a-disease-of-scienceyness-7b5571a34953
Van Norden, B. W. (2017). Taking Back Philosophy: A Multicultural Manifesto. Columbia University Press.
Weinberg, S. (1996). Sokal’s Hoax. The New York Review of Books, 43, 11–15. August 8, 1996. http://www.physics.nyu.edu/faculty/sokal/weinberg.html
Williams, C. (2017). Has Trump stolen philosophy’s critical tools? The New York Times. April 17, 2017. https://www.nytimes.com/2017/04/17/opinion/has-trump-stolen-philosophys-critical-tools.html
Wilson, E. O. (1998). Consilience: The Unity of Knowledge. Vintage.
Winburn, A. P. & Clemmons, C. M. J. (2021). Objectivity is a myth that harms the practice and diversity of forensic science. Forensic Science International: Synergy, 3, 100196. https://www.sciencedirect.com/science/article/pii/S2589871X21000668
Wittgenstein, L. (1961). Tractatus Logico-Philosophicus. (D. F. Pears & B. F. McGuinness, Trans.). Routledge. (Originally published in 1922). https://people.umass.edu/klement/tlp/tlp.html
1. Sellars (1963), p. 1.↩︎
4. Jacobs (2002).↩︎
5. Ladyman, Ross, Spurrett, & Collier (2007), p. TODO (find error correcting filters).↩︎
6. Maudlin (2007), p. 1.↩︎
7. Papineau (2007).↩︎
8. Jenkins (2016).↩︎
9. Prasetya (2021).↩︎
10. McEvilley (2002), p. 325–333.↩︎
11. Riepe (1961), p. TODO.↩︎
12. Krikorian (1944).↩︎
13. Costa (2012).↩︎
14. Descartes (1982), p. xxiv.↩︎
15. Reichenbach (1936), p. 142.↩︎
16. Russell (2003), p. TODO.↩︎
17. Lugg (2006).↩︎
18. Sellars (1963), p. 2.↩︎
19. Wilson (1998).↩︎
20. Hahn, Neurath, & Carnap (1973), §2.↩︎
21. Bunge (2010), p. xi.↩︎
22. Bunge (2012).↩︎
23. Priest (2020).↩︎
24. Carnap (2003), p. xvii.↩︎
25. Bunge (2001), p. 10.↩︎
26. Kant (1912), p. 2–3.↩︎
27. Parekh (2006).↩︎
28. Heidegger (1929).↩︎
29. Carnap (1959).↩︎
30. Friedman (2000).↩︎
31. Friedman (2002).↩︎
32. Gabriel (2003).↩︎
33. Putnam (1997).↩︎
34. Quine (1969), p. TODO.↩︎
35. Quine (1981b), p. 21 (emphasis added). A similar quote can be found in Quine (1981a), p. 474.↩︎
36. Ladyman et al. (2007), p. i.↩︎
37. Leibniz has similar quotes in several works. This quote is taken from a translation of his first book, Dissertatio de arte combinatoria, written in 1666 (Leibniz, 1989, p. 73). TODO: Actually, I haven’t found this yet. See also Leibniz (1951), p. 51. Supposedly it is also in Russell’s A Critical Exposition of the Philosophy of Leibniz.↩︎
38. Wittgenstein (1961), line 2.0201.↩︎
39. Carnap (1934).↩︎
40. Nagel (1961).↩︎
41. Nagel (2008).↩︎
42. Bunge (1991).↩︎
43. Morris (2019).↩︎
44. Ney (2018).↩︎
45. Ney (2021).↩︎
46. Rosaler (2019).↩︎
47. Fodor (1974).↩︎
48. Schweber (1993).↩︎
49. Laughlin & Pines (2000).↩︎
50. Wittgenstein (1961), 6.522.↩︎
51. Silvers (1997).↩︎
52. Quine (1969), pp. 26–68.↩︎
53. Quine (1969), pp. 114–138.↩︎
54. Ross, Ladyman, & Kincaid (2013).↩︎
55. Ladyman et al. (2007), p. 5.↩︎
56. Berenstain (2014).↩︎
57. Bird & Tobin (2015).↩︎
58. Bird (2018).↩︎
59. Tahko (2022).↩︎
60. Azzouni (2000).↩︎
61. Hennig (2015).↩︎
62. Haslanger (2012).↩︎
63. Ruja (1957).↩︎
64. Spurrett (1999).↩︎
65. Spurrett & Papineau (1999).↩︎
66. Ney (2021).↩︎
67. Ladyman et al. (2007), p. 37.↩︎
68. Blackmore (1996).↩︎
69. Blackmore (1997).↩︎
70. Jenkins (2014).↩︎
71. Lewis (1947).↩︎
72. Aaronson (2001).↩︎
73. Koons (2000).↩︎
74. Paul (2012).↩︎
75. Schopenhauer (2014), On thinking for yourself, section 7.↩︎
76. Sorell (1991).↩︎
77. Boudry & Pigliucci (2018).↩︎
78. Rosenberg (2011a).↩︎
80. Ney (2019).↩︎
81. Bennett (2015).↩︎
82. Bunge (1971).↩︎
83. Ross et al. (2013).↩︎
84. Feigl (1958).↩︎
85. Scruton (2015).↩︎
86. Boghossian (2006), p. 2.↩︎
87. Scruton (2015), p. 132-3.↩︎
88. Scruton (2015), p. 140-1.↩︎
89. Prescod-Weinstein (2017).↩︎
90. Poskett (2022).↩︎
91. Cañizares-Esguerra (2022).↩︎
92. Garfield & Van Norden (2016).↩︎
93. Van Norden (2017).↩︎
94. Winburn & Clemmons (2021).↩︎
96. Rosenberg (2011b).↩︎
97. Boghossian (2006).↩︎
98. Hicks (2011).↩︎
99. Sartwell (2015).↩︎
100. Pluckrose (2016).↩︎
101. Pluckrose (2017).↩︎
102. Sokal (1996a), Sokal (1996b), Sokal & Bricmont (1998).↩︎
103. Weinberg (1996).↩︎
104. Siegel (2016).↩︎
105. Pomerantsev (2016).↩︎
106. Heer (2017).↩︎
108. Williams (2017).↩︎
109. Leah (2018).↩︎
110. Perrin (2017).↩︎
111. McManus (2020).↩︎
112. Konnikova (2012).↩︎
113. Thomas (2015).↩︎
114. McCrea (2019).↩︎
115. Dang & Bright (2021).↩︎
116. Snow (1959).↩︎ | 2023-02-04 02:36:33 | {"extraction_info": {"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, "math_score": 0.4757551848888397, "perplexity": 7976.625692775034}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500080.82/warc/CC-MAIN-20230204012622-20230204042622-00179.warc.gz"} |
http://icfk.ibio.pw/rocket-fuel-efficiency.html | ## Rocket Fuel Efficiency
3 ounces of fuel per boil, these canisters are more efficient than homemade alcohol stoves and even Esbit fuel tablets on a fuel-per-boil basis. As others have said, see Scott Manley's tutorials. Medium stoves are suitable for heating small houses, medium-sized energy-efficient houses, and cottages used in winter. OLICAMP 328060 Olicamp Rocket Fuel 100G/3. Add as little rocket and fuel as possible. They are then blended together under carefully controlled conditions and poured into the prepared rocket case as a viscous semisolid. A review on Nissan Micra [2013-2018] XV Diesel [2013-2016] by lalit Singh - Amazing Fuel Efficient Diesel Rocket in India. It is an adaption of a 16-brick design by Dr Larry Winiarski, and has been specially modified by RIPPLE Africa for Malawi after lots of positive trials and community feedback. Researchers from the University of Illinois at Urbana-Champaign used a salt-based propellant that had already been proven successful in combustion engines, and demonstrated. If you want to: - Make product changes; - Create something not in stock. One unit of solid fuel contains 12MJ of energy, three times the energy value of coal. Standard screw-on valve fits a variety of appliances. The fuel efficiency of a rocket depends upon its exhaust velocity. Updated by Phil Gibbs 1998. Liquids are desirable because their reasonably high density allows the volume of the propellant tanks to be relatively low, and it is possible to use lightweight centrifugal turbopumps to pump the propellant from the tanks into the combustion chamber, which means that the propellants can be kept. All other values have been rounded up (meaning some buildings will be occassionally idle). Weighing 2. It is the second-most long-lasting of all the fuel types, beat only by Nuclear fuel. Outside use only. It states:. Mike Wehner @MikeWehner. Besides being useful as fuel in all burner devices, solid fuel is also used to produce rocket fuel, which is a component of rocket parts built in the rocket silo. (Retired) ----- 8th NH3 Fuel Assoc. Slosh Experiment Seeks to Improve Rocket Safety, Efficiency | NASA. High values of fuel propellant provide cold weather output and bring you quick and fuel efficient boil times. This means that Rocket Stoves are more efficient than open fires; they useless firewood therefore reduce long-term household expenditure. You could create a giant spreadsheet comparing thrust-to-weight ratios, specific impulses (efficiency metrics), and payload capabilities. All-New Ford EcoBlue Engine is Diesel Game Changer – Cleaner, More Fuel Efficient, More Power, More Torque 26-Apr-2016 | BIRMINGHAM, U. So my advice is if you want to increase the burn time without decreasing efficiency, increase the feed tube length rather than its cross sectional area. (Cost to operate) not to mention if speeds get high enough now the hull cost goes up for rocket due to heat not to mention cost of exotic material to go really fast. This solid is called the propellant grain. Each time, the rocket uses three different fuel types sequentially: Fuel 1 for the Stage 1 of the rocket, Fuel 2 for Stage 2 and Fuel 3 for Stage 3. Bowrider 28: a great tender. He said: “ROCKET FUEL: They’re going to cut rates and print money right as we march. The engine with the higher value of specific impulse is more efficient because it produces more thrust for the same amount of propellant. they may be extremely fuel efficient allowing for a greater change in speed (delta V) but because of their design, they could lack the thrust to get out of earths atmosphere. Piston/Propeller. The team that is able to lift the greatest payload (paper clips) into space (the ceiling) is the winner. Versa: Uses wood, dried biomass, or. (gas burned to. Another huge variable with any rocket engine is how efficient it is. The Pika stove is a passive air-laminar flow intake type stove, versus the pressurized Pepsi stove. Stove World now stock "THE WORLD'S MOST EFFICIENT WOOD BURNING STOVE" The Burley Fireball range of wood burning stoves are the result of thousands of hours of designing and prototyping, based and inspired by over 100 years of experience in the heating industry. I barely even noticed as I threw another stick in the rocket stove. Buick Fuel Efficiency Games are designed to provide relevant content to people. The fuel’s core ingredient responsible for significant increases in power and efficiency was a proprietary fuel catalyst that now serves as the foundation for all AFC Fuel Additives. Specific impulse refers to the units of thrust per the units of. Mining the moon could give us enough rocket fuel to get to Mars, student project discovers Our team also found a fuel-efficient way to get spacecraft from Earth orbit to the depot at L1. Word invented by psiscott once he saw this car with his little son. If you’re starting out, a small canister stove is probably your best bet. People often add too many engines; when you have more thrust than you need, you are carrying "dead weight" or you could be carrying more fuel. There are several types of combustion, including internal combustion, diesel combustion, low-temperature combustion, clean diesel combustion and other novel types. Used in the production of orbital rockets. A New Recipe for Rocket Fuel Researchers are using aluminum and frozen water to make a propellant that could allow rockets to refuel on the moon or even Mars. 93 range, as good as or better than most liquid propellant upper stages. Both have their pros and cons. How much will it cost to drive 2400 miles if gas is 4. The rockets that i make are around 4/5 inches high and weigh around 100g. com found similar results when testing a pickup truck, which got almost 10 percent better gas mileage with the windows down and the AC off when driving at 65 mph. "A rule of thumb is that for every ten-percent increase in efficiency for rocket fuel, the payload of the rocket can double. airlines, domestic. Enjoy the videos and music you love, upload original content, and share it all with friends, family, and the world on YouTube. ADVANTAGES OF ROCKET STOVES: A. Gathering all the terms together, TSFC is the mass of fuel burned by an engine in one hour divided by the thrust that the engine produces. Usually found in developing countries, the cost efficient stove produces almost no smoke and is a staple in areas with a low supply of fuel sources. THE HYDRO-REACTING MARINE SOLID FUEL ROCKET ENGINES I. STOVE EFFICIENCY Now that you have a general idea about your environment, conditions and group needs, consider your stove itself. Bottle Rockets that you can buy in many stores. It has less than 5-10% efficiency. First, PWR will work with Pennsylvania State University to create a high-efficiency gas turbine which uses supercritical fluids to cool the turbine blades. The shuttle's main engines would drain a swimming pool full of fuel in only 25 seconds! This gushing torrent of fuel is driven by a turbopump. Premade fuel cartridges are readily available at most hobby stores, but you might be surprised to. So the fuel to oxidizer ratio helps make Methalox’s fuel tanks a lot closer to RP-1 than it is to Hydrolox. At any rate, what you are going to be looking for is rocket formula. A text from around that time describes how the combustion efficiency and hence the rocket thrust could be improved by creating a cavity in the propellant along the centre line of the rocket tube to maximise the burning surface - a technique still used in solid fuelled rockets today. F1 rocket engine used on the Saturn V. A super-hot chimney above the fire draws the flames sideways and up, blending hot fuel and air into a quick, hot, clean-burning fire that takes little wood and leaves little residue. If you want to maximize delta-v, you want to save fuel. What I'm interested in is, assuming a rocket design similar to the Kerbal X, what would be the most efficient way of getting such a rocket into orbit, conserving as much fuel as possible? Note that whatever tips you suggest must work in 1. EzyStove’s modern, smart aesthetic, patented award-winning design, durable stainless steel construction and fun features combine to create the next generation in fuel-‐efficient. An artist's rendering of mining operations on the moon. [tex]v_f = v_i + v_p ln\left(\frac{m_i. Efforts to replicate the Gazan rocket threat have been nipped in the bud in the West Bank, and no effective rocket fire has occurred, he noted. • Field repairable, to keep the cost of replacement down. A mass M has been ejected from the rocket and is moving with velocity u as seen by the observer. OLICAMP 328060 Olicamp Rocket Fuel 100G/3. Efficient fuel types for heating your home will also vary, depending on where you live; what is cheaper at one end of the country, may not be at the opposite end. These self-sustaining businesses produce safe, affordable, fuel-efficient cookstoves to replace dangerous open cooking fires. Trump has been telling voters that his recipe of less regulation and lower taxes would be "rocket fuel" for employers. The wood. The low-thrust, high-efficiency plasma thrusters consume less propellant than conventional chemical rocket engines, allowing Eutelsat 7C to carry a lighter fuel load and giving Eutelsat access to. Most trawler type boats in this length category are much heavier, beamier and higher, have much more internal volume, weight, and therefore cost. Smoke is wood that has not combusted efficiently wasting time and energy and creating pollution”. com: msr pocket rocket fuel. A team of scientists and engineers at NASA's Kennedy Space Center is studying how to better understand propellant movement in fuel tanks to improve safety and efficiency. In three weeks we have used as much wood as we previously used in three days!” 12 lage A4 20/08/13_Lapknm 1 27/08/2013 14:16 Page 6. A very simple and inexpensive homemade rocket stove utilizes nothing more than common bricks to create a fully-functioning rocket stove — although not a portable one. Rocket propellant #3 Comments: Stinks like ammonia when mixed, and hardens faster than normal epoxy curing time. Fuller rearranged the traditional structure of a liquid motor, where the fuel and oxidizer are pumped from low pressure tanks into a thrust chamber, to look more like that of a hybrid rocket, where liquid oxidizer flows through a chamber that contains solid fuel. Rocket Stove Efficiency. This can open the door to a highly efficient wall-less Hall Thruster which can be utilized in deep space exploration missions in the future. NASA's Nuclear Thermal Engine Is a Blast From the Cold War Past Today's chemical propellant rocket engines may not be the fastest or most efficient way to send a crew to another planet. The fuel efficiency of a rocket depends upon its exhaust velocity. Fuel Efficiency. Fuel/Oxidizer Storage. It's so fuel-efficient that you can go from Earth's orbit to the moon's orbit with just about 30 gallons (113 liters) of gas [source: Charles]. Thrust is the force which moves the rocket through the air, and through space. Brimming with sophistication and individuality, the T100 Black delivers all of the Bonnevilles iconic design DNA and Triumphs signature precision handling and advanced rider technology and takes the styling to a different level with fully blacked-out high-quality details, including wheel rims. Vehicles with higher fuel capacities are more durable when travelling. Since about 1. Download Freeware Buick Fuel Efficiency Games. How to use fuel in a sentence. Synonyms for efficient at Thesaurus. A rocket has variable mass because its mass decreases over time, as a result of its fuel (propellant) burning off. Since they passed, the markets are down. Specific impulse is a measure of how fuel efficient an engine is (the greater the specific impulse, the more fuel efficient it is). Used fuel elements may be reprocessed to recover unconsumed fuel and the newly produced fissionable material. Fuel blend: 75% Isobutane 25% Propane. The fuel you use can range from simple wooden blocks, to coal, lava buckets or even blaze rods. And good gravity turn will leave you almost in orbit, so you can as well go for stable orbit. 4) FUEL TANKS A liquid-oxygen tank at the top of each core feeds the engines through a center tube; the lower portion of the tank contains rocket-grade kerosene. MSR does not recommend using any type of wind screen with this product. The shuttle's main engines would drain a swimming pool full of fuel in only 25 seconds! This gushing torrent of fuel is driven by a turbopump. It’s got to be kept under pressure at -253°C. Rocket Stove The Rocket Stove is a clean burning and fuel-efficient cooking stove which can use thinsticks as fuel. Pretty incredible for such a small setup. Specific impulse can be calculated in a variety of different ways with different units. Find rocket fuel ads. It was also the. Gathering all the terms together, TSFC is the mass of fuel burned by an engine in one hour divided by the thrust that the engine produces. Jetboil Minimo and MSR Pocket Rocket 2 are both excellent backpacking stoves. Differently, the thrust of the jet engine depends on velocity and declines as velocity increases, because of the ram drag. How to use fuel in a sentence. 2 ounces and takes an average of 4 minutes 45 seconds to boil a liter of water. Given the similarity of the results of the insulated stoves we could conclude that in terms of fuel efficiency , the insulation of the combustion chamber does not have a great impact on overall efficiency ie reduction in fuel consumption. systems is the increased specific impulse, or fuel efficiency brought about by the airbreathing part of the sys-tem as compared to an all rocket SSTO vehicle. A low displacement to length ratio is crucial for effortless cruising speeds and low fuel consumption. A home energy audit, also known as a home energy assessment, can help you understand the whole picture of your home's energy use. Converting it to rocket fuel with 4x t3 prod modules gives 1. The SunFire Rocket is a highly fuel efficient stove that turns small amounts of wood into large pots of food. 5 million, or $2. from Dave Lawton “There is a very simple way to doing this WHOLE water powered system on 0. The Rocket Equation is a harsh master. So, you might try designing the rocket from the lander down. Propellants are classified according to their state - liquid, solid, or hybrid. The 2017 tax cut that President Donald Trump claims has been "rocket fuel" for the economy has had a small, if any, effect in its first year, according to a newly released congressional report. Quicken Loans, Rocket Homes Real Estate LLC, Rocket Loans and Rocket HQ are separate operating subsidiaries of Rock Holdings Inc. Elon Musk: SpaceX's "BFR" Rocket Will Do More Than Take Us to Mars In other words, the BFR is really about efficiency: instead of spending SpaceX's resources on three separate rockets. This means that Rocket Stoves are more efficient than open fires; they useless firewood therefore reduce long-term household expenditure. MSR WindBurner Stove System for Fast Boiling Fuel-Efficient Cooking for Backpacking, Solo Travelers, and Minimalist Trips. Yuntenwi et al. More informally known as "The Rocket Equation". Rocket Stoves What is a Rocket Stove? Rocket stoves are a type of fuel-efficient device, named in the 70’s, but dating back millennia in concept. Ideally, if you want to be extremely resilient to the uncertainties of life – you should invest in a small portable rocket stove and a large rocket mass heater as well! 2 – They are insanely efficient – maximizing your fuel. Solid fuel rockets have lower specific impulse, a measure of propellant efficiency, than liquid fuel rockets. High values of fuel propellant provide cold weather output and bring you quick and fuel efficient boil times. “The Merlin 1D has a vacuum thrust-to-weight ratio exceeding 150, the best of any liquid rocket engine in history. Call (804) 746-2886 for more information. Common among campers and backpacking enthusiasts, rocket stoves are relatively new to urban dwellers or those that do not spend a lot of time outdoors. After all is said and done, what is the best backpacking stove for you? Most outdoor folk have two or three stoves, which allows them to take the right one for a specific adventure. The Bush and Obama recoveries not only generated many fewer jobs but also a lot fewer improvements in productivity than was the norm during post-World II decades of the. The Relativistic Rocket. A good measure of efficiency in rocket engines is specific impulse (Isp), which is the amount of thrust produced by an engine per unit of propellant consumed. It burns methane and liquid oxygen. It is the second-most long-lasting of all the fuel types, beat only by Nuclear fuel. What rocket propellant is the most efficient "mpg" and powerful "high thrust potential". Designed for low cost and efficiency from packaging to transport to household fuel savings, Ezystove doesn't sacrifice quality, aesthetic or aspirational value. This is a low-powered rocket fuel — which is to say, be very, very careful with it. A More Efficient Spacecraft Engine. Founded by Maine native Sascha Deri, bluShift. BONNEVILLE CHARACTER AND AUTHENTICITY The latest Bonneville T100 range stays faithful to these celebrated motorcycles’ evocative heritage and truly iconic silhouette, with sculpted Bonneville signature fuel tank, wire-spoked wheels, authentic peashooter silencers, and two-tone paint options with hand-painted gold coach lines. DETROIT (AP) — In a merger deal that appears to be heading for approval, Fiat Chrysler stands to gain electric vehicle technology while PSA Peugeot Citroen could benefit from a badly needed dealership network to reach its goal of selling vehicles in the U. Mining the moon could give us enough rocket fuel to get to Mars, student project discovers Our team also found a fuel-efficient way to get spacecraft from Earth orbit to the depot at L1. It's all about water electrolysis, a. A new generation of engines being developed by the world’s largest jet engine maker, CFM (a partnership between GE and Snecma of France), will allow aircraft to use about 15 percent less fuel. A good measure of efficiency in rocket engines is specific impulse (Isp), which is the amount of thrust produced by an engine per unit of propellant consumed. Ordinarily, in processing solid propellants the fuel and oxidizer components are separately prepared for mixing, the oxidizer being a powder and the fuel a fluid of varying consistency. Rocket Fuel. This is measured in specific impulse or ISP, but you can think of this like the fuel economy of a gas powered car. This application is intended to educate and inform users about the fuel-saving benefits of Buick s eAssist Technology through three interactive game experiences. Two different rocket engines have different values of specific impulse. It burns methane and liquid oxygen. Mining The Moon For Rocket Fuel To Get Us To Mars. The ability to personalize the Rocket III was integral to the design philosophy and asthe bike went. So far so good, but the real kicker is that the Project X rocket runs on natural gas. 5-second zero-to-60-mph time and a 12. Outside use only. We've prepared 20 tips to improve your gas mileage and see some savings into your bank account. The tube is known as the fuel grain. Crank was unable to reach 100 mph and sold the car in 1982 to Barber-Nichols Engineering Company in Colorado. March 12, 2015 By Guest 89 Comments This post may contain affiliate links which won't change your price but will share some commission. Top 10 most fuel efficient cars. Rocket Fuel turned in a better than expected second quarter, bolstered by enterprise sales of its DMP/DSP products, deal-making with holding companies, and cost savings from layoffs earlier this year. Started by Moroccan entrepreneur and engineer Moushine Serrar, Prakti Design designs, manufactures and distributes energy-efficient, low-cost and low-smoke cook stoves in rural India. Follow the Trump Administration’s Every Move. Mining the moon for rocket fuel to get us to Mars. Jetboil Minimo and MSR Pocket Rocket 2 are both excellent backpacking stoves. Tracking your usage over time can help you monitor changes to your driving habits and keep tabs on the health of your vehicle. NASA's Saturn V, the mighty rocket that launched men to the moon was first tested in 1967. The PocketRocket, on the other hand, burns "IsoPro", a mixture of Isobutane and Propane; you'll see this type of fuel in little round-top, flat bottom fuel canisters at any REI store. Sportfisher 32: a rocket ship. A mass M has been ejected from the rocket and is moving with velocity u as seen by the observer. The Rocket mud stove is an improvement on the previous Lorena stove design with other advantages including: Firewood fuel savings The rocket stoves have the potential to significantly reduce the amount of firewood used in cooking in comparison to the traditional 3 stone stoves. In aircraft and rocket design, overall propulsive efficiency is the efficiency with which the energy contained in a vehicle's propellant is converted into kinetic energy of the vehicle, to accelerate it, or to replace losses due to aerodynamic drag or gravity. What kind of fuel do rockets use and how does it give them enough power to get into space? A rocket works by exchanging momentum. Efficient Rockets Don’t Mean Sustainable Space Travel. The rocket's first flight, for the Apollo 4 mission, took place 50 years ago, on Nov. Recall that Maruti advertisement where a scientist talks to a crowd about latest rocket technology and its exclusive design and pat comes a question from an Indian. This richer fuel mixture, combined with the added restriction that the catalyst in the exhaust system represented adversely affected efficiency the net impact upon fuel economy was a 5% increase in fuel consumption and thus CO 2 emissions leapt up! (For more explanation on this, see the 'Lean Burn' box below. ESBIT solid fuel can also be used with the Caldera system which yields a slightly more efficient system in regards to weight of fuel needed to heat water compared an alcohol stove being used with the Caldera. From rocket fuel to clean cars (Nanowerk News) Austrian manufacturer MagnaSteyr has adapted technology developed for the Ariane rocket to build clean-burning cars that can use hydrogen instead of petrol for fuel. Swedish scientists have found the first new nitrogen-oxygen molecule since the 18th century. "It turns out driver skill and awareness contributes to about 30 percent of fuel efficiency, and so our driver's aware, our driver's comfortable and our driver is able to make good decisions on. The new models combine Top Fuel’s deep race heritage – harnessing the confident control of a longer-travel bike and a snappy XC race bike, that Trek claims descend like a rocket. Most of the fuel-cell vehicles coming to market in the next few years will be able to deliver close to 70 miles per. Clean-burning heat, adds a self-feeding wood box, and channels the exhaust. "A rule of thumb is that for every ten-percent increase in efficiency for rocket fuel, the payload of the rocket can double. It offers no real control beyond choosing when to ignite it, though tweakables enable altering the thrust limit and total fuel. And, instead of simply listing the most fuel efficient bikes, we'll be editorializing a bit and bringing you the ones we think make the best options for commuting, combined with that reported. Assuming 100% efficiency, complete burnup. More informally known as "The Rocket Equation". They rebuilt the car and tried for the record at El Mirage, California, reaching 111 mph. But you’re not going to get a locomotive-performance engine in a 1. Nuclear rockets are more fuel efficient and much lighter than chemical rockets. The Bonneville T100 is inspired by the legendary ’59 Bonneville and has been beautifully designed to incorporate many of the original motorcycle’s iconic features married to Triumph’s signature precision handling and advanced rider technology for an accessible and confidence inspiring ride. Get the wood burning stove DVDs. It's fuel efficient, effective, and has a stellar safety record. It measures the amount of heat actually delivered to the house compared to the amount of fuel that is supplied to the furnace. Advances are available in efficiency – from various ways to boost combustion pressure to the utilization of the great amount of waste heat. A low displacement to length ratio is crucial for effortless cruising speeds and low fuel consumption. Fuel-efficient engines are smaller than engines of typical cars and trucks, so they lose power. How much will it cost to drive 2400 miles if gas is 4. Updated by Phil Gibbs 1998. – 2004 Triumph Rocket 3 (Rocket III) The first ever production bike to break the 2-litres barrier, the Rocket III is the ultimate power cruiser. What are the correct ratios when making Solid Fuel or Rocket Fuel from oil? A setup with 25 refineries is ratio-perfect. It needs a structure, a backbone to support all this and it must survive the highly dynamic environment of launch (there is fire, shake, and force at work. I’m afraid to say the homemade stove we’ve been using is not as efficient as I previously thought, since using the StoveTec model. The Dow Is Up a Few Points Because Hope on Trade Isn't the Same as Rocket Fuel. Just add the dry wood fuel to the chamber below, light it up, and enjoy. Because the combustion chamber is either enclosed or insulated, radiant heat wood stoves will be much more efficient than an open burning fireplace, and the heat will be radiated into the room for a much longer time after the fire has gone out. Scientists measure the efficiency of rocket propellants by. Some months back I had this great idea for a fuel economy test. It is modeled after the real-world theoretical nuclear thermal rocket, which uses a fission reactor to heat its propellant and force it out of the rocket nozzle to generate thrust (as opposed to a normal rocket, which ignites the propellant and uses the energy released by the resulting chemical reaction for that. It states:. There are several types of combustion, including internal combustion, diesel combustion, low-temperature combustion, clean diesel combustion and other novel types. I’m afraid to say the homemade stove we’ve been using is not as efficient as I previously thought, since using the StoveTec model. Extra weight also increases a vehicle’s rolling resistance, which is a force that resists forward motion produced as the wheels roll over the road. For example, the non-vectoring stock engine LV-T30 has a vacuum specific impulse of 300 s. Soon will be buying a houseboat, but fuel consumption and gas prices are scaring me a little. The rocket must have engines, tanks, and plumbing. One unit of solid fuel contains 12MJ of energy, three times the energy value of coal. A propeller moves a large amount of air slowly; a jet or rocket moves a small amount of air quickly. Centaur is raised into the "J" Tower for testing at Point Loma, early 1960s. This would generate more fuel tax dollars with out raising the fuel tax, thus improving the roads. There is a corresponding brake specific fuel consumption (BSFC) for engines that produce shaft power. BUILD YOUR OWN ROCKET STOVE Why? – Rocket Stoves are: Fuel Efficient: Because they direct all the heat energy (from the flames) upward, most of the heat actually goes into heating the food, rather than being radiated outward (like most campfires). Researchers from the University of Illinois at Urbana-Champaign used a salt-based propellant that had already been proven successful in combustion engines, and demonstrated. A vacuum secondary carburetor is usually most fuel efficient when its used on street driven cars that have automatic transmissions. MSR does not recommend using any type of wind screen with this product. Small rocket stove makes for an efficient offgrid or camping stove Interest in rocket stoves is growing, and with good reason, as they're fast, efficient, and cleaner-burning than most other options. That is, come up with a lander design that will get you onto the surface, then put a transfer stage below it, and then get the two into orbit. My 10 year old son then followed up by frying an egg. Rand Lindsly reports that it generally takes around 2/3's of a 14 gram ESBIT tablet to bring 2 cups of water to a boil. In a liquid fuel rocket the fuel is a liquid. Fuel/Oxidizer Storage. The tax cuts were supposed to be ‘rocket fuel’ for the economy. The graph also shows us how much of difference fuel choice makes. ) There are two main downsides to liquid-fuel stoves:. this is not a real rocket stove , a nice little stove yes , but a rocket stove is built on the bases of the smoke being able to return for a reburn and that is what makes it smokeless , the emmissions get reburned and th air intake is hot and fullunburned gases , if you want this to work like a real smokeless rocket stove , make sure the air. ESBIT solid fuel can also be used with the Caldera system which yields a slightly more efficient system in regards to weight of fuel needed to heat water compared an alcohol stove being used with the Caldera. Rocket stoves are a type of fuel-efficient device, named in the 70s, but dating. We've prepared 20 tips to improve your gas mileage and see some savings into your bank account. The most basic type of rocket engine is the solid fuel rocket. The Boeing 747-400F freighter is the all-cargo transport variant of the Boeing 747-400 family of aircraft. Thrust and fuel efficiency have always seemed destined to remain mutually exclusive – the higher the one, the lower the other – inevitably forcing jet engine designers to make calculated trade-offs between the two. aircraft engine history. Whether you build your own or buy a commercial version, you should invest in one. systems is the increased specific impulse, or fuel efficiency brought about by the airbreathing part of the sys-tem as compared to an all rocket SSTO vehicle. Rand Lindsly reports that it generally takes around 2/3's of a 14 gram ESBIT tablet to bring 2 cups of water to a boil. You'll also improve aerodynamics by:. Garanin Moscow State Aviation Institute (Technical University), Russia. Overall, the top 10 fuel-efficient cars among the 2010 models are as follows: 1. At more than 120 new routes and counting, the 787 family is connecting cities and people around the world while meeting passengers’ expectations for direct flights. It is simply not wind resistant and in windy, exposed conditions its fuel efficiency will be significantly decreased. If you want to maximize delta-v, you want to save fuel. They are easy to use, easy to clean, and portable. In a rocket engine, stored fuel and stored oxidizer are ignited in a combustion chamber. Top 10 most fuel efficient cars. The life of fuel elements in power reactors is up to three years. “The auto industry in the U. Aug 09, 2019 · So the design of the rocket stove is extremely versatile. A mobile landing site could conceivably be placed where it is safest or most fuel-efficient for the rocket to come down. Get the wood burning stove DVDs. The use of such propellants simplifies the design of the rocket engines and is the simplest way to provide for engine restarts. Premade fuel cartridges are readily available at most hobby stores, but you might be surprised to. Just imagine, this camping stove only needs a handful of charcoal, or twigs, or wood. Much like actual rockets, the miniature versions launched by model rocketry enthusiasts require fuel to blast off. We add Rocket Fuel to your campaign!. But the specific impulse of chemical propellants — specific impulse is a measurement of an engine's efficiency taking into account a rocket or spacecraft's changing mass as it burns its fuel — is lower than the specific impulse of both ion and nuclear engines. Analysis of An After Burner in A Jet Engine 11 when installed, create too much drag. It still beats most of the other small canister stoves in the test but is no match for the fuel-sipping integrated canister stoves. Hydrogen is the smallest molecule in the Universe. Better yet, it has the potential of boosting future rocket fuel efficiency by 20 to 30%, compared to. BONNEVILLE CHARACTER AND AUTHENTICITY The latest Bonneville T100 range stays faithful to these celebrated motorcycles’ evocative heritage and truly iconic silhouette, with sculpted Bonneville signature fuel tank, wire-spoked wheels, authentic peashooter silencers, and two-tone paint options with hand-painted gold coach lines. Mining the Moon for Rocket Fuel Could Get Us to Mars Our team also found a fuel-efficient way to get spacecraft from Earth orbit to the depot at L1, requiring even less launch fuel and freeing. To get into orbit, a rocket has to be traveling 11 kilometers a second – 25,000 miles per hour!. The engine with the higher value of specific impulse is more efficient because it produces more thrust for the same amount of propellant. Distance between stars is huge, traveling said distance slower-than-light will take a huge amount of time, human beings have a very limited lifespan. 0-litre Ford EcoBlue engine; first in a new range of advanced Ford diesel engines delivers unrivalled package of fuel efficiency, performance and refinement. ? We have a science project in science. In this list of the 5 most fuel efficient non-hybrid cars in 2018; we present the most compelling reasons why you should forget about the complexity of hybrids or the limited range of full. Engineering Explained: High vs Low Octane Petrol. High values of fuel propellant provide cold weather output and bring you quick and fuel efficient boil times. A solid fuel rocket engine is just what it sounds like. From woods to charcoal, you can use a lot of natural elements for your fuel needs. After a long day of logging miles in the backcountry, there’s nothing better than sitting down for a warm meal when you reach basecamp. Trees Water and People (TWP) is a non-profit organisation established in 1998 and based in Colorado, USA. High-accuracy, Fuel-efficient System Places Satellites Right on Target PHOENIX, July 30, 2010 -- Honeywell (NYSE: HON) announced today that it has been selected by United Launch Alliance to provide primary avionics components for guidance and navigation of the Atlas V rocket in a follow-on contract worth up to$90 million over the life of. A conventional engine ignites fuel which then pushes on some pistons, and it turns a crank. Word invented by psiscott once he saw this car with his little son. I compare and test the weight efficiencies of a Trangia alcohol stove (Methyl Hydrate or Methanol) vs. A text from around that time describes how the combustion efficiency and hence the rocket thrust could be improved by creating a cavity in the propellant along the centre line of the rocket tube to maximise the burning surface - a technique still used in solid fuelled rockets today. It is largely useless if the engine speed approaches the exhaust velocity. Solid-fuel rockets consist of a fuel and oxidizer that are pre-mixed in a solid form. Solid fuel is a kind of fuel and is processed in a chemical plant. Modular Design: Our module’s flexibility allows it to be used for multiple missions – affordably and reliably. ESBIT solid fuel can also be used with the Caldera system which yields a slightly more efficient system in regards to weight of fuel needed to heat water compared an alcohol stove being used with the Caldera. This powerful unit can heat up to 100 m2, and the truly extraordinary performance of this heater has earned it a well-deserved place in the number one spot of our top 10 list of most efficient wood heaters. Tankless hot water heaters are an option, too. • Fuel efficient to give the aircraft the range the design requires. How to Make a Simple, Fuel-Efficient Rocket Stove for Cooking on the Patio It had to be simple to use, cost-efficient, fuel-efficient, clean, and DIY-friendly. 3 Rocket fuel [5 marks) You have just started a new job as a rocket scientist. | 2020-01-19 05:51:53 | {"extraction_info": {"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, "math_score": 0.458146333694458, "perplexity": 2854.517974823453}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250594209.12/warc/CC-MAIN-20200119035851-20200119063851-00235.warc.gz"} |
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January 20, 2021 4:21 pm 30 pts
For an integer n 2 2, let s, be the permutation group on n letters and, A, the alternating group. Let C* be the group of non-zero complex number under multiplication. Which of the following are correct statements? For every integern:2, there is a nontrivial homomorphism : S,-C* For every integer n 2 2, there is a unique nontrivial homomorphism : S, - C* For every integer n e3, there is a nontrivial homomorphism ¢: A C For every integern:5, there is no nontrivial homomorphism : A, - C | 2021-03-01 11:10:43 | {"extraction_info": {"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, "math_score": 0.9578129053115845, "perplexity": 929.0518180678712}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178362481.49/warc/CC-MAIN-20210301090526-20210301120526-00617.warc.gz"} |
https://www.physicsforums.com/threads/this-keyword-in-java.537912/ | # This keyword in Java
Hello everyone. I want explanation of the working of 'this' keyword in java. I read articles but couldn't understand. Please help me here. I don't understand what objects and variables they are talking about. Also give me an example of what object, variable, instance variable is.
Box(double w, double h, double d) {
this.width = w;
this.height = h;
this.depth = d;
In the above example, is width height depth an object or variable?? Explain me the 'this' keyword here please.
AlephZero
Homework Helper
Writing code like that is a common convention especially for constructor functions.
Actually your example is a bit silly. A better example would be
Box(double width, double height, double depth) {
this.width = width;
this.height = height;
this.depth = depth;
In that example there are two different variables called "width". One of them will have been declared in the class. The other one is the parameter to the constructor.
Within the constructor, the parameters are local variables and therefore the parameter "width" "hides" the class variable with the same name. "width" refers to the parameter, and "this.width" refers to the class variable. "this" just means "the name of the object of type Box that this function call refers to".
Your original example would work just as well without using "this":
Box(double w, double h, double d) {
width = w;
height = h;
depth = d;
because all the variable names are unique anyway. But using "this" makes it clear which variables belong to the class and which are local to the function.
Last edited:
An object is an instance of a class. A variable is a general term for a named "bucket" that you can keep a value inside (int number, float precise_number, String mySentence - these are all variables, and they all have a different type which is important, so the computer has some idea of what kind of variable you are talking about. Words are treated differently from numbers.) An instance variable, ok so you have a class (a template), and from this class you instantiate objects (you say, computer take this class design (of a box say), and build me an object (a box). Build as many boxes as you want. Box1, Box2, Box3 - all box objects created from the Box class. Each Box object has it's own copies of the variables that make up a box (width, height, depth). If you change one, you don't affect the other Boxes, because they are instance variables, related to the particular instance (Box1, Box2, Box3) of the class (Box)
Here's a small example:
Code:
public class Box
{
private:
int width, height, depth;
public Box(int width, height, depth)
{
this.width = width;
this.height = height;
this.depth = depth;
}
int getVolume()
{
volume = width * height * depth;
return volume;
}
public static void main()
{
Box Box1 = new Box(10, 10, 10);
Box Box2 = new Box(20, 20, 20);
Box Box3 = new Box(30, 30, 30);
int Vol1 = Box1.getVolume();
int Vol2 = Box2.getVolume();
}
}
Here I declare a class called Box. It contains a constructor, which is a special function to initialize the variables when you instantiate the class. What does that mean? In the main() function, refer to the line that says "Box Box1 = new Box(10, 10, 10);" This is an example of creating an object from a class definition, or instantiating the class, or creating an instance of the class. Before this, your class is just a template (a box design). After you create an object with the keyword new you have something useable, an actual Box.
You can see that we call getVolume() on two different instances of the Box class (Box1, Box2). They each have different width, height and depth, hence they will return a different volume, because their instance variables contain different values. Same class, different object (or instance).
The keyword this is a way of referring to the current object. See this page for a pretty good explanation with an example: | 2021-06-23 05:50:44 | {"extraction_info": {"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, "math_score": 0.17181770503520966, "perplexity": 2037.128113825213}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623488534413.81/warc/CC-MAIN-20210623042426-20210623072426-00199.warc.gz"} |
http://hackage.haskell.org/package/fclabels-2.0.0.3 | # fclabels: First class accessor labels implemented as lenses.
[ bsd3, data, lenses, library ] [ Propose Tags ]
This package provides first class labels that can act as bidirectional record fields. The labels can be derived automatically using Template Haskell which means you don't have to write any boilerplate yourself. The labels are implemented as lenses and are fully composable. Lenses can be used to get, set and modify parts of a data type in a consistent way.
See Data.Label for an introductory explanation or see the introductory blog post at http://fvisser.nl/post/2013/okt/1/fclabels-2.0.html
• Total and partial lenses
Internally lenses do not used Haskell functions directly, but are implemented as categories. Categories allow the lenses to be run in custom computational contexts. This approach allows us to make partial lenses that point to fields of multi-constructor datatypes in an elegant way.
See Data.Label.Partial for the use of partial labels.
• Monomorphic and polymorphic lenses
We have both polymorphic and monomorphic lenses. Polymorphic lenses allow updates that change the type. The types of polymorphic lenses are slightly more verbose than their monomorphic counterparts, but their usage is similar. Because monomorphic lenses are built by restricting the types of polymorphic lenses they are essentially the same and can be freely composed with eachother.
See Data.Label.Mono and Data.Label.Poly for the difference between polymorphic and monomorphic lenses.
• Using fclabels
To simplify working with labels we supply both a set of labels for Haskell's base types, like lists, tuples, Maybe and Either, and we supply a set of combinators for working with labels for values in the Reader and State monad.
- Support GHC 7.0. Note that there seems to be a problem with the
appicative syntax, see test cases. | 2018-12-16 02:37:37 | {"extraction_info": {"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, "math_score": 0.2668256163597107, "perplexity": 2510.7974681318683}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376827175.38/warc/CC-MAIN-20181216003916-20181216025916-00121.warc.gz"} |
https://www.physicsforums.com/threads/mistake-using-borel-cantelli-lemma.606171/page-2 | # Mistake using Borel-Cantelli Lemma
Ah, I see. In that case you are correct that $P(\overline{X}_n < c \text{ e.v.})=0$, but this is irrelevant, since we may have $\liminf_{n\rightarrow \infty} \overline{X}_n < c$ even if we do not have $\overline{X}_n < c \text{ e.v.}$ | 2021-01-21 06:14:18 | {"extraction_info": {"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, "math_score": 0.9631214141845703, "perplexity": 140.44354775493028}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703522242.73/warc/CC-MAIN-20210121035242-20210121065242-00114.warc.gz"} |
https://gamedev.stackexchange.com/questions/118630/tilemap-object-placement | # Tilemap Object Placement
I am looking for a way to select tiles within a bounding box for object placement within a scene/world, similar to Roller Coaster Tycoon and other simulation games.
(source: tumblr.com)
In the above screenshot of Theme Parkitect you can see that a 4x4 grid is used as the footprint for the object being placed. Given the bounding box and location of the mouse, how can I suitably determine the neighbouring tiles which fall into the bounding box?
I have implemented a graph like structure of tiles, corners and edges by following Amit's fantastic articles on Red Blob Games. As of such, I have a graph of all tiles and their neighbours as well as associated tile edges and corners. These tiles are laid out diagonally against the world axis so simply selecting all tiles within a rectangular region isn't as simple (or maybe it is?) as i'd expected. This layout can be changed if necessary however.
The main source of information used to steer my implementation has been from Amit's Thoughts on Grids and the graph structure implemented in his article about Procedural Map Generation
As a bonus, I would like to implement a way to rotate the object being placed such that a footprint of 1x5 could be rotated to instead be 5x1. I would imagine that once the basics of multiple tile picking has been implemented this should be reasonably simple.
Unfortunately, Amit's articles don't seem to cover this topic and the approaches I have found and tried so far haven't quite yielded the correct results. Any help that fits in with the graph like structure for connected tiles, edges and corners would be greatly appreciated.
I have tried to implement the approach outlined in Raytracing on a Grid with a mixture of both Broad-Phase collision and A* Path Finding but neither of these two approaches cover selection of multiple surrounding tiles.
I am not concerned with the viability of tiles for placing objects at this point (i.e.: sloped edges of tiles or obstructions from other objects). The ideal solution for now would simply be to find the tiles included within the bounds.
My initial forays into this have resulted in incorrect results whereby the bounding rect doesn't include all neighbouring tiles (sometimes missing entire rows/columns) or returning neighbours outside of the bounds. I was pretty sure that I was close to a winning implementation but my vector math isn't strong enough to validate my findings.
• Sharing your research helps everyone. What were the approaches you tried and how did their result differ from the desired results? Also, could you maybe post a direct link to the relevant article? You've linked to the frontpage of Red Blob Games, and the website has quite a lot of articles. This would help us to understand how you implemented your isometric engine which in turn would help us to propose a solution which fits your implementation. – Philipp Mar 21 '16 at 11:40
• I have updated my question with additional references to the articles used as the basis for my implementation thus far :] – CaptainRedmuff Mar 21 '16 at 12:05
• If i understand correctly you are looking to solve 2 problems? The first one trying to find the square the mouse is currently over and then selecting all grid locations covered by a template? When you solve the first problem the second one should be trivially easy as long as you can select your gridlocations by x-y-coordinates. Just don't try to select all the tiles at once, select one and place your building on that point, testing all locations covered by your template. edit: a question turned into half an awnser, I will work this out when i have more thime. – Niels Mar 22 '16 at 7:17
• I have already managed to implement tile picking for a single tile. I want to expand upon this to include neighbouring tiles for those within a bounding box. If it makes any difference to the answer, I am using Unity's built in ray tracing and mesh colliders to determine the tile being picked. – CaptainRedmuff Mar 22 '16 at 8:59
if your data is in a one dimensional array you can convert from index to coords and back easily. making it easy to determine what is beside a coord
public int TileIndex(int x, int y)
{
return y * Width + x;
}
public Vector2 TilePosition(int index)
{
float y = index / Width;
float x = index - Width * y;
return new Vector2(x, y);
}
so to check the tile to the east of 2,2 you can get the tile at index 3,2
if (Empty(TileAt(TileIndex(x + 1, y)))
MoveEast();
When you have your mouse-picking working all you need to do is overlap your map with the object you wish to place. This problem would be the same for an isometric grid as for a rectangular grid.
Take this exmaple map:
And this building:
On this map the green squares are meant to as grass, the brown squares as mountain and the blue squares as water. Although this is irrelevant to the actual solution
On the building the color of the squares determines the placemant restrictions, white can be placed anywhere, yellow can only be placed on green squares and orange can be placed on either green or brown squares.
After you've picked the square the mouse is over on the map you simply pick a square from your building-template, (this can be any square you want, top-left, best-effort middle or whatever provides the best experience to your players), let's use top-left for this example. and then check if all squares are compatible with the map.
This can be done by simply looping over the squares:
for each square in building_template do:
determine offset from cursor on template
pick tile from map with same offset
check if placement is allowed
if placement is not allowed, depending on your design either:
return false immediately and draw the entire template in red.
color that square from your template red and continue, keep track if any blocking squares were encountered
If we place the building on C4 for example We'd get the following result:
Where B2 of the building is blocked because that square could only be placed on green tiles.
Note that you can have the matching algorithm take into account any conditions you want.
I hope this is the problem you were trying to solve.
• This is an interesting answer but it doesn't quite cover how to determine which tiles are covered by the footprint of the object being placed. In your example, if you chose the top left corner of the object and find a tile on the map to place it, how do you then iterate over the remaining tiles, i.e.: all the tiles in the same row, and then those in the subsequent rows? – CaptainRedmuff Mar 22 '16 at 16:24
• @CaptainRedmuff a simple loop would cover that. for(int blueprint_x = 0; blueprint_x < blueprint.width; blueprint_x++){for (int blueprint_y = 0; blueprint_y < blueprint.width; blueprint_y++){ if(getBlueprintTile(blueprint_x, blueprint_y).canBePlacedOn(map.getTile(pickedTile_x + blueprint_x, pickedTile_y + blueprint_y) ){//do stuff}else{//show that tile cant be placed}}} – Niels Mar 23 '16 at 7:01
• @CaptainRedmuff I just realised the pseudocode for that was already in the awnser, so I assume I must be missing what you are actually asking. – Niels Mar 23 '16 at 7:08
• i.imgur.com/djq7UI2.jpg – Rakka Rage Mar 23 '16 at 22:01 | 2021-01-20 07:35:24 | {"extraction_info": {"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, "math_score": 0.27514025568962097, "perplexity": 999.7806009390342}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703519923.26/warc/CC-MAIN-20210120054203-20210120084203-00124.warc.gz"} |
https://poetry.openlibhums.org/article/id/706/ | Published by Rod Mengham’s Equipage press in 1997, For the Monogram is both typical and, arguably, inauguratory of what is sometimes called J. H. Prynne’s ‘late style’. The book is short, running to just 20 pp., and sits uneasily between the categories ‘poem’ and ‘sequence’, consisting of fourteen individually paginated blocks of text, each sixteen lines in length. (For convenience, and for a number of more important reasons which will be considered below, I will refer to the work as a sequence.) As in later books, the blocks are relatively consistent in width, and they are all untitled, marking the start of a wholesale avoidance of titles—which Prynne refers to derisively in a 1964 interview, echoing his interviewer, as ‘handles’—lasting until 2010’s Sub Songs.1 For the Monogram has not received significantly more critical attention than any of Prynne’s other books of this period, which is to say it has not received much attention at all. One of the most substantial and sustained attempts to tackle the sequence is Simon Jarvis’s 2003 article for the online magazine Jacket, a piece which runs to just 3,500 words and bills itself (over-modestly) as ‘a notable Failure’.2 What is it, then, other than the general obscurity of Prynne’s late work, that has prevented scholars from engaging seriously with For the Monogram? A clue might be found in the third poem in the sequence, which seems to have been quoted and cited more times than any other:
Select an object with no predecessors. Clip off its
roots, reset to zero and remove its arrows. At each
repeat decrement the loop to an update count for all
successors of the removed object ranking the loop body
at next successor to the array stack. Count back up
left and right scan, test for insert loops using
0 0 as sentinel pair. Such as of figures in space,
if (set if true) a product goto top list, an object
otherwise (if else) remaining the same. As a quantum
(put > zero) parse to occupy inner sense more by
recursion count to null, and reset. Match for error
to run output and restrict condition if at one
then also next. There is a bright blue light flashing
over the exit plaque. Connect atonal floats via
path initial to hydrated silica screen occlusion ice
batched out and bent through diode logic gates.3
For critics eager to demonstrate the supposedly alien character of Prynne’s late poetry, specifically its incorporation of conventionally non-poetic diction, this particular poem is a helpful tool—this is the use to which it is put in Ian Brinton’s Contemporary Poetry (2009), a book written largely for students.4 As Brinton points out, quoting Peter Middleton, the poem makes ‘the new communications technology part of [its] field of reference, semantic, visual, and lexical, all broadly at the level of content.’5 Unfortunately, as in so many readings of Prynne’s later poetry, this is where discussion stops. Scholars trained in the profession of English Literature are quick to identify what makes poems such as this unusual and are generally able to make educated guesses about the origins of their language; however, with very few exceptions, they are reluctant to engage with that language on the broad ‘level of content’ to which Middleton refers.6 When semantic judgements are made, they are very often of an abstract, second-order character: the deployment of a particular vocabulary is read as a statement, but the actual propositions made using that vocabulary are ignored. (In many cases, the supposed impenetrability of these baseline propositions is itself interpreted as a second-order statement.) What if ‘the new communications technology’ could itself help to bridge this gap?
Middleton contends that ‘the mobile phone, the internet, the personal computer, and digital image processing have had a greater impact on the way we live than any new set of technologies since the arrival of the motor car, telephone, and radio a century earlier and are genetically modifying the ways we remember and the way we read.’7 This is no doubt true, but the use of the perfect tense in the first part of this sentence risks concealing the way that these new technologies are themselves developing and superseding each other at an exponential rate. One example will serve to illustrate the pace of change. Middleton’s Distant Reading was published in 2005, but the essay ‘New Memoryism’ was first delivered at a conference at Birkbeck College in September 2000, roughly four years before Google formally announced the launch of its ‘Google Print’ project, later to be renamed ‘Google Books’.8 What began as a relatively small programme to make newly-published books visible in Google’s search engine has since expanded into a huge digitization effort; Google Books now hosts a searchable database of well over 30 million titles, a figure which, if a 2010 estimate is to believed, represents roughly one quarter of all books ever published.9
Google Books is undoubtedly one of the technologies most responsible for ‘genetically modifying […] the way we read’ in recent years, with poetry like Prynne’s representing prime material for the trial-and-error-based approach to literary interpretation that it facilitates. It is now possible, for instance, to identify a precise source for the sentence ‘Select an object with no predecessors’ simply by typing it into the search engine, which throws up a 1986 text, Data Structures, Algorithms and Programming Style, by the computer scientist James F. Korsh.1011 But what are the limits of this approach? Can it be used to gain a foothold on the elusive ‘level of content’ referred to earlier, or is it simply a quicker and more efficient way to make guesses about textual provenance? To test this thesis properly, it is necessary to accept the challenge posed by Prynne’s adoption of the language of computer programming and actually to write a computer program. Using JavaScript and the Google Books application programming interface (API), I put together a simple script intended, in the discourse of password cracking, to ‘brute force’ the poem. The script takes plain text as input—in this case, a copy of For the Monogram which had been scanned, run through optical character recognition (OCR) software and tidied up manually. It then strips the text of all punctuation and ‘whitespace’ and uses it to generate an overlapping list of four-word chunks: words 1, 2, 3 and 4, then words 2, 3, 4 and 5, and so on. (Four words was decided upon as the optimum length, being short enough to cast a wide net but not so short that clear sources would be impossible to identify.) Each chunk is passed to Google Books—if it returns results other than those books in which the poem itself is printed or quoted, their titles are listed and made available for inspection by hand.
One of the most useful functions of this program is to confirm sources that have already been identified or part-identified through conventional reading. In ‘The Incommunicable Silhouette’, Jarvis puts forward Immanuel Kant’s Critique of Pure Reason as an important precursor to the text, using as evidence Kant’s frequent use of the term ‘monogram’ in that work and the phrase ‘the scheme of a pure | sensible outline’ in the second poem—not a construction used by Kant, but one which Jarvis takes to be a reference to his transcendental schema.12 Jarvis’s argument is strengthened by a number of verbatim borrowings which are identified easily by the computer, but which, lacking a key vocabulary element such as ‘scheme’, are more difficult for human readers to pick up: ‘such as of figures in space’; ‘an object | otherwise […] remaining the same’; and ‘occupy inner sense more’.13 All three are taken from the chapter titled ‘The Schematism of the Pure Concepts of Understanding’ in the Kemp Smith translation, a chapter which Jarvis himself mentions.14 A similar pattern of repeated borrowing is identifiable in the case of the Korsh book, with the phrases ‘remove its arrows’ and ‘all | successors of the removed object’ also being taken from this source.15
While confirming previous educated guesses about source materials, the results listed here also shed light on For the Monogram’s formal organisation: specifically, the question of whether the text should be considered a sequence or a single poem. All direct quotations from the First Critique occur in Poem 3, as do all quotations from Korsh; this consistency seems to support the case for considering this particular text block as a more or less coherent whole in which epistemology and computer programming are set up to counterpoint each other.16 Coherence can also be identified through features other than the co-location of quotations from different texts, with Poem 7 being a clear example. When run through the Google Books API, this poem threw up a match which at first seemed accidental: the phrase ‘out | out to show’ matched a score for the composer Steve Reich’s 1966 piece Come Out, printed in an essay by Sumanth Gopinath in 2009, more than a decade after the publication of For the Monogram.1718 After further investigation, however, a pattern of reference to the piece became clear. Come Out uses a recording of Daniel Hamm, one of the ‘Harlem Six’ arrested for the murder of a police officer following the Harlem Riot of 1964. Describing an attempt to demonstrate to police that he had been beaten, Hamm says, ‘I had to, like, open the bruise up, and let some of the bruise blood come out to show them.’ Reich loops the final four words, first on two separate tape recorders, then on four, and lets them slip out of sync in playback, a process which at one point results in the phrase ‘out out to show’. Crucially, Prynne’s poem mimics both the vocabulary and the repetitious character of the piece: ‘come out to flay’; ‘fixture to show them’; ‘Clip act out | out to show’; ‘coals out to show’; ‘this vivid failed bruise’; ‘You got scarlet out to show it’; ‘come out for out now’.19
Before exploring some of the other sources exposed by For the Monogram’s computational disassembly, I would like to turn, deliberately, to an ‘actual proposition’ made by the text—one which seems particularly relevant to the question of how, exactly, the sequence is to be read. There is a regrettable tendency in criticism of Prynne’s poetry to search for coded acknowledgements of the work’s obscurity, as if the discovery of such an admission would absolve the critic of the task of actually reading the poems; the most famous example is the beginning of the 1966 poem ‘The Numbers’—‘The whole thing it is, the difficult | matter’—which, as Thomas Roebuck and Matthew Sperling note, has too often been used ‘to launch a somewhat sterile debate on “difficulty” in poetry’.2324 Even so, there are points in Prynne’s work at which the temptation to read for meta-language is practically irresistible. One such point is the beginning of Poem 12 in For the Monogram:
Prior guesswork loses the things in your power by
broken reach in seeking to verify the check-out
lag at the till. Glow to offence rating, narrow axis
tapers into counting lucky hits, rueful charge
card assent.25
A quick and admittedly speculative attempt at paraphrase: meaning is compromised (‘lost’ as a ‘thing in your power’) by any attempt to match up ‘prior guesswork’—the initial construction of some notion of what the poem or sequence is ‘about’—with the semantic state of affairs which obtains at the work’s conclusion, the ‘check-out’ or ‘till’. There is ‘lag’ between the two, making any attempt to reconcile them a ‘broken reach’. If the affronted reader, whatever their ‘offence rating’, persists with this sort of inadequate strategy, the already ‘narrow axis’ of their reading will ‘taper into counting lucky hits’, shaping incidental features of the work into patterns which confirm their ‘prior guesswork’.26 This is the interpretative equivalent of buying on credit, with a ‘charge card’, a fact of which the lazy but nevertheless ‘rueful’ reader can hardly be unaware.
If this reading is anything close to accurate—and as long as it remains sufficiently local to avoid being undermined by its own conclusions—then Prynne seems to be making an important statement about the way his late poetry is to be read, or not read. Concluding ‘The Incommunicable Silhouette’, Jarvis reveals a ‘hunch’ that the famed difficulty of this work
emerge[s] from an attempt to pay scrupulous attention to some single quite discrete object, experience or phenomenon and, instead of allowing the functional directives of divided intellectual labour to govern the presentation of that object, continually to exhibit the connexions and fissures between such languages from the demands made upon them by the complexity of the object itself.27
In Jarvis’s model, conventional reading strategies, trained precisely to obey ‘the functional directives of divided intellectual labour’, will clearly be inadequate to the task of reading For the Monogram—they will be exactly that ‘broken reach’ that cannot but end by ‘taper[ing] into counting lucky hits’. But if Prynne’s late poetry is not to be abandoned as absolutely unreadable, then some new strategy must be developed which is capable of approaching the work in a movement which mirrors its own ‘scrupulous attention’ to its ‘single quite discrete object’. It is my contention that this strategy is outlined, albeit negatively, in the pages of For the Monogram itself. Slightly earlier in the essay, Jarvis notes a congruence between Kant’s ‘absurdly optimistic’ notion that ‘an architectonic of all human knowledge would not even be difficult’ and the sanguinity of participants (Google among them) in ‘the current scramble for monopoly over information technologies.’ He goes on to argue that
[t]he drastic technical development of Prynne’s work since Brass, as well as its persistent, even absurd, determination to keep up with technical developments in the most disparate fields, such as computer languages and biochemistry, is what we might think of as a counter-architectonic working within and against the existing databanks.28
The point of Prynne’s late poetry, argues Jarvis, is ‘a search for the monogram, the sketch, outline, figure or programme of the engines of mutation themselves: so as to make their refiguring imaginable.’29 While I am ready to subscribe to the general tenor of this argument, I believe that Jarvis’s concept of the ‘counter-architectonic’ requires further clarification if it is to be of real explanatory value. Specifically, it is necessary to describe how exactly the work’s ‘determination to keep up with technical developments’ relates to the manifestation of those developments in the real, non-poetic realm of blood and silicon.
For the Monogram is, I would like to argue, a text which anticipates and even acts out its own mechanized dissection. The preoccupation with surgery in Poem 7 has already been described, and there is a similarly clear focus on bodily distortion and non-coherence as a result of what seems like military hardware in Poem 10. Weaponry is raised much more specifically in Poem 9, where the phrase ‘pressed candles require a first fire’ appears to have been lifted almost verbatim from A. Bailey and S.G. Murray’s 1989 Explosives, Propellants and Pyrotechnics.3031 So far, so much ‘prior guesswork’. Yet there are other, much more important indicators that For the Monogram is a work gazing forward to its own violent destruction. In the very first poem of the sequence, a text which might be expected to ‘set the tone’ for the remainder of the work, an object is described as ‘floating across bars in black? In green | flash scraping ionic burn?’32 Staying within the field of weaponry, this might be identified as the view through a sniper’s scope, with the ‘bars’ as a reticule and the colouring referring to the distinctive green tinge of night vision equipment. Once again, however, the computer is able to pick up a textual reference—‘in black, in green’—where a human reader might be thrown by the intervening question mark, or the apparent reference to the optical phenomenon of the ‘green flash’. The source—admittedly only potential—in this case is a text known popularly as ‘Mother Shipton’s Prophecy’, said to have been written in the 15th or 16th century by a legendary prophetess named Ursula Southeil, or Mother Shipton. The poem is reprinted in full below, in the version found in William H. Harrison’s Mother Shipton Investigated:
Carriages without horses shall go,
And accidents fill the world with woe.
Around the world thoughts shall fly
In the twinkling of an eye.
The world upside down shall be
And gold be found at the root of a tree.
Through hills man shall ride,
And no horse be at his side.
Under water men shall walk,
Shall ride, shall sleep, shall talk.
In the air men shall be seen,
In white, in black, in green;
Iron in the water shall float,
As easily as a wooden boat.
Gold shall be found and shown
In a land that’s now not known.
Fire and water shall wonders do,
England shall at last admit a foe.
The world to an end shall come,
In eighteen hundred and eighty one.33
For the modern reader, this text obviously bears the one key hallmark of spurious prophecy: verifiability built on vagueness. As such, it might be said to mirror the ‘charge card’ approach to poetic interpretation laid out earlier—disciples of Mother Shipton, like conspiracy theorists or devotees of Nostradamus, are ‘counters of lucky hits’ par excellence. Nevertheless, its implicit citation by Prynne 116 years after the supposed apocalypse serves to re-invest it with specific meaning. The third and fourth lines in particular—‘Around the world thoughts shall fly | In the twinkling of an eye’—having presumably already predicted the electric telegraph, the telephone and shortwave radio, now seem to refer unequivocally to the internet, and by extension to ‘the current scramble for monopoly over information technologies.’ For the Monogram is a text which understands the world’s direction of travel as it hurtles simultaneously towards the universal fungibility (though not accessibility) of information and an irresistible multiplication of the blacks and greens of night vision and military camouflage.
Further evidence of For the Monogram’s textual self-consciousness is provided in Poem 10, in which Prynne makes a clear reference to Wallace Stevens’s 1952 poem ‘Prologues to What is Possible’.34 Stevens’s lines ‘A boat carried forward by waves resembling the bright backs of rowers, | Gripping their oars, as if they were sure of the way to their destination’ are distorted into a violent and grotesque parody:
[…] enter green mourning tents in burned faces
arms and bodies charred resembling the bright backs
gripping and bending, to weep there.35
With the figure of the ‘green mourning tents’, we are once again in a military or refugee camp; given the tendency of Prynne’s poetry to react quickly to ‘current events’, we might even locate it more precisely in the former Yugoslavia. But what is the significance of this unacknowledged quotation in general, and particularly of the bathetic transformation enacted by its new context? A clue can be found in a recent prose work by Prynne, Concepts and Conception in Poetry (2014), which consists of a detailed commentary on the Stevens poem, as well as notes on two passages from Wordsworth. To provide a comprehensive account of Prynne’s argument in this pamphlet would be beyond the scope of this article; nevertheless, we may note his conclusion, which is that the poem is able to ‘pick up and even toy with its own strands’. More fully:
The ‘meaning’ of this poem, or even a meaning or meanings for it, is thus not overtly declared. The method of fluid abstraction prevails over particulate description, the landscape (seascape) and the schedule of colours and visual imagery, drawing these into a domain of conceptualised potential meaning […] which responds to ideation by constant promotion to higher levels of abstraction, so that motions of thought float closely between and within currents of the description but are not specified or captured by them.36
If this delicate and fine-tuned autonomy is the achievement of Stevens’s poem, then its ironic citation in For the Monogram seems to represent a deliberate and demonstrable failure of that achievement. To be sure, this is also a work in which meaning is ‘not overtly declared’—but only because there are so many pretenders to the title, pieces of language and other miscellaneous fragments of semiosis thrusting themselves in from all sides. The experience of reading For the Monogram is not, as Stevens has it, ‘like being alone in a boat at sea’, but more like swimming in a crowded shipping lane—and this is an experience of which the text itself is aware.
To put it bluntly, For the Monogram is a poem written by a human in the knowledge that it will, at some point, be read by a computer. This brings us back to what I believe is meant by Jarvis’s ‘counter-architectonic working within and against the existing databanks’—not a Wachowskian sci-fi cliché in which The Human Spirit is pitted against the ruthless logic of the machine, but a more ambivalent movement of strategic undermining and sabotage. One of the discourses in For the Monogram that has not yet been noted is that of mathematics, specifically graph theory, as when in Poem 13 we read about ‘a non-trivial path from the vertex back | to itself’.37 In mathematical terms, a graph is a group of objects (vertices) connected to each other in various configurations by a series of theoretical lines (edges); the first real work in the field is considered to be Euler’s paper on the Königsberg Bridge Problem, which was in turn spurred by the researches of Leibniz, whose Theodicy provides the epigraph for For the Monogram. Despite this historical pedigree, graph theory was in the late 1990s (and to some extent still is) an exciting field, due primarily to emerging practical applications which had the potential to revolutionize database technology, offering an alternative to the strictly hierarchical rows-and-columns model that had obtained up to that point. This new horizontal approach might be taken as a metaphor for the structure of For the Monogram, each poem, sentence or word a vertex and the strands of meaning a set of provisional and precarious edges tying them together.
Technological metaphors for what is essentially political resistance are, of course, inherently flawed—however symbolically or formally democratic a new technological development, capital’s effective control of research and development ensures that there is always a repressive application waiting in the wings. This is certainly true in the case of graph databases, whose suitability for modelling social relations has led to their adoption by intelligence agencies such as the US National Security Agency (NSA) and GCHQ in their efforts to interpret the vast swathes of metadata collected under programmes such as PRISM.38 Still, Prynne is not careless enough to remain bound by the restrictions of a single metaphor, and the fate of the graph database does take us close to the heart of For the Monogram’s sceptical understanding of technological development and its effect on poetic language. What a computational reading of this sequence reveals is, ultimately, its own insufficiency. The more data that is collected and the more efficiently the poem’s sources are identified, the more obvious it becomes that there is an interpretative gap at the centre of the reading corresponding to Jarvis’s ‘single quite discrete object, experience or phenomenon’. This is not to say that this object can be grasped through some sort of intuitive leap, nor that computational approaches to reading lack value altogether. The point is, ultimately, a phenomenological one, having to do with the process of interpretation as much as with the end result. Using a computer’s processor and memory to read is never quite the same as using one’s own brain, precisely because it implies the outsourcing of the act of processing and its separation from decisions about meaning.39
This point is derivable from any reading of the poem, even the most scrupulously hygienic, text-internal exercise in ‘practical criticism’. Yet it is bound to come across more forcefully in the course of a reading which accepts what For the Monogram tries so strenuously to say about itself: namely, that it is both one thing (a poem) and another (fragments of pre-existing language). From this perspective, the value of a computational interpretation is essentially performative, insofar as its separation of processing from understanding dramatizes a fundamental split which already exists in the text itself, as it does in every text. Facing the poem’s hidden centre, Jarvis describes it not only as ‘some single quite discrete object’, but also, potentially, as an ‘experience or phenomenon’.40 Is it possible that what sits at the heart of For the Monogram is the ‘experience or phenomenon’ of poetic creation itself, the presentation of which Prynne refuses to allow to be governed by ‘the functional directives of divided intellectual labour’? This would certainly give a new meaning to Jarvis’s contention that the poem ‘continually […] exhibit[s] the connexions and fissures between [its source] languages from the demands made upon them by the complexity of the object [i.e. poetic creation]’.41 In fact, those ‘connexions and fissures’ would themselves be the objects, rather than serving as phenomenal manifestations of some deeper truth. In this sense, For the Monogram really could be said to ‘pick up and toy with its own strands’, and the gap between processing and understanding, source text and deployment, would be something like the hinge between finger and thumb which makes it possible.
## Notes
1. Peter Orr interviews Jeremy Prynne, The Poet Speaks (London: British Council, Recorded Sound Department, 6 January 1964). [^]
2. Simon Jarvis, ‘The Incommunicable Silhouette’, Jacket, 24 (November 2003), <http://jacketmagazine.com/24/jarvis-tis.html> (accessed 15 July 2014). [^]
3. J. H. Prynne, ‘For the Monogram’, in Poems, 2nd edn (Tarset: Bloodaxe, 2005), 417–30 (p. 420). To facilitate the use of different editions, subsequent citations will be given in the form For the Monogram, [poem number].[line number(s)]. [^]
4. Ian Brinton, Contemporary Poetry: Poets and Poetry since 1990 (Cambridge: Cambridge University Press, 2009), p. 111. [^]
5. Peter Middleton, ‘New Memoryism’, in Distant Reading: Performance, Readership, and Consumption in Contemporary Poetry (Tuscaloosa, AL: University of Alabama Press, 2005), 137–59 (p. 154). [^]
6. Middleton is, admittedly, one of these exceptions—his essay ‘Dirigibles’, also included in Distant Reading (pp. 160–98), considers For the Monogram in much more detail. [^]
7. Middleton, p. 139. [^]
9. Leonid Taycher, ‘Books of the world, stand up and be counted! All 129,864,880 of you’, Google Books Search (5 August 2010), <http://booksearch.blogspot.co.uk/2010/08/books-of-world-stand-up-and-be-counted.html> (accessed 15 July 2014). The scale of the project is matched by the secrecy with which it is carried out, as a 2010 film by artist Andrew Norman Wilson makes abundantly clear (‘Workers Leaving the Googleplex’, Vimeo (14 October 2010), <http://vimeo.com/15852288> (accessed 15 July 2014)). [^]
10. For the Monogram, 3.1. [^]
11. James F. Korsh, Data Structures, Algorithms and Programming Style (Boston, MA: PWS, 1986). [^]
12. For the Monogram, 2.2–3. [^]
13. For the Monogram, 3.7, 3.8–9, 3.10. [^]
14. Immanuel Kant, The Critique of Pure Reason, trans. by Norman Kemp Smith (London: Macmillan, 1993). [^]
15. For the Monogram, 3.2, 3.3–4. [^]
16. There are, of course, indirect references to similar subject matter throughout For the Monogram, e.g. a use of the term ‘imperfect duty’, from Kant’s moral philosophy, in Poem 11 (2–3), and a series of references to mathematics and graph theory: ‘a shallow tree’ (9.5); ‘a non-trivial path from the vertex back | to itself’ (13.10–11). [^]
17. For the Monogram, 7.7–8. [^]
18. Sumanth Gopinath, ‘The Problem of the Political in Steve Reich’s Come Out’, in Sound Commitments, ed. by Robert Adlington (Oxford: Oxford University Press, 2009), 121–44 (p. 132). [^]
19. For the Monogram, 7.1, 7.5, 7.7–8, 7.9, 7.11, 7.12, 7.14. [^]
20. Middleton, ‘Dirigibles’, in Distant Reading, 160–98 (pp. 196–97). [^]
21. Middleton, for one, is well aware in 2000 of the archival potential afforded by the internet: ‘The internet, although at an early stage, appears to be on the way to offering an archive of all extant printed texts as well as accompanying commentaries, and however far short it falls of this expectation, it already creates a phenomenological strain on the individual’s capacity to work with it as a form of memory’ (‘New Memoryism’, p. 147). [^]
22. There is some evidence to suggest that one reference to the latter, ‘position and peel back’ (7.11), may have been taken from an instruction manual for breast implants, while the phrase ‘to show weft of’ (7.8) occurs in medical literature concerning the dissection of eyes; still, this level of precision is not necessary to identify the general surgical tenor of the poem. [^]
23. Prynne, ‘The Numbers’, in Poems, 10–12 (p. 10). [^]
24. Thomas Roebuck and Matthew Sperling, ‘“The Glacial Question, Unsolved”: A Specimen Commentary on Lines 1–31’, Glossator, 2 (2010), 39–78 (p. 43). The position of ‘The Numbers’ at the very start of every edition of the collected poems has no doubt encouraged this interpretation, despite the best efforts of the 1982 paperback’s blurb to pre-empt it: ‘Much early critical response to J. H. Prynne’s work mistakenly took its cue from the first line printed in this book: “The whole thing it is, the difficult”, failing to establish that difficulty as being the ardent “matter” and the accompanying breadth of imaginative and political reference’ (Poems (London: Agneau 2, 1982), back cover). [^]
25. For the Monogram, 12.1–5. [^]
26. ‘Glow to offence rating’ is grammatically ambiguous, potentially referring to something more like a ratio (glow:offence) than a steady movement towards a designated end-point (glow until the offence rating is reached). [^]
27. Jarvis, ‘The Incommunicable Silhouette’. [^]
28. Ibid. [^]
29. Ibid. [^]
30. For the Monogram, 9.15. [^]
31. A. Bailey and S. G. Murray, Explosives, Propellants and Pyrotechnics (London: Brassey’s, 1989). [^]
32. For the Monogram, 1.6–7. [^]
33. William H. Harrison, Mother Shipton Investigated (London: privately printed, 1881), pp. 12–13. [^]
34. Wallace Stevens, ‘Prologues to What is Possible’, The Hudson Review, 5.3 (Autumn 1952), pp. 330–31. [^]
35. For the Monogram, 10.4–6. [^]
36. Prynne, Concepts and Conception in Poetry (Cambridge: Critical Documents, 2014), pp. 43–44. [^]
37. For the Monogram, 13.10–11. [^]
38. Doug Henschen, ‘Defending NSA Prism’s Big Data Tools’, InformationWeek (11 June 2013), <http://www.informationweek.com/big-data/big-data-analytics/defending-nsaprisms-big-data-tools/d/d-id/1110318?> (accessed 19 August 2016). [^]
39. The interface between the two—in computing terms, the ‘bus’, as in ‘universal serial bus’ (USB)—is at present represented by a highly inefficient system of fingers, keyboards, eyes and screens, constituting a powerful bottleneck in comparison to copper wire, fibre optic cables or neurons. This is not necessarily an insurmountable technical problem, as new developments in biomechatronics constantly imply, but neither is it one whose supersession can be taken for granted—it is far from clear whether, even with a functionally perfect brain-computer interface (BCI), a biological organ would be able to recognise an artificial extension as ‘itself’. [^]
40. Jarvis, ‘The Incommunicable Silhouette’. [^]
41. Ibid. [^]
## Competing Interests
The author declares that they have no competing interests. | 2022-06-29 04:40:41 | {"extraction_info": {"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, "math_score": 0.40367990732192993, "perplexity": 4284.95716218829}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103620968.33/warc/CC-MAIN-20220629024217-20220629054217-00537.warc.gz"} |
https://socratic.org/questions/57728ac77c014920d8d46756 | # Is an ether a "Lewis base"?
Given the 2 lone pairs of electrons on the oxygen centre, ethers are $\text{Lewis bases}$. | 2021-10-16 11:11:32 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 1, "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, "math_score": 0.3942733705043793, "perplexity": 4738.695451985097}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323584567.81/warc/CC-MAIN-20211016105157-20211016135157-00195.warc.gz"} |
https://www.math-forums.com/threads/working-through-math-for-circular-motion.443158/ | # Working through math for circular motion
Discussion in 'Differentiation and Integration' started by biostartupguy, Jan 1, 2023.
1. ### biostartupguy
Joined:
Jan 1, 2023
Messages:
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0
Hey everyone,
I'm hoping to get some pointers on the following problem (screenshot of question and solution attached). The formula describes the motion of a particle in space with the position defined as r = r(cosωtˆi + sinωt ˆj)
It asks me to find the trajectory. In the solution, it solves for the trajectory by taking the magnitude of r
In a different problem, the trajectory is solved for by looking at limiting cases (i.e. in the limit where t -> 0 or infinity)
I don't get this first step and the logic in using one method over the other. I think this first step (setting up the problem) is critical to solving the question.
How does one normally go about this? What techniques does one use to determine how to tee up a problem like this?
Once the setup is complete, the math is relatively straight forward (e.g. substitution, taking the derivative of sin and cos)
#### Attached Files:
• ###### physics prob 1.png
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biostartupguy, Jan 1, 2023 | 2023-01-28 07:32:37 | {"extraction_info": {"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, "math_score": 0.8080148100852966, "perplexity": 993.5445660837668}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499524.28/warc/CC-MAIN-20230128054815-20230128084815-00865.warc.gz"} |
https://www.ideals.illinois.edu/handle/2142/16340/browse?rpp=20&order=ASC&sort_by=1&etal=-1&type=title&starts_with=O | # Browse Dissertations and Theses - Mathematics by Title
• (1996)
The purpose of this dissertation is to define homology functors for the category of definable sets and definable continuous maps in an o-minimal expansion of an ordered field. Both simplicial and singular homology functors ...
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• (1970)
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• (2012-06-27)
We utilize Floer theory and an index relation relating the Maslov index, Morse index and Conley-Zehnder index for a periodic orbit of the flow of a specific Hamiltonian function to state and prove some nonexistence results ...
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• (1983)
This dissertation is a study of the relationship between the structure of a group and that of its automorphism group. We examine the characteristics of groups G whose automorphism group Aut G has specified properties. If ...
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• (1973)
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Let C be a class of finite groups closed under the operations of taking subgroups, quotients, and extensions. Let H and K be pro-C-groups and let $G=H*K$ be their free pro-C-product. An open question in the theory of ...
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• (1955)
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In this thesis we give a generalization of a theorem of C. Fefferman and E. M. Stein on maximal operators on the Hardy classes of tempered distributions H('p)((//R)('n)) for 0 < p (LESSTHEQ) 1. Fix p, 0 < p (LESSTHEQ) ...
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In this thesis, we study congruence function fields, in particular those with many rational places. This thesis consists of three parts, the first two parts present our results in two different aspects of function fields ...
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• (1967)
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PDF (2Mb) | 2015-04-26 22:39:39 | {"extraction_info": {"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, "math_score": 0.8587074279785156, "perplexity": 1814.1488319500572}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1429246656747.97/warc/CC-MAIN-20150417045736-00147-ip-10-235-10-82.ec2.internal.warc.gz"} |
https://docs.krita.org/fr/untranslatable_pages/strokes_documentation.html | # Strokes queue¶
## Strokes, jobs… What it is all about? (theory)¶
### Structure of a stroke¶
An abstraction of a stroke represents a complete action performed by a user. This action can be canceled when it has not been finished yet, or can be undone after it's undo data has been added to the undo stack. Every stroke consists of a set of stroke jobs. Every job sits in a queue and does a part of work that the stroke as a whole must perform on an image. A stroke job cannot be canceled while execution and you cannot undo a single job of the stroke without canceling the whole stroke.
Example: Lets look at how the Freehand Tool works. Every time the user paints a single line on a canvas it creates a stroke. This stroke consists of several stroke jobs: one job initializes indirect painting device and starts a transaction, several jobs paint dabs of a canvas and the last job merges indirect painting device into the canvas and commit the undo information.
The jobs of the stroke can demand special order of their execution. That is the way how they will be executed on a multi-core machine. Every job can be either of the type:
CONCURRENT
concurrent job may be executed in parallel with any other concurrent job of the stroke as well as with any update job executed by the scheduler
Example: in Scale Image action each job scales its own layer. All the jobs are executed in parallel.
SEQUENTIAL
if the job is sequential, no other job may interleave with this one. It means that when the scheduler encounters a sequential job, it waits until all the other stroke jobs are done, starts the sequential job and will not start any other job until this job is finished. Note that a sequential job can be executed in parallel with update jobs those merge layers and masks.
Example: All the jobs of the Freehand Tool are sequential because you cannot rearrange the painting of dabs. And more than that, you cannot mix the creation of the transaction with painting of anything on a canvas.
BARRIER
barrier jobs are special. They created to allow stroke jobs to synchronize with updates when needed. A barrier job works like a sequential one: it does not allow two stroke jobs to be executed simultaneously, but it has one significant addition. A barrier job will not start its execution until all the updates (those were requested with setDirty() calls before) has finished their execution. Such behavior is really useful for the case when you need to perform some action after the changes you requested in previous jobs are done and the projection of the image does now correspond the changes you've just done.
Example: in Scale Image action the signals of the image like sigSizeChanged should be emitted after all the work is done and all the updates are finished. So it runs as a barrier job. See KisProcessingApplicator class for details.
Besides one of the types above a job may be defined as EXCLUSIVE. Exclusive property makes the job to be executed on the scheduler exclusively. It means that there will be no other jobs (strokes or updates) executed in parallel to this one.
### The queue of strokes¶
The strokes themselves are stored in a queue and executed one by one. This is important to know that any two jobs owned by different strokes cannot be executed simultaneously. That is the first job of a stroke starts its execution only after the last job of the previous stroke has finished.
The stroke is just a container for jobs. It stores some information about the work done, like id() and name(). Alongside storing this information it can affect the order of execution of jobs as well. The stroke can be defined exclusive. The meaning of this resembles the behavior of stroke job's exclusive property. Exclusive stroke is a stroke that executes its jobs with all the updates blocked. The execution of updates will start only after the stroke is finished.
## Implementation (practice)¶
### Implementation of a stroke¶
Overview of stroke classes
Each stroke is represented by a KisStroke object. It has all the basic manipulating methods like: addJob(), endStroke() and cancelStroke(). The behavior of a stroke is defined by a stroke strategy (KisStrokeStrategy class). This strategy is passed to the KisStroke object during construction and owned by the stroke.
Each stroke job is represented by KisStrokeJob object. The queue of KisStrokeJob objects is stored in every stroke object. This very object is used for actual running the job (KisUpdateJobItem calls KisStrokeJob::run() method while running). The behavior of the stroke job is defined by a strategy (KisStrokeStrategy) and a data (KisStrokeJobData). Those two objects are passed during the construction of the KisStrokeJob object.
A stroke can have four types of jobs:
• initialization
• canceling
• finishing
• actual painting (named as 'dab' in the code)
During construction the stroke asks its strategy to create strategies for all the four types of job. Then it uses these strategies on creation of jobs on corresponding events: initialization, canceling, finishing and when the user calls addJob() method.
The strategies define all the properties of strokes and stroke jobs we were talking above. The data class is used for passing information to the stroke by high-level code.
Example: FreehandStrokeStrategy::Data accepts such information as: node, painter, paintInformation, dragDistance
Other information that is common to the whole stroke like names of the paintOp, compositeOp are passed directly to the constructor of the stroke strategy.
### Execution of strokes by KisStrokesQueue¶
The key class of the strokes' execution is KisStrokesQueue. The most important method that is responsible for applying all the rules about interleaving of jobs mentioned above is KisStrokesQueue::processOneJob. This method is called by the update scheduler each time a free thread appears. First it gets the number of merge and stroke jobs currently executing in the updater context. Then it checks all the rules one by one.
### Canceling and undo information trick¶
It was stated above that a stroke can be canceled in each moment of time. That happens when a user calls KisStroke::cancelStroke() method. When it is requested the stroke drops all the jobs those are present in its queue and has not been started yet. Then it enqueues a special kind of job named cancel job that reverts all the work done by the stroke. This is used for interactive canceling of tools' strokes.
Taking into account that the strokes can be reverted, we cannot use QUndoStack capabilities directly. We should add commands to the stack after they have been executed. This resembles the way how KisTransactionData works: its first redo() method doesn't do anything because everything has already been painted on a device. Here in strokes this "after-effect-addition" is implemented in general way. Strokes work with a special kind of undo adapter: KisPostExecutionUndoAdapter. This adapter wraps the commands in a special wrapper that puts them into the stack without calling redo() and controls their threaded undo() and redo() operations. See information about KisPostExecutionUndoAdapter in a separate document.
### Queues balancing¶
So we ended up with a solution where our scheduler has two queues that it should spread between limited amount of threads. Of course there should be some algorithm that balances the queues. Ideally, we should balance them by the total area of image the queue should process. But we cannot achieve that currently. So the formula for size metrics is quite simple:
updatesMetric = <number of update jobs in the queue>
strokesMetric = <number of strokes> * <jobs in the first stroke>
Balancing formula:
balancingRatio = <updatesMetric> / <strokesMetric>
### Starting a stroke¶
The main entry point to strokes for the user is KisStrokesFacade interface. This interfaces provides four methods: startStroke(), addJob(), endStroke() and cancelStroke(). So every time you work with strokes you should work using this interface.
Note: KisImage and KisUpdateScheduler both implement this interface, so you can use them as a strokes facade. But please try not to store pointers to the whole image. Try store a link to interface only, if possible.
So if you want to start a stroke you should do the following:
1. Create a stroke strategy
2. Start a stroke with:
KisStrokeId strokeId = strokesFacade->startStroke(myStrategy);
Note: you'll get a KisStrokeId handle for the stroke you created. This handle will be used in all the other methods for controlling the stroke. This handle is introduced, because several users can access the strokes facade simultaneously, so there may be several strokes opened simultaneously. It's important to understand that even when several strokes are opened simultaneously, only one of them executes on the cpu. All the other strokes will be delayed until it is finished.
3. Create a data for your stroke job
4. Add a job to the execution queue:
strokesFacade->addJob(strokeId, myData);
5. You may add as many jobs as you wish
6. End or cancel the stroke:
strokesFacade->endStroke(strokeId);
or
strokesFacade->cancelStroke(strokeId);
# Strokes public API¶
## Simplified stroke classes¶
As you might noticed the internal strokes API is quite complex. If you decide to create your own stroke you need to create at least six new classes:
• stroke strategy class
• four stroke jobs strategies (init, finish, cancel, dab)
• data that will be passes to a dab-strategy-based job
That is not really a good solution for a public API, so we introduced an adapter that simplifies all these stuff. The class is called KisSimpleStrokeStrategy. It allows you to define all the jobs you need in a single class.
Simple stroke classes
This class has four virtual methods those you can use as callbacks. When you need to use one of them just override it in your own class and add activation of the corresponding callback to the constructor of your class:
class MyOwnStroke : public KisSimpleStrokeStrategy {
MyOwnStroke() {
enableJob(KisSimpleStrokeStrategy::JOB_INIT);
enableJob(KisSimpleStrokeStrategy::JOB_FINISH);
enableJob(KisSimpleStrokeStrategy::JOB_CANCEL);
enableJob(KisSimpleStrokeStrategy::JOB_DAB);
}
void initStrokeCallback()
{
}
void finishStrokeCallback()
{
}
void cancelStrokeCallback()
{
}
void doStrokeCallback(KisStrokeJobData *data)
{
Q_UNUSED(data);
}
};
Internally, KisSimpleStrokeStrategy creates all the job strategies needed for the lowlevel API. And these internal job strategies call the callbacks of the parental class.
Important: Notice that the job data passed to init, finish and cancel jobs is always null. It means that these jobs will always be sequential and non-exclusive. That is done intentionally to simplify the API. At the same time that is a limitation of the API. But currently, this is perfectly enough for us.
## Unit-testing of the strokes¶
One of the benefits of using the strokes is that you are able to test them separately from the UI using a common infrastructure.
### utils::StrokeTester class¶
That is a really simple class that you can use to test your own stroke. It test the following aspects of your stroke:
• canceling of the stroke
• working with indirect painting activated
• working with a layer that is not connected to any image
The result of the execution is compared against the reference png files those you create manually while writing your test.
### How to write your own test¶
You can check examples in MoveStrokeTest and FreehandStrokeTest tests.
1. You need to inherit your tester class from utils::StrokeTester. The constructor of that class accepts the name of your stroke (it'll be used for generating filenames), size of the image and a filename of the preset for the paintOp.
StrokeTester(const QString &name, const QSize &imageSize,
const QString &presetFileName = "autobrush_300px.kpp");
2. Then you need to override at least two methods:
KisStrokeStrategy* createStroke(bool indirectPainting,
KisResourcesSnapshotSP resources,
KisPainter *painter,
KisImageWSP image);
KisResourcesSnapshotSP resources,
KisPainter *painter);
If you thing you need it you may do some corrections for the image and active node in the following method:
void initImage(KisImageWSP image, KisNodeSP activeNode);
3. Run your test in a testing slot:
void MyStrokeTest::testStroke()
{
MyTester tester();
tester.test();
}
4. During the first run the test will report you many fails and will generate you several files with actual result of the test. You need to check these files, then move them into the tests' data folder: tests/data/<your_stroke_name>/
5. After you copied the files the tester will compare the actual result against these very files. That means it'll catch all the changes in the work of your stroke, so you'll be able to catch all the regressions automatically.
## Predefined classes for usage as base classes¶
### KisPainterBasedStrokeStrategy¶
This class can be used for the strokes those work with the node using a painter (or painters like in KisToolMultihand). This class accepts resources snapshot (KisResourcesSnapshot) and a painter (painters). Initialization, finishing and canceling callbacks of this class do all the work for dealing with indirect painting support, creation of transaction, reverting the stroke on canceling. This base class is used for FreehandStroke mostly.
### KisStrokeStrategyUndoCommandBased¶
It is obvious from the name of the class that it works with undo commands. In constructor you define which method of undo command should be used undo() or redo(). Afterwards, you just add commands to the stroke and they are executed with any the sequentiality constraints. This stroke strategy does all the work for adding the commands to the undo adapter and for canceling them if needed.
## Example classes¶
• KisPainterBasedStrokeStrategy
• FreehandStrokeStrategy
• KisStrokeStrategyUndoCommandBased
• MoveStrokeStrategy
# Internals of the freehand tool¶
Freehand tool classes
## Motivation for so many classes¶
We need to share the codebase between at least four classes: KisToolFreehand, KisToolMultihand, KisScratchPad. All these classes paint on a canvas with KisPainter, so they share quite much common code.
## KisResourcesSnapshot¶
After we introduced the strokes, the moments of time when user paints with mouse and when the line is actually painted on the canvas do not coincide. It means that by the time a thread starts actual changing the device, the contents of KoCanvasResourceProvider might have already changed. So before we start a stroke we should create a snapshot of all the resources we have and pass this snapshot to the stroke.
For this purpose we introduced KisResourcesSnapshot class. It solves two problems at the same time: first it stores all the resources we might have and second it encapsulates the algorithm of loading these resources into a KisPainter object. So this class is really easy to use. You just create the snapshot and then just load all the resources to the painter when needed.
KisResourcesSnapshotSP resources =
new KisResourcesSnapshot(image,
resourceManager);
KisPainter painter;
painter.begin(device, selection);
resources->setupPainter(&painter);
// paint something
painter.end();
In our implementation this class is usually created by KisToolFreehandHelper and passed to the KisPainterBasedStrokeStrategy class. The latter one creates painters and initializes them using setupPainter().
## KisToolFreehand and KisScratchPad¶
The freehand tool is split into four classes:
KisToolFreehand
highlevel tool class that get the mouse events form the Ko-classes and distributes events among internal classes.
KisToolPaintingInformationBuilder
converts mouse events represented by KoPointerEvent objects into KisPaintInformation objects.
KisRecordingAdapter
stays in charge of adding recording information into the image's action recorder. This class has two purposes: first we need to be able to disable recording for the scratch pad (then we just pass NULL instead of a recording adapter), second when the strokes are able to do their own recording, it'll be easier to port the freehand tool to it.
KisToolFreehandHelper
this is the main class that combines all the classes we were talking above. It accepts a mouse event, converts it using a painting information builder into the paint information, notifies recording adapter, takes the snapshot of resources and finally starts a stroke. Then it populates the stroke with stroke jobs, when the user moves the mouse (paint(event) method) and finishes the stroke in the end.
Such splitting allows us to use the same classes in both KisToolFreehand and KisScratchPad. The only difference between them is that the scratch pad doesn't have a recording adapter at all, and uses base class KisPaintingInformationBuilder instead of KisToolPaintingInformationBuilder. The latter differs from the former one in a way that it supports painting assistants (adjustDocumentPoint() method), complex coordinate transformations with KisCoordinatesConverter (documentToImage() method) and perspective painting (calculatePerspective() method). The rest of the code is shared.
## KisToolMultihand¶
Multihand tool uses the same classes. The only difference, it has a couple of modifications in its helper (KisToolMultihandHelper), those allow it to have several painters at the same time. The tool's class inherits the freehand tool's class and just substitutes the helper with its own (with resetHelper() method).
# Scheduled Undo/Redo¶
## Two ways of working with undo commands¶
The key problem of designing the undo system for strokes was that there are two ways of working with undo commands. That is we have two types of commands actually:
• Qt-like command - command's redo() method is executed while the command is added into the undo stack
• Transaction-like command - the command is added to the stack after its action has already been performed. It means that the first redo() of this command (the one that is called by undo stack) does nothing. That is a transaction-like command just saves undo data for the future and does not perform anything on addition.
You already know that our strokes can be reverted on the go, it means that the stroke's undo command should be added to the undo stack only after all the actions of the stroke have been performed. So it looks like the stroke's commands are transaction-like.
But there is another problem: the stroke should be able to execute regular undo commands those are not transaction-like (like is it done in KisStrokeStrategyUndoCommand). More than that, undo and redo of for such strokes should be performed with the same sequentiality properties (read "undo/redo operations should be threaded as well").
It follows that the undo commands generated by the stroke should be wrapped in a special wrapper command, lets call it KisSavedCommand, that hold the following properties:
• the wrapper skips the first redo(). It means the wrapped command's redo() method will not be called on its addition to the stack. Obviously, it is not needed, because the action has already been performed by the stroke itself.
• when undo stack calls to undo/redo methods of the wrapper-command, the command creates a stroke (KisStrokeStrategyUndoCommandBased) and runs the wrapped command in a context of this stroke.
• a special macro wrapper command, lets call is KisSavedMacroCommand, should be able to save all the commands executed by a stroke and undo/redo all of them in the original order with original sequentiality properties (concurrent, sequential, barrier, exclusive).
That is exactly what we have: KisSavedUndoCommand skips the first redo and runs undo()/redo() of an internal command in a separate stroke. We have KisSavedMacroCommand as well to save the contents of the whole stroke.
Scheduled commands
Well, it would be quite insane to ask all the users of strokes to wrap their commands into wrapper, so we introduced a separate undo adapter for strokes: KisPostExecutionUndoAdapter. This adapter wraps your command and puts it into the undo stack automatically. This is the only adapter we can use inside strokes, that is why all the strokes accept the pointer to it.
For the legacy code we still have KisUndoAdapter, but now we call it "legacy undo adapter". It works as usual: it adds a command to undo stack directly, so it gets executed right in the moment of addition. But there still is one trick. Stroke's commands come to the undo stack asynchronously, so if we try to simply add a command to the stack, we can catch a race condition easily. That's why the legacy undo adapter must guard itself from strokes with locking the strokes system. That is done with a special kind of lock barrierLock(). This barrier lock differs from a regular lock in a way that it ways for all the running strokes are finished, while a regular lock waits for all the running stroke jobs are done. That's the only difference.
The same race conditions problem applies to the undo()/redo() signals from the UI. The user may request the undo operation while the stroke is adding its commands. This will surely lead to a crash. We solved this problem in a bit hacky way: we hacked QUndoStack and made it's undo()/redo() slots virtual. After that we overridden the stack with our own, and changed these methods to block the strokes while undo()/redo() is happening. We use tryBarrierLock() there, because it is easier to cancel the undo than to wait until all the strokes are finished.
## Undo Adapters and Undo Stores¶
Well, we have two types of undo adapters now (not counting KisSurrrogateUndoAdapter). It's obvious that they should share some code. That is why we split the work with the actual undo stack into a separate class KisUndoStore. So now the undo store defines "where to store the undo data", and undo adapter defines "how to adapt krita's commands to qt's stack". There are additional types of store classes for using in tests and for special purposes.
# Processings framework¶
## Motivation¶
In Krita we have many actions which have common structure of execution. Take a look at actions like Scale Image, Rotate Image, Change Color Space - all of them have common phases:
1. Lock the image
2. Do the processing of nodes
3. Unlock the image
4. Emit setDirty() calls and update the projection of the nodes
5. Wait until all the setDirty()'es are finished
6. Emit image's signals like sigImageSizeChanged
More than that, you should pay attention to the fact that all these actions should support undo/redo operations. And the last two phases cannot be implemented as usual qt-commands inside a usual macro, because they should always be executed in the end of the action (in qt commands are executed in reverse order during undo operations, that is not what we want).
And, btw, it would be really good idea to have multithreading support for such actions, because some of them (like Scale Image) may be quite slow.
KisNodeVisitor cannot fit all these requirements, because it has important design limitations: first, walking through nodes is implemented inside the visitor itself and, second, emitting signals is put into visitors as well. These two limitations prevent the code to be shared between actions. That is why we introduced new shiny KisProcessingVisitor and a separate framework for them.
## Processing visitors¶
Processing framework
The key class of the processing framework is KisProcessingVisitor. Its main difference from the old visitor is that it is extremely simple. It performs one task only, it processes one node. And that is all. It does no locking, performs no updates, emits no signals. It just processes (that is, changes the content) a single node. You can look at the reference implementation of it in KisCropProcessingVisitor and KisTransformProcessingVisitor. The key idea of this framework is to keep the processings as simple as possible. So the rest of the work is done by external classes, those are shared between all the processings.
We have one such class. Its name is KisProcessingApplicator. This class performs several tasks:
• creates a stroke. So all the actions executed with this applicator will be undo/redo'able.
• applies a visitor to a requested node.
• applies a visitor recursively to a node and all its children. Note, that you can choose any sequentiality property for the execution of your visitor. It means that the visitors can be applied to nodes concurrently in multithreaded way.
• applies a usual qt-command to the image. Sequentiality properties may vary as well.
• emits setDirty() calls for all the nodes which need it. It is done in efficient way, so no nodes are updated twice.
• emits image signals after all the actions and updates are finished.
Lets look at an example:
void KisImage::resizeImageImpl(const QRect& newRect, bool cropLayers)
{
if(newRect == bounds()) return;
QString actionName = cropLayers ? i18n("Crop Image") : i18n("Resize Image");
(1) KisImageSignalVector emitSignals;
(2) emitSignals << SizeChangedSignal << ModifiedSignal;
(3) KisProcessingApplicator applicator(this, m_d->rootLayer,
KisProcessingApplicator::RECURSIVE,
emitSignals, actionName);
if(cropLayers || !newRect.topLeft().isNull()) {
(4) KisProcessingVisitorSP visitor =
new KisCropProcessingVisitor(newRect, cropLayers, true);
(5) applicator.applyVisitor(visitor, KisStrokeJobData::CONCURRENT);
}
(6) applicator.applyCommand(new KisImageResizeCommand(this, newRect.size()));
(7) applicator.end();
}
In lines (1) and (2) we create a list of signals we should emit after the execution of the applicator. This list should be passed to the constructor of the applicator (3) (the list is passed to the constructor instead of end() function, because we face a limitation connected with the internals of the implementation of undo for processings, I doubt it can create any troubles). In the line (3) we create a recursive applicator. In lines (4) and (5) we create a visitor and apply it to nodes recursively in a multithreaded way. Warning: the visitor is shared between all the threads so it should be written in a thread-safe way. In line (6) we apply a command sequentially, it means that it'll be executed right after all the threads with visitors has finished. Line (7) closes the stroke an tells it to perform all the updates and emit all the signals.
## Implementation of KisProcessingApplicator¶
The applicator is based on the "undo command"-based stroke (KisStrokeStrategyUndoCommandBased). It starts the stroke in the constructor and adds undo commands to it on every user request. The processings are inernally wrapped into a special command (KisProcessingCommand). This command has its own undo stack that collects the transactions executed by the processing. This can be easily achieved with our undo adapters interface. The command just defines its own KisSurrogateUndoAdapter and passes it to the processing. Processing adds its transactions to the fake adapter. And later, the command just uses the undo stack to undo/redo actions executed by the transaction.
The applicator defines several internal commands as well: UpdateCommand and EmitSignalsCommand. These commands are added to the beginning and to the end of every stroke, so that they can be executed in the end of both undo and redo operations. The parameter finalUpdate controls whether the command is executed during its redo() or undo() operation.
## Emission of signals trick¶
After actions have been moved to separate threads, problems with image signals appeared. When everything was executed in a single thread the connection of signals like sigAboutToAddNode and sigNodeHasBeenAdded worked as Qt::DirectConnection. So these signals were effectively function calls. After we moved the actions to a separate thread, all of them became Qt::QueuedConnection. I guess you know what it means. They simply lost all their sense. So we had to start to use Qt::BlockingQueuedConnection. But there is another problem with it. Some of the (old) code is still executed in a context of the UI thread and they emit signals as well. So all that code causes deadlocks when using Qt::BlockingQueuedConnection. That is why we had to introduce KisImageSignalRouter. This class checks which thread emits the signal and emits it either using Qt::DirectConnection or Qt::BlockingQueuedConnection. So no deadlocks are possible.
## Progress reporting¶
The fact that a processing visitor does a really simple task (processes a single node) that is very easy to report progress using progress bars in the layer box. We just need to use progress proxy of the node we process (KisNodeProgressProxy). Our processings framework provides an even easier way of doing this. You just need to instantiate a ProgressHelper object and ask it to create a KoUpdater object for you. And all is done. You can see an example in KisTransformProcessingVisitor class.
## Testing¶
Usage of a common framework makes testing really simple. There is a separate unittest in image's tests folder: KisProcessingsTest. To test a processing you need to write just a couple of lines. Everything is done by BaseProcessingTest helper class. This class will run your processing and compare results against reference png files those are stored in data folder. If there are some problems found, it'll dump result files to the current directory. | 2020-10-30 19:21:03 | {"extraction_info": {"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, "math_score": 0.19178342819213867, "perplexity": 2462.3734727483325}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107911229.96/warc/CC-MAIN-20201030182757-20201030212757-00317.warc.gz"} |
https://davidstutz.de/neural-networks-with-structural-resistance-to-adversarial-attacks-de-alfaro/ | # DAVIDSTUTZ
15thOCTOBER2019
Luca de Alfaro. Neural Networks with Structural Resistance to Adversarial Attacks. CoRR abs/1809.09262 (2018).
De Alfaro proposes a deep radial basis function (RBF) network to obtain robustness against adversarial examples. In contrast to “regular” RBF networks, which usually consist of only one hidden layer containing RBF units, de Alfaro proposes to stack multiple layers with RBF units. Specifically, a Gaussian unit utilizing the $L_\infty$ norm is used:
$\exp\left( - \max_i(u_i(x_i – w_i))^2\right)$
where $u_i$ and $w_i$ are parameters and $x_i$ are the inputs to the unit – so the network inputs or the outputs of the previous hidden layer. This unit can be understood as computing a soft AND operation; therefore, an alternative OR operation
$1 - \exp\left( - \max_i(u_i(x_i – w_i))^2\right)$
is used as well. These two units are used alternatingly in hidden layers in the conducted experiments. Based on these units, de Alfaro argues that the model is less sensitive to adversarial examples, compared to linear operations as commonly used in ReLU networks.
For training a deep RBF-network, pseudo gradients are used for both the maximum operation and the exponential function. This is done for simplifying training; I refer to the paper for details.
In their experiments, on MNIST, a multi-layer perceptron with the proposed RBF units is used. The network consists of 512 AND units, 512 OR units, 512 AND units and finally 10 OR units. Robustness against FGSM and I-FGSM as well as PGD attacks seems to improve. However, the used PGD attack seems to be weaker than usually, it does not manage to reduce adversarial accuracy of a normal networks to near-zero.
Also find this summary on ShortScience.org. | 2023-02-04 08:14:15 | {"extraction_info": {"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, "math_score": 0.79000324010849, "perplexity": 1032.8270305729054}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500095.4/warc/CC-MAIN-20230204075436-20230204105436-00486.warc.gz"} |
https://nigerianscholars.com/tutorials/chemistry-112/viscosity-density-and-intermolecular-forces/ | Chemistry » Organic Molecules » Physical Properties And Structure
# Viscosity and Density
## Viscosity
Viscosity is the resistance to flow of a liquid. Think how easy it is to pour water compared to syrup or honey. The water flows much faster than the syrup or honey.
### Definition: Viscosity
Viscosity is a measure of how much a liquid resists flowing. i.e. The higher the viscosity, the more viscous a substance is.
Pouring water versus pouring syrup.
You can see this if you take a cylinder filled with water and a cylinder filled with glycerol (propane-1,2,3-triol). Drop a small metal ball into each cylinder and note how easy it is for the ball to fall to the bottom (see figure below). In the glycerol the ball falls slowly, while in the water it falls faster.
The higher the viscosity (red) the slower the ball moves through the liquid.
As implied by the definition, substances with stronger intermolecular forces are more viscous than substances with weaker intermolecular forces. The stronger the intermolecular forces the more the substance will resist flowing. The greater the internal friction the more a substance will slow down an object moving through it.
If the glass sheet is tilted to either side during the resistance to flow activity the drops will run into each other and the activity will not work. The drops should reach the finish line in the order of: alcohol (least viscous), water, oil, syrup (most viscous).
## Optional Activity: Resistance to flow
Take a sheet of glass at least $$\text{10}$$ by $$\text{15}$$ $$\text{cm}$$ in size. Using a water-proof marker draw a straight line across the width of the glass about $$\text{2}$$ $$\text{cm}$$ from each end.
Place the glass flat on top of two pencils, then carefully put a drop of water on one end of the line. Leave at least $$\text{2}$$ $$\text{cm}$$ space next to the water drop, and carefully place a drop of alcohol. Repeat this with a drop of oil and a drop of syrup.
Slowly and carefully remove the pencil from the end opposite the drops (make sure you don’t tilt the glass to either side in the process).
• Which drop moves fastest and reaches the end line first?
• Which drop moves slowest?
The fastest moving substance has the least resistance to flow, and therefore has the least viscosity (is the least viscous). The slowest moving substance has the most resistance to flow, and therefore has the most viscosity (is the most viscous).
## Density
### Definition: Density
Density is a measure of the mass per unit of volume.
The solid phase is often the most dense phase (water is one noteworthy exception to this). This can be explained by the strong intermolecular forces found in a solid. These forces pull the molecules together, which results in more molecules in one unit of volume than in the liquid or gas phases. The more molecules in a unit volume the denser, and heavier that volume of the substance will be.
Density can be used to separate different liquids, with the more dense liquid settling to the bottom of the container, while the less dense liquid floats on top. If you throw a leaf into a river or pond the leaf will float. If you instead throw a rock (with the same surface area and volume) into the river or pond the rock will sink. This is due to the different densities of the two substances: rocks are more dense than water while leaves are less dense than water. | 2018-09-23 06:08:21 | {"extraction_info": {"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, "math_score": 0.42763379216194153, "perplexity": 770.7577400984331}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267159160.59/warc/CC-MAIN-20180923055928-20180923080328-00177.warc.gz"} |
https://optimization-online.org/2011/04/3005/ | # Exact Low-rank Matrix Recovery via Nonconvex Mp-Minimization
The low-rank matrix recovery (LMR) arises in many fields such as signal and image processing, statistics, computer vision, system identification and control, and it is NP-hard. It is known that under some restricted isometry property (RIP) conditions we can obtain the exact low-rank matrix solution by solving its convex relaxation, the nuclear norm minimization. In this paper, we consider the nonconvex relaxations by introducing $M_p$-norm (\$0
## Citation
Beijing Jiaotong University, April/2011
## Article
Download
View Exact Low-rank Matrix Recovery via Nonconvex Mp-Minimization | 2022-12-03 19:36:41 | {"extraction_info": {"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, "math_score": 0.9496963620185852, "perplexity": 623.5324279606336}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710936.10/warc/CC-MAIN-20221203175958-20221203205958-00541.warc.gz"} |
https://mathoverflow.net/questions/340215/graphs-weak-in-context-of-cutting-subgraphs | # Graphs “weak” in context of cutting subgraphs
Lately I've been looking into graphs (simple, undirected, finite) that are in some way weak when it comes to connectivity, that is:
Let $$G$$ be a graph of order $$n$$. We'll say that $$G$$ is $$k$$-weak if for every induced connected subgraph $$H$$ of order $$k$$, $$G \setminus H$$ is disconnected.
I wish to find a good (or any) characterization of such graphs. It is fairly easy to show that trees are $$2$$-weak iff. every leaf is a neighbour of a vertex of a degree greater than $$2$$, and the condition is necessary for all graphs, but it isn't sufficient in general. However, I haven't been able to find any publications on the topic, even for $$k=2$$. | 2019-09-19 00:52:25 | {"extraction_info": {"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": 10, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6656827330589294, "perplexity": 130.01457936683178}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514573385.29/warc/CC-MAIN-20190918234431-20190919020431-00043.warc.gz"} |
http://www.dlyj.ac.cn/CN/10.11821/dlyj201704012 | • 研究论文 •
### 青藏高原冬季积雪时空变化特征及其与北极涛动的关系
1. 1. 南京大学地理与海洋科学学院,南京 210023
2. 山东师范大学地理与环境学院,济南 250014
• 收稿日期:2016-11-09 修回日期:2017-02-12 出版日期:2017-04-20 发布日期:2017-05-04
• 作者简介:
作者简介：覃郑婕（1991- ）,女,广西宜州人,硕士,研究方向为积雪遥感与气候变化。E-mail:zhengjie129@sina.com
• 基金资助:
国家自然科学基金项目（41330526）
### Spatio-temporal variability of winter snow cover over the Tibetan Plateau and its relation to Arctic Oscillation
Zhengjie QIN1(), Shugui HOU1(), Yetang WANG2, Hongxi PANG1
1. 1. School of Geographic and Oceanographic Sciences, Nanjing University, Nanjing 210023, China
2. College of Geography and Environment, Shandong Normal University, Jinan 250014, China
• Received:2016-11-09 Revised:2017-02-12 Online:2017-04-20 Published:2017-05-04
Abstract:
The snow cover over the Tibetan Plateau (TP), as a sensitive indicator of climate change, has a significant impact on regional and even global climate. MODIS 8-day snow cover extent products and ERA-Interim reanalysis data were employed to study the spatial and temporal variability of the snow cover over the TP and its relation to Arctic Oscillation (AO) by climatological statistical diagnosis. The spatial distribution of winter snow cover over the TP is far from uniformity, with high snow cover fractions (SCF) at the western edge and the southeast part of the TP but scarce snow in the northern and central parts. It is found that the SCF is out of phase between the eastern and western parts of the TP with respect to the leading mode of empirical orthogonal functions (EOF1), namely, the positive (negative) anomalies in SCF over the eastern part of the TP are associated with negative (positive) anomalies in SCF over the western part. This pattern is positively correlated with AO. During the positive AO phase, the East Asian Trough weakens, together with intensive Southern Branch Trough. The warm moist flows easily lift to the eastern part of the TP because of intensive Subtropical High over the Western Pacific and result in excessive snowfall, while an anomalous anticyclone with its center to the southwest of the plateau leads to sinking dry air flows over the western TP, which is not prone to snowfall, and the corresponding higher surface temperature is also against maintaining the snow cover. During the negative AO phase, the East Asian Trough strengthens and so does the East Asian winter monsoon, with dry cold air flows over the eastern part of the TP, leading to less snowfall. On the other hand, an anomalous cyclone centered to the southwest of the plateau makes it easier for the warm moist flows from the Bay of Bengal and Arabian Sea to lift to the western part of the TP and meet the cold air from Siberia, thus prompting more snowfall over the western part of the TP. | 2020-01-27 10:28:27 | {"extraction_info": {"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, "math_score": 0.20446200668811798, "perplexity": 4356.441571543766}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579251696046.73/warc/CC-MAIN-20200127081933-20200127111933-00488.warc.gz"} |
https://socratic.org/questions/a-triangle-has-sides-a-b-and-c-sides-a-and-b-are-of-lengths-3-and-1-respectively | # A triangle has sides A, B, and C. Sides A and B are of lengths 3 and 1, respectively, and the angle between A and B is pi/12. What is the length of side C?
Jul 14, 2017
$c = 2.0505 u n i t s$
#### Explanation:
Use the law of cosine which is
${c}^{2} = {a}^{2} + {b}^{2} - 2 a b \cos C$
$c = \sqrt{{a}^{2} + {b}^{2} - 2 a b \cos C}$
$c = \sqrt{{3}^{2} + {1}^{2} - 2 \left(3\right) \left(1\right) \cos \left(\frac{\pi}{12}\right)}$
$c = \sqrt{9 + 1 - 6 \cos \left(\frac{\pi}{12}\right)}$
$c = \sqrt{10 - 6 \cos \left(\frac{\pi}{12}\right)}$
$c = 2.0505 u n i t s$ | 2019-01-19 03:53:03 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 7, "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, "math_score": 0.9197960495948792, "perplexity": 299.3547243922429}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547583662124.0/warc/CC-MAIN-20190119034320-20190119060320-00041.warc.gz"} |
https://docs.vespa.ai/documentation/writing-to-vespa.html | # Writing to Vespa
Vespa documents are created according to the Document JSON Format or constructed programmatically - options:
• RESTified Document Operation API: REST API for get, put, remove, update, visit.
• The Vespa HTTP client. A standalone jar which feeds to Vespa either by method calls in Java or from the command line. It provides a simple API while achieving high performance by using multiplexing and multiple parallel async connections. It is recommended in all cases when feeding from a node outside the Vespa cluster.
• The Document API. This provides direct read-and write access to Vespa documents using Vespa's internal communication layer. Use this when accessing documents from Java components in Vespa such as searchers and document processors.
• vespa-feeder is a utility to feed data with high performance. vespa-get gets single documents, vespa-visit gets multiple.
Refer to feed sizing guide for feeding performance.
CRUD operations:
Put Put is used to write a document. A document is a set of name-value pairs referred to as fields. The fields available for a given document is given by the document type, provided by the application's search definition - see field types. A document is overwritten if a document with the same document ID exists and without a test and set condition. Remove Remove removes a document. Later requests to access the document will not find it - read more about remove-entries. If the document to be removed is not found, this is returned in the reply. This is not considered a failure. Like the put and update operations, a test and set condition can be specified for remove operations, only removing the document when the condition is true. Update Update is also referred to as partial update as it updates parts of a document. If the document to update does not exist, the update returns a reply stating that no document was found. A test and set condition can be specified for updates. Example usage is updating only documents with given timestamps. Get Get returns the newest document instance. The get reply includes the last-updated timestamp of the document.
## Feed block
Feed operations fail when a cluster is at disk or memory capacity. Configure resource-limits to tune this - the defaults block feeding before disk or memory is full.
The attribute multivalue mapping and enum store can also go full and block feeding.
To remedy, add nodes to the content cluster or use nodes with higher capacity. The data will auto-redistribute, and feeding is unblocked. These metrics indicate whether feeding is blocked (set to 1 when blocked):
content.proton.resource_usage.feeding_blocked disk or memory attribute enum store or multivalue
When feeding is blocked, events are logged - examples:
Put operation rejected for document 'id:test:test::0': 'diskLimitReached: {
reason: \"disk used (0.85) > disk limit (0.8)\",
capacity: 100000000000,
free: 85000000000,
available: 85000000000,
diskLimit: 0.8
}'
## Batch delete
Options for batch deleting documents:
1. Find documents using search, delete, repeat. Pseudocode:
while True; do
query and read document ids, if empty exit
delete document ids using /document/v1
wait a sec
2. Like 1. but use the Java client. Instead of deleting one-by-one, stream remove operations to the API (write a Java program for this), or append to a JSON file and use the binary:
$java -jar$VESPA_HOME/lib/jars/vespa-http-client-jar-with-dependencies.jar --host document-api-host < deletes.json
3. Use a document selection. This deletes all documents not matching the expression. The content node will iterate over the corpus and delete documents (that are later compacted out):
<documents garbage-collection="true">
<document type="mytype" selection="mytype.version > 4" >
</documents>
## Ordering
The Document API uses the document identifier to implement ordering. Documents with the same identifier will have the same serialize id, and a Document API client will ensure that only one operation with a given serialize id is pending at the same time. This ensures that if a client sends multiple operations for the same document, they will be processed in a defined order.
Note: If sending two put operations to the same document, and the first operation fails, the second operation that was enqueued is sent. If the client chooses to just resend the failed request, the order of operations has been switched.
If different clients have pending operations on the same document, the order is undefined.
## Timestamps
Write operations like put, update and remove, have a timestamp assigned, passing through the distributor. The timestamp is guaranteed to be unique within the bucket where it is stored. The timestamp is used by the content layer to decide which operation is newest. These timestamps may be used when visiting, to only process/retrieve documents within a given timeframe. To guarantee unique timestamps, they are in microseconds, and the microsecond part may be generated or altered to avoid conflicts with other documents.
The internal timestamp is often referred to as the last modified time. This is the time of the last write operation going through the distributor. If documents are migrated from cluster to cluster, the target cluster will have new timestamps for their entries, and when reprocessing documents within a cluster, documents will have new timestamps even if not modified. | 2019-12-09 12:18:29 | {"extraction_info": {"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, "math_score": 0.24630434811115265, "perplexity": 4498.9765756011075}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540518882.71/warc/CC-MAIN-20191209121316-20191209145316-00245.warc.gz"} |
https://search.datacite.org/works/10.6092/ISSN.1973-2201/750 | ### On a method of estimating the expenditure elasticity from concentration curves
L. Dancelli
In 1960 N.S. Iyengar developed a new method of estimating the expenditure elasticity of a specific item from two concentration curves the Lorenz curve of total expenditure (or income) and a generalized Lorenz curve which relates the proportion of consumption of the specific item to the proportion of consumers spending up to a given level of the total expenditure. On the assumptions of log normality of total expenditure distribution and of constancy of elasticity, the... | 2017-12-16 03:39:40 | {"extraction_info": {"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, "math_score": 0.8933008909225464, "perplexity": 1385.6576152752866}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948581053.56/warc/CC-MAIN-20171216030243-20171216052243-00160.warc.gz"} |
http://www.ams.org/mathscinet-getitem?mr=2854559 | MathSciNet bibliographic data MR2854559 46L52 (46B20) Ueda, Yoshimichi On the predual of non-commutative $H\sp \infty$$H\sp \infty$. Bull. Lond. Math. Soc. 43 (2011), no. 5, 886–896. Article
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Privacy Statement | 2016-07-24 13:08:52 | {"extraction_info": {"found_math": true, "script_math_tex": 1, "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, "math_score": 0.9765235781669617, "perplexity": 10861.996354777948}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-30/segments/1469257824037.46/warc/CC-MAIN-20160723071024-00061-ip-10-185-27-174.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/females-are-not-dumb-i-just-need-help.55086/ | # Females are not dumb I just need help
1. Dec 2, 2004
### Astronomer107
Since one male in the class thinks the two females in my IB physics class are dumb (I being one of them), we are determined to do very well on tonight's problems. We don't have a teacher that actually teaches, I don't know how to approach some of the problems. Here is goes (please help! thanks)
There is an electric current in a coil (in a dc circuit), which is connected to a battery with an emf of 4 volts. The steady state value of the current is 2 Amperes. One of the questions asks to prove that the t (the greek t)= L/R given that the rate at which the current changes is V/L (L being inductance). I think I might have gotten that, but it then asks to determine the value of the inductance and I got 1 ohm second because I said that t was equal to the time constant ( or the time it would take to reach the 2 Amperes if it were to continue at the initial rate), which was 0.5. So, using the t= L/R, I got 0.5 = L/2 which was 1. Is that right?
I have more questions about other ones, but I don't want to ask too many on this thread.
2. Dec 2, 2004
### Dayle Record
Ask louder. But let me tell you this. You didn't get into IB on looks. Don't let guys do this to you, ignore guys like that. They have poor social skills, and will not be pleasant to have a relationship with, not anytime soon. So, they are unimportant. What that boy is practicing is called misogyny, a hatred of females. I am sorry he is such a sorry piece of work.
I am also sorry that I am not able to help you with your electrical problems, but find another girl in the class to discuss the problem with, and when you see that boy, use some sort of mental "remote" to just turn him off.
It is not masculine to make hateful generalizations about women, it is just pathetic.
3. Dec 2, 2004
### Integral
Staff Emeritus
Could you post more background for the problem. Do you have any other definitions of your $\tau$?
4. Dec 3, 2004
### jdstokes
Hi,
The steady state value of the current is reached in the limit as $t \rightarrow \infty$, not when $t=\tau$. Hmmm, I think there are too many variables to solve this problem. Could you post the question verbatim?
PS. What a loser, if he's stupid enough to judge someone's intelligence based on their sex, then his opinions clearly aren't worth an ounce of your attention.
Last edited: Dec 3, 2004
5. Dec 4, 2004
### Raza
I am a kind of a person who thinks women should stay at home but this is just stupid. Thinking females are dumb.
6. Dec 5, 2004
### Dayle Record
What you think about females has nothing to do with females, but has everything to do with your ability to relate, and your ability to be a competent human being. Humans vary in every ability, humans from every culture and religion. Lack of proper humanities education, results in crippling bias such as you manifest. The equation is sort of, Women do not absolutely equal Men, yet they are not less than Men, and Men are not greater than Women. This is the same as the problem Man1 does not absolutely equal Man2, but M1 is not less than Man2, and Man2 is not greater than Man1. The variances are so minor in the grand scheme, yet in the scheme of our survival, the nurturance of the unborn and newborn is given to the female, this is not because she is the weaker of the two. | 2017-12-13 15:45:43 | {"extraction_info": {"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, "math_score": 0.521538496017456, "perplexity": 1096.52636954581}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948527279.33/warc/CC-MAIN-20171213143307-20171213163307-00374.warc.gz"} |
http://comunidadwindows.org/standard-error/standard-error-beta-formula.php | Home > Standard Error > Standard Error Beta Formula
# Standard Error Beta Formula
Michael T · 7 years ago 0 Thumbs up 0 Thumbs down Comment Add a comment Submit · just now Report Abuse Add your answer How do I calculate the standard The variations in the data that were previously considered to be inherently unexplainable remain inherently unexplainable if we continue to believe in the model′s assumptions, so the standard error of the The fraction by which the square of the standard error of the regression is less than the sample variance of Y (which is the fractional reduction in unexplained variation compared to Different levels of variability in the residuals for different levels of the explanatory variables suggests possible heteroscedasticity. Check This Out
The deduction above is $\mathbf{wrong}$. In other words, we want to construct the interval estimates. That said, any help would be useful. This plot may identify serial correlations in the residuals. http://stats.stackexchange.com/questions/44838/how-are-the-standard-errors-of-coefficients-calculated-in-a-regression
The accompanying Excel file with simple regression formulas shows how the calculations described above can be done on a spreadsheet, including a comparison with output from RegressIt. That said, any help would be useful. The correlation coefficient is equal to the average product of the standardized values of the two variables: It is intuitively obvious that this statistic will be positive [negative] if X and
Contents 1 Linear model 1.1 Assumptions 1.1.1 Classical linear regression model 1.1.2 Independent and identically distributed (iid) 1.1.3 Time series model 2 Estimation 2.1 Simple regression model 3 Alternative derivations 3.1 Please try the request again. It was assumed from the beginning of this article that this matrix is of full rank, and it was noted that when the rank condition fails, β will not be identifiable. Please upload a file larger than 100x100 pixels We are experiencing some problems, please try again.
Estimation Suppose b is a "candidate" value for the parameter β. The least-squares estimate of the slope coefficient (b1) is equal to the correlation times the ratio of the standard deviation of Y to the standard deviation of X: The ratio of Similarly, the change in the predicted value for j-th observation resulting from omitting that observation from the dataset will be equal to [21] y ^ j ( j ) − y http://www.investopedia.com/ask/answers/070615/what-formula-calculating-beta.asp Step 6: Find the "t" value and the "b" value.
How to Find an Interquartile Range 2. Wooldridge, Jeffrey M. (2013). Classical linear regression model The classical model focuses on the "finite sample" estimation and inference, meaning that the number of observations n is fixed. The accuracy of the estimated mean is measured by the standard error of the mean, whose formula in the mean model is: This is the estimated standard deviation of the
This assumption may be violated in the context of time series data, panel data, cluster samples, hierarchical data, repeated measures data, longitudinal data, and other data with dependencies. For example, having a regression with a constant and another regressor is equivalent to subtracting the means from the dependent variable and the regressor and then running the regression for the s actually represents the standard error of the residuals, not the standard error of the slope. This is called the best linear unbiased estimator (BLUE).
N; Grajales, C. his comment is here That's it! Also when the errors are normal, the OLS estimator is equivalent to the maximum likelihood estimator (MLE), and therefore it is asymptotically efficient in the class of all regular estimators. The resulting value is multiplied by the correlation of the security's returns and the benchmark's returns.
By using this site, you agree to the Terms of Use and Privacy Policy. Thus a seemingly small variation in the data has a real effect on the coefficients but a small effect on the results of the equation. Practical Assessment, Research & Evaluation. 18 (11). ^ Hayashi (2000, page 15) ^ Hayashi (2000, page 18) ^ a b Hayashi (2000, page 19) ^ Hayashi (2000, page 20) ^ Hayashi http://comunidadwindows.org/standard-error/standard-error-of-beta-1-formula.php Note that the inner set of confidence bands widens more in relative terms at the far left and far right than does the outer set of confidence bands.
Is it good to call someone "Nerd"? The second formula coincides with the first in case when XTX is invertible.[25] Large sample properties The least squares estimators are point estimates of the linear regression model parameters β. So, I take it the last formula doesn't hold in the multivariate case? –ako Dec 1 '12 at 18:18 1 No, the very last formula only works for the specific
## In particular, this assumption implies that for any vector-function ƒ, the moment condition E[ƒ(xi)·εi] = 0 will hold.
For more general regression analysis, see regression analysis. You can only upload files of type PNG, JPG, or JPEG. You can only upload a photo (png, jpg, jpeg) or a video (3gp, 3gpp, mp4, mov, avi, mpg, mpeg, rm). Learn how debt affects a company's levered ...
ISBN0-387-95364-7. Is the ability to finish a wizard early a good idea? Read Answer >> Why should I register as a Limited Liability Company (LLC) if I am self-employed ... http://comunidadwindows.org/standard-error/standard-error-of-beta-formula.php You can only upload a photo or a video.
est. See also Bayesian least squares Fama–MacBeth regression Non-linear least squares Numerical methods for linear least squares Nonlinear system identification References ^ Hayashi (2000, page 7) ^ Hayashi (2000, page 187) ^ Harvard University Press. If the errors ε follow a normal distribution, t follows a Student-t distribution.
In light of that, can you provide a proof that it should be $\hat{\mathbf{\beta}} = (\mathbf{X}^{\prime} \mathbf{X})^{-1} \mathbf{X}^{\prime} \mathbf{y} - (\mathbf{X}^{\prime} \mathbf{X})^{-1} \mathbf{X}^{\prime} \mathbf{\epsilon}$ instead? –gung Apr 6 at 3:40 1 This contrasts with the other approaches, which study the asymptotic behavior of OLS, and in which the number of observations is allowed to grow to infinity. price, part 2: fitting a simple model · Beer sales vs. Investing How AQR Places Bets Against Beta Learn how the bet against beta strategy is used by a large hedge fund to profit from a pricing anomaly in the stock market
more stack exchange communities company blog Stack Exchange Inbox Reputation and Badges sign up log in tour help Tour Start here for a quick overview of the site Help Center Detailed Trending Now Kyrie Irving Charlize Theron Megan Mullally Statue of Liberty Kellyanne Conway 2016 Crossovers Real Madrid Auto Insurance Quotes Dating Sites Kendall Jenner Answers Best Answer: Compute the residuals e(i) However, as I will keep saying, the standard error of the regression is the real "bottom line" in your analysis: it measures the variations in the data that are not explained What's the bottom line?
Both matrices P and M are symmetric and idempotent (meaning that P2 = P), and relate to the data matrix X via identities PX = X and MX = 0.[8] Matrix | 2018-02-19 00:06:50 | {"extraction_info": {"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, "math_score": 0.5269661545753479, "perplexity": 1068.7224120804926}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891812293.35/warc/CC-MAIN-20180218232618-20180219012618-00366.warc.gz"} |
http://math.wikia.com/wiki/Hexagonal_prism | # Hexagonal prism
1,011pages on
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In geometry, the hexagonal prism is a prism with hexagonal base.
It is an octahedron. However, the term octahedron is mainly used with "regular" in front or implied, hence not meaning a hexagonal prism; in the general meaning the term octahedron it is not much used because there are different types which have not much in common except having the same number of faces.
If faces are all regular, the hexagonal prism is a semiregular polyhedron. This is the fourth in an infinite set of prisms formed by square sides and two regular polygon caps. The shape has 8 faces, 18 edges, and 12 vertices. It can also be formed by truncating a hexagonal hosohedron.
As in most prisms, the volume is found by taking the area of the base, with a side length of $a$, and multiplying it by the height $h$.
$V = \frac{3 \sqrt{3}}{2}a^2 \times h$
## Other images
A transparent model | 2017-02-26 21:20:28 | {"extraction_info": {"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": 3, "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, "math_score": 0.7228453755378723, "perplexity": 575.1701628426105}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-09/segments/1487501172077.66/warc/CC-MAIN-20170219104612-00258-ip-10-171-10-108.ec2.internal.warc.gz"} |
https://direct.mit.edu/neco/article/31/12/2390/95613/Bayesian-Filtering-with-Multiple-Internal-Models | ## Abstract
To exhibit social intelligence, animals have to recognize whom they are communicating with. One way to make this inference is to select among internal generative models of each conspecific who may be encountered. However, these models also have to be learned via some form of Bayesian belief updating. This induces an interesting problem: When receiving sensory input generated by a particular conspecific, how does an animal know which internal model to update? We consider a theoretical and neurobiologically plausible solution that enables inference and learning of the processes that generate sensory inputs (e.g., listening and understanding) and reproduction of those inputs (e.g., talking or singing), under multiple generative models. This is based on recent advances in theoretical neurobiology—namely, active inference and post hoc (online) Bayesian model selection. In brief, this scheme fits sensory inputs under each generative model. Model parameters are then updated in proportion to the probability that each model could have generated the input (i.e., model evidence). The proposed scheme is demonstrated using a series of (real zebra finch) birdsongs, where each song is generated by several different birds. The scheme is implemented using physiologically plausible models of birdsong production. We show that generalized Bayesian filtering, combined with model selection, leads to successful learning across generative models, each possessing different parameters. These results highlight the utility of having multiple internal models when making inferences in social environments with multiple sources of sensory information.
## 1 Introduction
One of the most notable abilities of biological creatures is their capacity to adapt their behavior to different contexts and environments (i.e., cognitive flexibility) (Mante, Sussillo, Shenoy, & Newsome, 2013; Dajani & Uddin, 2015) through learning. People can learn to call on various responses depending on the situation—for example, independently move the right and left hands when playing an instrument and speak several different languages. Such multitasking abilities are particularly crucial in communication with several people, who each demand subtly different forms of interaction (Taborsky & Oliveira, 2012; Parkinson & Wheatley, 2015). In this kind of situation, one needs to infer who has generated a heard voice—and infer that person's mental state—to respond in an appropriate manner. This is a requirement for exhibiting social intelligence—which usually indicates the ability of organisms to correctly recognize oneself and others—and behave adequately in the social environment with several conspecifics. This is an important challenge in understanding a key aspect of social intelligence. Experimental studies of primates have shown that the volumes of certain brain structures (e.g., the hippocampus) are correlated with the performance of cognitive and social tasks (Reader & Laland, 2002; Shultz & Dunbar, 2010) and that the ability to infer another's intentions increases with brain volume (Devaine et al., 2017). This speaks to a putative strategy for making inferences about several different conspecifics with a plurality of internal models, each associated with a particular of the community or econiche.
This ability of biological creatures contrasts with current notions of artificial general intelligence. The development of a synthetic system as flexible as the biological brain remains a challenge (LeCun, Bengio, & Hinton, 2015; Hassabis, Kumaran, Summerfield, & Botvinick, 2017). Here, we tried to understand how the brain might entertain distinct generative models in a context-sensitive setting. To do this, we focus on a social task, communication through birdsong, in which the conversational partner may change. This induces the dual task of inferring the identity of a conspecific and learning about that conspecific at the same time. Crucially, this learning should be specific to each partner.
To address this problem, we appeal to generalized Bayesian filtering, a corollary of the free-energy principle (Friston, 2008, 2010). We illustrate the behavior of the proposed scheme using artificial birdsongs and natural zebra finch songs. We consider a synthetic (student), whose generative model is based on a physiologically plausible model of birdsong production, and present the student bird with a song generated by one of several (teacher) conspecifics. During the exchange the student bird performs Bayesian model selection (Schwarz, 1978) to decide which teacher generated the heard song. Having accumulated sensory evidence under all hypotheses or models, the parameters of the generative models are updated in proportion to the evidence for each competing model.
We show that over successive interactions, our student is able to learn the individual characteristics of multiple teachers and recognize them with increasing confidence Finally, possible neurobiological implementations of the proposed scheme are discussed. Despite our emphasis on birdsong, our interest (and expertise) is not in the theoretical neurobiology of songbirds. We use birdsong as a vehicle to introduce a computational perspective on perceptual categorization and learning in communication (of any sort) that inherits from Bayesian model selection. We hope the scheme we showcase may be useful in areas like voice recognition and in other domains of social exchange.
### 1.1 Concept of Modeling
In formulating the generative model, we have to contend with a mixture of random variables in continuous time (i.e., latent states of each singing bird) and categorical variables (i.e., the identity of the bird) that constitute a perceptual categorization problem. In short, the listening bird (i.e., student) has to make inferences in terms of beliefs over both continuous and discrete random variables in order to recognize who is singing and what they are singing. In a general setting, this would call on mixed generative models with a mixture of continuous and discrete states of the sort considered in Roweis and Ghahramani (1999) and, more recently, Friston, Parr, and de Vries (2017). A complementary way of combining categorical and continuous latent states is to work within a continuous generative model that includes switching variables that have a discrete (i.e., categorical) probability distribution, with an accompanying conjugate prior such as the Dirichlet distribution. The most common example of this would be a gaussian mixture model (see Roweis & Ghahramani, 1999, for details).
Heuristically, this means the generative model can be constructed in one of two ways. We can select a singing bird to generate a song, leading to a hierarchical model with a categorical latent variable at the top and a continuous model generating outcomes. Alternatively, we could generate continuous outcomes from all possible birds and then select one to constitute the actual stimulus. In the second (switching variable) case, the categorical variable plays the role of a switch, basically switching from one possible sensory “channel” to another.
In terms of model inversion and belief propagation, both generative models are isomorphic and lead to the same update equations via minimization of variational free energy. However, the way in which the generative models play out in terms of requisite message passing can have different forms. We could use a generative model with a single bird and try to infer which bird was singing (and, implicitly, the parameters of its generative process). Alternatively, the student may entertain all possible teachers “in mind” and then select the best hypothesis or explanation for the sensory input. This would correspond to the second form of generative model, in which the dynamics are conditioned on the categorical variable (i.e., a student bird predicts songs under all possible hypotheses) and the best explanation is then selected. In this sense, the expectation about the identity of the singing bird acquires two complementary interpretations. In the first formulation, it is the posterior expectation about the bird that has been selected to generate the song. In the second interpretation, it becomes an expectation about the switching variable. This means the student (i.e., listening bird) effectively composes a Bayesian model average over all hypotheses (i.e., singing birds) entertained in providing posterior predictions of the song.
We can appeal to both forms when interpreting the results that follow. However, the second interpretation has some interesting aspects from a cognitive neuroscience perspective. In essence, the gating or selection of top-down predictions complements the gating or selection of ascending prediction errors usually associated with attention (Luck, Woodman, & Vogel, 2000; Green & Bavelier, 2003; Awh, Belopolsky, & Theeuwes, 2012). In other words, selecting (switching to) the best explanation from available hypotheses, when predicting sensory input, becomes a covert form of (mental) action. Examples of such attentional switching can be found in bistable visual illusions (Eagleman, 2001). This is in the sense that descending predictions are contextualized and selected on the basis of higher-order beliefs (i.e., expectations) about the most plausible hypothesis or context in play. The unique aspect of this gating rests on the fact that there are a discrete (categorical) number of competing hypotheses that are mutually exclusive. This is reminiscent of equivalent architectures in motor control (e.g., the MOSAIC architecture) and related mixture of experts (Roweis & Ghahramani, 1999; Lee, Lewicki, & Sejnowski, 2000; Haruno, Wolpert, & Kawato, 2003). In our case, a simple perceptual categorization paradigm mandates a selection among different possible categories and enforces a form of mental action through optimization of an implicit switching variable.
In what follows, we present the results of perceptual learning and inference using this form of model selection or structure learning, predicated on an ensemble or repertoire of generative models (using synthetic birds and real birdsongs). Using this setup, we show that Bayesian model averaging provides a plausible account of how multiple hypotheses can be combined to predict the sensorium, while Bayesian model selection enables perceptual categorization and selective learning. Crucially, all of these unsupervised processes conform to the same normative principle: the minimization of (the path integral of) a variational free energy bound on model evidence.
## 2 Results
### 2.1 Multiple Generative Models and Attentional Switching
Organisms continuously infer the causes of their sensations (unconscious inference and the Bayesian brain hypothesis: Helmholtz, 1925; Knill & Pouget, 2004) and thereby predict what will happen in the immediate future (e.g., predictive coding) (Rao & Ballard, 1999; Friston, 2005). This sort of perceptual inference rests on an internal generative model that expresses beliefs about how sensory inputs are generated, where perceptual inference is formulated as the minimization of surprise or prediction errors. These models typically assume that sensations are generated by latent or hidden (unobservable) causes in the external world. Such causes may themselves be generated by other causes in a hierarchical manner. In the setting of continuous state-space models, hierarchical Bayesian filtering can be used to perform inference under a hierarchical generative model (Friston, 2008; Friston, Trujillo-Barreto, & Daunizeau, 2008). This filtering uses variational message passing to furnish approximate posterior probability (recognition) densities over the hidden states. In what follows, we describe the process-generating sensory inputs. We assume that the same generative structure is used by the brain as an internal generative model; however the brain needs to learn underlying model parameters to infer the values of hidden states (Dayan, Hinton, Neal, & Zemel, 1995; Friston, Kilner, & Harrison, 2006; George & Hawkins, 2009). A detailed description of the generative models used in this study is provided in section 4.
Let us consider a generative model of birdsong. In brief, this model is a deep generative model with two levels, both levels based on attractor dynamics in the form of neural circuits (see section 4; see also Kiebel, Daunizeau, & Friston, 2008, and Friston & Kiebel, 2009, for details). The goal of an agent (student bird) is to learn about and categorize several different birdsongs, and hence reproduce particular songs depending on the currently heard song. Crucially, the state of a (slow) higher attractor, associated with neuronal dynamics in the high vocal center (HVC) in the songbird brain, provides a control parameter for a (fast) attractor at a lower level in the auditory hierarchy. The hidden or latent states of the lower attractor, associated with the robust nucleus of the archistriatum (RA), then drive fluctuations in the amplitude and frequency of birdsong. (For related songbird studies, see Laje & Mindlin, 2002; Long & Fee, 2008; Amador, Perl, Mindlin, & Margoliash, 2013; and Calabrese & Woolley, 2015.)
In our case, we are interested in multiple models (i.e., multiple teachers), each specified by $mi$ with $i=1,2,…∈M$ (see Figure 1, right). Each $mi$ indicates a specific model structure including certain functional forms and dimensions of latent variables and parameters. These models describe how sensory input (i.e., birdsong) $s$ is generated by a set of latent variables $ui$ that include hidden states $xi$ and hidden causes $vi$. Here, hidden states $xi$ are variables whose dynamics are determined by differential equation (as described below), while hidden causes $vi$ are variables generated at a higher level, with a probability distribution $pvi|mi$. In other words, hidden states are linked via dynamics within a hierarchical level, while hidden causes link successive hierarchical levels. Note that the bracketed superscript ($i)$ indicates they belong to model $i$ (or bird $i)$. These variables are associated with trajectories that are specified in generalized coordinates of motion: $s˜≡(s,s',s'',…)$, where dashes denote time derivatives. The processes that generate birdsong from these latent variables are parameterized by a set of parameters $θi$. We can represent the generation of birdsong under model $i$ by the following stochastic differential equations:
$s˜=g˜(i,1)x˜(i,1),v˜(i,1),θ(i,1)+ω˜(i,1),Dx˜(i,1)=f˜(i,1)x˜(i,1),v˜(i,1),θ(i,1)+z˜(i,1),v˜(i,1)=g˜(i,2)x˜(i,2),v˜(i,2),θ(i,2)+ω˜(i,2),Dx˜(i,2)=f˜(i,2)x˜(i,2),v˜(i,2),θ(i,2)+z˜(i,2),$
(2.1)
In the above, $ω˜(i,j)$ and $z˜(i,j)$ represent random fluctuations. They follow gaussian distributions with mean zero and precision (inverse covariance) of $Πv(i,j)$ and $Πx(i,j)$, respectively. $D$ is a block matrix operator that implements $Dx˜=x˜'≡(x',x'',...)T$. The superscript (ij) indicates the $j$th level of model $i$ (i.e. latent variables at level 2 generate latent variables at level 1 that generate sensory input). The model specified by these equations can be interpreted in terms of a set of probability distributions (see Figure 1 middle left), and their product provides the $i$th generative model (see Figure 1 top).
Figure 1:
This figure illustrates how random (stochastic) differential equations of motion (see equation 2.1) can be interpreted as a probabilistic generative model. This probabilistic model comprises a joint distribution over latent variables and observable sensory input (upper panel) that can be factorized into the marginal distributions shown in the middle left panel. The large lower right panel depicts two generative models in the form of a normal (Forney) factor graph (Friston, Parr et al., 2017; Forney, 2001; Dauwels, 2007). This graphical form shows that the sensory input (birdsong) may be generated by one of two teacher birds, each represented by its own hierarchical generative model. Here, $η$ at the top of the graph indicates the (prior) expectations of hidden causes. A switcher placed in the center determines which bird generates the sensory input as described in the bottom left panel. The bottom left panel shows that the switcher state $γi$ corresponds to the probability of model $i$ being selected, where only $γc$ takes a value of one (i.e., $mc$ is the present model), while the remaining $γi$ with $i≠c$ are zero. Importantly, regardless of the switcher state, all models generate dynamics; for example, $pv˜(i,2)|mi$ indicates the probability of the hidden (generalized motion of) cause $v˜(i,2)$, under the $i$th model structure $mi$, while the selected model is denoted by $mc$. The task of our synthetic (student) bird that hears the sensory input (birdsong) is to infer which (teacher) bird generated the song (i.e., to infer $γi)$. Having done so, the parameters $θi$ associated with bird $i$ are updated in proportion to the evidence that bird $i$ was, in fact, singing. (See also section 4 for details.)
Figure 1:
This figure illustrates how random (stochastic) differential equations of motion (see equation 2.1) can be interpreted as a probabilistic generative model. This probabilistic model comprises a joint distribution over latent variables and observable sensory input (upper panel) that can be factorized into the marginal distributions shown in the middle left panel. The large lower right panel depicts two generative models in the form of a normal (Forney) factor graph (Friston, Parr et al., 2017; Forney, 2001; Dauwels, 2007). This graphical form shows that the sensory input (birdsong) may be generated by one of two teacher birds, each represented by its own hierarchical generative model. Here, $η$ at the top of the graph indicates the (prior) expectations of hidden causes. A switcher placed in the center determines which bird generates the sensory input as described in the bottom left panel. The bottom left panel shows that the switcher state $γi$ corresponds to the probability of model $i$ being selected, where only $γc$ takes a value of one (i.e., $mc$ is the present model), while the remaining $γi$ with $i≠c$ are zero. Importantly, regardless of the switcher state, all models generate dynamics; for example, $pv˜(i,2)|mi$ indicates the probability of the hidden (generalized motion of) cause $v˜(i,2)$, under the $i$th model structure $mi$, while the selected model is denoted by $mc$. The task of our synthetic (student) bird that hears the sensory input (birdsong) is to infer which (teacher) bird generated the song (i.e., to infer $γi)$. Having done so, the parameters $θi$ associated with bird $i$ are updated in proportion to the evidence that bird $i$ was, in fact, singing. (See also section 4 for details.)
In our task design, although every generative model is running simultaneously, only the signal generated by a specific bird is selected as the sensory input (i.e., a teacher song) that the student can actually hear—as an analogy to social communication with several distinct conspecifics. This selection is controlled by a switcher (see also Figure 1 right). Suppose the currently selected model is indexed by $c$. We represent the switcher state by a set of binary variables $Pi=c=γi∈{0,1}$ where only $γc=1$, while the remaining variables are zero to ensure $∑i∈Mγi=1$. Note that $γi$ indicates the probability of model $i$ being selected but takes only either 0 or 1 by design. When this switching process is used, the probability of sensory input is $ps˜|mc=EP(i=c)p(s˜|mi)=∑i∈Mγip(s˜|mi)$, where $p(s˜|mi)$ is the conditional probability of the sensory input when model $i$ is selected (see Figure 1 bottom left). We suppose that the switcher $γ$ is sampled from a categorical prior distribution $Pγ=Cat(Γ)$ for each epoch of singing. In sum all models ($mi)$ are generating fluctuations in hidden states, while only the output from $mc$ is selected as the sensory input.
### 2.2 Update Rules for Inference, Model Selection, and Learning
The inversion of a generative model corresponds to inferring the unknown variables and parameters, which we will treat as perceptual inference and learning respectively. Formally, in variational Bayes, this rests on optimizing an approximate posterior belief over unknown quantities by minimizing the variational free energy (and its path integral) under each model. This comprises three steps, as shown at the top of Figure 2: (1) in the inference step, latent variables under all models are updated over an epoch of birdsong; (2) in the model selection step, a softmax function of variational free action, under each model, gives the model posterior (i.e., model evidence); and (3) in the learning step, this posterior plays the role of an adaptive learning rate when updating model parameters using a descent on variational free action. This ensures that only models that are likely to be generating the birdsong (sensory data) are updated, while the remaining models retain their current parameters. In what follows, we derive the associated update rules to illustrate their general form (section 4 for details).
The latent variables and parameters under model $i$ are given by $u(i)≡x˜i,1,x˜i,2,v˜i,1,v˜i,2$ and $θi≡θi,1,θi,2$, respectively. The internal energy $Uis˜,ui,θi≡-logps˜,ui|θi,mi$ quantifies the amount of (squared) prediction error induced by sensory data for a given generative model $i$, that is, the likelihood of $(s˜,ui)$ when $θi$ and $mi$ are given. Using this, the conditional free energy of model $i$ is given by
$Fit≡EquiqθiUis˜,ui,θi+logqui≈Uis˜,ui,θi+const.$
(2.2)
Here, $qu(i)$ and $qθ(i)$ are approximate posterior (i.e., recognition) densities over the latent variables and parameters of each model. The expression $Eq(u(i))q(θ(i))·$ denotes the expectation over these posterior beliefs, and bold symbols $ui$ and $θi$ denote their posterior expectations (i.e., the means of $qu(i)$ and $qθ(i))$, respectively. Thus, $ui$ and $θi$ are maximum a posteriori estimates of the latent variables and parameters under model $i$. If we ignore the second-order derivative of $Ui$, we can express $Fi$ by simply substituting $ui$ and $θi$ into $Ui$ up to a constant term. In neurobiological process theories, $ui$ and $θi$ are usually associated with neural activities and synaptic strengths, respectively (Bastos et al., 2012; Friston, FitzGerald, Rigoli, Schwartenbeck, & Pezzulo, 2017).
Inference optimizes the approximate posterior beliefs (expectations) about the latent variables. This can be expressed as a gradient flow in generalized coordinates of motion (noting that the solution satisfies the variational principle of least action):
$Inference:u˙(i)-Dui∝-∂∂uiFi(t).$
(2.3)
Special cases of this Bayesian filtering reduce to Kalman filtering. The implicit optimization of $ui$ allows for inference to take place under every model. In addition, our agent needs to infer which model is currently generating its sensory input. This involves minimization of the free action (denoted by a bar) over models, given by
$F¯≡EQi=cF¯iθ(i)+DQγ||Pγ+∑i∈MDqθ(i)||pθ(i)|mi.$
(2.4)
Note that the first term is the weighted sum of the path integral of the conditional free energies, where $F¯i(θ(i))≡∫0TFitdt$ is the conditional free action of model $i$. In this expression, $Qi=c=γi∈[0,1]$ with $∑i∈Mγi=1$ denotes the posterior expectation about model $i$ being selected, which is equivalent to the posterior belief about the switcher state $Qγ=Catγ$. The second and third terms are complexity terms relating to the switcher and parameters, expressed by Kullback-Leibler divergence (Kullback & Leibler, 1951). When the prior distribution of the switcher state $Pγ$ is the same for each model (i.e., all birds are equally likely), we obtain the posterior expectation of the switcher state $γi$ that minimizes the total free action as
$Modelselection:γi=σ-F¯iθ(i).$
(2.5)
This means that the posterior expectation (i.e., evidence) that model $i$ generated the song (denoted by $γi)$ can be computed by taking a softmax $σ·$ (normalized exponential) of the conditional free actions for each model, analogous to a post hoc Bayesian model selection (Friston & Penny, 2011) and a discrete categorical model (Friston, FitzGerald et al., 2017). We also refer to $γi$ as the model plausibility since this quantifies how likely model $i$ is to have generated the current sensory input.
Finally, learning entails updating posterior expectations about the parameters $θi$ to minimize the total free action. Taking the gradient of the total free action with respect to the parameters furnishes the learning update rule. When the prior density of parameters $pθi|mi$ is flat for every model (i.e., no prior knowledge about parameters), this optimization is given by the minimization of the conditional free action weighed by $γi$:
$Learning:θ˙(i)∝-γi∂∂θiF¯iθi.$
(2.6)
The novel aspect of this update rule is the weighting of its learning rate by the model evidence or plausibility. This means that only plausible models will change their parameters, which enables the learning of several different generative models in a (soft) winner-takes-all manner. (Detailed derivations of the above equations are in section 4.)
The posterior distribution of the switch $Qγ$ can be considered an attentional filter (Luck et al., 2000; Green & Bavelier, 2003; Awh et al., 2012). According to this view, an attended generative model and its associated posterior beliefs correspond to the marginal distributions over models and posteriors (i.e., because the attended model is more plausible than all others, the posteriors conditioned on this model will approximate those obtained through a Bayesian model average over all models). Let $u!$ and $θ!$ be the marginal beliefs over latent variables and parameters, respectively. These may be thought of as Bayesian model averages over each of the internal models. When each model has the same structure and dimensions, these marginals are given by $ps˜,u!,θ!≡∑i∈Mγips˜,ui=u!,θi=θ!|mi$, $qu!≡∑i∈Mγiqui=u!$, and $qθ!≡∑i∈Mγiqθi=θ!$. Thus, our model is formally analogous to a gaussian-mixture-model version of a Bayesian filter. On a more anthropomorphic note, the marginal beliefs over latent variables $u!$ and parameters $θ!$ are fictive (i.e., they do not exist in the external real world). One could imagine that they underwrite some conscious inference, with several competing generative models (i.e., hypotheses) running at a subpersonal or unconscious level in the brain.
Interestingly the above formulation can be applied to generative models that have different structures and dimensions because there is no direct interaction between generative models and the switcher receives only the output from each generative model. This property may be particularly pertinent for recognizing conspecifics, since conspecifics may not be best modeled using the same generative model structure.
### 2.3 Demonstrations of Multiple Internal Models Using Artificial and Natural Birdsongs
A birdsong has a hierarchical structure that enables the expression of complicated narratives using a finite set of notes (Suzuki, Wheatcroft, & Griesser, 2016). Young songbirds are known to learn such a song by mimicking adult birds' song (Tchernichovski, Mitra, Lints, & Nottebohm, 2001; Woolley, 2012; Lipkind et al., 2013; Yanagihara & Yazaki-Sugiyama, 2016; Lipkind et al., 2017). Previous studies have developed a songbird model that infers the dynamics of another's song based on a deep (two-layer) generative model (Kiebel et al., 2008; Friston & Kiebel, 2009). Perceptual inference requires an internal model of how the song was generated. However, in a social situation, several birds may produce different songs generated by different brain states (or generative models). In the simulations that follow, we consider a case where two birds (denoted by teacher 1, 2) sang two different songs in turn, as illustrated in Figure 2 (left). A song $s=s1,s2T$ is given by a 4 s sequence of a two-dimensional vector, where $s2$ and $s1$ represent the mode of sound frequency and its power, respectively—analogous to a physiological model of birdsong vocalizations (Laje, Gardner, & Mindlin, 2002; Perl, Arneodo, Amador, Goller, & Mindlin, 2011). Here, we supposed that the generative model had two layers of three-neuron circuits (or circuits comprising three neural populations) for birdsong generation, the so-called Laje-Mindlin style model (Laje & Mindlin, 2002). In preliminary simulations, we confirmed that when a student with a single generative model heard their songs, it was unable to learn either teacher 1 or 2's song (see Figure 6 in appendix A). This is because a single generative model cannot generate two songs. Thus, the student tried to learn a spurious intermediate model of the two songs and failed to learn either.
Figure 2:
Schematics illustrating the variational update scheme (top): the models that our synthetic student (right) uses to make inferences about the songs generated by two teachers (left). A flowchart at the top summarizes the inference, model selection, and learning processes that the student must implement. Here, $σ·$ is a softmax (normalized exponential) function that converts the conditional free actions to a model plausibility $γi$, and $ξi,sξi,jξa$ are error-encoding units that encode between actual and expected error in sensation, hidden states, and action, respectively (see Friston, 2008, for details). Our learning process is weighted by the model plausibility, ensuring that the model most likely to have generated the heard song updates its parameters during learning. See section 4 for further details.
Figure 2:
Schematics illustrating the variational update scheme (top): the models that our synthetic student (right) uses to make inferences about the songs generated by two teachers (left). A flowchart at the top summarizes the inference, model selection, and learning processes that the student must implement. Here, $σ·$ is a softmax (normalized exponential) function that converts the conditional free actions to a model plausibility $γi$, and $ξi,sξi,jξa$ are error-encoding units that encode between actual and expected error in sensation, hidden states, and action, respectively (see Friston, 2008, for details). Our learning process is weighted by the model plausibility, ensuring that the model most likely to have generated the heard song updates its parameters during learning. See section 4 for further details.
This limitation can be overcome using a repertoire of generative models (see section 4 for details). We found that a student with two generative models ($m1$, $m2$; see Figure 2, right) can solve this unsupervised learning problem efficiently. We trained the model by providing two (alternating) teacher songs by updating an unknown parameter of both generative models. Posterior densities over parameters (i.e., synaptic strengths) were updated over learning and successfully converged to the true values used in the simulation (see Figure 3A). As a result, the student was able to make perceptual inferences about latent states generating both songs (see Figure 3B). In each session of training, the free action (i.e., average free energy) was computed for both internal models. The trajectories of free action evince a process of specialization, where each model becomes an expert for one of two songs (see Figure 3C). At the beginning of each exposure, the probability of each model was around 0.5, which led to parameter updates in both models (see Figure 3D). Following learning, the difference in model plausibility became significantly larger—and only the most likely model updated its parameter following the appropriate song. In Figure 3, the hidden states of teachers were reset at the beginning of each session, which made the song sequence periodic and easy to learn. When the hidden states of teachers were not reset, the song sequence became chaotic and was more difficult to learn. However, even in this chaotic case, our model successfully learned from two distinct teachers (see Figure 7 in appendix A).
Figure 3:
Simulation results when learning two birdsongs using multiple generative models. Teacher bird 1 generated a song in odd sessions, and teacher bird 2 sang in even sessions. At the beginning of each session, the initial latent variables of both teachers were reset to their initial values to ensure they generated quasi-periodic dynamics. The parameters of both teachers were fixed over sessions. A student bird was equipped with two generative models ($m1$, $m2)$. (A) Trajectories of the posterior of a parameter that was optimized. The parameters for $m1$ and $m2$ (red and blue curves, respectively) were initialized from the middle point ($≈$ 0.5) and updated according to the variational scheme in the main text. After training, $m1$ and $m2$'s parameter approximated the true parameter value of teacher 1 ($=$ 0; red dashed line) and 2 ($=$ 1; blue), respectively, reflecting veridical learning. Shaded areas indicate the standard deviation of the posterior density. (B) Comparisons between true (teacher) hidden states and their posterior expectations inferred by the student (left: teacher 1 versus $m1$, right: teacher 2 versus $m2)$. (C) Trajectories of conditional free actions for $m1$ (red) and $m2$ (blue). When teacher 1 sang (odd sessions), $F¯1$ was lower than $F¯2$, and vice versa. A free action difference of about three corresponds to strong evidence for the presence of a song—that is, a log odds ratio of 3 ($=$ 20 to 1). (D) Trajectories of model plausibility used for parameter updates. Simulation results with different experimental setup are provided in appendix A.
Figure 3:
Simulation results when learning two birdsongs using multiple generative models. Teacher bird 1 generated a song in odd sessions, and teacher bird 2 sang in even sessions. At the beginning of each session, the initial latent variables of both teachers were reset to their initial values to ensure they generated quasi-periodic dynamics. The parameters of both teachers were fixed over sessions. A student bird was equipped with two generative models ($m1$, $m2)$. (A) Trajectories of the posterior of a parameter that was optimized. The parameters for $m1$ and $m2$ (red and blue curves, respectively) were initialized from the middle point ($≈$ 0.5) and updated according to the variational scheme in the main text. After training, $m1$ and $m2$'s parameter approximated the true parameter value of teacher 1 ($=$ 0; red dashed line) and 2 ($=$ 1; blue), respectively, reflecting veridical learning. Shaded areas indicate the standard deviation of the posterior density. (B) Comparisons between true (teacher) hidden states and their posterior expectations inferred by the student (left: teacher 1 versus $m1$, right: teacher 2 versus $m2)$. (C) Trajectories of conditional free actions for $m1$ (red) and $m2$ (blue). When teacher 1 sang (odd sessions), $F¯1$ was lower than $F¯2$, and vice versa. A free action difference of about three corresponds to strong evidence for the presence of a song—that is, a log odds ratio of 3 ($=$ 20 to 1). (D) Trajectories of model plausibility used for parameter updates. Simulation results with different experimental setup are provided in appendix A.
We next provided six distinct natural zebra finch songs to our model to see if it could learn and recognize six different teachers (see Figure 4; see also section 4 for details). Here we assumed a generative model with realistic song generation capacity comprising two layers of four-neuron circuits (or circuits comprising four neural populations), based on the Laje-Mindlin-style model (see section 4). The posteriors of the parameters of the student's internal models were randomized. However, for simplicity, the posteriors over hidden states at time $t=0$, and the time constants of the differential equations, were optimized a priori (i.e., initialized) to be consistent with one of the six teacher songs. Before training, we tested the responses of the student to the teacher songs as a reference (movie 1; see appendix B).
Figure 4:
A demonstration of learning six natural zebra finch songs using the multiple generative models. The dynamics of teacher and student states before, during, and after training are provided in appendix B. This figure shows a snapshot of a movie after training. (A) A teacher song (right) and underlying dynamics of hidden states and causes (left and middle; arbitrary scale; they were estimated from the sensory data). Six real zebra finch songs were processed and used as teacher songs (illustrated in six colors). A song is given by a 10s sequence of $s$, illustrated by the mode of sound frequency and its amplitude (right; arbitrary scale; amplitude is plotted in both positive and negative sides). The currently selected song is indexed by $c$. (B) Six internal generative models in a student listening to a teacher, making inferences about latent states (centre), model plausibility (right), and the predicted trajectories of sensory input $gi,1$ (left). These constitute the expected song sequences (output) under each model. The color of trajectories indicates the song for which each model is specialized. (C) To evaluate prediction capability, a student generates song (action) in the absence of sensory input. Action is given by the average of predictions of the six models weighted by model plausibility, the Bayesian model average. (D) Posterior expectations of the switcher state (or model plausibility) were updated in each model selection step (see equation 2.5). They were initialized from a uniform distribution and converged to a definitive identity matrix, suggesting that each model became specialized for a specific teacher song. (E) Posterior expectations of the parameters of six internal models (circles) and optimal parameters for six teacher songs (plus marks) plotted in a subspace of the first and second principal components (PC1, PC2) of parameter space. During learning, only internal models with high model plausibility enjoy parameter updates (see equation 2.6). The initial parameter values were adjusted in the absence of model selection to ensure their initial values generated an averaged song (see section 4 for details).
Figure 4:
A demonstration of learning six natural zebra finch songs using the multiple generative models. The dynamics of teacher and student states before, during, and after training are provided in appendix B. This figure shows a snapshot of a movie after training. (A) A teacher song (right) and underlying dynamics of hidden states and causes (left and middle; arbitrary scale; they were estimated from the sensory data). Six real zebra finch songs were processed and used as teacher songs (illustrated in six colors). A song is given by a 10s sequence of $s$, illustrated by the mode of sound frequency and its amplitude (right; arbitrary scale; amplitude is plotted in both positive and negative sides). The currently selected song is indexed by $c$. (B) Six internal generative models in a student listening to a teacher, making inferences about latent states (centre), model plausibility (right), and the predicted trajectories of sensory input $gi,1$ (left). These constitute the expected song sequences (output) under each model. The color of trajectories indicates the song for which each model is specialized. (C) To evaluate prediction capability, a student generates song (action) in the absence of sensory input. Action is given by the average of predictions of the six models weighted by model plausibility, the Bayesian model average. (D) Posterior expectations of the switcher state (or model plausibility) were updated in each model selection step (see equation 2.5). They were initialized from a uniform distribution and converged to a definitive identity matrix, suggesting that each model became specialized for a specific teacher song. (E) Posterior expectations of the parameters of six internal models (circles) and optimal parameters for six teacher songs (plus marks) plotted in a subspace of the first and second principal components (PC1, PC2) of parameter space. During learning, only internal models with high model plausibility enjoy parameter updates (see equation 2.6). The initial parameter values were adjusted in the absence of model selection to ensure their initial values generated an averaged song (see section 4 for details).
A student bird with six internal models inferred latent states (with a small update rate) and calculated the accompanying free energy and model evidence (see Figures 4A and 4B). After exposure, the student generated a song to predict (or imitate) the current teacher song by running the generative models in a forward or active mode. In this mode, the bird reproduces its predicted sensory input based on a Bayesian model average of the dynamics generating a particular song. This Bayesian model average is the mixture of model-specific predictions weighted by model evidence or plausibility. However, prior to learning, the student could not reproduce the teacher song because it has not yet learned the teacher's parameters and could not categorize the teachers. During training, we randomly provided one of the six teacher songs for 60 sessions (movie 2; see appendix B). The student listened to the song and evaluated model plausibility for each of its six internal models. It then learned (924-dimensional) unknown model parameters, with a learning rate determined by model plausibility, to ensure only plausible models were updated. These parameters controlled a nonlinear (polynomial) mapping from latent states expressing the dynamics of the deep generative models to fluctuations in amplitude and peak frequency of the sensory input.
We found that the student's internal models became progressively specialized for one of six teacher songs (movie 3; see appendix B). After learning, only the most plausible model (with veridical parameters) contributed to the Bayesian model average, so that the student could reproduce the teacher songs in a remarkably accurate way (see Figure 4C). These results are particularly pleasing because they also suggest that real songbirds (zebra finches) learn and generate songs (in their RA and HVC) using dynamics with the form we have assumed. Indeed, to compare inferred and true hidden states (and parameters), the real zebra finch songs were learned separately and regenerated under the appropriate model to provide stimuli. Learning success was further confirmed by a specialization of each model for a specific teacher (see Figure 4D) and a convergence of posterior parameter expectations, under each model, to the teacher-specific values (see Figure 4E). These results suggest that the proposed scheme works robustly, even with natural data and a large number of songs.
Finally, we illustrate how inference is affected by either the absence of attentional switching or by a discrepancy between the number of internal models and teacher songs presented (see Figure 5). A standard Bayesian filter, lacking attentional switching, failed to find optimal internal models—to track six teacher songs separately—even when equipped with six internal models. In Figures 5A and 5B, this is evidenced by the absence of free-energy reduction (black line). Conversely, the current scheme, with attentional switching, was able to reduce free energy (red line). This is due to the suppression of the learning rate in implausible models during model inversion. The resulting difference is especially evident when the generative model comprises four-neuron circuits. The mixture model learned six different songs with a high degree of accuracy, as shown in Figure 4, thereby reducing the free energy substantially. When there were equal numbers of internal models and teachers, only one of six internal models was plausible for each session (see Figure 5C). When the number of internal models was greater than that of teachers, several internal models came to represent a teacher song, while the superfluous models were never considered plausible (see Figure 5D), indicating the continued success of the agent in categorizing and learning multiple songs. However, the confidence in these categorizations diminished relative to the correct model. Conversely, when the number of internal models was fewer than that of teachers, each internal model came to represent a mixture of teacher songs, thereby failing to recognize distinct teacher songs (see Figure 5E). Collectively, these findings highlight the potential utility of equipping an adequate number of generative models with attentional selectivity for learning (and inverting) context-sensitive models of the social world.
Figure 5:
Comparison of learning with multiple internal models in the presence or absence of attentional switching (A, B) and when the number of internal models matches or differs from the teacher songs (C, D, E). (A) This figure illustrates the mean trajectories (lines) of averaged free energy, or free action, in each session and their standard errors (shaded areas). In all cases, a student bird has six internal models. Simulations were conducted 20 times, using different random initial states and song orders. The generative model had two layers of three-neuron circuits for generating birdsong. (B) The generative model had two layers of four-neuron circuits. (C) Trajectories of model plausibility for six internal models. The agent heard one of six teacher songs at random. (D) The same information as panel is displayed, but in the case where the agent received one of three teacher songs. (E) Trajectories of model plausibility for three internal models. In this case, the agent had only three internal models, while hearing one of six teacher songs.
Figure 5:
Comparison of learning with multiple internal models in the presence or absence of attentional switching (A, B) and when the number of internal models matches or differs from the teacher songs (C, D, E). (A) This figure illustrates the mean trajectories (lines) of averaged free energy, or free action, in each session and their standard errors (shaded areas). In all cases, a student bird has six internal models. Simulations were conducted 20 times, using different random initial states and song orders. The generative model had two layers of three-neuron circuits for generating birdsong. (B) The generative model had two layers of four-neuron circuits. (C) Trajectories of model plausibility for six internal models. The agent heard one of six teacher songs at random. (D) The same information as panel is displayed, but in the case where the agent received one of three teacher songs. (E) Trajectories of model plausibility for three internal models. In this case, the agent had only three internal models, while hearing one of six teacher songs.
## 3 Discussion
The brain may use multiple generative models and select the most plausible explanation for any given context. Findings from comparative neuroanatomy (Reader & Laland, 2002; Shultz & Dunbar, 2010; Devaine et al., 2017) suggest that as the brain becomes larger, it can entertain more hypotheses, or internal models, about how its sensations were caused. This strategy can be used to learn and recognize particular conspecifics in a communication or social setting. A key question here is how the brain separately establishes distinct generative models before it recognizes which model is fit for purpose, and vice versa. In this study, we introduce a novel learning scheme for updating the parameters of the multiple internal models that are themselves being used to filter continuous data. First, several alternative generative models run in parallel to explain the sensory input in terms of inferred latent variables; this enables the free action (under each model) and associated model plausibility to be evaluated; finally, the parameters of each model are updated with a learning rate that is proportional to the model plausibility. This ensures that only models with high model plausibility or evidence are informed by sensory experience. The proposed scheme allows an agent to establish and maintain several different generative models (or hypotheses) and to perform an adaptive online Bayesian model selection (i.e., switching) of generative models depending on the provided input.
The definition of social intelligence varies greatly. The term could be applied broadly to species able to engage in coordinated behaviors with other conspecifics (e.g., swarm behaviors or shoals of fish; Mann & Garnett, 2015). For an account of the sort of inferences required for a creature to “know its place” in this sort of society, Friston, Levin, Sengupta, and Pezzula (2015) illustrate how this can be achieved in the absence of inferences about other individuals in an ensemble. The sort of intelligence we are interested in here is of a more sophisticated sort: the capacity to recognize oneself and others. We are interested in creatures that interact with their conspecifics at an individual level and can tailor their behavior to whomever they interact with. This requires not just (a minimal) theory of mind but a theory of multiple minds and is closer to the sorts of social intelligence thought to be impaired in conditions like autism (Happé & Frith, 1995).
The ideas presented here address a key challenge for social systems: that of disambiguating between and learning about other conspecifics or members of a society. Our hope is that this takes us a step closer to a formal theory of social intelligence. A complete formal theory would entail computational approaches to solving other aspects of social behavior, including those addressing behavioral economic and trust games (Moutoussis, Trujillo-Barreto, El-Deredy, Dolan, & Friston, 2014), and approaches to understanding the optimal depth of recursive sophistication for social interactions (Devaine, Hollard, & Daunizeau, 2014).
The Bayesian filtering or (sensory) evidence accumulation simulated in this letter offers proof of concept that biologically plausible schemes can be used to recognize the source of dynamically rich sensory streams. The simulations show how neuronal-like message passing can solve two key problems: (1) abstracting or deconvolving a time-invariant representation of how fluctuating sensations are generated and (2) disambiguating among alternative sources. The particular message passing used in this study and in a number of previous publications (Friston, Adams, Perrinet, & Breakspear, 2012; Friston & Frith, 2015b; Friston & Herreros, 2016) can be regarded as a generalization of predictive coding that has growing empirical support as a scheme that the brain might use (Kok, Rahnev, Jehee, Lau, & de Lange, 2012; Brodski-Guerniero et al., 2017; Heilbron & Chait, 2017). A review of the evidence for the basic architecture and ideas can be found in several papers (Bastos et al., 2012; Adams, Shipp, & Friston, 2013; Shipp, 2016). A more technical treatment based on message passing on factor graphs can be found in other publications (Friston, Parr et al., 2017). This letter pursues the biological plausibility of belief updating and, in particular, shows the formal similarities between neuronal message passing required under generative models of both discrete and continuous state spaces.
One might ask if there are alternative schemes that could perform equally well—for example, classification schemes from machine learning (LeCun et al., 2015). This is a potentially important question that would speak to different computational architectures and neurophysiological implementation. However, current machine learning approaches would probably converge on the Bayesian filtering scheme under the deep temporal models used above. This follows from the fact that high-end machine learning schemes use exactly the same (variational free energy) objective function used in Bayesian filtering (and generalized predictive coding). We have in mind here variational autoencoders based on a deep bottleneck architecture, for example (Suh, Chae, Kang, & Choi, 2016). Our model is a mixture model for generalized Bayesian filtering. In this sense, our model can be viewed as a time-domain extension of an autoencoder mixture model (Aljundi, Chakravarty, & Tuytelaars, 2017). A theoretical comparison also supports a close link between learning mechanisms in predictive coding and backpropagation (Whittington & Bogacz, 2017). At the present time, most variational autoencoders do not deal with time-varying data. The implication is that extending current deep learning and variational inference in machine learning to solve the inference problem in a dynamic setting will produce the same scheme as the one used in our simulations.
From a technical perspective, one of the key contributions to the literature of this study is the evaluation of the evidence for competing hypotheses about the sources of sensory input. This evaluation can be cast in terms of Bayesian model selection. From a psychological perspective, this sheds light on our capacity for perceptual categorization, where the underlying selective processes may be cast in terms of attentional selection (Deubel & Schneider, 1996; Itti & Koch, 2001; Bosman et al., 2012). From the perspective of optimal control theory and machine learning, this has clear homologues with selection from mixtures of experts in motor control (Tani & Nolfi, 1999; Haruno et al., 2003) and, indeed, any selective process that involves mutual or lateral inhibition leading to a winner-takes-alll-ike behavior (Zelinsky & Bisley, 2015).
Recent machine learning studies show that task-specific synaptic consolidation can protect the network from forgetting previously learned associations while learning new associations (Kirkpatrick et al., 2017; Zenke, Poole, & Ganguli, 2017). We have focused on disambiguating between, and learning with, multiple internal models. Therefore, we do not consider the explicit protection of previously learned associations. However, a combination of attentional switching and synaptic consolidation would be a potentially interesting extension. We would like to address this issue in the future work.
When an agent encounters an environment that generates data in several possible ways, it can model the environment as either a single generative model (with distinct contextual levels) or multiple generative models. In contrast, when an agent encounters an environment with multiple conspecifics, explaining data with a single model is not straightforward because there are multiple sources of sensory data, calling for a mixture of generative models. In this sense, our model is particularly useful when an agent encounters several different agents in a social context.
Neurobiologically, our learning update rule might be implemented by associative (Hebbian) plasticity modulated by a third factor, a concept that has recently received attention (Pawlak, Wickens, Kirkwood, & Kerr, 2010; Frémaux & Gerstner, 2016; Kuśmierz, Isomura, & Toyoizumi, 2017). While Hebbian plasticity occurs depending on the spike timings of pre- and postsynaptic neurons (Hebb, 1949; Bliss & Lømo, 1973; Markram, Lübke, Frotscher, Sakmann, 1997; Bi & Poo, 1998; Froemke & Dan, 2002; Malenka & Bear, 2004; Feldman, 2012), recent studies have reported that various neuromodulators (Reynolds, Hyland, & Wickens, 2001; Seol et al., 2007; Zhang, Lau, & Bi, 2009; Salgado, Köhr, & Treviño, 2012; Yagishita et al., 2014; Johansen et al., 2014), GABAergic inputs (Paille et al., 2013; Hayama et al., 2013), and glial factors (Ben Achour & Pascual, 2010) can modulate Hebbian plasticity in various ways. Our learning update rule consists of the product of the (conditional) free action gradient providing a Hebbian-like term (see Friston, 2008 for details) and the posterior belief of the switcher state, in which the latter might be implemented by such additional neurobiological factors.
Previous studies have modeled communication between agents in analogy with the mirror neuron system (Kilner, Friston, & Frith, 2007; Friston, Mattout, & Kilner, 2011; Friston & Frith, 2015a, 2015b). These simulations involve two birds that make inferences about each other, converging onto the same internal state and generating the same song. This has been used as a model of hermeneutics, cast in terms of generalized synchrony. Heuristically, both birds come to sing from the same “hymn sheet” and thereby come to “know each other” through knowing themselves. Such a synchronous exchange minimizes the joint free energy of both birds because both birds become mutually predictable. This might be related to an experimental observation that a birdsong can propagate emotional information to another bird and have influence on its behavior (Schwing, Nelson, Wein, & Parsons, 2017). This setup can be generalized when more than two birds are singing the same song. However, when several conspecifics generate different songs (or speak different languages), an agent with a single generative model is no longer fit for purpose. We addressed this limitation by equipping synthetic birds with alternative attractors or hypotheses. Interestingly, it has been reported that some birdsongs display such a learning flexibility—for example, white-crowned sparrows learn multiple songs during the vocal development stage and later switch between learned songs (Hough, Nelson, & Volman, 2000). In our case, generative models are explicitly decomposed and their learning rates are tuned by an attentional switch, allowing an agent to optimize a specific model for each context. In other words, inference about a particular correspondent's “state of mind” can be modeled by synchronous dynamics during conversation, where one of the listener's internal models converges to an attractor representing the speaker's. In this view, empathic capacity may be quantified by how many attractors the agent can deploy and how well it can optimize each attractor. In future work, it will be interesting to consider the relationship between this conceptual model and recent experimental work that suggests that the human brain uses dissociable activity patterns to separately represent self and other (Ereira, Dolan, & Kurth-Nelson, 2018).
In terms of relating the dynamics of learning and inference to empirical observations, the ability to simulate learning and inference, implicit in our simulations, raises the possibility of using empirical data to constrain the scheme's parameters. In other words, there is, in principle, an opportunity to use the simulations of the birdsong recognition above as an observation model to explain the empirical time course of perceptual categorization, learning, and their neuronal correlates. For example, the time course of learning in Figures 3 to 5 suggests that a unit of time (the number of sessions) would correspond to a range from a few hours to a day, given the results reported in Tchernichovski et al. (2001). In our simulation, the first few sessions exhibited a small free-energy reduction because model plausibility was almost uniform and the associated learning rates were similarly small. After a difference in model evidence emerged, the rate of free-energy reduction reached a peak and then gradually returned to zero. This might correspond to the learning process of songbirds that learn the prototypes of songs early and later learn the details of respective songs (Tchernichovski et al., 2001). In the songbird brain, a population of neurons in the auditory association areas exhibits an experience-dependent selective response to one of several learned songs (Gentner & Margoliash, 2003), suggesting that neurons encode posterior expectations of individual songs based on experience. Moreover, the HVC plays an important role not only in song production but also in the formation of associations between a song and a conspecific that emits that song (Gentner, Hulse, Bentley, & Ball, 2000). Again, this is consistent with our generative model that is used for both generation and recognition. The existence of neurons with a teacher-specific activity has been reported in the higher-level auditory cortex of the songbird, the caudomedial nidopallium (NCM; Yanagihara & Yazaki-Sugiyama, 2016). In our model, the switcher exhibits teacher-specific activity by accumulating model evidence. One can imagine that neurons in the NCM might encode the posterior belief about the switcher state. In addition, the accuracy of song recognition might be bounded by the memory capacity of the neural circuit encoding songs (Gentner, 2004), which is similar to the memory capacity of our model, as determined by the number of generative models that can be supported by the neuronal infrastructure. In short, these empirical observations support the neurobiological plausibility of our model and speak to the empirical tests.
While we have randomly interspersed the order of presentation from each teacher, it would be interesting to examine the influence of more systematic changes in presentation order. In a social neuroscience context, this could be important in understanding things like multiple language acquisition in children who speak to different family members in different languages. Specifically, there appear to be differences in bilingualism when languages are learned simultaneously (i.e., interspersed like the model here) or successively (Klein, Mok, Chen, & Watkins, 2014). This might imply different mechanisms for the latter compared to the former and require an extension of the generative model employed here.
In terms of the questions and challenges for empirical neuroscience, the picture that emerges from the current solution raises the following considerations: the computational anatomy in Figure 1 communicates with a deep (temporal) architecture in which there are multiple, competing attractor networks in the brain. These effectively compete to explain the sensory data, and their ability to do so determines the rate of perceptual learning. In turn, this means that one would predict distinct autonomous dynamics corresponding to competing hypotheses about the current dynamical form of sensory input. This means that there should be neuronal correlates of distinct pattern generators that are engaged contemporaneously during perceptual synthesis. Second, it suggests a convergence of descending projections to lower (e.g., primary) sensory systems. This brings an interesting and complementary perspective on the divergent neuroanatomy of descending backward connections in cortical hierarchies in the brain (Zeki & Shipp, 1988; Angelucci & Bressloff, 2006). We mean this in the sense that usually one interprets the asymmetry between convergent and divergent zones in terms of things like extra classical receptive field effects, particularly in the visual cortex (Angelucci & Bressloff, 2006). A complementary perspective is that the divergence of descending efferents can also be looked upon as a convergence of descending afferents. This is precisely the architecture described in Figure 1. More interesting, the factor graph representation of neuronal architectures speaks to a selective (Bayesian model selection) modulation of the messages converging on any given lower level. Physiologically, this means that there must be a neuromodulatory mechanism in play that can handle multiple convergent inputs to a postsynaptic neuron or population. In effect, this implies a winner-takes-all-like mechanism at the level of synaptic efficacy, as opposed to synaptic activity. One could speculate about the neurotransmitter basis of this selection process—for example, appealing to the neuromodulatory effects of neurotransmitter systems targeting cholinergic and 5HT receptors (Everitt & Robbins, 1997; Collerton, Perry, & McKeith, 2005; Vossel, Bauer, Bauer, & Mathys, 2014; Hedrick & Waters, 2015; Doya, 2002; Yu & Dayan, 2005; Dayan, 2012)—on inhibitory interneurons in superficial layers and deep pyramidal cells in deep layers. Many of these speculations have often been rehearsed in relation to the deployment of attention in the context of predictive coding and usually implicate synchronous gain mechanisms via the action of inhibitory interneurons and the current connections with pyramidal cells (Fries, 2005; Womelsdorf & Fries, 2006; Saalmann & Kastner, 2009; Feldman & Friston, 2010; Buschman & Kastner, 2015). Finally, one key message of this theoretical work is that the rate of perceptual learning is determined by the evidence for competing models of sensory input. In principle, this predicts that the rate of sensory learning under ambiguity should be sensitive to the relative probability ascribed to different explanations for sensory input. In turn, this relates to psychophysical studies of perceptual learning under ambiguity with, for example, ambiguous figures or other forms of multistable perception (Tani & Nolfi, 1999; Wurtz, 2008; Hohwy, Paton, & Palmer, 2016).
In summary, we have introduced a novel learning scheme that integrates Bayesian filtering and model selection to learn and deploy multiple generative models. We assumed that a switching variable selects a particular model to generate current sensory input (like switching to a particular radio channel from a repertoire of radio programs), while many alternative generative models are running in the background. To deal with the problem of context-sensitive learning, the proposed scheme calculates the model plausibility (i.e., model evidence) of each generative model based on conditional free actions and updates parameters only in models with a convincing degree of evidence. Our synthetic agents were able to both learn and recognize different artificial and natural birdsongs. These results highlight the potential utility of equipping agents with multiple generative models to make inferences in context-sensitive environments.
## 4 Methods
The proposed variational update scheme is described in section 2. Further details are provided in this section.
### 4.1 Generative Model
Formally, the multiple generative models are defined as the following. Hierarchical Bayesian filtering supposes a model that consists of latent variables $u$ (a set of hidden states $x$ and hidden causes $v)$ and parameters $θ$ and infers and learns their approximate probability (recognition) densities. To extend this for a multiple-model version, we express the $i$th generative model consisting of two layers as $mi$ with $i∈M≡1,2,3,…$. This $mi$ indicates a specific model structure including certain forms of functions and dimensions of latent variables and parameters. Let $s˜$ be sensory inputs (i.e., teacher song) generated by $mi$, and $x˜i,j$, $v˜i,j$, and $θi,j$ be hidden states, hidden causes, and parameters in the $j$th layer ($j=$ 1,2) of $mi$, respectively. The tilde over a symbol denotes a set of a variable and its time derivatives $s˜≡s,s',s'',…$. Throughout this letter, $i$ indices the model while $j$ indices the level of layers. The $i$th generative model is given by equation 2.1. The corresponding probabilities are defined as gaussian distributions and written as
$s˜=g˜i,1+ω˜i,1⟺ps˜|x˜i,1,v˜i,1,θi,1,mi≡Ns˜;g˜i,1,Πsi,Dx˜i,1=f˜i,1+z˜i,1⟺px˜i,1|v˜i,1,θi,1,mi≡NDx˜i,1;f˜i,1,Πxi,1,v˜i,1=g˜i,2+ω˜i,2⟺pv˜i,1|x˜i,2,v˜i,2,θi,2,mi≡Nv˜i,1;g˜i,2,Πvi,1,Dx˜i,2=f˜i,2+z˜i,2⟺px˜i,2|v˜i,2,θi,2,mi≡NDx˜i,2;f˜i,2,Πxi,2,pv˜i,2|mi≡Nv˜i,2;η˜i,Πvi,2,pθi,j|mi≡Nθi,j;Θi,j,Πθi,j,j=1,2.$
(4.1)
Note that $Dx˜i,j$ is the derivative of $x˜i,j$ with respect to time; $ω˜i,1∼N[ω˜i,1;0,Πsi],ω˜i,2∼N[ω˜i,2;0,Πvi,1]$ and $z˜i,j∼N[z˜i,j;0,Πxi,j]$ are background gaussian noises; $g˜i,j≡g˜i,jx˜i,j,v˜i,j,θi,j$ and $f˜i,j≡f˜i,jx˜i,j,v˜i,j,θi,j$ are arbitrary functions of $x˜i,j$ and $v˜i,j$ parameterized by $θi,j$;$Πsi,Πxi,j$, and $Πvi,j$ are precision matrices; and $pv˜i,2|mi$ and $pθi,j|mi$ are gaussian priors parameterized by the mean and the precision matrix. To simplify notation, we define $ui,j≡x˜i,j,v˜i,j$ as latent variables, $ui≡ui,1,ui,2$ as a set of latent variables in all layers of $mi$, and $θi≡θi,1,θi,2$ as a set of parameters in all layers of $mi$. By multiplying all equations on the right-hand side of equation 4.1, the $i$th generative model is expressed as
$ps˜,ui,θi|mi=ps˜|x˜i,1,v˜i,1,θi,1,mipv˜i,1|x˜i,2,v˜i,2,θi,2,mi·pv˜i,2|mi∏j=1,2px˜i,j|v˜i,j,θi,j,mipθi,j|mi,$
(4.2)
as shown in the top panel in Figure 1.
#### 4.1.1 Sensory Inputs
The sensory input that an agent (i.e., a student bird) actually receives is selected by one of the models in $M≡1,2,…$, where we index the currently selected model by $c$. The sensory input is expressed by the sum of the product of the conditional probability of $s˜$ under each model and the probability of each model being selected:
$ps˜|mc=EPi=cps˜|mi=∑i∈MPi=cps˜|mi=∑i∈Mγips˜|mi=∑i∈Mps˜|miγi.$
(4.3)
Note that the sufficient statistics $γi≡Pi=c∈0,1$ satisfies $∑i∈Mγi=1$ by design, where only $γc$ takes a value of one (and the others zero). In this setting, $γi$ plays a role of a switcher that switches which model has generated the current sensory input, while all models are running in the background. The probability of $γ=γ1,γ2,…$ follows a categorical prior distribution $Pγ=CatΓ$.
Interestingly, this definition of the multiple generative models and the switcher is slightly different from supposing a large generative model with switcher-dependent parameters. This idea is rather an assumption that an agent has a set of hypotheses (generative models) about sensory inputs from which to select in a given context. Each generative model is running independently from the others. They interact only via sensory inputs through the model switching scheme, but this interaction does not change their latent variables or parameters. Because of this conditional independence, each model can have different forms and dimensions, although in this work, we assume models based on the same model structure and dimension with different latent variables and parameters.
#### 4.1.2 Free Energy and Free Action
The negative log of $ps˜|mc$ denotes the surprise (aka surprisal) associated with sensory inputs. Variational free energy is defined as an upper bound on surprise. In this work, we slightly modify the derivation of variational free energy to include the model selection procedure. First, we show that Bayesian model averaging of the conditional surprises provides an upper bound of surprise. Suppose $Qi=c≡γi∈0,1$ with $∑i∈Mγi=1$ is the posterior expectation of the switcher state. This is equivalent to a categorical posterior distribution of $γ$ given by $Qγ=Catγ$. Since model $c$ is selected, the following inequality holds from the nonnegativity of Kullback-Leibler divergence (Kullback & Leibler, 1951):
$Eps˜|mclogps˜|mc-∑i∈Mγilogps˜|mi=∑i∈MγiEps˜|mclogps˜|mc-logps˜|mi=∑i∈MγiDKLps˜|mc||ps˜|mi≥0,$
(4.4)
where $logps˜|mi$ is a conditional surprise under model $i$ and $DKL·||·$ denotes the Kullback-Leibler divergence between two distributions. The expectation over sensory inputs $Eps˜|mc·$ can be approximated using the time average of surprise. From equation 4.4, we have
$Eps˜|mcQi=c-logps˜|mi=Eps˜|mc-∑i∈Mγilogps˜|mi≥Eps˜|mc-logps˜|mc$
(4.5)
and
$EQi=c-∫0Tlogps˜|midt=-∑i∈Mγi∫0Tlogps˜|midt≥-∫0Tlogps˜|mcdt,$
(4.6)
where $T$ is the measurement time within a session. Finally, we define total free action (the path integral of free energy) as an upper bound of $EQ(i=c)[-∫0Tlogp(s˜|mi)dt]$:
$F¯≡EQi=c∫0T-logps˜|mi+EqθiDKLqui||pui|s˜,θi,midt+DKLQγ||Pγ+∑i∈MDKLqθi||pθi|mi=EQi=cF¯i+logγi-logΓi+∑i∈MEqθilogqθi-logpθi|mi≥EQi=c-∫0Tlogps˜|midt.$
(4.7)
Note that $qui$ and $qθi$ are the posterior densities over latent variables and parameters under model $i$, respectively. In this expression, the total free energy is defined as the weighed sum of conditional free actions $F¯1,F¯2,…$ plus the Kullback-Leibler divergence (i.e., complexity) associated with the switcher state and parameters. The free action is defined by
$F¯i≡∫0TEquiqθi-logps˜,ui|θi,mi+logquidt=∫0TFitdt,$
(4.8)
where $Fit≡Equiqθi-logps˜,ui|θi,mi+logqui$ is free energy given model $i$. The first term of $Fit$ is the negative log of the generative model (see equation 4.2) divided by $pθi|mi$ and is referred to as internal energy under model $i$: $Uis˜,ui,θi≡-logps˜,ui|θi,mi$. Note that $Ft≡EQi=cFit=∑i∈MγiFit$ denotes total free energy.
#### 4.1.3 Posteriors
From the Laplace assumption, the posterior density of latent variables $ui$ is approximated as a gaussian distribution $qui=N[ui;ui,Pui]$ with an expectation (or mode) vector $ui$ and a precision matrix $Pui$. The posterior density of parameters $θi$ is approximated as a gaussian distribution $qθi=N[θi;θi,Pθi]$ with an expectation vector $θi$ and a precision matrix $Pθi$. As described above, the posterior distribution of the switcher state (i.e., model plausibility) has been defined as a categorical distribution $Qi=c=γi$ with $∑i∈Mγi=1$, which is equivalent to $Qγ=Catγ.$
### 4.2 Variational Update Rules
Updates of the posteriors of the latent variables, the switcher state, and the parameters are conducted in the inference, model selection, and learning steps, respectively. In the simulation, these three steps are repeated in order for each session. In what follows, we formally derive update rules from the minimization of free energy or free action.
#### 4.2.1 Inference (Neural Activity)
The optimal $qui$ is obtained by solving the variation of $Fi$, $δFi=∫{Eqθi[Uis˜,ui,θi]+logqui+1}δquidui$. To satisfy $δFi=0$, the posterior density should be $qui∝exp[-Eqθi[Uis˜,ui,θi]]≈exp[-Ui(s˜,ui,θi)]$, where $Ui(s˜,ui,θi)$ is the first-order approximation of the variational energy for latent variables. When $ui$ has been optimized, the path of the mode $u˙i$ should be equal to the mode of the path $Dui$, $u˙i=Dui$, in addition to minimizing $Ui(s˜,ui,θi)$ (see Friston, 2008, and Friston et al., 2008, for details). Thus, the gradient descent rule to minimize $Fit$ with respect to $ui$ is given by
$u˙i-Dui∝-∂∂uiUis˜,ui,θiui=ui=-∂∂uiUis˜,ui,θi≈-∂∂uiFit.$
(4.9)
Moreover, $Pui$ that minimizes $Fit$ is given by the Hessian:
$Pui=∂2∂ui2Uis˜,ui,θiui=ui≈∂2∂ui2Fit.$
(4.10)
Hence, we find equation 2.3. Equation 4.9 is usually supposed to be the dynamics of state coding neurons.
#### 4.2.2 Model Selection (Attentional Switch)
This step performs online Bayesian model selection analogous to post hoc Bayesian model selection (Friston & Penny, 2011). Since the switcher state comprises discrete variables, this step is similar to a Markov decision process scheme (Friston, FitzGerald et al., 2017). From equation 4.7, the derivative of $F¯$ with respect to $γi$ is given by $δF¯=F¯iθi+logγi-logΓi+1δγi$. From $δF¯=0$, we find $γi$ that minimizes $F¯$ as
$γi=σ-F¯iθi+logΓi.$
(4.11)
Here $σ·$ is a softmax function defined by $σ·i≡exp·i/∑k∈Mexp·k$. This $γi$ expresses the model plausibility of model $i$. When $Pγi$ is the flat (uniform) prior distribution, equation 4.11 becomes equation 2.5.
#### 4.2.3 Learning (Synaptic Plasticity)
Estimation of parameters is based on a conventional gradient descent approach. To satisfy $δF¯=∫{γi∫0TEquiUis˜,ui,θidt+logqθi-logpθi|mi+1}δqθidθi=0$, from $EquiUis˜,ui,θi≈Uis˜,ui,θi$, the density should be $qθi∝exp[-γi∫0TUis˜,ui,θidt+logpθi|mi]$, where $∫0TUis˜,ui,θidt$ is the approximate variational action for parameters. The gradient descent rule to minimize $F¯$ with respect to $θi$ is given by
$θ˙i∝-∂∂θiγi∫0TUis˜,ui,θidt-logpθi|miθi=θi≈-∂∂θiγiF¯iθi-logpθi|mi.$
(4.12)
Moreover, $Pθi$ that minimizes $F¯$ is given by the Hessian:
$P˙θi∝-Pθi+∂2∂θi2γi∫0TUis˜,ui,θidt-logpθi|miθi=θi≈-Pθi+∂2∂θi2γiF¯iθi-logpθi|mi.$
(4.13)
When $pθi|mi$ is the flat (uniform) prior distribution, equation 4.12 becomes equation 2.6.
Accordingly, we obtain posterior beliefs of the latent variables, the switcher state, and the parameters that minimize free action. The difference in learning rate mediated by the model plausibility enables that only the parameters in the most plausible models are updated, while the parameters in the remaining models are maintained in a winner-takes-all manner, whereas the latent variables in all models are updated with a fixed update rate. Therefore, inference occurs for all generative models, while learning occurs only for the most plausible generative models. This mechanism enables the agent to make inferences and learning with several different generative models.
#### 4.2.4 Action
Action $a$ is generated to minimize the total free energy $Ft=∑i∈MγiFit$: $a˙∝-∂F/∂a$. In the absence of the external sensory input, action directly induces sensory input, that is, $s˜=a$. Suppose all internal models use the same precision matrix. In this special case, the optimal action is approximately solved as
$a˙∝-∂F∂a≈∑i∈Mγigi,1ui,1,θi,1.$
(4.14)
### 4.3 Songbird Model
A generative model for birdsong generation is defined as a two-layer hierarchical generative model as mentioned in previous studies (Kiebel et al., 2008; Friston & Kiebel, 2009), in which each layer has three or four hidden states that express biological neural circuits for birdsong generation. For simulation purposes, several different teacher songs and the same number of internal models were used.
#### 4.3.1 For Figure 3
Two teacher songs were generated from two generative models with different parameters. A student was supposed to have two internal models. A generative model with two layers was defined. Each layer has three hidden states $x1(i,j),x2(i,j),x3(i,j)$ that recapitulate the Laje-Mindlin-style three-neuron circuit model for birdsong production (Laje & Mindlin, 2002). Layer 1 has one hidden cause $v1(i,1)$ and one parameter $θ1(i,1)$, while layer 2 has no hidden cause or parameter. Several functions for the generative model were defined as follows:
$gi,1≡&x1i,1x3i,1,fi,1≡1τ1-x1i,1x2i,14x3i,1︸intrinsicdynamics+sig10x1i,1-10x2i,1sig8.5x1i,1+2x2i,1+2x3i,1-5.5-θ1i,1+2.7xvi,14sig-20x2i,1+4x3i,1+6︸synapticinput,gi,2≡x3i,2,fi,2≡1τ2-x1i,2x2i,24x3i,2+sig10x1i,2-10x2i,2sig8.5x1i,2+2x2i,2+2x3i,2-5.54sig-20x2i,2+4x3i,2+6.$
(4.15)
Here, $sigx≡1/(1+e-x)$ is the sigmoid function. Neurobiologically, $x1(i,j)$ and $x3(i,j)$ correspond to excitatory neurons, while $x2(i,j)$ corresponds to an inhibitory neuron. Different teacher songs use different parameter $θ1(i,1)$ (0 for teacher 1 and 1 for teacher 2). This parameter was learned by a student without supervision. Training was repeated for 32 sessions. Each session was a 4 s sequence. Time resolution $dt=1/64[s]$ and the time constants $τ1=32/3[s],τ2=128/3[s]$ were used.
#### 4.3.2 For Figures 4 and 5
Six natural zebra finch songs were used as teacher songs, and a student that has six internal models was supposed. These models share the same structure while using different latent variables and parameters. To enable the model to accurately imitate natural zebra finch songs, we extended the original three-neuron Laje-Mindlin model to a four-neuron circuit by adding an inhibitory neuron (see Figure 8 in appendix A). This model was used in Figures 4 and 5B. This addition served to introduce a delay in the attractor, whereas in Figure 5A, the original three-neuron attractor was used for comparison. We also supposed a nonlinear mapping from the four-neuron attractor to the outputs (songs), which afforded the capability to generate (imitate) complex songs. Each layer has four hidden states: $x1(i,j)$ and $x4(i,j)$ correspond to excitatory neurons, whereas $x2(i,j)$ and $x3(i,j)$ correspond to inhibitory neurons. Layer 1 has two hidden causes $v1(i,1),v2(i,1)$ and 924-dimensional parameters (a 2 $×$ 462 matrix $θ(i,1))$, while layer 2 has no hidden cause or parameter. As before, several functions for the generative model were defined as follows:
$gi,1≡θ1,1i,1θ1,2i,1⋯θ1,462i,1θ2,1i,1θ2,2i,1⋯θ2,462i,11ui,1ui,1⊗ui,1ui,1⊗ui,1⊗ui,1ui,1⊗ui,1⊗ui,1⊗ui,1ui,1⊗ui,1⊗ui,1⊗ui,1⊗ui,1,fi,j≡1τj-ρx1i,1x2i,1x3i,1x4i,1︸intrinsicdynamics+-x2i,jψx1i,j-εx3i,j-εx3i,j-εx2i,j+x4i,j-λx3i,j︸synapticinput,j=1,2,gi,2≡x1i,2x4i,2.$
(4.16)
Here $u(i,1)=x1(i,1),x2(i,1),x3(i,1),x4(i,1),v1(i,1),v2(i,1)T$ is a vector of latent variables $⊗$ expresses an outer product (arranging all pair-wise products of elements of two vectors in vertical line except for duplicate terms), and $ψx≡-x+x3$ is a nonlinear activation function that exhibits bistable neural dynamics. This modification was implemented to ensure that this circuit exhibited chaotic dynamics with separatrix crossing (Fukuda, Petrosky, & Konishi, 2016; Ogawa et al., 2016), which is caused by the bistable dynamics of an excitatory neuron ($x1(i,j))$. For simplification, the other three neurons were supposed to be linear functions instead of sigmoid functions. In this circuit model, two excitatory-inhibitory couplings ($x1(i,j)-x2(i,j)$ and $x3(i,j)-x4(i,j))$ exhibited oscillatory dynamics, whereas a weak mutual connection between two inhibitory neurons $x2(i,j)$ and $x3(i,j)$ made the attractor chaotic.
In the simulations, the leakage parameter $ρ=0.1$ (to ensure stability), a small inhibitory synaptic weight $ε=0.2$ (which controlled the coupling strength between the first and second oscillators), and a large inhibitory synaptic weight $λ=1.2$ (which determined the period of the second neural oscillator) were used. We supposed that the hidden causes $v(i,1)$ converged to $x(i,2)$ to simplify the simulation. The definition of $g(i,1)$ was chosen to ensure that it could express a general quintic function by a linear product of a 2 $×$ 462 matrix and a 462-dimensional vector. These parameters were learned by a student without supervision. When updating the posterior belief of hidden states, we smoothed the posterior trajectory by adding small amounts of components of the prior (i.e., a trajectory of an attractor without perturbation) to avoid a divergence of variables induced by a large perturbation. Training was repeated for 60 sessions. Each session had a 10 s sequence. Time resolution $dt=10-3[s]$ was used.
The following procedure was applied before training: (1) the student's initial states of $x(i,1),x(i,2)$ at time $t=0$ and the time constants $τ1,τ2[s]$ were optimized in relation to one of six teacher songs (these were in the range of $-1≤x1i,j,x2i,j,x3i,jx4i,j≤1$, $1/60≤τ1≤1/40$, and $1/6≤τ2≤1/4)$; (2) the posterior expectation of parameters $θ(i,1)$ was randomly generated; and (3) $θ(i,1)$ were modified by pretraining, in which each model randomly received one of six teacher songs, made an inference, and updated the parameters without model selection for 18 sessions to ensure that each internal model initially represented an averaged song. Then the response songs of a student were tested with different teacher songs (movie 1; see appendix B). For training, we randomly selected one of six teacher songs and provided it to a student (movie 2; see appendix B). Training was repeated for 60 sessions. After training, we tested the response songs again (see Figure 4 and movie 3; see appendix B).
### 4.4 Preprocessing for Natural Birdsong Data
The birdsong data used for Figures 4 and 5 and the supplementary movies were downloa-ded from http://ofer.sci.ccny.cuny.edu/song_database/zebra-finch-song-library-2015/view. This data set was recorded by the Tchernichovski group (see Tchernichovski et al., 2001). We treated the data as follows. First, we acquired a spectrogram of the song by performing a Fourier transform with a 23.2 ms time window. As an analogy to a physiological model of vocal coda that generates a birdsong by sequences of power and tone (frequency) of the voice (Laje, 2002), we defined the leading frequency ($s2)$ and the amplitude ($s1)$ of a song by the mode of its frequency and the power of the mode frequency for each time step, respectively. They were normalized and introduced as sensory inputs $s=s1,s2T$.
## Appendix A: Supplementary Figures
Figure 6:
A schematic illustrating an experimental procedure (A) and simulation results of a synthetic bird with a single generative model (B,C). (A) Experimental procedure. Teacher bird 1 sings in odd sessions, while teacher bird 2 sings in even sessions. Our synthetic bird (student) listens to either song in turn. (B) Trajectory of the posterior expectation of a parameter ($θ)$ of the student that employs a single generative model (black sold line). A black dashed line shows another trajectory where $θ$ started from a different initial value. Shaded areas indicate the standard deviation. Red and blue dashed lines express the true parameter of teachers 1 and 2, respectively. The student tried to infer either parameter of teachers 1 or 2, but it failed to learn either parameter; even its posterior belief was initialized to the same value as either teacher 1 or 2's parameter. This is because the student inferred the intermediate value of teacher 1 and 2's parameters. (C) Transition of free action (filled circles). Open circles show the transition of free action with another initial $θ$.
Figure 6:
A schematic illustrating an experimental procedure (A) and simulation results of a synthetic bird with a single generative model (B,C). (A) Experimental procedure. Teacher bird 1 sings in odd sessions, while teacher bird 2 sings in even sessions. Our synthetic bird (student) listens to either song in turn. (B) Trajectory of the posterior expectation of a parameter ($θ)$ of the student that employs a single generative model (black sold line). A black dashed line shows another trajectory where $θ$ started from a different initial value. Shaded areas indicate the standard deviation. Red and blue dashed lines express the true parameter of teachers 1 and 2, respectively. The student tried to infer either parameter of teachers 1 or 2, but it failed to learn either parameter; even its posterior belief was initialized to the same value as either teacher 1 or 2's parameter. This is because the student inferred the intermediate value of teacher 1 and 2's parameters. (C) Transition of free action (filled circles). Open circles show the transition of free action with another initial $θ$.
Figure 7:
Simulation results when learning two birdsongs using multiple generative models. Simulation setup and layout are the same as Figure 3, but initial hidden states of teachers 1 and 2 were not reset for each session. This yielded the chaotic dynamics in their songs. Even in this case, a student bird employing the proposed scheme could learn from two distinct teachers. In this figure, to generate various song trajectories, a chaotic attractor considered in Kiebel et al. (2008) and Friston and Kiebel (2009) was used as the generative model instead of the Laje-Mindlin style model.
Figure 7:
Simulation results when learning two birdsongs using multiple generative models. Simulation setup and layout are the same as Figure 3, but initial hidden states of teachers 1 and 2 were not reset for each session. This yielded the chaotic dynamics in their songs. Even in this case, a student bird employing the proposed scheme could learn from two distinct teachers. In this figure, to generate various song trajectories, a chaotic attractor considered in Kiebel et al. (2008) and Friston and Kiebel (2009) was used as the generative model instead of the Laje-Mindlin style model.
Figure 8:
A birdsong generative model with two-layer four neuron circuits. This model is defined by extending the Laje-Mindlin style model to facilitate the song generation capability. The lower layer (level 1) corresponds to RA, while the higher layer (level 2) corresponds to HVC. In each layer, $x1$ and $x4$ are associated with excitatory neurons, while $x2$ and $x3$ are associated with inhibitory neurons. Signals of $x1$ and $x4$ in the HVC are translated into $v1$ and $v2$ in the RA. The sensory input (i.e., song) is generated through nonlinear functions $g1$ and $g2$ which receive inputs from $x1$, $x2$, $x3$, $x4$, $v1$, and $v2$ (see section 4 for details).
Figure 8:
A birdsong generative model with two-layer four neuron circuits. This model is defined by extending the Laje-Mindlin style model to facilitate the song generation capability. The lower layer (level 1) corresponds to RA, while the higher layer (level 2) corresponds to HVC. In each layer, $x1$ and $x4$ are associated with excitatory neurons, while $x2$ and $x3$ are associated with inhibitory neurons. Signals of $x1$ and $x4$ in the HVC are translated into $v1$ and $v2$ in the RA. The sensory input (i.e., song) is generated through nonlinear functions $g1$ and $g2$ which receive inputs from $x1$, $x2$, $x3$, $x4$, $v1$, and $v2$ (see section 4 for details).
## Appendix B: Supplementary Movies 1–3
Movies 1 to 3 are available online at https://www.mitpressjournals.org/doi/suppl/10.1162/neco_a_01239. These movies show the dynamics of teacher and student birds before, during, and after training. The details are described in the caption to Figure 4.
### Acknowledgments
This work was supported by RIKEN Center for Brain Science (T.I.) and Tateisi Science and Technology Foundation (T.I.). T.P. is supported by the Rosetrees Trust (award 173346). K.J.F. is funded by a Wellcome Trust Principal Research Fellowship (088130/Z/09/Z). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
### Author Contributions
K.J.F. conceptualized the free-energy principle; T.I. conceived and designed the method using the multiple internal models and performed the simulations. T.I., T.P., and K.J.F. wrote the paper.
## References
,
R. A.
,
Shipp
,
S.
, &
Friston
,
K. J.
(
2013
).
Predictions not commands: Active inference in the motor system
.
Brain Structure and Function
,
218
,
611
643
. doi:10.1007/s00429-012-0475-5. PMID:23129312.
Aljundi
,
R.
,
Chakravarty
,
P.
, &
Tuytelaars
,
T.
(
2017
).
Expert gate: Lifelong learning with a network of experts
. In
Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition
(pp.
3366
3375
).
Piscataway, NJ
:
IEEE
. https://arxiv.org/abs/1611.06194.
,
A.
,
Perl
,
Y. S.
,
Mindlin
,
G. B.
, &
Margoliash
,
D.
(
2013
).
Elemental gesture dynamics are encoded by song premotor cortical neurons
.
Nature
,
495
,
59
65
. doi:10.1038/nature11967.
PMID:23446354
.
Angelucci
,
A.
, &
Bressloff
,
P. C.
(
2006
).
Contribution of feedforward, lateral and feedback connections to the classical receptive field center and extra-classical receptive field surround of primate V1 neurons
.
Progress in Brain Research
,
154
,
93
120
. doi:10.1016/S0079-6123(06)54005-1.
PMID:17010705
.
Awh
,
E.
,
Belopolsky
,
A. V.
, &
Theeuwes
,
J.
(
2012
).
Top-down versus bottom-up attentional control: A failed theoretical dichotomy
.
Trends in Cognitive Sciences
,
16
,
437
443
. doi:10.1016/j.tics.2012.06.010.
PMID:22795563
.
Bastos
,
A. M.
,
Usrey
,
W. M.
,
,
R. A.
,
Mangun
,
G. R.
,
Fries
,
P.
, &
Friston
,
K. J.
(
2012
).
Canonical microcircuits for predictive coding
.
Neuron
,
76
,
695
711
. doi:10.1016/j.neuron.2012.10.038.
PMID:23177956
.
Ben Achour, S., &
Pascual
,
O.
(
2010
).
Glia: the many ways to modulate synaptic plasticity
.
Neurochemistry International
,
57
,
440
445
. doi:10.1016/j.neuint.2010.02.013.
PMID:20193723
.
Bi
,
G. Q.
, &
Poo
,
M. M.
(
1998
).
Synaptic modifications in cultured hippocampal neurons: Dependence on spike timing, synaptic strength, and postsynaptic cell type
.
Journal of Neuroscience
,
18
,
10464
10472
. doi:10.1523/JNEUROSCI.18-24-10464.1998.
PMID:9852584
.
Bliss
,
T. V.
, & Lømo, T.
(
1973
).
Long-lasting potentiation of synaptic transmission in the dentate area of the anaesthetized rabbit following stimulation of the perforant path
.
Journal of Physiology
,
232
,
331
356
. doi:10.1113/jphysiol.1973.sp010273.
PMID:4727084
.
Bosman
,
C. A.
,
Schoffelen
,
J. M.
, Brunet, N., Oostenfeld,
Bastos
,
A. M.
,
Womelsdorf
,
T.
, …
Fries
,
P.
(
2012
).
Attentional stimulus selection through selective synchronization between monkey visual areas
.
Neuron
,
75
,
875
888
. doi:10.1016/j.neuron.2012.06.037.
PMID:22958827
.
Brodski-Guerniero
,
A.
,
Paasch
,
G. F.
,
,
P.
,
Özdemir
,
I.
,
Lizier
,
J. T.
, &
Wibral
,
M.
(
2017
).
Information-theoretic evidence for predictive coding in the face-processing system
.
Journal of Neuroscience
,
37
,
8273
8283
. doi:10.1523/JNEUROSCI.0614-17.2017.
PMID:28751458
.
Buschman
,
T. J.
, &
Kastner
,
S.
(
2015
).
From behavior to neural dynamics: An integrated theory of attention
.
Neuron
,
88
,
127
144
. doi:10.1016/j.neuron.2015.09.017.
PMID:26447577
.
Calabrese
,
A.
, &
Woolley
,
S. M.
(
2015
).
Coding principles of the canonical cortical microcircuit in the avian brain
.
Proceedings of the National Academy of Sciences of the USA
,
112
,
3517
3522
. doi:10.1073/pnas.1408545112.
PMID:25691736
.
Collerton
,
D.
,
Perry
,
E.
, &
McKeith
,
I.
(
2005
).
Why people see things that are not there: A novel perception and attention deficit model for recurrent complex visual hallucinations
.
Behavioral and Brain Sciences
,
28
,
737
757
. doi:10.1017/S0140525X05000130.
PMID:16372931
.
Dajani
,
D. R.
, &
Uddin
,
L. Q.
(
2015
).
Demystifying cognitive flexibility: Implications for clinical and developmental neuroscience
.
Trends in Neurosciences
,
38
,
571
578
. doi:10.1016/j.tins.2015.07.003.
PMID:26343956
.
Dauwels
,
J.
(
2007
).
On variational message passing on factor graphs
. In
Proceedings of the 2007 IEEE International Symposium on Information Theory
(pp.
2546
2550
).
Piscataway, NJ
:
IEEE
. doi:10.1109/ISIT.2007.4557602.
Dayan
,
P.
(
2012
).
Twenty-five lessons from computational neuromodulation
.
Neuron
,
76
,
240
256
. doi:10.1016/j.neuron.2012.09.027.
PMID:23040818
.
Dayan
,
P.
,
Hinton
,
G. E.
,
Neal
,
R. M.
, &
Zemel
,
R. S.
(
1995
).
The Helmholtz machine
.
Neural Compututation
,
7
,
889
904
. doi:10.1162/neco.1995.7.5.889.
PMID:7584891
.
Deubel
,
H.
, &
Schneider
,
W. X.
(
1996
).
Saccade target selection and object recognition: Evidence for a common attentional mechanism
.
Vision Research
,
36
,
1827
1837
. doi:10.1016/0042-6989(95)00294-4.
PMID:8759451
.
Devaine
,
M.
,
Hollard
,
G.
, &
Daunizeau
,
J.
(
2014
).
Theory of mind: Did evolution fool us
?
PLoS One
,
9
,
e87619
. doi:
10.1371/journal.pone.0087619
.
PMID:24505296
.
Devaine
,
M.
,
San-Galli
,
A.
,
Trapanese
,
C.
,
Bardino
,
G.
,
Hano
,
C.
,
Saint Jeime
,
M.
,
Daunizeau
,
J.
(
2017
).
Reading wild minds: A computational assay of theory of mind sophistication across seven primate species
.
PLoS Computational Biology
,
13
,
e1005833
. doi:10.1371/journal.pcbi.1005833.
PMID:29112973
.
Doya
,
K.
(
2002
).
Metalearning and neuromodulation
.
Neural Networks
,
15
,
495
506
. doi:10.1016/S0893-6080(02)00044-8.
PMID:12371507
.
Eagleman
,
D. M.
(
2001
).
Visual illusions and neurobiology
.
Nature Reviews Neuroscience
,
2
,
920
926
. doi:10.1038/35104092.
PMID:11733799
.
Ereira
,
S.
,
Dolan
,
R. J.
, &
Kurth-Nelson
,
Z.
(
2018
).
Agent-specific learning signals for self–other distinction during mentalising
.
PLoS Biology
,
16
,
e2004752
. doi:10.1371/journal.pbio.2004752.
PMID:29689053
.
Everitt
,
B. J.
, &
Robbins
,
T. W.
(
1997
).
Central cholinergic systems and cognition
.
Annual Review of Psychology
,
48
,
649
684
. doi:10.1146/annurev.psych.48.1.649.
PMID:9046571
.
Feldman
,
D. E.
(
2012
).
The spike-timing dependence of plasticity
.
Neuron
,
75
,
556
571
. doi:10.1016/j.neuron.2012.08.001.
PMID:22920249
.
Feldman
,
H.
, &
Friston
,
K. J.
(
2010
).
Attention, uncertainty, and free-energy
.
Frontiers in Human Neuroscience
,
4
,
215
. doi:10.3389/fnhum.2010.00215.
PMID:21160551
.
Forney
,
G. D.
(
2001
).
Codes on graphs: Normal realizations
.
IEEE Transactions on Information Theory
,
47
,
520
548
. doi:10.1109/18.910573.
Frémaux, N., &
Gerstner
,
W.
(
2016
).
Neuromodulated spike-timing-dependent plasticity, and theory of three-factor learning rules
.
Frontiers in Neural Circuits
,
9
,
85
. doi:10.3389/fncir.2015.00085.
PMID:26834568
.
Fries
,
P.
(
2005
).
A mechanism for cognitive dynamics: Neuronal communication through neuronal coherence
.
Trends in Cognitive Sciences
,
9
,
476
480
. doi:10.1016/j.tics.2005.08.011.
PMID:16150631
.
Friston
,
K.
(
2005
).
A theory of cortical responses
.
Philosophical Transactions of the Royal Society London B Biological Sciences
,
360
,
815
836
. doi:10.1098/rstb.2005.1622.
PMID:15937014
.
Friston
,
K.
(
2008
).
Hierarchical models in the brain
.
PLoS Computational Biology
,
4
,
e1000211
. doi:10.1371/journal.pcbi.1000211.
PMID:18989391
.
Friston
,
K.
(
2010
).
The free-energy principle: A unified brain theory
?
Nature Reviews Neuroscience
,
11
,
127
138
. doi:10.1038/nrn2787.
PMID:20068583
.
Friston
,
K.
,
,
R. A.
,
Perrinet
,
L.
, &
Breakspear
,
M.
(
2012
).
Perceptions as hypotheses: Saccades as experiments
.
Frontiers in Psychology
,
3
,
151
. doi:10.3389/fpsyg.2012.00151.
PMID:22654776
.
Friston
,
K.
,
FitzGerald
,
T.
,
Rigoli
,
F.
,
Schwartenbeck
,
P.
, &
Pezzulo
,
G.
(
2017
).
Active inference: A process theory
.
Neural Computation
,
29
,
1
49
. doi:10.1162/neco_a_00912.
PMID:27870614
.
Friston
,
K. J
, &
Frith
,
C. D.
(
2015a
).
Active inference, communication and hermeneutics
.
Cortex
,
68
,
129
143
. doi:10.1016/j.cortex.2015.03.025.
PMID:25957007
.
Friston
,
K.
, &
Frith
,
C.
(
2015b
).
A duet for one
.
Consciousness and Cognition
,
36
,
390
405
. doi:10.1016/j.concog.2014.12.003.
PMID:25563935
.
Friston
,
K.
, &
Herreros
,
I.
(
2016
).
Active inference and learning in the lerebellum
.
Neural Computation
,
28
,
1812
1839
. doi:10.1162/neco_a_00863.
PMID:27391681
.
Friston
,
K.
, &
Kiebel
,
S.
(
2009
).
Cortical circuits for perceptual inference
.
Neural Networks
,
22
,
1093
1104
. doi:10.1016/j.neunet.2009.07.023.
PMID:19635656
.
Friston
,
K.
,
Kilner
,
J.
, &
Harrison
,
L.
(
2006
).
A free energy principle for the brain
.
Journal of Physiology Paris
,
100
,
70
87
. doi:10.1016/j.jphysparis.2006.10.001.
PMID:17097864
.
Friston
,
K.
,
Levin
,
M.
,
Sengupta
,
B.
, &
Pezzulo
,
G.
(
2015
).
Knowing one's place: A free-energy approach to pattern regulation
.
J. Royal Soc. Interface
,
12
,
20141383
. doi:10.1098/rsif.2014.1383.
PMID:25788538
.
Friston
,
K.
,
Mattout
,
J.
, &
Kilner
,
J.
(
2011
).
Action understanding and active inference
.
Biological Cybernetics
,
104
,
137
160
. doi:10.1007/s00422-011-0424-z.
PMID:21327826
.
Friston
,
K. J.
,
Parr
,
T.
, &
de Vries
,
B. D.
(
2017
).
The graphical brain: Belief propagation and active inference
.
Network Neuroscience
,
1
,
381
414
. doi:10.1162/NETN_a_00018.
PMID:29417960
.
Friston
,
K.
, &
Penny
,
W.
(
2011
).
Post hoc Bayesian model selection
.
NeuroImage
,
56
,
2089
2099
. doi:10.1016/j.neuroimage.2011.03.062.
PMID:21459150
.
Friston
,
K. J.
, Trujillo-Barreto, N., &
Daunizeau
,
J.
(
2008
).
DEM: A variational treatment of dynamic systems
.
NeuroImage 41
,
849
885
. doi:10.1016/j.neuroimage.2008.02.054.
PMID:18434205
.
Froemke
,
R. C.
, &
Dan
,
Y.
(
2002
).
Spike-timing-dependent synaptic modification induced by natural spike trains
.
Nature
,
416
,
433
438
. doi:10.1038/416433a.
PMID:11919633
.
Fukuda
,
H.
,
Petrosky
,
T.
, &
Konishi
,
T.
(
2016
).
Analytical investigation of an isomerization system using the resonance overlap criterion
.
Progress of Theoretical and Experimental Physics
,
2016
,
093A01
. doi:10.1093/ptep/ptw122.
Gentner
,
T. Q.
(
2004
).
Neural systems for individual song recognition in adult birds
.
Annals of the New York Academy of Sciences
,
1016
,
282
302
. doi:10.1196/annals.1298.008.
PMID:15313781
.
Gentner
,
T. Q.
,
Hulse
,
S. H.
,
Bentley
,
G. E.
, &
Ball
,
G. F.
(
2000
).
Individual vocal recognition and the effect of partial lesions to HVc on discrimination, learning, and categorization of conspecific song in adult songbirds
.
Journal of Neurobiology
,
42
,
117
133
. doi:10.1002/(SICI)1097-4695(200001)42:1<117::AID-NEU11>3.0.CO;2-M.
PMID:10623906
.
Gentner
,
T. Q.
, &
Margoliash
,
D.
(
2003
).
Neuronal populations and single cells representing learned auditory objects
.
Nature
,
424
,
669
674
. doi:10.1038/nature01731.
PMID:12904792
.
George
,
D.
, &
Hawkins
,
J.
(
2009
).
Towards a mathematical theory of cortical micro-circuits
.
PLoS Computational Biology
,
5
,
e1000532
. doi:10.1371/journal.pcbi.1000532.
PMID:19816557
.
Green
,
C. S.
, &
Bavelier
,
D.
(
2003
).
Action video game modifies visual selective attention
.
Nature
,
423
,
534
537
. doi:10.1038/nature01647.
PMID:12774121
.
Happé
,
F.
, &
Frith
,
U.
(
1995
).
Theory of mind in autism
. In
E.
Schopler
&
G. B.
Mesibov
(Eds.),
Learning and cognition in autism
.
Boston
:
Springer
.
Haruno
,
M.
,
Wolpert
,
D. M.
, &
Kawato
,
M.
(
2003
).
Hierarchical MOSAIC for movement generation
.
International Congress Series
,
1250
,
575
590
. doi:10.1016/S0531-5131(03)00190-0.
Hassabis
,
D.
,
Kumaran
,
D.
,
Summerfield
,
C.
, &
Botvinick
,
M.
(
2017
).
Neuroscience-inspired artificial intelligence
.
Neuron
,
95
,
245
258
. doi:10.1016/j.neuron.2017.06.011.
PMID:28728020
.
Hayama
,
T.
,
Noguchi
,
J.
,
Watanabe
,
S.
,
Takahashi
,
N.
,
Hayashi-Takagi
,
A.
,
Ellis-Davies
,
G. C.
, …
Kasai
,
H.
(
2013
).
GABA promotes the competitive selection of dendritic spines by controlling local CA2 signaling
.
Nature Neuroscience
,
16
,
1409
1416
. doi:10.1038/nn.3496.
PMID:23974706
.
Hebb
,
D. O.
(
1949
).
The organization of behavior: A neuropsychological theory
.
New York
:
Wiley
.
Hedrick
,
T.
, &
Waters
,
J.
(
2015
).
Acetylcholine excites neocortical pyramidal neurons via nicotinic receptors
.
Journal of Neurophysiology
,
113
,
2195
2209
. doi:10.1152/jn.00716.2014.
PMID:25589590
.
Heilbron
,
M.
, &
Chait
,
M.
(
2017
).
Great expectations: Is there evidence for predictive coding in auditory cortex
?
Neuroscience
pii:S0306-4522(17)30547-X. doi:10.1016/j.neuroscience.2017.07.061.
PMID:28782642
.
Helmholtz
,
H.
(
1925
).
Treatise on physiological optics
(vol.
3
).
Washington, DC
:
Optical Society of America
.
Hohwy
,
J.
,
Paton
,
B.
, &
Palmer
,
C.
(
2016
).
Distrusting the present
.
Phenomenology and the Cognitive Sciences
,
15
,
315
335
. doi:10.1007/s11097-015-9439-6.
Hough,
II
,
G. E.
,
Nelson
,
D. A.
, &
Volman
,
S. F.
(
2000
).
Re-expression of songs deleted during vocal development in white-crowned sparrows, Zonotrichia leucophrys
.
Animal Behaviour
,
60
,
279
287
. doi:10.1006/anbe.2000.1498.
PMID:11007636
.
Itti
,
L.
, &
Koch
,
C.
(
2001
).
Computational modelling of visual attention
.
Nature Reviews Neuroscience
,
2
,
194
203
. doi:10.1038/35058500.
PMID:11256080
.
Johansen
,
J. P.
,
Diaz-Matair
,
L.
,
Hamanaka
,
H.
,
Ozawa
,
T.
,
Ycu
,
E.
,
Koivumaa
,
J.
, … Le
Doux
,
J. E.
(
2014
).
Hebbian and neuromodulatory mechanisms interact to trigger associative memory formation
.
Proceedings of the National Academy of Sciences of the USA
,
111
,
E5584
E5592
. doi:10.1073/pnas.1421304111.
PMID:25489081
.
Kiebel
,
S. J.
,
Daunizeau
,
J.
, &
Friston
,
K. J.
(
2008
).
A hierarchy of time-scales and the brain
.
PLoS Computational Biology
,
4
,
e1000209
. doi:10.1371/journal.pcbi.1000209.
PMID:19008936
.
Kilner
,
J. M.
,
Friston
,
K. J.
, &
Frith
,
C. D.
(
2007
).
Predictive coding: An account of the mirror neuron system
.
Cognitive Processing
,
8
,
159
166
. doi:10.1007/s10339-007-0170-2.
PMID:17429704
.
Kirkpatrick
,
J.
,
Pascanu
,
R.
,
Rabinowitz
,
N.
,
Veness
,
J.
,
Desjardins
,
G.
,
Rusu
,
A. A.
, …
,
R.
(
2017
).
Overcoming catastrophic forgetting in neural networks
.
Proceedings of the National Academy of Sciences of the USA
,
114
,
3521
3526
. doi:10.1073/pnas.1611835114.
PMID:28292907
.
Klein
,
D.
,
Mok
,
K.
,
Chen
,
J. K.
, &
Watkins
,
K. E.
(
2014
).
Age of language learning shapes brain structure: A cortical thickness study of bilingual and monolingual individuals
.
Brain Lang.
,
131
,
20
24
. doi:10.1016/j.bandl.2013.05.014.
PMID:23819901
.
Knill
,
D. C.
, &
Pouget
,
A.
(
2004
).
The Bayesian brain: The role of uncertainty in neural coding and computation
.
Trends in Neurosciences
,
27
,
712
719
. doi:10.1016/j.tins.2004.10.007.
PMID:15541511
.
Kok
,
P.
,
Rahnev
,
D.
,
Jehee
,
J. F. M.
,
Lau
,
H. C.
, &
de Lange
,
F. P.
(
2012
).
Attention reverses the effect of prediction in silencing sensory signals
.
Cerebral Cortex
,
22
,
2197
2206
. doi:10.1093/cercor/bhr310.
PMID:22047964
.
Kullback
,
S.
, &
Leibler
,
R. A.
(
1951
).
On information and sufficiency
.
Annals of Mathematical Statistics
,
22
,
79
86
. doi:10.1214/aoms/1177729694.
Kuśmierz
,
Ł.
,
Isomura
,
T.
, &
Toyoizumi
,
T.
(
2017
).
Learning with three factors: modulating Hebbian plasticity with errors
.
Current Opinion in Neurobiology
,
46
,
170
177
. doi:10.1016/j.conb.2017.08.020.
PMID:28918313
.
Laje
,
R.
,
Gardner
,
T. J.
, &
Mindlin
,
G. B.
(
2002
).
Neuromuscular control of vocalizations in birdsong: A model
.
Physical Review E
,
65
,
051921
. doi:10.1103/PhysRevE.65.051921.
PMID:12059607
.
Laje
,
R.
, &
Mindlin
,
G. B.
(
2002
).
Diversity within a birdsong
.
Physical Review Letters
,
89
,
288102
. doi:
10.1103/PhysRevLett.89.288102
.
PMID:12513182
.
LeCun
,
Y.
,
Bengio
,
Y.
, &
Hinton
,
G.
(
2015
).
Deep learning
.
Nature
,
521
,
436
444
. doi:10.1038/nature14539.
PMID:26017442
.
Lee
,
T. W.
,
Lewicki
,
M. S.
, &
Sejnowski
,
T. J.
(
2000
).
ICA mixture models for unsupervised classification of non-gaussian classes and automatic context switching in blind signal separation
.
IEEE Transactions on Pattern Analysis and Machine Intelligence
,
22
,
1078
1089
. doi:10.1109/34.879789.
Lipkind
,
D.
,
Marcus
,
G. F.
,
Bemis
,
D. K.
,
Sasahara
,
K.
,
Jacoby
,
N.
,
Takahasi
,
M.
, &
Tchernichorski
,
O.
(
2013
).
Stepwise acquisition of vocal combinatorial capacity in songbirds and human infants
.
Nature
,
498
,
104
108
. doi:10.1038/nature12173.
PMID:23719373
.
Lipkind
,
D.
,
Zai
,
A. T.
,
Hanuschkin
,
A.
,
Marcus
,
G. F.
,
Tchernichovski
,
O.
, &
Hahnloser
,
R. H.
(
2017
).
Songbirds work around computational complexity by learning song vocabulary independently of sequence
.
Nature Communications
,
8
,
1247
. doi:10.1038/s41467-017-01436-0.
PMID:29089517
.
Long
,
M. A.
, &
Fee
,
M. S.
(
2008
).
Using temperature to analyse temporal dynamics in the songbird motor pathway
.
Nature
,
456
,
189
194
. doi:10.1038/nature07448.
PMID:19005546
.
Luck
,
S. J.
,
Woodman
,
G. F.
, &
Vogel
,
E. K.
(
2000
).
Event-related potential studies of attention
.
Trends in Cognitive Sciences
,
4
,
432
440
. doi:10.1016/S1364-6613(00)01545-X.
PMID:11058821
.
Malenka
,
R. C.
, &
Bear
,
M. F.
(
2004
).
LTP and LTD: An embarrassment of riches
.
Neuron
,
44
,
5
21
. doi:10.1016/j.neuron.2004.09.012.
PMID:15450156
.
Mann
,
R. P.
, &
Garnett
,
R.
(
2015
).
The entropic basis of collective behaviour
.
J. Royal Soc. Interface.
,
12
,
20150037
. doi:10.1098/rsif.2015.0037.
PMID:25833243
.
Mante
,
V.
,
Sussillo
,
D.
,
Shenoy
,
K. V.
, &
Newsome
,
W. T.
(
2013
).
Context-dependent computation by recurrent dynamics in prefrontal cortex
.
Nature
,
503
,
78
84
. doi:10.1038/nature12742.
PMID:24201281
.
Markram
,
H.
,
Lübke
,
J.
,
Frotscher
,
M.
, &
Sakmann
,
B.
(
1997
).
Regulation of synaptic efficacy by coincidence of postsynaptic APs and EPSPs
.
Science
,
275
,
213
215
. doi:10.1126/science.275.5297.213.
PMID:8985014
.
Moutoussis
,
M.
,
Trujillo-Barreto
,
N. J.
,
El-Deredy
,
W.
,
Dolan
,
R. J.
, &
Friston
,
K. J.
(
2014
).
A formal model of interpersonal inference
.
Front. Hum. Neurosci.
,
8
,
160
. doi:10.3389/fnhum.2014.00160.
PMID:24723872
.
Ogawa
,
S.
,
Cambon
,
B.
,
Leoncini
,
X.
,
Viltot
,
M.
,
Castillo-Negrete
,
D.
,
,
G.
, &
Garbet
,
X.
(
2016
).
Full particle orbit effects in regular and stochastic magnetic fields
.
Physics of Plasmas
,
23
,
072506
. doi:10.1063/1.4958653.
Paille
,
V.
,
Fino
,
E.
,
Du
,
K.
,
Morera-Herreras
,
T.
,
Perez
,
S.
,
Kotaleski
,
J. H.
, &
Venance
,
L.
(
2013
).
GABAergic circuits control spike-timing-dependent plasticity
.
Journal of Neuroscience
,
33
,
9353
9363
. doi:10.1523/JNEUROSCI.5796-12.2013.
PMID:23719804
.
Parkinson
,
C.
, &
Wheatley
,
T.
(
2015
).
The repurposed social brain
.
Trends in Cognitive Sciences
,
19
,
133
141
. doi:10.1016/j.tics.2015.01.003.
PMID:25732617
.
Pawlak
,
V.
,
Wickens
,
J. R.
,
Kirkwood
,
A.
, &
Kerr
,
J. N.
(
2010
).
Timing is not everything: Neuromodulation opens the STDP gate
.
Frontiers in Synaptic Neuroscience
,
2
,
146
. doi:10.3389/fnsyn.2010.00146.
PMID:21423532
.
Perl
,
Y. S.
,
Arneodo
,
E. M.
,
,
A.
,
Goller
,
F.
, &
Mindlin
,
G. B.
(
2011
).
Reconstruction of physiological instructions from zebra finch song
.
Physical Review E
,
84
,
051909
. doi:10.1103/PhysRevE.84.051909.
PMID:22181446
.
Rao
,
R. P.
, &
Ballard
,
D. H.
(
1999
).
Predictive coding in the visual cortex: A functional interpretation of some extra-classical receptive-field effects
.
Nature Neuroscience
,
2
,
79
87
. doi:10.1038/4580.
PMID:10195184
.
,
S. M.
, &
Laland
,
K. N.
(
2002
).
Social intelligence, innovation, and enhanced brain size in primates
.
Proceedings of the National Academy of Sciences of the USA
,
99
,
4436
4441
. doi:10.1073/pnas.062041299.
PMID:11891325
.
Reynolds
,
J. N.
,
Hyland
,
B. I.
, &
Wickens
,
J. R.
(
2001
).
A cellular mechanism of reward-related learning
.
Nature
,
413
,
67
70
. doi:10.1038/35092560.
PMID:11544526
.
Roweis
,
S.
, &
Ghahramani
,
Z.
(
1999
).
A unifying review of linear gaussian models
.
Neural Computation
,
11
,
305
345
. doi:10.1162/089976699300016674.
PMID:9950734
.
Saalmann
,
Y.
, &
Kastner
,
S.
(
2009
).
Gain control in the visual thalamus during perception and cognition
.
Current Opinion in Neurobiology
,
19
,
408
414
. doi:10.1016/j.conb.2009.05.007.
PMID:19556121
.
,
H.
,
Köhr
,
G.
, &
Treviño
,
M.
(
2012
).
Noradrenergic “tone” determines dichotomous control of cortical spike-timing-dependent plasticity
.
Scientific Reports
,
2
,
417
. doi:10.1038/srep00417.
PMID:22639725
.
Schwarz
,
G.
(
1978
).
Estimating the dimension of a model
.
Annals of Statistics
,
6
,
461
464
. doi:10.1214/aos/1176344136.
Schwing
,
R.
,
Nelson
,
X. J.
,
Wein
,
A.
, &
Parsons
,
S.
(
2017
).
Positive emotional contagion in a New Zealand parrot
.
Current Biology
,
27
,
R213
R214
. doi:10.1016/j.cub.2017.02.020.
PMID:28324733
.
Seol
,
G. H.
, Ziburkus,
Huang
,
S.
,
Song
,
L.
,
Kim
,
I. T.
,
Takamiya
,
K.
, …
Kirkwood
,
A.
(
2007
).
Neuromodulators control the polarity of spike-timing-dependent synaptic plasticity
.
Neuron
,
55
,
919
929
. doi:10.1016/j.neuron.2007.08.013.
PMID:17880895
.
Shipp
,
S.
(
2016
).
Neural elements for predictive coding
.
Frontiers in Psychology
,
7
,
1792
. doi:10.3389/fpsyg.2016.01792.
PMID:27917138
.
Shultz
,
S.
, &
Dunbar
,
R. I. M.
(
2010
).
Species differences in executive function correlate with hippocampus volume and neocortex ratio across nonhuman primates
.
Journal of Comparative Psychology
,
124
,
252
260
. doi:10.1037/a0018894.
PMID:20695656
.
Suh
,
S.
,
Chae
,
D. H.
,
Kang
,
H. G.
, &
Choi
,
S.
(
2016
).
Echo-state conditional variational autoencoder for anomaly detection
. In
Proceedings of the International Joint Conference on Neural Networks
(pp.
1015
1022
).
Piscataway, NJ
:
IEEE
. doi:10.1109/IJCNN.2016.7727309.
Suzuki
,
T. N.
,
Wheatcroft
,
D.
, &
Griesser
,
M.
(
2016
).
Experimental evidence for compositional syntax in bird calls
.
Nature Communications
,
7
,
10986
. doi:10.1038/ncomms10986.
PMID:26954097
.
Taborsky
,
B.
, &
Oliveira
,
R. F.
(
2012
).
Social competence: An evolutionary approach
.
Trends in Ecology and Evolution
,
27
,
679
688
. doi:10.1016/j.tree.2012.09.003.
PMID:23040461
.
Tani
,
J.
, &
Nolfi
,
S.
(
1999
).
Learning to perceive the world as articulated: An approach for hierarchical learning in sensory-motor systems
.
Neural Networks
,
12
,
1131
1141
. doi:10.1016/S0893-6080(99)00060-X.
PMID:12662649
.
Tchernichovski
,
O.
,
Mitra
,
P. P.
,
Lints
,
T.
, &
Nottebohm
,
F.
(
2001
).
Dynamics of the vocal imitation process: How a zebra finch learns its song
.
Science
,
291
,
2564
2569
. doi:10.1126/science.1058522.
PMID:11283361
.
Vossel
,
S.
,
Bauer
,
M.
, &
Mathys
,
C.
(
2014
).
Cholinergic stimulation enhances Bayesian belief updating in the deployment of spatial attention
.
Journal of Neuroscience
,
34
,
15735
15742
. doi:10.1523/JNEUROSCI.0091-14.2014.
PMID:25411501
.
Whittington
,
J. C.
, &
Bogacz
,
R.
(
2017
).
An approximation of the error backpropagation algorithm in a predictive coding network with local Hebbian synaptic plasticity
.
Neural Computation
,
29
,
1229
1262
. doi:10.1162/neco_a_00949.
PMID:28333583
.
Womelsdorf
,
T.
, &
Fries
,
P.
(
2006
).
Neuronal coherence during selective attentional processing and sensory-motor integration
.
Journal of Physiology Paris
,
100
,
182
193
. doi:10.1016/j.jphysparis.2007.01.005.
PMID:17317118
.
Woolley
,
S.
(
2012
).
Early experience shapes vocal neural coding and perception in songbirds
.
Developmental Psychobiology
,
54
,
612
631
. doi:10.1002/dev.21014.
PMID:22711657
.
Wurtz
,
R. H.
(
2008
).
Neuronal mechanisms of visual stability
.
Vision Research, 48
,
2070
2089
. doi:10.1016/j.visres.2008.03.021.
PMID:18513781
.
Yagishita
,
S.
,
Hayashi-Takagi
,
A.
,
Ellis-Davies
,
G. C.
,
Urakubo
,
H.
,
Ishii
,
S.
, …
Kasai
,
H.
(
2014
).
A critical time window for dopamine actions on the structural plasticity of dendritic spines
.
Science
,
345
,
1616
1620
. doi:10.1126/science.1255514.
PMID:25258080
.
Yanagihara
,
S.
, &
Yazaki-Sugiyama
,
Y.
(
2016
).
Auditory experience-dependent cortical circuit shaping for memory formation in bird song learning
.
Nature Communications
,
7
,
11946
. doi:10.1038/ncomms11946.
PMID:27327620
.
Yu
,
A. J.
, &
Dayan
,
P.
(
2005
).
Uncertainty, neuromodulation and attention
.
Neuron
,
46
,
681
692
. doi:10.1016/j.neuron.2005.04.026.
PMID:15944135
.
Zeki
,
S.
, &
Shipp
,
S.
(
1988
).
The functional logic of cortical connections
.
Nature
,
335
,
311
317
. doi:10.1038/335311a0.
PMID:3047584
.
Zelinsky
,
G. J.
, &
Bisley
,
J. W.
(
2015
).
The what, where, and why of priority maps and their interactions with visual working memory
.
Annals of the New York Academy of Sciences
,
1339
,
154
164
. doi:10.1111/nyas.12606.
PMID:25581477
.
Zenke
,
F.
,
Poole
,
B.
, &
Ganguli
,
S.
(
2017
).
Continual learning through synaptic intelligence
.
In Proceedings of the International Conference on Machine Learning
(pp.
3987
3995
). http://proceedings.mlr.press/v70/zenke17a.html.
Zhang
,
J. C.
,
Lau
,
P. M.
, &
Bi
,
G. Q.
(
2009
).
Gain in sensitivity and loss in temporal contrast of STDP by dopaminergic modulation at hippocampal synapses
.
Proceedings of the National Academy of Sciences of the USA
,
106
,
13028
13033
. doi:10.1073/pnas.0900546106.
PMID:19620735
.
## Competing Interests
Competing Interests: The authors declare that they have no competing interests. | 2021-07-31 18:52:46 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 396, "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, "math_score": 0.7008049488067627, "perplexity": 2020.9462232180576}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046154099.21/warc/CC-MAIN-20210731172305-20210731202305-00126.warc.gz"} |
https://www.physicsforums.com/threads/a0-0-0-of-a-vector-space.701911/ | # A0 = 0 (0 of a vector space)
1. Jul 17, 2013
### 1MileCrash
1. The problem statement, all variables and given/known data
Prove that a0 = 0
2. Relevant equations
3. The attempt at a solution
Let V be a vector space on a field F. Let x be a member of V and a be a member of F.
Consider that the 0 vector is the unique vector such that
x + 0 = x
Now, apply a scalar multiplication by a to both sides of the equation. Because scalar multiplication is distributive in all vector spaces,
ax + a0 = ax
Thus, we see that a0 has the same property of the 0 vector in V. Since the 0 vector in V is also unique, it must be the case that
a0 = 0
QED
Can I do that?
2. Jul 17, 2013
### LCKurtz
I don't think you quite have it. Have you shown that for ALL y you have y + a0 = y?
3. Jul 17, 2013
### 1MileCrash
But if
x + 0 = x
is true for all x (which it is by definition of the zero vector) then
ax + a0 = ax
is also true for all x, and it is true for all a (because scalar multiplication is distributive in all vector spaces), so then it is always true, so it is true for any ax (or y).
Right?
Last edited: Jul 17, 2013
4. Jul 17, 2013
### LCKurtz
It may be trivial, but you haven't shown that any y can be written in the form ax.
5. Jul 17, 2013
### 1MileCrash
Oh, ok.
So, I need to justify that ax is also an element of V? Am I understanding that correctly?
If so, then can I just say that in all vector spaces, for each element a in F and each element x in V, there is a unique element ax in V, thus, let ax = y, which is a known element of V (and then the result follows)?
Last edited: Jul 17, 2013
6. Jul 17, 2013
### LCKurtz
You have shown that a0 works as an additive identity on all elements of the form ax. If every y in V can be written that way you have it as an additive identity on all of V, which is what you want. Think about $y = a(\frac 1 a y)$.
As an alternative approach to the problem think about a(0+0) in the first place.
7. Jul 17, 2013
### 1MileCrash
Ok, instead of saying "the result follows" I'll just write it out.
"for each element a in F and each element x in V, there is a unique element ax in V, thus, let ax = y, which is a known element of V (and then the result follows)?"
ax = y
=>
x = (1/a)y
(1/a) is the multiplicative inverse of a and is therefore a member of F. y is already known as a member of V. Therefore any element x of V can be expressed as a scalar multiplication of a member of the field and another member of V.
That's what I was getting at, would you say that is adequate?
Thanks again.
8. Jul 17, 2013
### Zondrina
Using LC's hint :
a0 = a(0+0) = a0 +a0
So you have :
a0 = a0 + a0
Hmm what property of vector spaces would help you here now? There aren't too many to consider so I'm sure you will see it any moment.
9. Jul 17, 2013
### 1MileCrash
I'd rather just complete the attempt I've done myself, this is the proof my book gives.
10. Jul 17, 2013
### LCKurtz
No, I don't think so. At the very least, it is confusing. Summarizing your argument so far you have shown that for any $a\in F$ and ${\bf x} \in V$ that $a{\bf x} + a{\bf 0} = a{\bf x}$. What you need to show is that for any ${\bf y}\in V$ you have ${\bf y} + a{\bf 0} = {\bf y}$. You have to start with $\bf y$. You can't start with $a{\bf x}$ and call it $\bf y$. Look at my suggestion in post #6 again.
11. Jul 17, 2013
### Zondrina
Well you know that $a \in F$ and I would recommend writing $0$ as $0_V$ as to be more explicit.
So really you want to show $a0_V = 0_V$.
$F$ is a field of scalars. Since $V$ is a vector space, it is closed under addition and scalar multiplication. You know that $\forall v \in V$ and $\forall a \in F$ that $av \in V$ because $V$ is closed.
Now, we've shown there are basically infinitely many vectors of the form $av$ inside of $V$. So if you want to proceed your way from before, add $av$ to both sides :
$a0_V + av = 0_V + av$
Now because $0_V$ is the unique vector such that $0_V + v = v, \forall v \in V$ ( Prove this if you havent! ) we can proceed :
$a0_V + av = av$
I'll let you finish that one if you want to. Your argument in your last post wouldn't work though. All you did was set y=ax and say y was in V. LC stated if you can show every $y \in V$ can be written as some $ax$, then you know that $a0$ will work perfectly as an identity.
12. Jul 17, 2013
### 1MileCrash
Why can't I call ax, y? It's just an element of V.
Looking at y=a((1/a)y) makes me fall back to the same reasoning. If y is an element of V then so is (1/a)y, so a((1/a)y) is an expression for y that is a scalar multiplication of an element of F and an element of V. But that's exactly what I did before. What is wrong with this reasoning?
13. Jul 18, 2013
### verty
1MileCrash: I think you can't say for each $a$ because you are given $a$. Given $a$, you must show that a$0_v = 0_v$. I can see a much quicker proof using components.
-edit- Hmm, I suppose this component idea is limited (without choice) to finite-dimensional vector spaces, better to prove this in general.
Last edited: Jul 18, 2013
14. Jul 18, 2013
### LCKurtz
Because you have to start with an arbitrary ${\bf y}\in V$. You have all the pieces but your argument should be written like this:
Suppose ${\bf y}\in V$ and $a \in F$.
Let ${\bf x} =\frac 1 a{\bf y}$
${\bf x} + {\bf 0} = {\bf x}$
$a{\bf x} + a{\bf 0} = a{\bf x}$
$a\frac 1 a{\bf y} +a{\bf 0} = a\frac 1 a{\bf y}$
${\bf y} +a{\bf 0} ={\bf y}$
Therefore $a{\bf 0}$ is the unique additive identity.
15. Jul 18, 2013
### 1MileCrash
Oh, I see, that comes out very nicely. I just thought that since x was an arbitrary vector, then so is ax or y. I don't see a difference between letting ax=y and letting x=(1/a)y.
Thanks again.
16. Jul 18, 2013
### 1MileCrash
EDIT: nevermind, got it backwards.
17. Jul 19, 2013
### verty
There is still a mistake here. Given a, we aren't told that it isn't 0.
Gotta keep your apples in order. If you start with a given $a \in F$, you must be faithful to that.
18. Jul 19, 2013
### 1MileCrash
I have no idea what you are saying. Could you explain how I have not done that? Or what that means?
All I did was define a to be an element of F so that I could use it in an expression without the reader not knowing what it is. What do you mean by "starting" with it? | 2017-08-19 15:56:08 | {"extraction_info": {"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, "math_score": 0.7824442386627197, "perplexity": 437.5558255621848}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886105455.37/warc/CC-MAIN-20170819143637-20170819163637-00054.warc.gz"} |
https://mathcracker.com/receivables-turnover-calculator | # Receivables Turnover Calculator
Instructions: You can use this Receivables Turnover Calculator, by providing the Sales, the current accounts receivables and the previous accounts receivables in the form below:
Sales =
Current Accounts Receivable =
Previous Accounts Receivable =
## Receivables Turnover Calculator
More about the Receivables Turnover so you can better use the results provided by this solver. The Receivables Turnover is the ratio between sales and the average accounts receivables. This ratio is a measure of asset management, and it roughly indicates how many times a year a firm collects its outstanding credit accounts. In order to calculate the Receivables Turnover, we use the following formula:
$\text{Receivables Turnover} = \displaystyle \frac{\text{Sales}}{\text{Average Accounts Receivables}}$
The Receivables Turnover is a broadly used financial ratio to measure efficiency in credit accounts management. We provide many other financial ratio calculators in our site, including our current ratio, quick ratio, our days' sales in receivables, and our inventory turnover calculator.
In case you have any suggestion, or if you would like to report a broken solver/calculator, please do not hesitate to contact us. | 2020-04-10 08:17:48 | {"extraction_info": {"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, "math_score": 0.18652279675006866, "perplexity": 4755.807517720333}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585371893683.94/warc/CC-MAIN-20200410075105-20200410105605-00091.warc.gz"} |
http://tex.stackexchange.com/tags/longtable/new | # Tag Info
2
This uses the titlesec package to build on David's suggestion and converts the use of \newenvironment to \newcommand since you want a command rather than an environment judging by the syntax you wish to use. \documentclass[12pt]{report} \usepackage{titlesec} \titleformat{\section}[block]{\centering\bfseries\sffamily\large}{\thesection}{0pt}{}{} ...
5
Use the @ operator for setting the margin: \documentclass{memoir} \usepackage{longtable} \usepackage{lipsum} \setlength{\footmarkwidth}{\leftmargin} \setlength{\footmarksep}{0em} \footmarkstyle{#1\hfill} \begin{document} \lipsum[1-4] \begin{longtable}[l]{ @{\hspace*{\leftmargin}} ll} Chapters & Description\\ 1--13 & ...
2
Package ltxtable allows the combination of longtable with tabularx column type X. The latter is useful for the last column with the description text. The first columns can be set via column type l. The indentation in the description part can be achieved via \hangindent and \hangafter (assuming there is only one paragraph). Since the package ltxtable ...
4
It is a common misunderstanding about m it does not mean place the content in the middle of the vertical space for that cell. It means: place the reference point for the content in the middle of the content. I added a line to your image showing the reference points of the cells in the second row, the reference point of column 1 is the middle of the 40 ...
2
I redefined \LT@makecaption from longtbale so as to typeset the captions in a similar way that svjour3 typesets table captions: \documentclass{svjour3} \usepackage{longtable} \makeatletter \def\LT@makecaption#1#2#3{% \LT@mcol\LT@cols c{\hbox to\z@{\hss\parbox[t]\LTcapwidth{% \captionstyle ...
0
You were doubling some vertical rules inside the \multicolumns and this was giving you thicker rules: \documentclass{article} \usepackage[table]{xcolor} \usepackage{multirow} \usepackage{longtable} \usepackage{array} \begin{document} \begin{longtable}{| >{\centering}p{4cm} | p{2cm} | p{7cm} |} \hline \multicolumn{3}{| >{\centering}p{14cm} ...
0
This response was possible with the help of @UlrikeFischer comments. Remove caption package is the solution. To conserve the caption's option, use the babel option like captionsfrench. % !TEX encoding = UTF-8 Unicode % !TEX TS-program = arara \documentclass{scrbook} %\usepackage{caption}\captionsetup[table]{name=\scshape Tableau, position=bottom} ...
2
You can use p-width command line option to save width of p columns: htlatex filename "xhtml,p-width" This doesn't support m columns by default, but you can add support for them easily with this configuration file, my.cfg: \Preamble{xhtml,p-width} \catcode`\:=11 \Configure{halignTD} {}{} {<}{\HCode{ style="white-space:nowrap; text-align:left;"}} ...
0
As mentioned in the comments, a tabular construct is not designed to allow text to flow from one column to another. Tables have a fixed number of columns and the text is in one of those columns. Thus you will need to rethink either your input format (a multicolumn approach rather than a tabular one) or your output expectations.
3
Some suggestions: The main issue that's making the longtable wider than the text block is the material in the header row. To shrink the width of the cells (without outright deletion of any material...), you need to reduce the value of \thickmuskip, the parameter that governs the amount of whitespace that's inserted around the < symbols. In the example ...
Top 50 recent answers are included | 2015-07-05 17:58:06 | {"extraction_info": {"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, "math_score": 0.8833115696907043, "perplexity": 4368.347298169662}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-27/segments/1435375097546.28/warc/CC-MAIN-20150627031817-00022-ip-10-179-60-89.ec2.internal.warc.gz"} |
https://www.lessonplanet.com/teachers/math-fact-cafe-fractions-common-denominator-2 | # Math Fact Cafe; Fractions - Common Denominator #2
Give your students the chance to practice their basic fraction addition skills with this activity, which consists of fifteen problems (all with common denominators). Denominators and numerators are all single digit numbers. The answers may be proper or improper fractions, which allows your students to try simplifying their answers. | 2017-05-25 21:18:24 | {"extraction_info": {"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, "math_score": 0.8231298923492432, "perplexity": 1962.9431180463123}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-22/segments/1495463608416.96/warc/CC-MAIN-20170525195400-20170525215400-00310.warc.gz"} |
https://dspace.kaist.ac.kr/handle/10203/27426 | #### Function and regulation of D-xylose operons in escherichia coli K-12 = 대장균 D-자일로즈 오페론의 기능과 발현조절
Cited 0 time in Cited 0 time in
• Hit : 309
Mutants defective in the high-affinity transport for D-ribose are still able to utilize the sugar as a sole carbon and energy source, suggesting that other low-affinity transport systems for the sugar exist in the cells. In order to search for these transport systems, transposon mutagenesis for the high affinity transporter-defective cell was performed and several dozens of candidates showing different phenotypes on D-ribose minimal growth were isolated with a frequency of $1 \times 10^{-5}$. The precise locations of insertions on the chromosome were determined by in vivo cloning and analysis using polA mutation. By genetic and biochemical studies of the mutants it was revealed that two transport systems for D-xylose (XylFGH) and D-allose (AlsBAC) are involved in the uptake of D-ribose into cells with low affinity. In case of Als system, its contribution to D-ribose growth is elevated by a point mutation in the negative regulator alsR. Atypical insertions showing an enhanced D-ribose growth were appeared and mapped at xylA and its promoter region. The mutations elevated not only the uptake rate for D-ribose but also the expression of xylFGH encoding high-affinity transporter for D-xylose. Suppressor mutations resulting in the basal growth were obtained in xylF, xylG, and xylR. The elevated level of D-ribose uptake was lowered to basal level both in the presence of D-xylose and by the suppressor mutations. Also, a constitutive expression of xylFGH under $P_{trc}$ promoter enhanced the growth for D-ribose. These results indicate that XylFGH transports D-ribose into cell. The metabolism of D-xylose in Escherichia coli K-12 was known to be mediated by xylAB gene. However, the xylFGH and xylR genes were recently found by genome sequencing and predicted for transport and regulation based on their sequence similarities. The functions and regulation of the xyl genes were also investigated. An analysis with transposon insertions in xyl genes revealed that the xyl genes ...
Park, Chan-Kyuresearcher박찬규researcher
Description
한국과학기술원 : 생물과학과,
Publisher
한국과학기술원
Issue Date
1998
Identifier
135089/325007 / 000935191
Language
eng
Description
학위논문(박사) - 한국과학기술원 : 생물과학과, 1998.2, [ viii, 136 p. ]
URI
http://hdl.handle.net/10203/27426 | 2021-06-21 23:36:28 | {"extraction_info": {"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, "math_score": 0.3133993148803711, "perplexity": 8532.1677977754}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623488504838.98/warc/CC-MAIN-20210621212241-20210622002241-00293.warc.gz"} |
https://physics.stackexchange.com/questions/523434/if-the-running-coupling-constant-alpha-mu-of-qed-becomes-of-order-one-at-hi | # If the running coupling constant $\alpha(\mu)$ of QED becomes of order one at high $\mu$, why not changing $\mu$?
In the (modified) MS renormalization scheme, after dimensional regularization, we introduce some parameter $$\mu$$ with power of mass to keep the dimensionality of integrals under control. The parameters in the lagrangian end up being functions of $$\mu$$, and from the beta functions we can compute how they change when we change the "scale" $$\mu$$.
Now that's the point: we say that perturbation theory "breaks down" at high energy in QED because, if we identify $$\mu^2 = s$$, we end up with an effective fine structure constant which is of order one, hence of course we can no longer make use of perturbation theory.
If that's the case, why do we choose such a value of $$\mu$$? If it's a fake parameter that we put in our theory to "fix" an issue we have with dimensional regularization, why don't we pick a different value of $$\mu$$ which is very different from $$s$$, rendering $$\alpha(\mu)$$ very small?
Forget about the RG flow for a moment. Pick a value of $$\mu$$ so that $$\alpha(\mu)$$ is small. Now, fixing that small $$\alpha$$, take a look at a scattering process at high enough energy $$M$$ so that $$\alpha(M)$$ would be large (but still use your small $$\alpha$$). You will find that the 1-loop correction to your scattering process is large compared to the tree level diagram. This must happen because calculating 1-loop corrections is exactly how we derived the beta function after all. So perturbation theory is still breaking down even if you keep a small $$\alpha(\mu)$$.
• So if we choose $\mu$ such that $\alpha(\mu)$ is small, the actual contributions from loop diagrams (coefficients multiplying $\alpha$, $\alpha^2$, ...) become large? – user35319 Jan 6 at 20:58
• @user35319, Exactly, you typically get something like a factor $\log (M/\mu)$ that ends up making perturbation theory still break down – octonion Jan 7 at 19:15
It's the $$\alpha(Q)$$ at the high momentum/energy scale $$Q$$ (not the renormalization scale $$\mu$$!) of the physical process that invalidates the perturbation theory.
"Running" is actually in terms of the tangible scale of the scattering/physical process, which is $$Q$$ rather than some obscure "renormalization scale $$\mu$$" (text book QFT view) or "cutoff scale $$\Lambda$$" (Wilsonian view). See “Running with momentum p” v.s. “running with renormalization scale μ” for more details.
In QFT, there are 5 different mass scales (See here for more details), namely,
• m: the mass of the particle in concern
• $$\Lambda$$: the UV cutoff scale of the regularization scheme
• Q: the energy scale of the incoming/outgoing particles involved in a scattering process.
• $$\mu$$: the renormalization scale
• M:the mass scale where beyond standard model physics effect comes into the picture.
If you manage to appreciate the differences and relationships between these 5 mass scales, you would have a clearer picture of QFT. | 2020-09-20 15:32:42 | {"extraction_info": {"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": 23, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8975611329078674, "perplexity": 460.9760156887399}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400198213.25/warc/CC-MAIN-20200920125718-20200920155718-00298.warc.gz"} |
https://cs.stackexchange.com/questions/119245/from-2in3sat-to-nae3sat-not-all-equal | # From 2IN3SAT to NAE3SAT ( not all equal )
We know that 1IN3SAT is np-complete , I will define an ne form of the SAT problem 2IN3SAT (exactly two true variables ). 2IN3SAT is np-complete because (x,y,z) is in 1IN3SAT iff (!x,!y,!z) is in 2IN3SAT ,
Let c be a new variable. For each clause (x,y,z) in a 2IN3SAT problem let the 7 clauses below forming a NAE3SAT problem. (x,y,z) ,(c,y,z) ,(x,c,z) ,(x,y,c) ,(!c,!y,!z) ,(!x,!c,!z) ,(!x,!y,!c) (*)
I think that, if we use the same c for all clauses of the initial 2IN3SAT the correspendant NAE3SAT will have the property. 2IN3SAT has a solution if and only if NAE3SAT has a solution.
Do you agree with this statement ? | 2020-02-22 22:35:47 | {"extraction_info": {"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, "math_score": 0.8460163474082947, "perplexity": 4350.406212337886}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875145729.69/warc/CC-MAIN-20200222211056-20200223001056-00502.warc.gz"} |
http://physicshelpforum.com/special-general-relativity/13273-what-does-following-equations-prove-when-equated.html | Physics Help Forum What does the following equations prove when equated?
Special and General Relativity Special and General Relativity Physics Help Forum
Jun 23rd 2017, 07:58 PM #1 Senior Member Join Date: Feb 2017 Posts: 205 What does the following equations prove when equated? We know that einsteins formula states E=mc2. Now this is the potential energy. So PE= mgh= mc2. So gh=c2. Does it prove that speed of light is gravitational constant into height of the object?
Jun 23rd 2017, 08:56 PM #2 Senior Member Join Date: Apr 2017 Posts: 486 Well it's true to say , in a way, that E=mc2 shows the energy potential of matter ... But this is not the same as the potential energy matter has by having height in a gravitational field .. the two are not the same, so it's not true to say mgh = mc2
Jun 23rd 2017, 09:07 PM #3 Senior Member Join Date: Nov 2013 Location: New Zealand Posts: 550 PE = mgh is an approximate formula for gravitational potential energy. More accurate would be PE = -GmM/r where M is mass of the earth. Most accurate is Einstein's general theory of relativity and gravity. E = mc^2 is more about how mass changes to a "stationary" object if it radiates or loses energy. See this video: and In brief it appears like you're trying to work out the effect of relativistic mass in earths gravitational field but the formulas E=mc^2 and PE = mgh don't seem to me to be in the same league so I'm not sure why someone might do that.
Jun 24th 2017, 11:01 PM #4
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Originally Posted by avito009 We know that einsteins formula states E=mc2. Now this is the potential energy. So PE= mgh= mc2. So gh=c2. Does it prove that speed of light is gravitational constant into height of the object?
In general relativity the energy of a body is given by the time component of the time component of the canonical momentum, P[sub]0[/sub]. For a particle of proper mass m[sub]0[/sub] it has the value
P[sub]0[/sub] = m[sub]0[/sub]g[sub]0v[/sub]P[sup]v[/sup] where you sum over v = 0, ..., 3.
The gravitational potential is part of g[sub]0v[/sub]. However its not true that P[sub]0[/sub] = mc[sup]2[/sup]. In fact E = mc[sup]2[/sup] is not an expression which holds in general.
Jun 25th 2017, 05:05 AM #5 Senior Member Join Date: Feb 2017 Posts: 205 Can we equate these? If E=mc2 =mgh cant be true then can GMm/r2=mc2 be true?, now m cancels out so we are left with GM/r2=c2. But we know that speed of light is 300000 km/s. If we solve left hand side answer is GM/r2= 9.79811147 m / s2 which doesnt match to speed of light so we cant equate these formulas also. So which formula of potential energy can we equate with mc2?
Jun 25th 2017, 01:09 PM #6
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Originally Posted by avito009 If E=mc2 =mgh cant be true then can GMm/r2=mc2 be true?
Don't try to understand physics as though you were simply trying to understand how to understand algebra. Understand the physical concepts first and then let maths follow. I'm a believer in that mathematics is a tool, a "servant" rather than an "oracle". (Maybe everyone may not agree with me on this but it has helped me). In other words the understanding of the physical processes should lead the maths, not the other way around.
Did you see the video above that E=mc^2 is incomplete. Its only half the story. It constitutes only the energy radiated from mass, not the kinetic energy. What you are probably after is Total Energy E = Kinetic energy + potential energy and total energy is always conserved (can't be created or destroyed). So a more useful equation is mv^2/2 = mgh or mv^2/2 = GMm/r (not over r^2, one is a force, the other is an energy potential).
This video animation is the best one I found that explained energy.
Jun 25th 2017, 02:35 PM #7 Forum Admin Join Date: Apr 2008 Location: On the dance floor, baby! Posts: 2,684 $\displaystyle mc^2$ isn't a potential energy. As PMB and kiwiheretic were saying, $\displaystyle mc^2$ is the energy of a mass that is at rest. If the particle moves you have to account for any momentum. In this respect $\displaystyle E^2 = (mc^2)^2 + (pc)^2$. This reduces to $\displaystyle E = \pm mc^2$ when p = 0. -Dan kiwiheretic likes this. __________________ Do not meddle in the affairs of dragons for you are crunchy and taste good with ketchup. See the forum rules here.
Jun 26th 2017, 12:43 AM #8 Senior Member Join Date: Feb 2017 Posts: 205 My interpretation So E is the hypotenuse and the base is pc. So base cannot be more than hypotenuse. So let us assume E= pc. so after substituting mc2 in E which gives us mc2=pc now substitute mv into p this gives us mc2= mvc after cancelling what remains is v=c. But we know that velocity of an object cant be equal speed of light. so E is not equal to pc. But in that case E=pc in case of light? Am i right? Last edited by avito009; Jun 26th 2017 at 01:24 AM.
Jun 26th 2017, 06:34 PM #9
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Originally Posted by avito009 So E is the hypotenuse and the base is pc. So base cannot be more than hypotenuse. So let us assume E= pc. so after substituting mc2 in E which gives us mc2=pc now substitute mv into p this gives us mc2= mvc after cancelling what remains is v=c. But we know that velocity of an object cant be equal speed of light. so E is not equal to pc. But in that case E=pc in case of light? Am i right?
In terms of high speeds near speed of light we need to take into account relativistic mass so $\displaystyle p = mv$ becomes $\displaystyle p = \frac{ m_{0}v}{\sqrt{1 - \frac{v^{2}}{c^{2}}}}$ where m0 is rest mass. If you are talking about massless particles like photons then it is correct to use E=pc. Otherwise, if the particle has mass, then $\displaystyle E=m c^2$ becomes $\displaystyle E=\frac{m_{0} c^{2}}{\sqrt{1 - \frac{v^{2}}{c^{2}}}}$ and the m0 is again rest mass moving at velocity v. (Note: The E we are talking about here is kinetic, not potential energy.)
Now for low speeds much slower than light we can approximate $\displaystyle E=\frac{m_{0} c^{2}}{\sqrt{1 - \frac{v^{2}}{c^{2}}}} \approx \frac{m_{0} v^{2}}{2} + m_{0} c^{2} + \mathcal{O}\left(v^{4}\right)$ using Taylor's series expansion. This shows that for low speeds $\displaystyle \frac{m_{0} v^{2}}{2}$ is a good approximation of kinetic energy.
Hence for low speeds and for ordinary physical conditions like dropping a ball off a cliff then using $\displaystyle E_{kinetic} = \frac{m_{0} v^{2}}{2}$ with $\displaystyle E_{potential} = m g h$ along with conservation of energy principles makes sense. However. trying to use relativistic formulae in such cases is "overkill" and is a pain to calculate minor increases in inertial mass for low velocity when $\displaystyle \frac{m_{0} v^{2}}{2}$ does just as well. So remember when you see E=mc^2 it is not a formula about rest mass (m0) but the inertial mass ($\displaystyle m = \frac{m_{0}}{\sqrt{1 - \frac{v^{2}}{c^{2}}}}$ ) which is not a constant for relativistic velocities and is generally not a formula we would use for low velocities when the difference between rest mass and inertial mass are negligible.
Last edited by kiwiheretic; Jun 26th 2017 at 06:40 PM.
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https://math.stackexchange.com/questions/2130763/structure-of-z-nz/2131778 | # Structure of $Z/nZ$
I am trying to improve my understanding of structure of $Z/nZ$.
Facts I know so far:
1. $Z/nZ \cong \oplus_i Z/p_i^{k_i}$ - CRT(1).
2. $(Z/nZ)^* \cong \oplus_i (Z/p_i^{k_i})^*$. Not quite sure about my proof of this fact.(2)
Proof: We have $Z/nZ \cong \oplus_i Z/p_i^{k_i}$ and its isomorphism $\phi$ : $r \to (r_1, \dotso, r_m)$. Since homomorphism preserves inverses then $r^* \to (r_1^*, \dotso, r_n^*)$, where each $r_i$ must be a unit. Hence, every $r^*$ has its own unique (because of CRT isomorphism) inverse element.
3. $(Z/pZ)^*$ is cyclic of order $p - 1$. I don't understand this at all. Why its order is $p - 1$ and why it is cyclic?(3)
4. From previous statement we know $(Z/pZ)^*$ is cyclic: $(Z/p^{k}Z)^* = C_{p^{k} \ (p - 1)}$ , since $\phi(p^k) = p^{k}\cdot (p - 1)$.(4)
There is also a special case, when $p = 2$.
$(Z/2^{k}Z)^* = C_2 \times C_{2^{k - 2}}$ . It can be proven using the fact $ord\left<-1\right> = 2$, $ord\left<5\right> = 2^{k - 2}$ and $\left<5\right> \cap \left<-1\right> = {e}$.
From these facts I can conclude:
1. $(Z/pZ)^* \cong C_{p_2^{k_2} \ (p_2 - 1)} \times \dotso \times C_{p_n ^{k _ n} \ (p_n - 1)}\ \text{, if } (n \mod 2 = 0) \land (n \mod 4 \neq 0) \text{ note that in this case } C_{p_1 ^{k _ 1} \ (p_1 - 1)} = C_1 \text{ ,so the first term cancels out.}$.
2. $(Z/pZ)^* \cong C_2 \times C_{2^{k_1 - 2}} \times C_{p_2 ^{k _ 2} \ (p_2 - 1)} \times \dotso \times C_{p_n ^{k _ n} \ (p_n - 1)}$, since $(Z/2^{k}Z)^* = C_2 \times C_{2^{k - 2}}$, when $(n \mod 8 = 0)$
3. $(Z/pZ)^* \cong C_{p_1^{k_1} \ (p_1 - 1)} \times \dotso \times C_{p_n ^{k _ n} \ (p_n - 1)}$ , when $(n \mod 2 \neq 0) \ \lor ((n \mod 8 \neq 0) \land (n \mod 4 = 0))$, because of (2).
My questions are:
Can you please verify my conclusion?
Why $(Z/pZ)^*$ is cyclic of order $p - 1$?
How we can prove the fact: $(Z/pZ)^*$ is cyclic $\Rightarrow n = 2, 4, p^k, 2p^k$?
• By Euler criterion in number theory we have $a^{p-1}\equiv 1(mod p)$ – Mustafa Feb 5 '17 at 20:05
• Why there is no $m \in N\ m < p - 1$, such that $a^m = 1 \mod p$? – False Promise Feb 5 '17 at 20:23
• In general if $(a,m)=1$ then $a^{\phi(m)} \equiv 1(mod m)$ where $\phi(m)$ is Euler function – Mustafa Feb 5 '17 at 20:38
• Your statement in 4) is incorrect : We have that $\phi(p^k) = p^{k-1}(p-1)$. $\phi(n)$ is obviously smaller than $n$. – Marc Bogaerts Feb 6 '17 at 12:36
• @FalsePromise There are : if $p-1 = mn$ with $m < p-1$ and $b = a^n$ then $b^m = 1$.. In $C_2 \times C_2$ we have for each element $a^4$ but the group is not cyclic. – Marc Bogaerts Feb 6 '17 at 14:08
Why $(\Bbb Z/p\Bbb Z)^∗$ is cyclic of order $p−1$?
It should be clear that $p$ is prime. First of all note that $\Bbb Z/p\Bbb Z$ is a field, so every nonzero element is inverible. Let $k < p-1$ be the maximal order of the elements of $\Bbb Z/p\Bbb Z^*$. Then $\forall a \in \Bbb Z/p\Bbb Z^* : a^k = 1$. But this means that the polynomial $x^k-1$ of degree $k$ has $p-1 >k$ distinct roots, a contradiction, so $k = p-1$. So there is an element $a$ of order $p-1$. Such an element is called a primitive root.
How we can prove the fact: $\Bbb Z/p\Bbb Z$ is cyclic $\implies$ $n=2,4,p^k,2p^k$?
This question is rather ambiguous. I suppose that you ask for a proof that $\Bbb Z/n\Bbb Z$ is cyclic for $n=2,4,p^k,2p^k$ for an odd prime $p$. The cases 2 and 4 are easily proved by hand. In the case $n=p^k$ we have the following : If $a \in \Bbb Z_n^*$ (a simpler way of writing $\Bbb Z/n\Bbb Z^*$) is a primitive root of $\Bbb Z_p$ then either $a$ or $a+p$ is a primitive root of $\Bbb Z_{p^2}$, and if $a$ is a primitive root of $\Bbb Z_{p^2}$ then $a$ is a primitive root of $\Bbb Z_{p^k}$ for $k >2$. You can find these proofs here.
• It does /not/ imply that the two polynomials are identical (which I'm assuming means term-by-term equal). It means that the two polynomial have the same roots. But the larger one has all distinct roots by construction, contradiction. – Stella Biderman Feb 6 '17 at 13:56
• I agree, but that was implicitely implied: a polynomial of degree $< p-1$ having $p-1$ distinct roots. – Marc Bogaerts Feb 6 '17 at 14:01
• I think that the way it's phrased now is misleading, and that the term "identical" specifically should be removed. You could also say "... but this means that $x^k-1$ has $p-1>k$ distinct roots, contradiction." But that's not what I get out of reading what you've written. – Stella Biderman Feb 6 '17 at 14:18
• @StellaBiderman: I've edited it so there will be less confusion. – Marc Bogaerts Feb 6 '17 at 16:55
• @MarcBogaerts Actually the fact that the order of $(Z/nZ)^*$ is $p - 1$ clearly follows from the formula $\phi(p^k) = p^{k-1}(p-1) = p^k - p^{k - 1} < p^k$ (about your comment), since we define $\phi$ as the order of $(Z/nZ)^*$. In our case we got $\phi(p) = p^0\cdot (p - 1) = p - 1$. The fact that this is cyclic can be proven using the definition of exponent of a group (I'm not sure, but it seems to me that you're using this concept in your proof as $k$). And then, yes, I have to make a conclusion about number of roots etc. It all ends up that exponent of a group is its order. – False Promise Feb 7 '17 at 2:37 | 2019-09-21 15:58:27 | {"extraction_info": {"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, "math_score": 0.9213683009147644, "perplexity": 140.36049314792677}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514574532.44/warc/CC-MAIN-20190921145904-20190921171904-00391.warc.gz"} |
http://mathhelpforum.com/discrete-math/185789-relative-sizes-sets-classes.html | # Thread: relative sizes of sets and classes
1. ## relative sizes of sets and classes
One often talks of the cardinality of a model (or more exactly, the cardinality of the universe of the model), such as when one is referring to countable models. Since the universe is, in this model, not a set to which one can assign cardinality, presumably this is referring to the cardinality of this universe when it is embedded in another universe in which the erstwhile proper class becomes a set. So far, so good?
Then one says that a class that has the same cardinality with the universe must be a proper class. For example, the class of ordinals. However, what does one do about the infinite set in a countable model of ZF? The universe and the set are both countable, yet the infinite set is not a proper class.
2. ## Re: relative sizes of sets and classes
One often talks of the cardinality of a model (or more exactly, the cardinality of the universe of the model), such as when one is referring to countable models. Since the universe is, in this model, not a set to which one can assign cardinality [...]
Usually, the universe for a model IS a set.
3. ## Re: relative sizes of sets and classes
Sorry for the delay in my response.
Usually, the universe for a model IS a set.
From the point of view of another model and another theory, yes. However, if <M,R> |= T, with M being the universe and T being a first-order theory, then the sets designated by T are elements of M, and M itself is not a member of itself. Hence, with respect to T, M is not designated as a set; hence it is a proper class.
4. ## Re: relative sizes of sets and classes
From the point of view of another model and another theory, yes. However, if <M,R> |= T, with M being the universe and T being a first-order theory, then the sets designated by T are elements of M, and M itself is not a member of itself. Hence, with respect to T, M is not designated as a set; hence it is a proper class.
I don't know see a rigorous notion at work in "universe M of a model <M R> has property P with respect to theory T".
I'll restrict the context of the following to ordinary set theory (such as Z set theory and its ordinary extensions) and ordinary class theory (such as NBG).
[notation: '0' for the empty set, 'A' for the universal quantifier, 'E' for the existential quantifier, 'e' for 'is a member of']
x is a class <-> (x=0 or Ey yex)
x is a set <-> (x is a class & Ey xey)
x is a proper class <-> (x is a class ~Ey xey)
Now in oridinary set theory we may prove
Ax x is a set
and in ordinary class theory we may prove
Ex x is a proper class.
So, in set theory, when we refer to certain proper classes, that is only informal for schemata in the meta-theory in which we refer to formulas. For example, the locution 'the class of ordinals' refers really to the formula 'x is an ordinal'.
Now, suppose we're using a formal (or even informal) set theory as our theory H in which to do mathematical logic. In H, we define 'model for a language', 'model of a theory', etc. And in so doing, we have no choice but that the universe of the model is a set, because ANYTHING we refer to in that theory is a set.
Now, suppose we're using a formal (or even informal) class theory as our theory J in which to do mathematical logic. Now we may refer to things that are not sets, but still, if I'm not mistaken, the definition of 'model' requires that the universe is a set. Indeed even saying <U R>, where U is the universe, requires that U is a set, since U can't be a proper class while also being the first coordinate of the ordered pair <U R>.
Now, there might be a way to develop models in a class theory so that the universe of a model may be a proper class, but I don't happen to know how it would be done. Also, I don't know how one would express and then prove things (such as the completeness theorem) using models with universes that may be a proper class.
What I've said also does not contradict the informal notion of proper classes as universes for proving such things as relative consistency theorems. For example, using L (the constructible universe) and things like that. Again, in such cases, this resolves to mention of formulas (such as relativization of formulas to a "proper class" as the "proper class" is not itself an object but rather a locution for the relativizing formula "x is constructible").
Finally, as to your notion of a universe being a proper class with regard to a theory T, recall that a theory is syntactical and a consistent theory admits of lots of DIFFERENT models. So while the universe U of a given model of T is not a member of itself (given the axiom of regularity), it is not precluded that that U may be a member of a different universe for a different model of T. And, back to the original point, the cardinality of U is not affected by whether it is or is not a universe of any given theory T.
In sum, where we use either set theory or class theory as our theory in which to do mathematical logic (including the subject of models), a proper class does not have a cardinality, while every set (with the axiom of choice) does have a cardinality equinumerous with that set. For example, suppose w (the set of natural numbers) is the universe of some given model. The fact that w is not in w does not in the least detract from the theorem that w has a cardinality, and irrespective of any theory of which w may be the universe of a model of that theory.
5. ## Re: relative sizes of sets and classes
The explanation was very clear, thanks. (I have been rather busy this last month, so I only now am getting around to thanking you (MoeBlee) for the explanation.) | 2016-09-25 04:26:02 | {"extraction_info": {"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, "math_score": 0.813387393951416, "perplexity": 460.0066784724233}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-40/segments/1474738659833.43/warc/CC-MAIN-20160924173739-00053-ip-10-143-35-109.ec2.internal.warc.gz"} |
https://lifewithdata.com/2022/06/25/how-to-generate-random-integers-between-0-and-9-in-python/ | # How to Generate Random Integers Between 0 and 9 in Python?
## Problem –
You want to generate random integers between 0 and 9 (inclusive) in Python.
## Solution –
### Using random.randrange –
To generate random integers between 0 and 9, you can use random.randrange
In [1]: from random import randrange
In [2]: randrange(10)
Out[2]: 5
In [3]: randrange(10)
Out[3]: 5
In [4]: randrange(10)
Out[4]: 1
### Using random.randint –
You can also use random.randint
In [5]: from random import randint
In [6]: randint(0, 9)
Out[6]: 5
In [7]: randint(0, 9)
Out[7]: 1
Rating: 1 out of 5. | 2022-12-05 18:50:00 | {"extraction_info": {"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, "math_score": 0.20494312047958374, "perplexity": 7761.839817876292}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711042.33/warc/CC-MAIN-20221205164659-20221205194659-00764.warc.gz"} |
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# Jill has received 8 of her 12 evaluation scores. So far, Ji
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Jill has received 8 of her 12 evaluation scores. So far, Ji [#permalink] 23 May 2017, 02:50
Expert's post
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Question Stats:
51% (01:48) correct 48% (02:02) wrong based on 31 sessions
Jill has received 8 of her 12 evaluation scores. So far, Jill’s average (arithmetic mean) is 3.75 out of a possible 5. If Jill needs an average of 4.0 points to get a promotion, which list of scores will allow Jill to receive her promotion?
Indicate all such sets.
❑ 3.0, 3.5, 4.75, 4.75
❑ 3.5, 4.75, 4.75, 5.0
❑ 3.25, 4.5, 4.75, 5.0
❑ 3.75, 4.5, 4.75, 5.0
[Reveal] Spoiler: OA
_________________
Last edited by Carcass on 27 Dec 2018, 04:26, edited 1 time in total.
Edited by Carcass
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Joined: 03 Sep 2017
Posts: 521
Followers: 1
Kudos [?]: 344 [1] , given: 66
Re: Jill has received 8 of her 12 evaluation scores. So far, Ji [#permalink] 10 Oct 2017, 08:33
1
KUDOS
The mean of 8 grades being 3.75 permits us to compute the sum of those 8 grades as 8*3.75 = 30. Then, we can write an equation which gives us the sum of the last four grades, as $$\frac{30+x}{12} = 4$$ from which we know x = 18, the sum of the last four grades.
Now we just have to check which choices sum to 18. B and D sum to 18, thus they are the answers!
Re: Jill has received 8 of her 12 evaluation scores. So far, Ji [#permalink] 10 Oct 2017, 08:33
Display posts from previous: Sort by | 2019-03-25 16:16:56 | {"extraction_info": {"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, "math_score": 0.4513339102268219, "perplexity": 4207.447035252274}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912204077.10/warc/CC-MAIN-20190325153323-20190325175323-00263.warc.gz"} |
https://docs.botto.com/details/voting-mechanism | Docs
Search…
Voting
Botto utilizes a voting mechanism to train the algorithm that creates the art. Every week, 350 art pieces (called fragments) are presented to the community. Individuals from the community independently vote between two pieces, stating which art fragment they believe to be more aesthetically pleasing.
When you stake $BOTTO and/or provide liquidity for the first time, you are automatically rewarded with 100VP. ## VP Regeneration VP regenerate over time for as long as you have your$BOTTO or LP tokens staked. The regeneration formula for VP is as follows:
$VP_{earning} = VP_{base} + VP_{staked}$
Where the base rate earned for all stakers (
$VP_{base}$
) and the additional VPs earned per amount staked (
$VP_{staked}$
) are:
$VP_{base} = 25 * stakedDays$
$VP_{staked} = 25 * log^{10}(stakedAmount) * stakedDays$
For LP tokens, the stakedAmount is:
$stakedAmount_{LPs}=\frac{LP staked}{LP price in BOTTO}$
## Voting Mechanism
### Voting Page
Botto's voting page displays two art fragments. The voter is required to choose which of the two art fragments is their preferred one. They can also choose to skip if they really don’t like either one, however even finding a preferable element or characteristic can be good for training.
Pieces are given a score for the proportion of how often they are picked:
$score_{views<50}=100$
$score_{views>50}=(uniqueVotes / views ) * 100$
80% of the time, the voting mechanism will show a fragment with a top score
20% of the time it will show a low score to ensure they still have a chance to be seen and maintain variety
The score of 100 if seen less than 50 times ensures all pieces have a fair chance of being seen and selected before getting a lower score. It also means that the best time to see all the new pieces, and the greatest diversity, is at the beginning of the voting round.
On 17/03/22 (DD/MM/YY) proposal BIP-04 to cull the voting pool passed in a snapshot vote. As a result, the the required views to determine True Score have been reduced from 100 to 50.
### Voting Pool Fragments Cull
At the end of each voting round, 349 fragments with the lowest True Score are removed from the voting pool. With the weekly mint removing another fragment, the resulting voting pool size is 1,535 fragments.
The cull mechanism aims to reduce the size of the fragment pool in order to improve the voting experience and engagement.
To better enable this strategy, the required views to determine True Score have been reduced from 100 to 50.
On 17/03/22 (DD/MM/YY) proposal BIP-04 to cull the voting pool passed in a snapshot vote. As a result, the above culling mechanism was implemented.
### Weight Mechanism
Users are able to adjust the weight of their votes by using the slider at the bottom of the voting screen
Vote amplification maxes out at 100 VP per choice. This does not add weight to the training, it only adds weight to the total votes for picking the round's winner to be minted for auction. | 2022-05-29 01:58:35 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 8, "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, "math_score": 0.1751169115304947, "perplexity": 2153.3596780477756}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652663035797.93/warc/CC-MAIN-20220529011010-20220529041010-00765.warc.gz"} |
https://zbmath.org/?q=an:0657.93024 | # zbMATH — the first resource for mathematics
Linear periodic systems: Eigenvalue assignment using discrete periodic feedback. (English) Zbl 0657.93024
This note discusses the eigenvalue assignment problem of a T-periodic linear system using discrete periodic state feedback gains. For controllable systems, an explicit formula for the feedback law is given that can be used for the arbitrary assignment of the eigenvalues of $$\Phi_{cl}(T,0)$$, the closed-loop state transition matrix from 0 to T. Also, for the special case of periodic systems controllable over one period, this control law can be used to obtain any desired $$\Phi_{cl}(T,0)$$.
##### MSC:
93B55 Pole and zero placement problems 93C05 Linear systems in control theory 34C25 Periodic solutions to ordinary differential equations 49M99 Numerical methods in optimal control
time-dependent
Full Text: | 2021-03-07 11:56:08 | {"extraction_info": {"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, "math_score": 0.8522489070892334, "perplexity": 920.0528278128064}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178376467.86/warc/CC-MAIN-20210307105633-20210307135633-00557.warc.gz"} |
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15 30 50 per page | 2015-08-30 08:01:47 | {"extraction_info": {"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": 1, "x-ck12": 0, "texerror": 0, "math_score": 0.9515022039413452, "perplexity": 4813.540514184494}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-35/segments/1440644064951.43/warc/CC-MAIN-20150827025424-00043-ip-10-171-96-226.ec2.internal.warc.gz"} |
http://openstudy.com/updates/5112befae4b07c1a5a648051 | ## Yumira I Forgot How To Convert A fractions into whole numbers They Decided To Bring old Things Up Great -.- Helpp Please ! one year ago one year ago
1. stamp
Example problem?
2. Yumira
484/14
3. stamp
$484\div14$There are a few ways to do this, do you want to do long division and see what your remainder is? Your answer will be in the form of$W\frac{r}{14}$14 goes into 484 some W whole times with r remainder.
4. Yumira
Ohh Ok I got It Now Than :D
5. Yumira
Thanks :D * | 2014-03-10 05:17:40 | {"extraction_info": {"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, "math_score": 0.35997235774993896, "perplexity": 3046.74409918851}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1394010650250/warc/CC-MAIN-20140305091050-00073-ip-10-183-142-35.ec2.internal.warc.gz"} |
http://taggedwiki.zubiaga.org/new_content/bd055748f3cb405b14be8211cd07be92 | Chi-square distribution
Parameters Probability density function Cumulative distribution function $k > 0\,$ degrees of freedom $x \in [0; +\infty)\,$ $\frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}\,$ $\frac{\gamma(k/2,x/2)}{\Gamma(k/2)}\,$ $k\,$ approximately $k-2/3\,$ $k-2\,$ if $k\geq 2\,$ $2\,k\,$ $\sqrt{8/k}\,$ $12/k\,$ $\frac{k}{2}\!+\!\ln(2\Gamma(k/2))\!+\!(1\!-\!k/2)\psi(k/2)$ $(1-2\,t)^{-k/2}$ for $2\,t<1\,$ $(1-2\,i\,t)^{-k/2}\,$
In probability theory and statistics, the chi-square distribution (also chi-squared or χ2 distribution) is one of the most widely used theoretical probability distributions in inferential statistics, e.g., in statistical significance tests.[1][2][3][4] It is useful because, under reasonable assumptions, easily calculated quantities can be proven to have distributions that approximate to the chi-square distribution if the null hypothesis is true.
The best-known situations in which the chi-square distribution are used are the common chi-square tests for goodness of fit of an observed distribution to a theoretical one, and of the independence of two criteria of classification of qualitative data. Many other statistical tests also lead to a use of this distribution, like Friedman's analysis of variance by ranks.
Definition
If Xi are k independent, normally distributed random variables with mean 0 and variance 1, then the random variable
$Q = \sum_{i=1}^k X_i^2$
is distributed according to the chi-square distribution with k degrees of freedom. This is usually written
$Q\sim\chi^2_k.\,$
The chi-square distribution has one parameter: k - a positive integer that specifies the number of degrees of freedom (i.e. the number of Xi)
The chi-square distribution is a special case of the gamma distribution.
Characteristics
Probability density function
A probability density function of the chi-square distribution is
$f(x;k)= \begin{cases}\displaystyle \frac{1}{2^{k/2}\Gamma(k/2)}\,x^{(k/2) - 1} e^{-x/2}&\text{for }x>0,\\ 0&\text{for }x\le0, \end{cases}$
where Γ denotes the Gamma function, which has closed-form values at the half-integers.
Cumulative distribution function
$F(x;k)=\frac{\gamma(k/2,x/2)}{\Gamma(k/2)} = P(k/2, x/2)$
where γ(k,z) is the lower incomplete Gamma function and P(k,z) is the regularized Gamma function.
Tables of this distribution — usually in its cumulative form — are widely available and the function is included in many spreadsheets and all statistical packages.
Characteristic function
The characteristic function of the Chi-square distribution is [5]
$\chi(t;k)=(1-2it)^{-k/2}.\,$
Expected value and variance
If $X\sim\chi^2_k$ then the mean is given by
$\frac{}{} \mathrm{E}(X)=k,$
and the variance is given by
$\frac{}{} \mathrm{Var}(X)=2k.$
Median
The median of $X\sim\chi^2_k$ is given approximately by
$k-\frac{2}{3}+\frac{4}{27k}-\frac{8}{729k^2}.$
Information entropy
The information entropy is given by
$H = \int_{-\infty}^\infty f(x;k)\ln(f(x;k)) dx = \frac{k}{2} + \ln \left( 2 \Gamma \left( \frac{k}{2} \right) \right) + \left(1 - \frac{k}{2}\right) \psi(k/2).$
where ψ(x) is the Digamma function.
Noncentral moments
The moments about zero of a chi-square distribution with k degrees of freedom are given by[6][7]
\begin{align} E(X^m) &= k (k+2) (k+4) \cdots (k+2m-2) \\ &= 2^m \frac{\Gamma(m+k/2)}{\Gamma(k/2)}. \end{align}
Derivation of the pdf for one degree of freedom
Let random variable Y be defined as Y = X2 where X has normal distribution with mean 0 and variance 1 (that is X ~ N(0,1)).
Then if $y<0, ~ P(Y and if $y\geq0, ~ P(Y
$f_y(y) = f_x(\sqrt{y})\frac{\partial(\sqrt{y})}{\partial y}-f_x(-\sqrt{y})\frac{\partial(-\sqrt{y})}{\partial y}$
$= \frac{1}{\sqrt{2\pi}}e^{\frac{-y}{2}}\frac{1}{2y^{1/2}} + \frac{1}{\sqrt{2\pi}}e^{\frac{-y}{2}}\frac{1}{2y^{1/2}}$
$= \frac{1}{2^{\frac{1}{2}} \Gamma(\frac{1}{2})}y^{\frac{1}{2} -1}e^{\frac{-y}{2}}$
Then $Y = X^2 \sim \chi^2_1$.
Related distributions and properties
The chi-square distribution has numerous applications in inferential statistics, for instance in chi-square tests and in estimating variances. It enters the problem of estimating the mean of a normally distributed population and the problem of estimating the slope of a regression line via its role in Student's t-distribution. It enters all analysis of variance problems via its role in the F-distribution, which is the distribution of the ratio of two independent chi-squared random variables divided by their respective degrees of freedom.
• If $X\sim\chi^2_k$, then as k tends to infinity, the distribution of $(X-k)/\sqrt{2k}$ tends to a standard normal distribution: see asymptotic distribution. This follows directly from the definition of the chi-squared distribution, the central limit theorem, and the fact that the mean and variance of $\chi^2_1$ are 1 and 2 respectively. However, convergence is slow as the skewness is $\sqrt{8/k}$ and the excess kurtosis is 12 / k.
• If $X\sim\chi^2_k$ then $\sqrt{2X}$ is approximately normally distributed with mean $\sqrt{2k-1}$ and unit variance (result credited to R. A. Fisher).
• If $X\sim\chi^2_k$ then $\sqrt[3]{X/k}$ is approximately normally distributed with mean 1 − 2 / (9k) and variance 2 / (9k) (Wilson and Hilferty,1931)
• $X \sim \mathrm{Exponential}(\lambda = \tfrac{1}{2})$ is an exponential distribution if $X \sim \chi_2^2$ (with 2 degrees of freedom).
• $Y \sim \chi_{\nu}^2$ is a chi-square distribution if $Y = \sum_{m=1}^{\nu} X_m^2$ for $X_i \sim N(0,1)$ independent that are normally distributed.
• If $\boldsymbol{z}'=[Z_1,Z_2,\cdots,Z_n]$, where the Zis are independent Normal(0,σ2) random variables or $\boldsymbol{z}\sim N_p(\boldsymbol{0},\sigma^2 \mathrm{I})$ and $\boldsymbol{A}$ is an $n\times n$ idempotent matrix with rank nk then the quadratic form $\frac{\boldsymbol{z}'\boldsymbol{A}\boldsymbol{z}}{\sigma^2}\sim \chi^2_{n-k}$.
• If the $X_i\sim N(\mu_i,1)$ have nonzero means, then $Y = \sum_{m=1}^k X_m^2$ is drawn from a noncentral chi-square distribution.
• The chi-square distribution $X\sim\chi^2_\nu$ is a special case of the gamma distribution, in that $X \sim {\Gamma}(\frac{\nu}{2}, \theta=2)$.
• $Y \sim \mathrm{F}(\nu_1, \nu_2)$ is an F-distribution if $Y = \frac{X_1 / \nu_1}{X_2 / \nu_2}$ where $X_1 \sim \chi_{\nu_1}^2$ and $X_2 \sim \chi_{\nu_2}^2$ are independent with their respective degrees of freedom.
• $Y \sim \chi^2(\bar{\nu})$ is a chi-square distribution if $Y = \sum_{m=1}^N X_m$ where $X_m \sim \chi^2(\nu_m)$ are independent and $\bar{\nu} = \sum_{m=1}^N \nu_m$.
• if X is chi-square distributed, then $\sqrt{X}$ is chi distributed.
• in particular, if $X \sim \chi_2^2$ (chi-square with 2 degrees of freedom), then $\sqrt{X}$ is Rayleigh distributed.
• if $X_1, \dots, X_n$ are i.i.d. N(μ,σ2) random variables, then $\sum_{i=1}^n(X_i - \bar X)^2 \sim \sigma^2 \chi^2_{n-1}$ where $\bar X = \frac{1}{n} \sum_{i=1}^n X_i$.
• if $X \sim \mathrm{SkewLogistic}(\tfrac{1}{2})\,$, then $\mathrm{log}(1 + e^{-X}) \sim \chi_2^2\,$
• The box below shows probability distributions with name starting with chi for some statistics based on $X_i\sim \mathrm{Normal}(\mu_i,\sigma^2_i),i=1,\cdots,k,$ independent random variables:
Name Statistic
chi-square distribution $\sum_{i=1}^k \left(\frac{X_i-\mu_i}{\sigma_i}\right)^2$
noncentral chi-square distribution $\sum_{i=1}^k \left(\frac{X_i}{\sigma_i}\right)^2$
chi distribution $\sqrt{\sum_{i=1}^k \left(\frac{X_i-\mu_i}{\sigma_i}\right)^2}$
noncentral chi distribution $\sqrt{\sum_{i=1}^k \left(\frac{X_i}{\sigma_i}\right)^2}$
References
1. ^ Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 26", Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover, ISBN 0-486-61272-4 .
2. ^ NIST (2006). Engineering Statistics Handbook - Chi-Square Distribution
3. ^ Jonhson, N.L.; S. Kotz, , N. Balakrishnan (1994). Continuous Univariate Distributions (Second Ed., Vol. 1, Chapter 18). John Willey and Sons. ISBN 0-471-58495-9.
4. ^ Mood, Alexander; Franklin A. Graybill, Duane C. Boes (1974). Introduction to the Theory of Statistics (Third Edition, p. 241-246). McGraw-Hill. ISBN 0-07-042864-6.
5. ^ M.A. Sanders. "Characteristic function of the central chi-square distribution". Retrieved on 2009-03-06.
6. ^ Chi-square distribution, from MathWorld, retrieved Feb. 11, 2009
7. ^ M. K. Simon, Probability Distributions Involving Gaussian Random Variables, New York: Springer, 2002, eq. (2.35), ISBN 978-0-387-34657-1 | 2020-12-01 21:09:28 | {"extraction_info": {"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": 77, "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, "math_score": 0.9898567795753479, "perplexity": 647.8250848933492}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141681524.75/warc/CC-MAIN-20201201200611-20201201230611-00028.warc.gz"} |
https://www.esaral.com/q/if-f-x-is-continuous-and-f-9-2-2-9-then-21620 | # If f(x) is continuous and f(9/2)=2/9, then
Question:
If $f(x)$ is continuous and $f(9 / 2)=2 / 9$, then $\lim _{x \rightarrow 0} f\left(\frac{1-\cos 3 x}{x^{2}}\right)$ is equal to:
1. $9 / 2$
2. 0
3. $2 / 9$
4. $2 / 9$
Correct Option: , 3
Solution: | 2023-02-06 19:44:26 | {"extraction_info": {"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, "math_score": 0.9159355163574219, "perplexity": 4500.626885964783}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500357.3/warc/CC-MAIN-20230206181343-20230206211343-00686.warc.gz"} |
http://mathhelpforum.com/calculus/92243-unit-tangent-normal-vectors.html | # Math Help - unit tangent and normal vectors
1. ## unit tangent and normal vectors
r(t)=cos(t)i + sin(t)j + (t)k. Find the unit tangent and normal vectors as well as the curvature. Sketch the graph.
r(t)=cos(t)i + sin(t)j + (t)k. Find the unit tangent and normal vectors as well as the curvature. Sketch the graph.
You should start by reviewing and then applying the mathematical formulas for these three things. Where do you get stuck?
3. I get how to use the forumals Im just not sure how to apply when its in the i + j + k form
4. Given a curve in the form $\vec{r}(t)= f(t)\vec{i}+ g(t)\vec{j}+ h(t)\vec{k}$, a tangent vector but not necessarily the unit tangent, is $\vec{r}'(t)= f'(t)\vec{i}+ g'(t)\vec{j}+ h'(t)\vec{k}$. | 2014-07-30 22:41:46 | {"extraction_info": {"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": 2, "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, "math_score": 0.8313347101211548, "perplexity": 477.48223682947184}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1406510271654.40/warc/CC-MAIN-20140728011751-00359-ip-10-146-231-18.ec2.internal.warc.gz"} |
http://modeltheory.wikia.com/wiki/Groups_of_finite_Morley_rank | ## FANDOM
78 Pages
A group of finite Morley rank is a group $(G,\cdot)$, usually with extra structure, whose Morley rank is less than $\omega$.
The Cherlin-Zilber conjecture asserts that every simple group of finite Morley rank is an algebraic group over a field. This remains open as of 2014.
However, a considerable amount is known about groups of finite Morley rank. See for example, Bruno Poizat's book Stable Groups, as well as…[more recent books]
• Morley rank and Lascar rank coincide, and are definable. In particular, Morley rank satisfies the Lascar inequalities.
• If $G$ is a group of finite Morley rank, then the connected component $G^0$ exists, and is definable, rather than merely being type-definable. There is a unique type in $G^0$ of maximal Morley rank, i.e., $G^0$ has Morley degree 1. The translates of $G^0$ are called the generics of $G$, and have many good properties. They are the unique types which are translation invariant.
• Any field of finite Morley rank is algebraically closed, but may have additional structure.
• A group of finite Morley rank is simple (in the group theoretic sense) if and only if it is definable simple. That is, if $G$ is not simple as an abstract group, then $G$ has a definable normal subgroup.
• Every infinite group of finite Morley rank contains an infinite abelian definable subgroup.
• Every simple group of finite Morley rank is almost strongly minimal, i.e., is algebraic over a strongly minimal set.
• Groups of finite Morley rank are "dimensional." This falls out of the Lascar analysis.
• Every type-definable subgroup of a group of finite Morley rank is, in fact, definable.
## Transitive action on a strongly minimal set Edit
One rather strong result about groups of finite Morley rank is the following:
Let $G$ be a group of finite Morley rank, acting transitively and faithfully on a strongly minimal set $S$. Then we are in one of the following three situations:
• $G$ has rank 1, is commutative, and $S$ is a $G$-torsor.
• $G$ has rank 2, $S$ is the affine line over a definable field $K$, and $G$ is the group of affine linear transformations over $K$
• $G$ has rank 3, $G$ is $PSL_2(K)$ for a definable field $K$, and $S$ is the projective line over $K$, with the usual action.
In cases 2 or 3, $K$ is algebraically closed. $G$ cannot have rank greater than 3.
Under the hypothesis that there are no bad groups, it can be shown that this implies that the Cherlin-Zilber conjecture holds for groups of Morley rank at most 3: any simple group of Morley rank at most 3 must be $PSL_2(K)$ for a definable field $K$.
It also implies that if $G$ is a simple group of finite Morley rank, containing a definable subgroup $H$ such that $RM(H) = RM(G) - 1$, then $G$ has rank 3 and is $PSL_2(K)$ over an algebraically closed definable field. | 2019-01-21 05:12:31 | {"extraction_info": {"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, "math_score": 0.8963979482650757, "perplexity": 173.69228311832654}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547583763149.45/warc/CC-MAIN-20190121050026-20190121072026-00619.warc.gz"} |
https://factionrock.blogspot.com/2012/ | ## Friday, December 21, 2012
### Beatshifting remix stems
Earlier today, I explained how I created a remix. In the tradition of open source music, I'm giving away the remix stems for Beatshifting, so that you can try remixing it yourself!
Just for reference, here's the original song:
And my remix:
### The Creation of Beatshifting (re-shifted)
To give you a little insight into the underlying creative process I'll highlight some of the coolest things I did on this remix of my own song.
• Use Ableton Live to deconstruct the original song into its component sections. Merge similar tracks from the original into a smaller set of remix stems.
• Study The Glitch Mob's music and become entranced by their use of selective silences to create spooky rhythmic effects. Take this idea totally overboard to create rhythms that as awkward as Napoleon Dynamite by routing some of your tracks into the "Gator" track which has a clips to automate muting of your submix in a rhythmic way.
• Record your own percussion instrument samples and apply the stochastic music principles pioneered by Iannis Xenakis to create a custom music digital instrument in Kontakt to add randomized chaotic musical noise to your remix.
• Use Ableton Live's Beat Repeat to sound like every other kid sitting in his bedroom making a remix with Ableton Live in the 2010's. Also automate a low pass filter to get that sweet 'whole-mix filtered' sound.
• Use the built-in OSX voice to create custom robot lady voice samples, from the terminal.
• Use Processing to create song art by algorithmically kit-bashing fonts together, blending sections of letters to create a smooth transition between two words.
• Write a blog post about the process so that your audience can enjoy a full multi-media experience. And so that you can be a pretentious stuck-up geek.
## Wednesday, December 19, 2012
### Why Looking for a Job is Fundamentally Demoralising
Looking for a job begets misery because job hunters face brutal free-market competition, the expected time until employment is constant even if a hunter has been unemployed for a long time, and getting a good job means turning down adequate jobs. Ultimately, I aim to encourage my fellow job seekers by helping them to understand their predicament more clearly so they take the bad aspects less personally.
Many people find that looking for a job is a long, frustrating process. No, duh. I'm not here to talk about the beaten-path reasons for why job hunting sucks, I want to convince you that job hunting is fundamentally demoralising. Why?
1) You are un-jacked from The Matrix. Welcome to the real world of free-market competition. At school, or in your last job, someone was telling you what to do so that you could add value to yourself. Now you have to show someone else how you can add value to them. You are truly master of your own destiny. You feel the full weight of your freedom, but also the opposing force of everyone else's freedom.
2) Finding a job can take forever. Think about it like this: you either have a job or you don't. Each day you actively look for a job, there's a chance you might get a ready-to-sign offer. That chance is the same, regardless of how long you've been looking. So the length of your periods unemployment are drawn from Poisson distribution. (A better model would be that they're fractally distributed; read about Black Swan theory.) This has horrible psychological implications because you seem to be making no progress while facing a constant flow of rejection. But, while we're breaking illusions, the rejection is not personal. Employers aren't looking to objectively evaluate prospective employees, they're just trying to find ones that they like without doing much work themselves. Read Paul Graham's essay on judgement.
3) Getting a good job requires reckless bravery. Different job offers have different values to you. You're probably looking for more value than a job at McDonalds, so you might reject some jobs. Some of those jobs would be good enough. But you're trying to do what you love
not just survive. Read Paul Graham's essay on doing what you love. So you have to voluntarily stay unemployed until you get that amazing job offer that you can't refuse. It takes guts to do that.
Misery is in the very essence of the job hunt process. Once you realise that, hopefully you will view your search in a different light. You're not a loser, you're just facing self-interested employers in the free market. Don't give up, you get out of the unemployment rut not by searching for a fixed time but by getting just one lucky break. Be brave, you might just have to turn down an offer. To get practical help on finding a job, read the excellent book by Richard Bolles, "What Colour is Your Parachute?"
## Friday, November 23, 2012
### My PhD dissertation summary
Today I submitted my PhD dissertation for examination. The title of my dissertation is: ‘Measuring and Influencing Sequential Joint Agent Behaviours.’
The essential thesis of my research is that:
Algorithmically designed reward functions can influence groups of learning agents toward measurable desired sequential joint behaviours.
The thesis is demonstrated with research explaining how to measure a particular sequential joint behaviour, turn-taking, how to identify rewards that are conducive (or prohibitive) to turn-taking by learning agents in a simulated context and how to design rewards that incentivise arbitrary sequential joint behaviours in multi-agent stochastic games.
Informally, the thesis is about activities performed together through time by a group of agents that figure out how to do things better as they go. An agent could be a person, a robot or a computer program. We mathematically explain how to get the overall outcomes we want by telling the agents what they should individually want. Because we do this mathematically, we need to measure the things we want our group of agents to do. This dissertation explains some new ideas about how we can measure how well a group of agents is taking turns, how we can guess whether or not pairs of a certain kind of robot-like computer programs will take turns, and how we can tell individual agents what they should want so that they collectively end up doing something that we want, for some situations.
My dissertation includes most of two journal papers that I published, plus other bits that I’m planning to submit as another journal.
One of the things I studied was simulated agents communicating and learning from rewards.
## Friday, November 16, 2012
### Nash and Bourne
Violent intrigue can be analysed mathematically
Recently, I finished reading "The Bourne Ultimatum" by Robert Ludlum, an energetic, intriguing tale of violent, manipulative men head-to-head in a struggle of life and death. My definite impression was that Robert Ludlum's Bourne series is a realistic portrayal of the total opposition that can be mathematically modelled as a 'Nash equilibrium.' Read on to get the details.
Many of you will know of Jason Bourne by the series of movies where Matt Damon plays the amnesic protagonist. However, the book series has a different plot and a different mood. In both cases, Jason Bourne is an intelligent man of violence that outguns and outwits his opponents in deadly struggles.
Matt Damon plays Jason Bourne
At the same time as I read the Bourne books, I was studying mathematical ways to understand groups of opposing agents. The concept of a 'Nash equilibrium' is one mathematical way to understand the theory behind violent strategies. A Nash equilibrium is a situation where each person is acting to as to maximise his or her own good, given the actions of everyone else. A Nash equilibrium can be a good thing, where everyone is helping each other, or a Nash equilibrium can be a bad thing, where each person is at the others' throats. In the case of Jason Bourne, the Nash equilibrium is always one where the two masters of intrigue are trying to kill each other. There can only be one winner in the Bourne series.
Without giving too much of the books away, Jason Bourne opposes Carlos the Jackal, the leading international assassin. One will set a trap for the other and the other will 'reverse' the trap, and the one narrowly escapes. Neither man's appearance is clearly known by the other and they both enlist pawns to fight against each other. Jason Bourne threatens, bribes and takes every possible extreme measure in order to defeat the Jackal. If you have only seen the movies, then consider how Bourne tricks and outwits his opponents for his own ends.
John Nash was a revolutionary mathematician
The overall impression I got from the books was 'Oh, this is what a purely competitive Nash equilibrium really looks like.' The winner is not the strongest, the fastest or the man with the best weapons, but the man who thinks the extra step ahead. If you can foresee what your opponent will do, then you can defeat him. But your opponent will try to foresee what you will do. So you must think N+1 steps ahead. In fact, to win, you must be unpredictable. The mathematical solution of a purely competitive game is to randomise over all of your possible actions. In practise, you become an unstable psychopath who commits apparently arbitrary acts of violence without any discernible pattern. I read a lot of action books, but most of the bad guys are stupid or have some other drastic failings. Jason Bourne's opponents seem much more closely matched.
Life can approach the theoretical abstraction of a Nash equilibrium, but game theoretic methods often provide exact answers to slightly the wrong questions. This can make game theory blindingly addictive to some, as Venkatesh Rao observes. In the areas of game theory that I've studied, people often assume that the people have a finite number of actions available to them. However, in practice, the number of devious things that you can do to someone else is limited only by your imagination. Game theory can give us some intuition, but it's probably best to let Robert Ludlum fill in the details. (Actually, stop that thought process right there, just in case you think of a new way of inflicting harm.)
In "How the Mind Works," Steven Pinker gives some good explanations of why people are emotional and do crazy, violent things that seem irrational from some perspectives. In "The Better Angels of Our Nature: Why Violence Has Declined," Pinker explains why people have become much less violent with time. Contrary to the constant whine of the alarmists, not all our morals are bad and getting worse. Part of the reason for the decline in violence is a change in the prevailing Nash equilibrium. We are now more incentivised to be peaceful and non-violent, which is good for all of us. Centrally administrated justice helps us all be more charitable. The real life Carlos the Jackal is serving time in a French prison and Jason Bourne is best approximated by Matt Damon, who co-founded Water.org to help the poor get better access to water.
## Monday, November 5, 2012
### How to be an Awesome Postgrad Student
Here are a few tips on getting through a PhD or a Master’s, based on my experience doing a PhD in Electrical Engineering at the University of Canterbury. I’ve had a good time so far, and I hope that you can have the best experience possible. This document is based on ideas that I gathered from other written works on how to be an awesome postgrad, my supervisors, friends, parents, wife and various talks that I attended as a postgrad. I hope you can translate any discipline-specific details into your own field. Your PhD experience probably be unlike mine, especially if you are not doing electrical engineering. Get advice from wise scholars in your discipline on how to do be an awesome postgrad student in your context.
## Sunday, October 28, 2012
### When can I expect dinner?
Imagine that I am driving from my in-laws house in Ashburton to my own house in Christchurch and that afternoon is waning closer and closer to dinner time. (Recently, this really happened to me.) I start to wonder: how much time do I expect between now and dinner? At first glance, this is an innocent question that can be mentally approximated with enough accuracy to be useful. I can formalise the process happen in my head by expressing the expected time until dinner as the sum of the delays caused by events that may occur between now and dinner (Eq. 1):$$t = \sum_{e \in E} p(e)d(e)$$where $$t$$ is the time until dinner, $$E$$ is the set of all possible events between now and dinner, $$p(e)$$ is the probability that event $$e$$ will occur and $$d(e)$$ is the additional delay caused by event $$e$$. The probability of each event depends on my own choices so I can use Eq. 1 to run 'what if' scenarios in my head. If I choose to stop my car by the side of the road, then I expect dinner to be later than if I continue driving at the speed limit until I reach home. On the other hand, if I stop at a cafe along the way, then I am likely get tea sooner. In general, I only need approximations that are accurate enough for making decisions in life, and I need not worry about the precise philosophy of mathematics that stands behind the approximation. But there are hidden catches in Eq. 1.
Suppose a drunk driver going south swerves onto the wrong side of the road and kills me in a head-on collision. (Keeping to one's side of the road is a life-and-death argument for multi-agent coordination!) This event has non-zero probability of occurring before my dinner time. But for the purposes of Eq. 1, how long would the event of my death delay my dinner? Because I never get dinner, arguably the delay $$d(\textrm{my death})$$ is infinite. But even one infinite delay paired with any non-zero probability means that the expected time until my dinner is also infinite. If I claim that the 'real answer' is always infinity, then I lose the ability to make judgements based on my calculations; the expected time until dinner is the same if I drive home at 10 kilometers per hour over the speed limit or if I stop to take a nap in a nearby paddock.
Instead of assigning $$d(\textrm{my death}) = \infty$$ maybe I could say that dinner is delayed only up until the point of my death; that would mean that I am effectively computing the expected time until dinner or my death, whichever comes first. That seems like a reasonable way to resolve the problem of an infinity in the equation and it will produce finite answers because I am sure to die eventually, notwithstanding events that could prevent my death, such as the return of Christ (The Bible, Matthew 24:31) or a technology singularity. However, the technology singularity is another thing to ponder in conjunction with the mathematical difficulties surrounding infinity and the probability of Christ's return before dinner or before my otherwise certain death is theologically guaranteed to be uncomputable: 'concerning that day and hour no one knows' (Matthew 24:36, ESV).
Alternatively, consider another situation that is slightly less unfortunate: I doze off behind the wheel and crash my car into a power pole on the side of the road, but I survive. I might miss dinner entirely while the doctors put my bones back in their original places but get fed breakfast in hospital the next day. Does that $$d(\textrm{I miss dinner that day})$$ count as infinity if I get dinner, or say breakfast, the next day? Do I actually mean to compute the expected time until my next meal? If I keep the original question, should I count until dinner the next day as the 'time until dinner' or should I count situations like that as also being an infinite time until tonight's dinner? Perhaps a better way to resolve these dilemmas is to use Bayes' theorem and ask a more restricted question: 'Assuming that I will eat dinner tonight, when do I expect to eat dinner?' Again, this will avoid the infinities, but the question has been changed.
I could look at my question in other ways, too. Perhaps the informal use of the word 'expect' should actually be interpreted as referring to the mode of the distribution of times that I might wait until I get dinner. Or maybe the median. Maybe I should exclude all outliers that have $$p(e) < \phi$$ for some arbitrary, small event probability $$\phi$$, replacing the original question with 'How long until dinner, assuming nothing too unexpected happens?' Because $$\phi$$ can take any value, I now have an infinite number of possibilities. Before I can start computing an answer, I face the difficult problem of how to refine the question.
While this discussion may seem excessively philosophical, in fact this kind of problem must be solved every day by professional computer programmers. A program's requirements may state a subroutine computes the 'time until my dinner.' How does one deal with unusual cases? Many programmers will simply ignore their existence out of laziness or error. Others will write code that explicitly identifies odd cases but still fails to function in a meaningful way. Maybe the limitations of floating point numbers and finite sets in a computer will shine serendipitously on the problem, but maybe not. Maybe the rest of the program has no way of considering the possibility of my death and therefore the subroutine will not encounter that case, but maybe that is a flaw in the rest of the program. How does one meaningfully substitute answerable questions for unanswerable ones? Is that even possible in general?
## Tuesday, July 10, 2012
### New open source projects
My research has generated two new open source projects recently:
A wave digital filter-based emulation of a famous stereo tube limiter from the 1950's.
Tools for measuring the quantity of turn-taking present in the behaviour of a group of agents.
The links also give the references for the associate research. Part of the idea is that academic research, open source and Wikipedia are all on a massive collision course for Knowledge 2.0. Or maybe more accurately, Knowledge N+1, because open source and Wikipedia are constantly changing. Soon the academic topics will be as open and fluid as say, the Linux kernel. Some smart person will probably be the spearhead of each topic, but there will be many contributors and the development of ideas will be out in the open.
Peter | 2018-03-23 22:02:10 | {"extraction_info": {"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, "math_score": 0.32071903347969055, "perplexity": 1412.8924424108766}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257649095.35/warc/CC-MAIN-20180323220107-20180324000107-00053.warc.gz"} |
https://community.wolfram.com/groups/-/m/t/2442916 | # How to convert string to date object to sort the data ?
Posted 9 months ago
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Hello, I have been struggling with how to sort my data by date.I have a CSV file with one attribute as date. I have tried importing the file using Sematic Import in order to obtain the date data as a date object but it has been futile.Here is an example of my data:A.Name 2022-01-01 1:02 Subject MessageWhenever i import the data as a dataset, it is imported as a string. How can i convert it into a date object so that i can sort the entire dataset by date.I have lost hope and will try to use some other language if this is not possible with Mathematica.Any help would be much appreciated. Been struggling with this for 4 weeks.
6 Replies
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Posted 9 months ago
Hi Rashmi,Could you please attach a sample (first 10 rows) of your CSV file to your question. Not obvious why SemanticImport would fail. do = DateObject["2022-01-01 1:02"] do // InputForm (* DateObject[{2022, 1, 1, 1, 2}, "Minute", "Gregorian", -6.] *)
Posted 9 months ago
Posted 9 months ago
Alternatively, using Eric's sample data file rawData = Import["~/sampledata.csv", "Dataset", HeaderLines -> 1] data = rawData[All, <|#, "date" -> DateObject[#date]|> &] To sort by the date column data[SortBy["date"]]
Posted 15 days ago
The last suggestion proposed by Rohit has solved a problem I am/was having. My problem is the following. I have a csv file that contains the typical stock data (Symbol, Sector, Date, Open, High, Low, Close, Volume) and a few indicators (e.g. Relative Strength, moving average). There are 49 columns of which the Symbol and Sector are Strings. The 3rd columns ("Date") a date represented as "Month/Day/Year". When I used SemanticImport as in: rawData = SemanticImport[filename, HeaderLines -> 1] Things worked well until the file size got to about 200 rows. I first noticed the problem when I tried to do the SemanticImport on a file of 257 rows with 49 columns (as mentioned above). The cell where I executed the code above simply ran for ever (meaning hours) until I had to abort the evaluation. However, if I was doing the same thing with a file of 128 rows, the SemanticImport worked and returned a Dataset. Not knowing what was wrong, I thought that there might be a bad field in the file but simply using: rawData = Import[filename,"Dataset",HeaderLines->1] data = rawData[All, <|#, "Date" -> DateObject[DateList[{#Date, {"Month","Day", "YearShort"}}]]|> &] Worked without a hitch. So, I figured that the size of the file was not a problem since Import[] was able to load the data and I could then convert all 256 dates strings to date objects. But SemanticImport simply stopped working once the file had about 200 rows. Though your suggestion has allowed me to solve my problem, I still would like to understand why SemanticImport stopped working once my file reached a certain size whereas Import[] was able to read the file without any issues.
Posted 14 days ago
Hi Henrick,Since the "Date" column imports fine, SemanticImport is probably failing or taking a long time to interpret data in some other column. Can you share a file that demonstrates the problem?
Posted 14 days ago
I am providing the file. The problem is that others seem to be able to read that file using SemanticImport[]. When I edit the sample file so that it has 200 rows rather than the 257, I am able to read it directly with SemanticImport[] but beyond that, SemanticImport just hangs where I have to Abort the evaluation.In case it helps, my specs are:Edition Windows 11 Home Version 21H2 Installed on 2/4/2022 OS build 22000.856 Experience Windows Feature Experience Pack 1000.22000.856.0 Attachments: | 2022-09-27 08:11:12 | {"extraction_info": {"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, "math_score": 0.28501075506210327, "perplexity": 2247.3666839384005}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030334992.20/warc/CC-MAIN-20220927064738-20220927094738-00585.warc.gz"} |
https://forum.snap.berkeley.edu/t/ogg-sound-files/681 | # Ogg sound files
Hi,
This is just a simple request for support of .ogg sound files.
Why... "ogg" format, because it is opensource & good quality I believe, where as mp3 & wave are not.
Am using Snap! v5.0.3 in Firefox v64.0 on a Mac v10.12.6 and can't seem to get Snap! to recognize .ogg sound files.
Thanks.
I believe we support any sound format your browser supports. Can you play ogg files from other sites in your browser? Or maybe you can get a Firefox extension to support them.
Firefox seems to play ogg's just fine...
just can't seem to load them into Snap! either by drag & drop or through Sounds... menu.
Hmm I don't know then. We'll look into it...
importing ogg audio files into Snap! and using them as sounds works fine for me in Chrome on MacOS. Firefox doesn't seem recognize them as audio files. This looks like a Firefox issue, because Snap isn't even checking for certain filetypes, it's looking at whether a dropped file is an audio mime type.
Oh… tried Vivaldi which I think is based on Chrome & ogg’s work fine.
Still no luck with Firefox though…
So when I said I thought “Firefox seems to play ogg's just fine” that was only because I tried playing some some samples from a web page and they seemed to work, but maybe the page just had some default player or code or somthing?
Looked for plugin for playing ogg’s on Firefox but couldn’t find anything specific.
Maybe I should ask the Firefox people on how to make it recognize ogg’s “audio mime” type?
Except I’m not quite sure if I know what I’m talking about.
It would be interesting to know whether the web page you tried is HTML-based or Flash-based. It has to be HTML-based to be a relevant test.
Sound effects from…
Sorry, I’m not qualified to know if it is HTML-based or Flash-based.
Thought all web pages were HTML-based?
That's actually a bug in Firefox, but it's actually quite easy for us to circumvent.
Apparently Firefox provides the wrong video/ogg MIME type instead of the correct audio/ogg one, and since we're expecting the MIME type to start with the audio word in morphic.js, we never get to process the file.
I'm sending a PR with a simple fix, give me a minute
Jens has pulled this fix and it is now live, so you should be able to import OGGs normally from Firefox, @perspexsphinx
Well, it turned out to be a bug, just not our bug. :~) | 2022-06-29 20:16:36 | {"extraction_info": {"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, "math_score": 0.34662243723869324, "perplexity": 3714.8592274486186}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103642979.38/warc/CC-MAIN-20220629180939-20220629210939-00535.warc.gz"} |
https://www.lidolearning.com/questions/m-bb-selina7-ch10-ex10-q13/q13-a-sum-amounts-to-rs-2652-i/ | Selina solutions
# Question 13
Q13) A sum amounts to Rs. 2,652 in 6 years at 5% p.a. simple interest.
Find :
(i) the sum
(ii) the time in which the same sum will double itself at the same rate of interest.
Solution:
(i) In first case, Let principal (P) = Rs. 100
Rate ( r ) = 5% p.a , Time (T) = 6 years
∴ SI = \frac{PRT}{100}=\frac{100\times6\times5}{100}=Rs.30
And, amount = Rs. 100 + 30 = Rs. 130
If amount is Rs. 130 , then principal = Rs. 100
And, if amount is Rs. 2652 then principal
=\frac{100\times2652}{130}=Rs.2040
(ii)In second case, let sum (P) = Rs. 100
Amount (A) = 100 x 2 = Rs. 200
SI = A - P = 200 - 100 = 100 Rs.
Rate = 5% p.a
Time = \frac{SI\times100}{P\times R}=\frac{100\times100}{100\times5}=20\ years
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Chemistry | 2020-09-19 00:41:27 | {"extraction_info": {"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, "math_score": 0.958994448184967, "perplexity": 8990.437551713756}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400189264.5/warc/CC-MAIN-20200918221856-20200919011856-00476.warc.gz"} |
https://math.stackexchange.com/questions/2087926/meaning-of-the-continuous-spectrum-and-the-residual-spectrum | Meaning of the continuous spectrum and the residual spectrum
Let's consider an infinite dimensional space $X$ and a linear operator $T$. The resolvent operator of $T$ is $R_\lambda (T) = (T-\lambda I)^{-1}$. A regular value $\lambda$ of $T$ is a complex number such that:
(R1) $R_\lambda(T)$ exists
(R2) $R_\lambda(T)$ is bounded
(R3) $R_\lambda(T)$ defined on a set which is dense in $X$
The resolvent set $\rho(T)$ consists of all regular values $\lambda$ of $T$. The complement $\sigma(T)=C-\rho(T)$ is the spectrum of $T$ and we may distinguish parts of the spectrum:
point spectrum (eigenvalues) $\sigma_p(T)$: (R1) isn't satisfied
continuous spectrum $\sigma_c(T)$: (R2) isn't satisfied, but (R1) and (R3) are satisfied
residual spectrum $\sigma_r(T)$: (R3) isn't satisfied, (R1) is satisfied, (R2) - doesn't matter
Please, help me to clarify a couple of points:
Question 1: The point spectrum consists of eigenvalues and exists in finite dimensional case. So its meaning seems to be the same as in a finite dimensional case (scaling of eigenvectors that roughly represent orientation of the distortion by $T$). What is the meaning of the continuous and the residual spectrum?
Question 2: Why do we care about dense in the definitions? I have found a related question but didn't get the exact answer from it.
Question 1: If $\lambda$ is in the continuous spectrum, then $R_{\lambda}(T)$ is unbounded. That means there exists a sequence $\{ x_n \}$ such that $\|x_n\|=1$ and $\|R_{\lambda}(T)x_n\|\rightarrow\infty$. So, $y_n=\frac{1}{\|R_{\lambda}(T)x_n\|}R_{\lambda}(T)x_n$ is a sequence of unit vectors in $X$ such that $(T-\lambda I)y_n\rightarrow 0$ as $n\rightarrow\infty$. It is typical to refer to this sequence $\{ y_n \}$ of unit vectors as an "approximate eigenvector."
The reason one refers to this as "continuous spectrum" Historically had nothing to do with continuity; such spectrum was found to fill a continuum, rather than being discrete. For example, the multiplication operator $(Mf)(x)=xf(x)$ defined on $L^2[0,1]$ has no eigenvalues, but $[0,1]$ is in the continuous spectrum of $M$. You can see this because the step function $f_{\lambda,\delta}=\frac{1}{\sqrt{2\delta}}\chi_{[\lambda-\delta,\lambda+\delta]}$ is a unit vector for which $$Mf_{\lambda,\delta} \approx \lambda f_{\lambda,\delta},$$ in the sense that $$\|(M-\lambda I)f_{\lambda,\delta}\|^2=\int_{\lambda-\delta}^{\lambda+\delta}(x-\lambda)^2f_{\lambda,\delta}^2dx \le 2\delta^2\|f_{\lambda,\delta}\|^2.$$ So $\|(M-\lambda I)f_{\lambda,\delta}\|\rightarrow 0$ as $\delta\rightarrow 0$ while $\|f_{\lambda,\delta}\|=1$.
The same type of spectrum occurs for the differentiation operator $\frac{1}{i}\frac{d}{dx}$ on $L^2(\mathbb{R})$ where you would like to identify $e^{isx}$ as an eigenvector with eigenvalue $s$, but it's not in the space. However, $s$ is in the continuous spectrum, a.k.a. approximate point spectrum. This is one of the earliest cases that motivated these definitions, where you could think of $$F(x)=\int_{-\infty}^{\infty}c(s)e^{isx}ds$$ as a "continuous" (i.e. integral) sum of approximate eigenfunctions of the differentiation operator, with a coefficient function $c(s)$ defined on the continuum. And, when considered in $L^2$, the operator $\frac{1}{i}\frac{d}{dx}$ has continuous spectrum $\mathbb{R}$.
Question 2: For your second question, you don't care about a dense domain for $R_{\lambda}(T)$ if it is bounded because you can then extended it to the whole space. And, if $T$ is closable, then the extension of $R_{\lambda}(T)$ and the closure of $T$ remain inverses of each other. However, if $\lambda$ is an approximate eigenvalue and $T$ is closed, then $R_{\lambda}(T)$ must be unbounded, and you're basically stuck with a dense domain for $R_{\lambda}(T)$ because of the closed graph theorem.
• Thank you for the reply! Can you please explain why the continuous spectrum is defined on a dense set in $X$? Could you tell a couple of words about the meaning of the residual spectrum? – Konstantin Jan 8 '17 at 19:09
• @Konstantin : The continuous spectrum requires that you have an inverse that is unbounded. If $X$ is a complete space, then the inverse cannot be defined on the full space. It is standard to require the inverse to be defined on a dense subspace. If it is defined on a non-dense subspace, that falls into the miscellaneous category of residual spectrum. Residual should be regarded as the "left over" part of the spectrum. – DisintegratingByParts Jan 8 '17 at 21:06 | 2019-09-17 19:35:44 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.920220136642456, "perplexity": 79.44760320973413}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514573105.1/warc/CC-MAIN-20190917181046-20190917203046-00335.warc.gz"} |
https://leicester.figshare.com/articles/journal_contribution/Stochastic_Separation_Theorems/10208396/1 | 1703.01203v3.pdf (167.14 kB)
Stochastic Separation Theorems
journal contribution
posted on 21.08.2017, 09:52 by A. N. Gorban, I. Y. Tyukin
The problem of non-iterative one-shot and non-destructive correction of unavoidable mistakes arises in all Artificial Intelligence applications in the real world. Its solution requires robust separation of samples with errors from samples where the system works properly. We demonstrate that in (moderately) high dimension this separation could be achieved with probability close to one by linear discriminants. Surprisingly, separation of a new image from a very large set of known images is almost always possible even in moderately high dimensions by linear functionals, and coefficients of these functionals can be found explicitly. Based on fundamental properties of measure concentration, we show that for $M1-\vartheta$, where $1>\vartheta>0$ is a given small constant. Exact values of $a,b>0$ depend on the probability distribution that determines how the random $M$-element sets are drawn, and on the constant $\vartheta$. These {\em stochastic separation theorems} provide a new instrument for the development, analysis, and assessment of machine learning methods and algorithms in high dimension. Theoretical statements are illustrated with numerical examples.
Citation
Neural Networks, 2017, 94, pp. 255-259
Author affiliation
/Organisation/COLLEGE OF SCIENCE AND ENGINEERING/Department of Mathematics
Version
AM (Accepted Manuscript)
Neural Networks
Publisher
Elsevier for European Neural Network Society (ENNS), International Neural Network Society (INNS), Japanese Neural Network Society (JNNS)
0893-6080
1879-2782
21/07/2017
2017
31/07/2018
Publisher version
http://www.sciencedirect.com/science/article/pii/S0893608017301776?via=ihub
Notes
The file associated with this record is under embargo until 12 months after publication, in accordance with the publisher's self-archiving policy. The full text may be available through the publisher links provided above.
en
Exports
figshare. credit for all your research. | 2021-10-22 04:18:48 | {"extraction_info": {"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, "math_score": 0.7146749496459961, "perplexity": 1984.3071286859001}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585450.39/warc/CC-MAIN-20211022021705-20211022051705-00292.warc.gz"} |
https://calculus7.org/2017/07/07/the-kolakoski-cantor-set/ | # The Kolakoski-Cantor set
A 0-1 sequence can be interpreted as a point in the interval [0,1]. But this makes the long-term behavior of the sequence practically invisible due to limited resolution of our screens (and eyes). To make it visible, we can also plot the points obtained by shifting the binary sequence to the left (Bernoulli shift, which also goes by many other names). The resulting orbit is often dense in the interval, which doesn’t really help us visualize any patterns. But sometimes we get an interesting complex structure.
The vertical axis here is the time parameter, the number of dyadic shifts. The 0-1 sequence being visualized is the Kolakoski sequence in its binary form, with 0 and 1 instead of 1 and 2. By definition, the n-th run of equal digits in this sequence has length ${x_n+1}$. In particular, 000 and 111 never occur, which contributes to the blank spots near 0 and 1.
Although the sequence is not periodic, the set is quite stable in time; it does not make a visible difference whether one plots the first 10,000 shifts, or 10,000,000. The apparent symmetry about 1/2 is related to the open problem of whether the Kolakoski sequence is mirror invariant, meaning that together with any finite word (such as 0010) it also contains its complement (that would be 1101).
There are infinitely many forbidden words apart from 000 and 111 (and the words containing those). For example, 01010 cannot occur because it has 3 consecutive runs of length 1, which implies having 000 elsewhere in the sequence. For the same reason, 001100 is forbidden. This goes on forever: 00100100 is forbidden because it implies having 10101, etc.
The number of distinct words of length n in the Kolakoski sequence is bounded by a power of n (see F. M. Dekking, What is the long range order in the Kolakoski sequence?). Hence, the set pictured above is covered by ${O(n^p)}$ intervals of length ${2^{-n}}$, which implies it (and even its closure) is zero-dimensional in any fractal sense (has Minkowski dimension 0).
The set KC apparently does not have any isolated points; this is also an open problem, of recurrence (whether every word that appears in the sequence has to appear infinitely many times). Assuming this is so, the closure of the orbit is a totally disconnected compact set without isolated points, i.e., a Cantor set. It is not self-similar (not surprising, given it’s zero-dimensional), but its relation to the Bernoulli shift implies a structure resembling self-similarity:
Applying the transformations ${x\mapsto x/2}$ and ${x\mapsto (1+x)/2}$ yields two disjoint smaller copies that cover the original set, but with some spare parts left. The leftover bits exist because not every word in the sequence can be preceded by both 0 and 1.
Applying the transformations ${x\mapsto 2x}$ and ${x\mapsto 2x-1}$ yields two larger copies that cover the original set. There are no extra parts within the interval [0,1] but there is an overlap between the two copies.
The number ${c = \inf KC\approx 0.146778684766479}$ appears several times in the structure of the set: for instance, the central gap is ${((1-c)/2, (1+c)/2)}$, the second-largest gap on the left has the left endpoint ${(1-c)/4}$, etc. The Inverse Symbolic Calculator has not found anything about this number. Its binary expansion begins with 0.001 001 011 001 001 101 001 001 101 100… which one can recognize as the smallest binary number that can be written without doing anything three times in a row. (Can’t have 000; also can’t have 001 three times in a row; and 001 010 is not allowed because it contains 01010, three runs of length 1. Hence, the number begins with 001 001 011.) This number is obviously irrational, but other than that…
In conclusion, the Python code used to plot KC.
```import numpy as np
import matplotlib.pyplot as plt
n = 1000000
a = np.zeros(n, dtype=int)
j = 0
same = False
for i in range(1, n):
if same:
a[i] = a[i-1]
same = False
else:
a[i] = 1 - a[i-1]
j += 1
same = bool(a[j])
v = np.array([1/2**k for k in range(60, 0, -1)])
b = np.convolve(a, v, mode='valid')
plt.plot(b, np.arange(np.size(b)), '.', ms=2)
plt.show()``` | 2018-10-15 14:05:43 | {"extraction_info": {"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": 10, "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, "math_score": 0.8647265434265137, "perplexity": 425.9062493568152}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-43/segments/1539583509196.33/warc/CC-MAIN-20181015121848-20181015143348-00159.warc.gz"} |
http://math.stackexchange.com/questions/344536/curious-sum-n-1-infty-frac1n2-x2-identity | # Curious $\sum _{n=1}^{\infty} \frac{1}{n^2 - x^2}$ identity [duplicate]
This question already has an answer here:
Let $$F(x) = \sum _{n=1}^{\infty} \frac{1}{n^2 - x^2}$$
It seems that for odd integer $k$ $$F\left(\frac{k}{2}\right) = \frac{2}{k^2}$$ My evidence is strictly computational, and I have no idea how to approach a proper proof. So standard questions are:
• is it a known fact
• is it a fact
• is it curious
• what is a proof strategy
math.stackexchange.com/questions/314986/… and put $a= ix$ – Cortizol Mar 28 '13 at 8:39 | 2015-07-04 07:29:19 | {"extraction_info": {"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, "math_score": 0.8230317831039429, "perplexity": 1132.247675346955}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-27/segments/1435375096579.52/warc/CC-MAIN-20150627031816-00120-ip-10-179-60-89.ec2.internal.warc.gz"} |
https://epirecip.es/epicookbook/chapters/ob18/c5/r | # Chapter 5 from Bjornstad et al. (2018): Seasonality
### Environmental drivers
We can illustrate various types of seasonality using four diseases in Pennsylvania contained in the paili, palymes, pagiard and pameasle datasets. We first write a simple function to extract and plot weekly average incidence (and standard errors) through the year from time series. Weekly incidence data occasionally has a 53 reporting week (because years are 52.14 weeks, and leap-years are 52.28 weeks). The function omits these extras.
library(epimdr)
data(paili)
data(palymes)
data(pagiard)
data(pameasle)
ppp <- function(wk, x){
require(plotrix)
x = x[wk<53]
wk = wk[wk<53]
ses = sapply(split(x, wk), mean, na.rm = TRUE)
sesdv = sapply(split(x, wk), sd, na.rm = TRUE)/sqrt(length(split(x, wk)))
plotCI(x = c(1:52), y = ses, ui = ses+sesdv, li = ses-sesdv, xlab = "Week", ylab = "Incidence")
}
We apply the function to reported incidence of Lyme’s disease, giardiosis, measles and mortality from influenza-like illness (ILI) (Fig. [fig:c5-paseas]):
par(mfrow = c(2, 2)) #A four panel plot
ppp(paili[, "WEEK"], paili[, "PENNSYLVANIA"])
title("ILI mortality (1972-98)")
ppp(palymes[, "WEEK"], palymes[, "PENNSYLVANIA"])
title("Lymes (2006-14)")
ppp(pagiard[, "WEEK"], pagiard[, "PENNSYLVANIA"])
title("Giardia (2006-14)")
ppp(pameasle[, "WEEK"], pameasle[, "PENNSYLVANIA"])
title("Measles (1928-69)")
### The seasonally-forced SEIR model
To study the effect of seasonality in transmission we modify the SEIR model. We first define the gradient functions for the ‘undriven’ system:
seirmod = function(t, y, parms) {
S = y[1]
E = y[2]
I = y[3]
R = y[4]
mu = parms["mu"]
N = parms["N"]
beta = parms["beta"]
sigma = parms["sigma"]
gamma = parms["gamma"]
dS = mu * (N - S) - beta * S * I/N
dE = beta * S * I/N - (mu + sigma) * E
dI = sigma * E - (mu + gamma) * I
dR = gamma * I - mu * R
res = c(dS, dE, dI, dR)
list(res)
}
We simulate 10-years of dynamics. The seasonally-forced SEIR model has been very successfully applied to understand the dynamics of measles (and other immunizing childhood infections). To simulate measles-like dynamics we assume a latent period of 8 days and an infectious period of 5 days. We assume the initial host population to be $0.1\%$ infectious, $6\%$ susceptibles and the rest immune; The $R_0$ of measles is typically quoted in the 13 to 20 range, which means that the equilibrium fraction of susceptibles is somewhere around $5\%$. For simplicity we assume a host life-span of 50 years and set $N=1$ to model the fraction in each compartment.
require(deSolve)
times = seq(0, 10, by=1/120)
paras = c(mu = 1/50, N = 1, beta = 1000,
sigma = 365/8, gamma = 365/5)
start = c(S = 0.06, E = 0, I = 0.001, R = 0.939)
As discussed previously, the $R_0$ for this system – assuming disease induced mortality is negligible – is $\frac{\sigma}{\sigma +\mu} \frac{\beta}{\gamma+\mu}$. We can verify that our choice of $\beta$ places $R_0$ in the ‘measles-like’ range. We use expression to define the equation for $R_0$. We use with(as.list(…)) to evaluate the expression using the definitions in the paras-vector.
R0 = expression(sigma/(sigma + mu) * beta/(gamma + mu))
with(as.list(paras), eval(R0))
13.688877511744
We can integrate the ODEs and plot the time series in time and the phase plane (fig. [fig:seir]). As is the case of the SIR model, the unforced SEIR model predicts dampened oscillations toward the endemic equilibrium.
out = as.data.frame(ode(start, times, seirmod, paras))
par(mfrow = c(1,2)) #Two plots side by side
plot(times, out$I, ylab = "Prevalence", xlab = "Time", type = "l") plot(out$S, out$I, ylab = "Prevalence", xlab = "Susceptible", type = "l") ## Seasonality in$\beta$The predicted dampend oscillations toward an equilibrium is at odds with the recurrent outbreaks seen in many immunizing infections. Sustained oscillations require either additional predictable seasonal drivers or stochasticity. An important driver in human childhood infections is seasonality in contact rates because of aggregation of children during the school term . For simplicity we can analyze the consequences of seasonality by assuming sinusoidal forcing on the transmission rate according to$\beta(t) = \beta_0 (1 + \beta_1 cos(2 \pi t))$. The mean transmission rate is$\beta_0$but the realized transmission varies cyclically with a period of one time unit, and the magnitude of the seasonal variation is controlled by the parameter$\beta_1$. The modified gradient function is: seirmod2 = function(t, y, parms){ S = y[1] E = y[2] I = y[3] R = y[4] with(as.list(parms),{ dS = mu * (N - S) - beta0 * (1+beta1 * cos(2 * pi * t)) * S * I / N dE = beta0 * (1 + beta1 * cos(2*pi * t)) * S * I / N - (mu + sigma) * E dI = sigma * E - (mu + gamma) * I dR = gamma * I - mu * R res = c(dS, dE, dI, dR) list(res) }) } With no seasonality the model predicts dampened oscillation, with moderate seasonality the prediction is low-amplitude annual outbreaks. However, as seasonality increases (to$\beta_1 = 0.2$, say) we start seeing some surprising consequences of the seasonal forcing: the appearance of harmonic resonance between the internal cyclic dynamics of the SEIR clockwork and the annual seasonal forcing function. times = seq(0, 100, by=1/120) paras = c(mu = 1/50, N = 1, beta0 = 1000, beta1 = 0.2, sigma = 365/8, gamma = 365/5) start = c(S = 0.06, E = 0, I = 0.001, R = 0.939) out = as.data.frame(ode(start, times, seirmod2, paras)) par(mfrow = c(1,2)) #Side-by-side plot plot(times, out$I, ylab="Infected", xlab="Time",
xlim = c(90, 100), ylim = c(0,
max(out$I[11001:12000])), type = "l") plot(out$S[11001:12000], out$I[11001:12000], ylab = "Infected", xlab = "Susceptible", type = "l") The emergent pattern of recurrence in the forced SEIR is the result of an interaction between the internal periodic clockwork (the ‘damping period’) of the SEIR flow and the externally-imposed periodic forcing. The damping period is the focus of chapter [chap:c10], however we can use the results previewed in [sec:c1peri]: When working with a continuous-time ODE model which results in cyclic behavior like the SEIR model, the dominant eigenvalues of the the – when evaluated at the equilibrium – are a conjugate pair of complex numbers ($a \pm b \imath$) that determines the period of the cycle according to$2 \pi / b$. The endemic equilibrium of the SEIR model (ignoring the absorbing R compartment$R^* = 1^-S^-E^-I^$) is$S^* = 1/R_0$,$E^= (\mu+\gamma) I^/\sigma$and$I^* = \mu (1-1/R_0) R_0 / \beta$: mu = paras["mu"] N = paras["N"] beta0 = paras["beta0"] beta1 = paras["beta1"] sigma = paras["sigma"] gamma = paras["gamma"] R0 = sigma/(sigma + mu) * beta0/(gamma + mu) # Equilibria Sstar = 1/R0 Istar = mu * (1 - 1/R0) * R0/beta0 Estar = (mu + gamma) * Istar/sigma eq = list(S = Sstar, E = Estar, I = Istar) We next use Rs inbuilt D-function to carry out symbolic differentiation, to generate and evaluate the Jacobian matrix (ignoring the absorbing R compartment): dS = expression(mu * (N - S) - beta0 * S * I / N) dE = expression(beta0 * S * I / N - (mu + sigma) * E) dI = expression(sigma*E - (mu + gamma) * I) #Elements of Jacobian j11 = D(dS, "S"); j12 = D(dS, "E"); j13 = D(dS, "I") j21 = D(dE, "S"); j22 = D(dE, "E"); j23 = D(dE, "I") j31 = D(dI, "S"); j32 = D(dI, "E"); j33 = D(dI, "I") #Jacobian J = with(eq, matrix(c(eval(j11),eval(j12),eval(j13), eval(j21),eval(j22), eval(j23), eval(j31),eval(j32), eval(j33)), nrow=3, byrow=TRUE)) We finally calculate the eigenvalues. The dominant pair of complex conjugates are at the 2nd and 3rd place in the vector of eigenvalues. The resultant resonant period is: round(eigen(J)$values, 3)
2 * pi/(Im(eigen(J)$values[2])) <ol class=list-inline> • -118.725+0i • -0.107+2.667i • -0.107-2.667i • </ol> 2.35589072899997 So the recurrent biennial epidemics are sustained because the internal epidemic clock-work cycles with a period of 2.3 year, but it is forced at an annual time scale, so as a ‘compromise’ the epidemics are locked on to the annual clock, but with alternating major and minor epidemics such as seen, for example, in pre-vaccination measles in New York 1944-58 and London 1950-1965. ### Bifurcation analysis We can make a more comprehensive summary of the consequences of seasonality on the SEIR-flow using a bifurcation analysis: a systematic search across a range of$\beta_1$values. For annually forced models we study the dynamics by ‘strobing’ the system once each year. To study the long-term (asymptotic) dynamics we discard the initial transient part of the simulation. In the below we hence use one data point per year for the last 42 years of simulation – which the sel variable flags – so that an annual cycle produces a single value (so will a fixed-point equilibrium), biannual cycles two values, etc. The resultant bifurcation plot shows when annual epidemics gives way to biannual cycles and finally chaotic dynamics as seasonality increases. The irregular dynamics with strong seasonality comes about because there is no simple resonant compromise between the internal clock and the external forcing function. We may think of it as ‘resonance’ giving place to ‘dissonance’ in the dynamical system. That stronger seasonality pushes measles from regular to irregular epidemics has been predicted by the theoretical literature and is supported by an empirical comparison of measles in pre-vaccination UK versus US. We define initial conditions and the sequence of parameter values to be considered for$\beta_1$and then do the numerical integration for each parameter set: times = seq(0, 100, by = 1/120) start = c(S = 0.06, E = 0, I = 0.001, R = 0.939) beta1 = seq(0,0.25, length=101) #Matrix to store infecteds Imat = matrix(NA, ncol = 12001, nrow = 101) #Loop over beta1's for(i in 1:101){ paras = c(mu = 1/50, N = 1, beta0 = 1000, beta1 = beta1[i], sigma = 365/8, gamma = 365/5) out = as.data.frame(ode(start, times, seirmod2, paras)) Imat[i,] = out$I
}
For the visualization we select one observation per year for the last 42 years of simulation and plot the values against the associated $\beta_1$ values (Fig. [fig:bifplot]).
sel = seq(7001, 12000, by = 120)
plot(NA, xlim = range(beta1), ylim = c(1E-7,
max(Imat[,sel])), log = "y", xlab = "beta1",
ylab = "prevalence")
for(i in 1:101){
points(rep(beta1[i], length(sel)),
Imat[i, sel], pch=20)
}
### Stroboscopic section
It is interesting to integrate the model with parameters leading to chaotic dynamics for a very long time (in this case for 10,000 years) to better understand/visualize the meaning of quasi-periodic chaos. The following shows a time series of prevalence and the dynamics in the S-I phase plane strobed at the annual time scale – the annual ‘stroboscopic section’ of the S-I plane. The time series is erratic, but the paired S-I series trace out a very intricate pattern. The four-armed shape corresponds to the propensity of the chaotic pattern to adhere to a wobbly (‘quasiperiodic’) 4-year recurrence. We will revisit on this attractor and its role in facilitating ‘chaotic stochasticity’ and ‘stochastic resonance’ in disease dynamics later.
times = seq(0, 100000, by = 1/120)
start = c(S = 0.06, E = 0, I = 0.001, R = 0.939)
paras = c(mu = 1/50, N = 1, beta0 = 1800, beta1=0.28,
sigma = 35.84, gamma = 100)
out = as.data.frame(ode(start, times,seirmod2, paras))
sel = seq(7001, 12000000, by = 120)
par(mfrow = c(1,2))
plot(out$time[7001:13001], out$I[7001:13001],
type = "l", xlab = "Year", ylab = "Prevalence")
plot(out$S[sel], out$I[sel], type = "p", xlab = "S",
ylab = "I", log = "y", pch = 20, cex = 0.25)
### Susceptible recruitment
The patterns of recurrent epidemics is also shaped by other characteristics of the host and pathogen. studied how susceptible recruitment affects dynamics of the seasonally forced SEIR model by doing a bifurcation analysis over $\mu$. As concluded by , reduced susceptible recruitment in the seasonally-forced SEIR model leads to a cascade from annual to biennial to coexisting annual and complex attractors[2]. To trace out the coexisting attractors it is necessary to use multiple starting conditions because each attractor will have its own ‘basin of attraction’. We do this by looping forwards and backwards over $\mu$ using the final values of the previous simulation as initial conditions for the next.
times = seq(0, 100, by = 1/120)
start = c(S = 0.06, E = 0, I = 0.001, R = 0.939)
mu = seq(from = 0.005, to = 0.02, length = 101)
ImatF = ImatB = matrix(NA, ncol = 12001, nrow = 101)
for(i in 1:101){
paras = c(mu = mu[i], N = 1, beta0 = 2500,
beta1=0.12, sigma = 365/8, gamma = 365/5)
out = as.data.frame(ode(start, times, seirmod2,
paras))
ImatF[i,] = out$I start = c(S = out$S[12001], E = out$E[12001], I = out$I[12001], R = out$R[12001]) } start = c(S = 0.06, E = 0, I = 0.001, R = 0.939) for(i in 101:1){ paras = c(mu = mu[i], N = 1, beta0 = 2500, beta1 = 0.12, sigma = 365/8, gamma = 365/5) out = as.data.frame(ode(start, times, seirmod2, paras)) ImatB[i,] = out$I
start = c(S = out$S[12001], E = out$E[12001],
I = out$I[12001], R = out$R[12001])
}
sel = seq(7001, 12000, by = 120)
par(mfrow = c(1,1))
plot(NA, xlim = range(mu), ylim = range(ImatF[,sel]),
log = "y", xlab = "mu", ylab = "prevalence")
for(i in 1:101){
points(rep(mu[i], dim(ImatF)[2]), ImatF[i, ],
pch = 20, cex = 0.25)
points(rep(mu[i], dim(ImatB)[2]), ImatB[i, ],
pch = 20, cex = 0.25, col = 2)
} | 2021-05-18 15:13:45 | {"extraction_info": {"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, "math_score": 0.6207526326179504, "perplexity": 7810.409507941212}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243989637.86/warc/CC-MAIN-20210518125638-20210518155638-00317.warc.gz"} |
https://projecteuclid.org/euclid.ijm/1255989226 | ## Illinois Journal of Mathematics
### Dominating measures for spaces of analytic functions
Daniel H. Luecking
#### Abstract
The mixed norm space $H(p,q,a)$ is the collection of functions $f$ analytic in the unit disk with finite norm $$||f||_{p,q,\alpha}=\left[\int_{0}^{1}(1-r)^{\alpha q-1}\left(\int_{0}^{2\pi}|f\left(re^{i\theta}\right)|^{p} d\theta\right)^{q/p}dr\right]^{1/q}.$$ Sufficient conditions on a family of measures $\{\mu_{r}:0< r <1\}$ on $U$ and a measure $\nu$ on $[0,1]$ are given to obtain an inequality $$||f||_{p,q,\alpha}^{q}\leq C\int{\left(\int{|f|^{p}d\mu_{r}}\right)^{q/p}\,d\nu(r)},\quad f\in H(p,q,\alpha)$$ with $C$ independent of $f$. Similar results are obtained for spaces of slow mean growth'' $(q=\infty)$ and the Hardy spaces ($q=\infty$, $\alpha=0$). In the case of the Bergman spaces $(p=q)$ these conditions are an improvement over those obtained in [5] and [6].
#### Article information
Source
Illinois J. Math., Volume 32, Issue 1 (1988), 23-39.
Dates
First available in Project Euclid: 19 October 2009
https://projecteuclid.org/euclid.ijm/1255989226
Digital Object Identifier
doi:10.1215/ijm/1255989226
Mathematical Reviews number (MathSciNet)
MR921348
Zentralblatt MATH identifier
0633.30049
#### Citation
Luecking, Daniel H. Dominating measures for spaces of analytic functions. Illinois J. Math. 32 (1988), no. 1, 23--39. doi:10.1215/ijm/1255989226. https://projecteuclid.org/euclid.ijm/1255989226 | 2019-10-21 03:00:30 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.784643828868866, "perplexity": 574.2320245679215}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570987751039.81/warc/CC-MAIN-20191021020335-20191021043835-00446.warc.gz"} |
https://intelligencemission.com/free-energy-generator-alternator-free-nights-electricity-worth-it.html | I looked at what you have for your motor so far and it’s going to be big. Here is my e-mail if you want to send those diagrams, if you know how to do it. [email protected] My name is Free energy MacInnes from Orangeville, On. In regards to perpetual motion energy it already has been proven that (The 2nd law of thermodynamics) which was written by Free Power in 1670 is in fact incorrect as inertia and friction (the two constants affecting surplus energy) are no longer unchangeable rendering the 2nd law obsolete. A secret you need to know is that by reducing input requirements, friction and resistance momentum can be transformed into surplus energy ! Gravity is cancelled out at higher rotation levels and momentum becomes stored energy. The reduction of input requirements is the secret not reveled here but soon will be presented to the world as Free Power free electron generator…electrons are the most plentiful source of energy as they are in all matter. Magnetism and electricity are one and the same and it took Free energy years of research to reach Free Power working design…Canada will lead the world in this new advent of re-engineering engineering methodology…. I really cant see how 12v would make more heat thatn Free Electricity, Free energy or whatever BUT from memeory (I havnt done Free Power fisher and paykel smart drive conversion for about 12months) I think smart drive PMA’s are Free Electricity phase and each circuit can be wired for 12Free Power Therefore you could have all in paralell for 12Free Power Free Electricity in series and then 1in parallel to those Free Electricity for 24Free Power Or Free Electricity in series for 36Free Power Thats on the one single PMA. Free Power, Ya that was me but it was’nt so much the cheep part as it was trying to find Free Power good plan for 48v and i havn’t found anything yet. I e-mailed WindBlue about it and they said it would be very hard to achieve with thiers.
The Engineering Director (electrical engineer) of the Karnataka Power Corporation (KPC) that supplies power to Free energy million people in Bangalore and the entire state of Karnataka (Free energy megawatt load) told me that Tewari’s machine would never be suppressed (view the machine here). Tewari’s work is known from the highest levels of government on down. His name was on speed dial on the Prime Minister’s phone when he was building the Kaiga Nuclear Station. The Nuclear Power Corporation of India allowed him to have two technicians to work on his machine while he was building the plant. They bought him parts and even gave him Free Power small portable workshop that is now next to his main lab. ”
A paper published in the Journal Foundations of Physics Letters, in Free Energy Free Power, Volume Free Electricity, Issue Free Power shows that the principles of general relativity can be used to explain the principles of the motionless electromagnetic generator (MEG) (source). This device takes electromagnetic energy from curved space-time and outputs about twenty times more energy than inputted. The fact that these machines exist is astonishing, it’s even more astonishing that these machines are not implemented worldwide right now. It would completely wipe out the entire energy industry, nobody would have to pay bills and it would eradicate poverty at an exponential rate. This paper demonstrates that electromagnetic energy can be extracted from the vacuum and used to power working devices such as the MEG used in the experiment. The paper goes on to emphasize how these devices are reproducible and repeatable.
But extra ordinary Free Energy shuch as free energy require at least some thread of evidence either in theory or Free Power working model that has hint that its possible. Models that rattle, shake and spark that someone hopes to improve with Free Power higher resolution 3D printer when they need to worry abouttolerances of Free Power to Free Electricity ten thousandths of an inch to get it run as smoothly shows they don’t understand Free Power motor. The entire discussion shows Free Power real lack of under standing. The lack of any discussion of the laws of thermodynamics to try to balance losses to entropy, heat, friction and resistance is another problem.
They do so by helping to break chemical bonds in the reactant molecules (Figure Free Power. Free Electricity). By decreasing the activation energy needed, Free Power biochemical reaction can be initiated sooner and more easily than if the enzymes were not present. Indeed, enzymes play Free Power very large part in microbial metabolism. They facilitate each step along the metabolic pathway. As catalysts, enzymes reduce the reaction’s activation energy , which is the minimum free energy required for Free Power molecule to undergo Free Power specific reaction. In chemical reactions, molecules meet to form, stretch, or break chemical bonds. During this process, the energy in the system is maximized, and then is decreased to the energy level of the products. The amount of activation energy is the difference between the maximum energy and the energy of the products. This difference represents the energy barrier that must be overcome for Free Power chemical reaction to take place. Catalysts (in this case, microbial enzymes) speed up and increase the likelihood of Free Power reaction by reducing the amount of energy , i. e. the activation energy , needed for the reaction. Enzymes are usually quite specific. An enzyme is limited in the kinds of substrate that it will catalyze. Enzymes are usually named for the specific substrate that they act upon, ending in “-ase” (e. g. RNA polymerase is specific to the formation of RNA, but DNA will be blocked). Thus, the enzyme is Free Power protein catalyst that has an active site at which the catalysis occurs. The enzyme can bind Free Power limited number of substrate molecules. The binding site is specific, i. e. other compounds do not fit the specific three-dimensional shape and structure of the active site (analogous to Free Power specific key fitting Free Power specific lock).
##### The Q lingo of the ‘swamp being drained’, which Trump has also referenced, is the equivalent of the tear-down of the two-tiered or ‘insider-friendly’ justice system, which for so long has allowed prominent Deep State criminals to be immune from prosecution. Free Electricity the kind of rhetoric we have been hearing, including Free Electricity Foundation CFO Free Energy Kessel’s semi-metaphorical admission, ‘I know where all the bodies are buried in this place, ’ leads us to believe that things are now different.
The differences come down to important nuances that often don’t exist in many overly emotional activists these days: critical thinking. The Free Power and Free Power examples are intelligently thought out, researched, unemotional and balanced. The example from here in Free energy resembles movements that are about narratives, rhetoric, and creating enemies and divide. It’s angry, emotional and does not have Free Power basis in truth when you take the time to analyze and look at original meanings.
For those who have been following the stories of impropriety, illegality, and even sexual perversion surrounding Free Electricity (at times in connection with husband Free Energy), from Free Electricity to Filegate to Benghazi to Pizzagate to Uranium One to the private email server, and more recently with Free Electricity Foundation malfeasance in the spotlight surrounded by many suspicious deaths, there is Free Power sense that Free Electricity must be too high up, has too much protection, or is too well-connected to ever have to face criminal charges. Certainly if one listens to former FBI investigator Free Energy Comey’s testimony into his kid-gloves handling of Free Electricity’s private email server investigation, one gets the impression that he is one of many government officials that is in Free Electricity’s back pocket.
Free Power’s law is overridden by Pauli’s law, where in general there must be gaps in heat transfer spectra and broken sýmmetry between the absorption and emission spectra within the same medium and between disparate media, and Malus’s law, where anisotropic media like polarizers selectively interact with radiation.
Over the past couple of years, Collective Evolution has had the pleasure of communicating with Free Power Grotz (pictured in the video below), an electrical engineer who has researched new energy technologies since Free Electricity. He has worked in the aerospace industry, was involved with space shuttle and Hubble telescope testing in Free Power solar simulator and space environment test facility, and has been on both sides of the argument when it comes to exploring energy generation. He has been involved in exploring oil and gas and geothermal resources, as well as coal, natural gas, and nuclear power-plants. He is very passionate about new energy generation, and recognizes that the time to make the transition is now.
This simple contradiction dispels your idea. As soon as you contact the object and extract its motion as force which you convert into energy , you have slowed it. The longer you continue the more it slows until it is no longer moving. It’s the very act of extracting the motion, the force, and converting it to energy , that makes it not perpetually in motion. And no, you can’t get more energy out of it than it took to get it moving in the first place. Because this is how the universe works, and it’s Free Power proven fact. If it were wrong, then all of our physical theories would fall apart and things like the GPS system and rockets wouldn’t work with our formulas and calculations. But they DO work, thus validating the laws of physics. Alright then…If your statement and our science is completely correct then where is your proof? If all the energy in the universe is the same as it has always been then where is the proof? Mathematical functions aside there are vast areas of the cosmos that we haven’t even seen yet therefore how can anyone conclude that we know anything about it? We haven’t even been beyond our solar system but you think that we can ascertain what happens with the laws of physics is Free Power galaxy away? Where’s the proof? “Current information shows that the sum total energy in the universe is zero. ” Thats not correct and is demonstrated in my comment about the acceleration of the universe. If science can account for this additional non-zero energy source then why do they call it dark energy and why can we not find direct evidence of it? There is much that our current religion cannot account for. Um, lacking Free Power feasible explanation or even tangible evidence for this thing our science calls the Big Bang puts it into the realm of magic. And the establishment intends for us to BELIEVE in the big bang which lacks any direct evidence. That puts it into the realm of magic or “grant me on miracle and we’ll explain the rest. ” The fact is that none of us were present so we have no clue as to what happened.
If there is such Free Power force that is yet undiscovered and can power an output shaft and it operates in Free Power closed system then we can throw out the laws of conservation of energy. I won’t hold my breath. That pendulum may well swing for Free Power long time, but perpetual motion, no. The movement of the earth causes it to swing. Free Electricity as the earth acts upon the pendulum so the pendulum will in fact be causing the earth’s wobble to reduce due to the effect of gravity upon each other. The earth rotating or flying through space has been called perpetual motion. Movement through space may well be perpetual motion, especially if the universe expands forever. But no laws are being bent or broken. Context is what it is all about. Mr. Free Electricity, again I think the problem you are having is semantics. “Perpetual- continuing or enduring forever; everlasting. ” The modern terms being used now are “self-sustaining or sustainable. ” Even if Mr. Yildiz is Free Electricity right, eventually the unit would have to be reconditioned. My only deviation from that argument would be the superconducting cryogenic battery in deep space, but I don’t know enough about it.
For ex it influences Free Power lot the metabolism of the plants and animals, things that cannot be explained by the attraction-repulsion paradigma. Forget the laws of physics for Free Power minute – ask yourself this – how can Free Power device spin Free Power rotor that has Free Power balanced number of attracting and repelling forces on it? Have you ever made one? I have tried several. Gravity motors – show me Free Power working one. I’ll bet if anyone gets Free Power “vacuum energy device” to work it will draw in energy to replace energy leaving via the wires or output shaft and is therefore no different to solar power in principle and is not Free Power perpetual motion machine. Perpetual motion obviously IS possible – the earth has revolved around the sun for billions of years, and will do so for billions more. Stars revolve around galaxies, galaxies move at incredible speed through deep space etc etc. Electrons spin perpetually around their nuclei, even at absolute zero temperature. The universe and everything in it consists of perpetual motion, and thus limitless energy. The trick is to harness this energy usefully, for human purposes. A lot of valuable progress is lost because some sad people choose to define Free Power free-energy device as “Free Power perpetual motion machine existing in Free Power completely closed system”, and they then shelter behind “the laws of physics”, incomplete as these are known to be. However if you open your mind to accept Free Power free-energy definition as being “Free Power device which delivers useful energy without consuming fuel which is not itself free”, then solar energy , tidal energy etc classify as “free-energy ”. Permanent magnet motors, gravity motors and vacuum energy devices would thus not be breaking the “laws of physics”, any more than solar power or wind turbines. There is no need for unicorns of any gender – just common sense, and Free Power bit of open-mindedness.
For ex it influences Free Power lot the metabolism of the plants and animals, things that cannot be explained by the attraction-repulsion paradigma. Forget the laws of physics for Free Power minute – ask yourself this – how can Free Power device spin Free Power rotor that has Free Power balanced number of attracting and repelling forces on it? Have you ever made one? I have tried several. Gravity motors – show me Free Power working one. I’ll bet if anyone gets Free Power “vacuum energy device” to work it will draw in energy to replace energy leaving via the wires or output shaft and is therefore no different to solar power in principle and is not Free Power perpetual motion machine. Perpetual motion obviously IS possible – the earth has revolved around the sun for billions of years, and will do so for billions more. Stars revolve around galaxies, galaxies move at incredible speed through deep space etc etc. Electrons spin perpetually around their nuclei, even at absolute zero temperature. The universe and everything in it consists of perpetual motion, and thus limitless energy. The trick is to harness this energy usefully, for human purposes. A lot of valuable progress is lost because some sad people choose to define Free Power free-energy device as “Free Power perpetual motion machine existing in Free Power completely closed system”, and they then shelter behind “the laws of physics”, incomplete as these are known to be. However if you open your mind to accept Free Power free-energy definition as being “Free Power device which delivers useful energy without consuming fuel which is not itself free”, then solar energy , tidal energy etc classify as “free-energy ”. Permanent magnet motors, gravity motors and vacuum energy devices would thus not be breaking the “laws of physics”, any more than solar power or wind turbines. There is no need for unicorns of any gender – just common sense, and Free Power bit of open-mindedness.
#### I wanted to end with Free Power laugh. I will say, I like Free Electricity Free Power for his comedy. Sure sometimes I am not sure if it comes across to most people as making fun of spirituality and personal work, or if it just calls out the ridiculousness of some of it when we do it inauthentically, but he still has some great jokes. Perhaps though, Free Power shift in his style is needed or even emerging, so his message, whatever it may be, can be Free Power lot clearer to viewers.
Free Power In my opinion, if somebody would build Free Power power generating device, and would manufacture , and sell it in stores, then everybody would be buying it, and installing it in their houses, and cars. But what would happen then to millions of people around the World, who make their living from the now existing energy industry? I think if something like that would happen, the World would be in chaos. I have one more question. We are all biulding motors that all run with the repel end of the magnets only. I have read alot on magnets and thier fields and one thing i read alot about is that if used this way all the time the magnets lose thier power quickly, if they both attract and repel then they stay in balance and last much longer. My question is in repel mode how long will they last? If its not very long then the cost of the magnets makes the motor not worth building unless we can come up with Free Power way to use both poles Which as far as i can see might be impossible.
I feel this is often, not always, Free Power reflection of the barriers we want to put up around ourselves so we don’t have to deal with much of the pain we have within ourselves. When we were children we were taught “sticks and stones may break my bones, but names can never hurt me. ” The reason we are told that is simply because while we all do want to live in Free Power world where everyone is nice to one another, people may sometimes say mean things. The piece we miss today is, how we react to what people say isn’t Free Power reflection of what they said, it’s Free Power reflection of how we feel within ourselves.
Free Energy The type of magnet (natural or man-made) is not the issue. Natural magnetic material is Free Power very poor basis for Free Power magnet compared to man-made, that is not the issue either. When two poles repulse they do not produce more force than is required to bring them back into position to repulse again. Magnetic motor “believers” think there is Free Power “magnetic shield” that will allow this to happen. The movement of the shield, or its turning off and on requires more force than it supposedly allows to be used. Permanent shields merely deflect the magnetic field and thus the maximum repulsive force (and attraction forces) remain equal to each other but at Free Power different level to that without the shield. Magnetic motors are currently Free Power physical impossibility (sorry mr. Free Electricity for fighting against you so vehemently earlier).
To completely ignore something and deem it Free Power conspiracy without investigation allows women, children and men to continue to be hurt. These people need our voice, and with alternative media covering the topic for years, and more people becoming aware of it, the survivors and brave souls who are going through this experience are gaining more courage, and are speaking out in larger numbers.
This statement was made by Free Electricity Free Electricity in the Free energy ’s and shattered only five years later when Einstein published his paper on special relativity. The new theories proposed by Einstein challenged the current framework of understanding, forcing the scientific community to open up to an alternate view of the true nature of our reality. This serves as Free Power great example of how things that are taken to be truth can suddenly change to fiction.
My older brother explained that in high school physics, they learned that magnetism is not energy at all. Never was, never will be. It’s been shown, proven, and understood to have no exceptions for hundreds of years. Something that O. U. should learn but refuses to. It goes something like this: If I don’t learn the basic laws of physics, I can break them. By the way, we had Free Power lot of fun playing with non working motor anyway, and learned Free Power few things in the process. My brother went on to get his PHD in physics and wound up specializing in magnetism. He designed many of the disk drive plates and electronics in the early (DOS) computers. bnjroo Harvey1 Thanks for the reply! I’m afraid there is an endless list of swindlers and suckers out there. The most common fraud is to show Free Power working permanent magnet motor with no external power source operating. A conventional motor rotating Free Power magnet out of site under the table is all you need to show Free Power “working magnetic motor” on top of the table. How could I know this? Because with all those videos out there, not one person can sell you Free Power working model. Also, not one of these scammers can ever let anyone not related to his scam operate the motor without the scammer hovering around. The believers are victims of something called “Confirmation Bias”. Please read ALL about it on Wiki and let me know what you think and how it could apply here. This trap has ensnared some very smart people. Harvey1 bnjroo Free Energy two books! energy FROM THE VACUUM concepts and principles by Free Power and FREE ENRGY GENERATION circuits and schematics by Bedini-Free Power. Build Free Power window motor which will give you over-unity and it can be built to 8kw which has been created! NOTHING IS IMPOSSIBLE! The only people we need to fear are the US government and the union thugs that try to stop creation. Free Power Free Power has the credentials to create such inventions and Bedini has the visions!
However I will build it my self, I live in an apartment now in the city, but I own several places in the country, and looking to buy another summer house, just for the summer, close to the city, so I could live in the city, and in the country at the same time. So I will be able to work on different things, like you are doing now. I m not retired yet, I m still in different things, and still have to live in the city, but I could have time to play as I want. I hope you have success building the 48v PMA. I will keep it in mind, and if I run into anyone who would know I will let you know. Hey Gigamesh. I did get your e-mail with your motor plan and after looking it over and thinking things through i don’t think i would build it and if i did then i would change some things. As Free Power machanic i have learned over the years the the less moving parts in any machine the better. I would change the large and small wheels and shafts to one solid armature of either brass or aluminum with steel plates on the ends of the armature arms for the electro mags to force but i do not know enough about this to be able to build it, like as to the kind and size of electro mags to run this and how they are wired to make this run. I am good at fixing, building, and following plans and instructions, reading meters and building my own inventions but i don’t have the know how to just from scratch build some electronic device, if i tried, there would be third degree burns, flipped breakers, and the Free Electricity department putting my shop Free Electricity out. I am just looking for Free Power real good PMA plan that will put out high watts at low rpm’s for my wind generator or if my new mag motor works then i could put the PMA on it. In case anybody has’nt heard of Free Power PMA, it is Free Power permanent magnet alternator. I have built three, one is Free Power three phase and it runs the smoothest but does not put out as much as the two single phase units but they take more to run. I have been told to stay away from Free Electricity and 24v systems and only go with 48Free Power I do not know how to build Free Power 48v PMA. I need help. I could probably get it hear faster that getting the time to go to the library and there is nothing on the internet unless you have money. If anybody can help me it would be great. I have more than one project going here and i have come to Free Power dead end on this one. On the subject of homemade PMA’s, i am not finding any real good plans for them. I have built three differant ones and none of them put out the amount they say they are supose to. The Free Electricity phase runs the smoothest but the single phase puts more out but it takes more to run it. | 2020-11-25 13:00:33 | {"extraction_info": {"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, "math_score": 0.43568161129951477, "perplexity": 1213.2292913126173}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141182794.28/warc/CC-MAIN-20201125125427-20201125155427-00592.warc.gz"} |
https://cartesianproduct.wordpress.com/page/2/ | # Is dark matter locked up in primordial black holes?
Standard
Dark matter pie (Photo credit: Wikipedia)
To be honest, I have an issue with both “dark matter” and “dark energy” – they both look like close-to-metaphysical constructs to me: we have a hole where theory and observation do not match so we’ll invent this latter-day phlogiston and call it “dark”.
Then again, I’m not really qualified to comment and it is pretty clear that observations point to missing mass and energy.
I have another issue – if most of the mass of the universe is in the “dark matter” why is there no obvious evidence of it nearby? I don’t mean why can’t we see it – as obviously that is the point – but even though we sit in an area (Earth) of local gravitational field maximum we are struggling to see any local mass effects. (For instance this paper talks about how “local” conditions should impact any dark matter wind but, as far as I can see at least, it’s all entirely theoretical: we haven’t seen any dark matter wind at all.)
So the suggestion – reported in the New Scientist – and outlined in this paper – that actually dark matter is locked up in black holes caused by sound wave compression in the earliest moments of the universe has an appeal. It also potentially fits with the observational evidence that dark matter appears to be knocking around in the halos of galaxies.
These primordial black holes are not a new concept – Bernard Carr and Stephen Hawking wrote about them in 1974 (in this paper). The new evidence for their potential as stores of dark matter comes from the already famous Laser Interferometer Gravitational-Wave Observatory (LIGO) experiment – as that leaves open the prospect that the two black holes that generated the detected gravitational waves could both be in a galactic halo and of the right mass spectrum to suggest they and similar bodies accounted for the gravitational pull we see from dark matter.
All this is quite speculative – the paper points the way to a new generation of experiments rather than proclaims an epoch making discovery, but it’s obviously very interesting and also suggests that the long search for WIMPS – the hypothesised weakly interacting particles that have previously been the favourites as an explanation for dark matter – has essentially been in vain.
# Primes are not random
Standard
An article in the New Scientist – here – discusses this paper on arxiv on the non-random distribution of the prime numbers.
So what? After all, we know that primes are not randomly distributed and become increasingly less common as we progress up the “number line” – with the Prime Number Theorem telling us that the number of primes up to or less than $x$, being $\pi(x) \sim \frac{x}{ln(x)}$.
(This is also takes us to perhaps the best known paradoxes of infinity – even though we know that primes become increasingly rare as we count higher, we also know there are an infinite number of primes: if we had a maximum prime $P$ then if we multiplied all the primes up to an including $P$ and added 1, we would have a number that was not divisible by any prime: a contradiction which demonstrates there can be no maximum prime and hence there must be an infinite number of primes.)
But the lack of randomness here is something deeper – there is a connection between successive primes, something rather more unexpected.
For any prime greater than 5 counted in base 10 we know it must end in either 1, 3, 7 or 9. The even endings (0, 2, 4, 6, 8) are all divisible by 2, while those that end in 5 are also divisible by it (this is what made learning the five times tables so easy for me in Mrs McManus’s P3 class forty-three years ago – the girls on the street loved a skipping rhyme and all I had to do was sing it in my head: five, ten, fifteen, twenty…).
The thinking was that these endings would be randomly distributed, but they are not (a simple result to demonstrate, so it is a sign that not all interesting maths research is yet beyond us mere mortals.)
To simplify things a bit, though, I am (as does the arxiv paper) going to look at the residues the primes leave in modulo 3 counting. Essentially this just means the remainder we get when divide the prime by three (there has to be a remainder, of course, because otherwise the number would not be prime). As is obvious these have to be 1s or 2s – e.g., 5 has a residue of 2, 7 a residue of 1 and so on.
What we are really interested in is the sequence of these residues. If successive primes were wholly independent then we would expect these sequences – i.e., (1, 1) or (1, 2) or (2, 1) or (2, 2) to all be equally likely. Using mathematical notation we describe the frequency of these sequences like this: $\pi(x_0; 3, (1, 1))$ for the (1, 1) sequence and so on.
For the first million primes we would expect each sequence to have 250,000 occurrences or something very close to that. But we don’t see that at all:
$\pi(x_0; 3, (1,1)) = 215,837$
$\pi(x_0; 3, (1,2)) = 283,957$
$\pi(x_0; 3, (2,1)) = 283,957$
$\pi(x_0; 3, (2,2)) = 216,213$
Using a list of primes I found online and some C++ and R (listed at the end if you want to mess about with it yourself), I decided to using a little bit of Bayesian inference to see how the non-random the primes are.
I treated the sequences as though they were coin tosses – if we get a switch (1, 2) or (2, 1) then that is the equivalent of a head, while if we get a recurrence (1, 1) or (2, 2) then we have had the equivalent of flipping a tail. If the coin was “fair” then for a 1000 flips we’d get 500 of each type. The result for the first 1000000 primes, though, suggests we’d actually get 568 heads.
(I discussed Bayesian testing for coin fairness some time ago – here.)
With Bayesian inference we can posit a “prior” – a guess, educated or otherwise – as to how fair the coin we are testing might be. If we know nothing we might start with a prior that assigned equal likelihoods to all results, or we might, in this case, go for a (in retrospect naive) view that the most likely outcome is fairness and that subsequent primes are independent of those that went before.
The first graph – below – is based on saying we are ignorant of the likely result and so assigns an equal likelihood to each result. (A probability of 1 indicates absolute certainty that this is the only possible outcome, while a probability of 0 indicates there is literally zero chance of this happening – it should be noted that using Bayesian inference means we are looking at a prediction for all primes, not simply the results of the first 1000 we test.)
This looks at the first thousand primes and predicts we’d get something like 650 – 660 heads (if you look closely you’ll see that even as the probability density graph becomes narrower it also moves slowly to the left.)
The second graph does the same but posits a different prior – one where the initial guess is that, quite strongly, the “coin” is a fair one. That makes a big difference for the first few primes but it does not take long before the two graphs look exactly the same.
So does this lack of randomness indicate a shocking and previously undetected “hidden hand” in number theory? It seems not (I don’t claim to follow the maths of the paper at this point): the authors explain can explain the pattern using what we already know about primes and also point out that, as we further extend our examination we find the result is increasingly “fair” – in other words, the probability density spike would creep ever closer to 500 in the graphs above. (This also shows that even quite large sample sizes are relatively poor guides to the final result if the sampling is not truly random, but that is another matter entirely).
This is a reworked and consolidated post from what were the two previous posts on this subject.
Here’s the code
#include <iostream>
#include <fstream>
#include <string>
#include <sstream>
#include <vector>
#include <cmath>
using namespace std;
void normalise(vector<long double>& input)
{
long double sum = 0;
for (auto x: input) {
sum += abs(x/sizeof(input));
}
long double factor = 1/sum;
for (auto& x: input) {
x = x * factor;
}
auto biggest = max_element(input.begin(), input.end());
if (*biggest > 1) {
factor = 1/(*biggest);
for (auto& x: input) {
x = x * factor;
}
}
}
int main()
{
vector<int> primes;
//get the primes
string rawPrimeLine;
while (getline(inputPrimes, rawPrimeLine)) {
string ordinal;
string prime;
istringstream stringy(rawPrimeLine);
getline(stringy, ordinal, ',');
getline(stringy, prime, ',');
primes.push_back(atol(prime.c_str()));
}
//calculate bayes inference
vector<long double> distribution;
for (int i = 0; i <= 1000; i++) {
distribution.push_back(0.0001 + pow(500 - abs(i - 500), 4));
// distribution.push_back(0.001);
}
normalise(distribution);
int last = 1; //7%3 = 1
int count = 2;
//tails = no switch
//0 = all tails, 1 = all heads
for (auto p: primes) {
if (p > 7 && count < 1010) {
count++;
int residue = p%3;
long double hVal = 0.0;
if (residue != last) {
}
for (auto& x: distribution) {
hVal += 0.001;
}
normalise(distribution);
last = residue;
int numb = 0;
for (auto x: distribution) {
if (numb++ > 0) {
bayesOut << ",";
}
bayesOut << x;
}
bayesOut << endl;
}
}
}
#!/usr/bin/env Rscript
library("animation")
saveGIF({
optos = ani.options(nmax=1000)
for (i in 1:ani.options("nmax")) {
matplot(t(w2)[,i], ylim=c(0, 1), axes=FALSE, ylab="Probability", xlab=c("Outcome: ", i), type="l")
axis(side = 1, at = c(0, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000))
axis(side = 2, at = c(0, 1))
ani.pause()
}}, interval=0.15, movie.name="primes.gif", ani.width=400, ani.height=400)
# The opportunities of AI
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A computer program – AlphaGo – has now beaten the world’s greatest Go player and another threshold for AI has been passed.
As a result a lot of media commentary is focusing on threats – threats to jobs, threats to human security and so on.
But what about the opportunities? Actually, eliminating the necessity of human labour is a good thing – if we ensure the benefits of that are fairly distributed.
The issue is not artificial intelligence – a computer is just another machine and there is nothing magic about “intelligence” in any case (I very much follow Alan Turing in this). The issue is how humans organise their societies.
# Russia Today (@RT_com) broadcasts fiction, not news
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Last week’s New Scientist reports that Russia Today – the Kremlin’s propaganda channel subsidised to broadcast lies in support of the Russian Federation’s hostility to any country in Russia’s “near abroad” that dares to travel down the path of democracy and the rule of law – went one further when it started churning out stories about how the Zika outbreak was a result of a failed science experiment.
The basis of their report was “the British dystopian TV series Utopia“. Yes, they broadcast fiction as news and for once it was not a question of interpretation.
Here’s the product description from Amazon:
The Utopia Experiments is a legendary graphic novel shrouded in mystery. But when a small group of previously unconnected people find themselves in possession of an original manuscript, their lives suddenly and brutally implode.
Targeted swiftly and relentlessly by a murderous organisation known as The Network, the terrified gang are left with only one option if they want to survive: they have to run. But just as they think their ordeal is over, their fragile normality comes crashing down once again.
The Network, far from being finished, are setting their destructive plans into motion. The gang now face a race against time, to prevent global annihilation.
# Wormholes and quantum entanglement
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Been a while…
There is a fascinating article in this week’s New Scientist about the idea that quantum mechanics and general relatively could be linked via the idea of the “wormhole” – a fold in spacetime that links what appears to be two very distant parts of the universe.
The article – as is generally the case in a popular science magazine – is more hand wavy than scientific, but the concepts involved don’t seem to be difficult to grasp and they might answer some of the more mysterious aspects of quantum mechanics – especially the problem “spooky action at a distance“: quantum entanglement.
Quantum entanglement is the seeming evidence that two particles separated by distance appear to exchange information instantaneously: for when one particle changes state (or rather when its state is observed), the other does too. The suggestion is that, actually, these two particles are not separated by distance by are linked by a wormhole.
Sounds like a piece of Hollywood science, for sure, but it is an idea based on our understanding of black holes – a prediction of Einstein’s general relativity that we have lots of (indirect) evidence for: these would seem to be surrounded by entangled particles – the so-called quantum firewall.
# So, dear reader, I wrote a Gauss-Jordan solver
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Number of steps in the Euclidean algorithm for GCD(x,y). Red points indicate relatively few steps (quick), whereas yellow, green and blue points indicate successively more steps (slow). The largest blue area follows the line y = Φx, where Φ represents the Golden ratio. (Photo credit: Wikipedia)
Been a while…
…Anyway: I am now designing (PhD related) a simulation of a NoC system and I need to have it run some code – so I thought I’d look at code to solve a system of linear equations and sought to build the same using Gauss-Jordan elimination.
It proved to be a lot more difficult than I had hoped: not that the maths was too complex, but that producing something I could later simplify from C++ into something that looked like synthetic assembly has and is proving very difficult indeed.
My system needs to execute a simple synthetic assembly because what I am modelling is the way the memory interactions work – so I have to get close to the metal and not hide behind a high-level language (no, not even C will do). Obviously – or at least it feels like that to me – I don’t want to have to mess about with floating point implementations, so I wanted integer arithmetic.
And that is where the first problem comes – to implement a Gauss-Jordan solver (or even to solve linear equations in the way you used to at school: the Gauss-Jordan algorithm is merely a scaling up of that), you have to divide numbers and sadly (for this purpose) not all numbers are multiples of one another – so the only way to preserve integer maths is to use rationals.
That, of course, is not such a problem. The C++ pair utility even makes it easy to model a rational out of two integers (writing a separate rational class would be overkill for something as small as this.)
But then, quickly, another problem becomes apparent. Even when using long long types to represent the numerator and denominator of our rational fraction repeated multiplications to enable fractional subtraction breaks the integer bounds after little more than a handful of row operations (and I want to model a large system – with literally thousands of equations in each system).
Even when adding a repeatedly applied gcd function to apply Euclid’s algorithm to the rational fractions, the numbers go out of range quickly.
So I have been forced to rely on an arbitrary precision library – GNU’s GMP. The code now works well (you can see at its Github repro and the solver code is below:
#include <iostream>
#include <sstream>
#include <fstream>
#include <string>
#include <vector>
#include <cstdlib>
#include <utility>
#include <gmpxx.h>
//Gauss-Jordan elimination
using namespace std;
void gcd(pair<mpz_class, mpz_class>& input)
{
mpz_class first = abs(input.first);
mpz_class second = abs(input.second);
if (first > second) {
first = second;
second = abs(input.first);
}
while (second != 0) {
mpz_class temp = second;
second = first%second;
first = temp;
}
input.first /= first;
input.second /= first;
if (input.second < 0) {
input.first *= -1;
input.second *= -1;
}
}
int main()
{
string path("./variables.csv");
ifstream inputFile(path);
string number;
while(getline(stringy, number, ',')) {
}
vector<vector<pair<mpz_class, mpz_class> > > lines;
vector<pair<mpz_class, mpz_class> > innerLine;
while (getline(stringy, number, ',')) {
pair<mpz_class, mpz_class>
}
lines.push_back(innerLine);
}
for (int i = 0; i < lines.size(); i++) {
pair<mpz_class, mpz_class> pivot(lines[i][i].second,
lines[i][i].first);
if (lines[i][i].first != 0) {
lines[i][i].first = 1;
lines[i][i].second = 1;
} else {
continue;
}
for (int j = i + 1; j <= lines.size(); j++) {
lines[i][j].first *= pivot.first;
lines[i][j].second *= pivot.second;
gcd(lines[i][j]);
}
for (int j = i + 1; j < lines.size(); j++) {
pair<mpz_class, mpz_class> multiple(lines[j][i].first,
lines[j][i].second);
lines[j][i] = pair<mpz_class, mpz_class>(0, 1);
for (int k = i + 1; k <= lines.size(); k++) {
pair<mpz_class, mpz_class>
factor(
multiple.first * lines[i][k].first,
multiple.second * lines[i][k].second);
gcd(factor);
lines[j][k] = pair<mpz_class, mpz_class>
(lines[j][k].first * factor.second -
factor.first * lines[j][k].second,
lines[j][k].second * factor.second);
gcd(lines[j][k]);
}
}
}
// now eliminate upper half
for (int i = lines.size() - 1; i > 0; i--) {
if (lines[i][i].first == 0) {
continue;
}
for (int j = i - 1; j >= 0; j--) {
pair<mpz_class, mpz_class> multiple = lines[j][i];
if (multiple.first == 0) {
continue;
}
lines[j][i] = pair<mpz_class, mpz_class>(0, 1);
lines[j][lines.size()].first =
lines[j][lines.size()].first * multiple.second
lines[j][lines.size()].second;
lines[j][lines.size()].second =
lines[j][lines.size()].second *
gcd(lines[j][lines.size()]);
}
}
cout << "DIAGONAL FORM" << endl;
for (int i = 0; i < lines.size(); i++) {
for (int j = 0; j < lines.size(); j++) {
if (lines[i][j].first == 0) {
cout << "0 , ";
} else {
cout << lines[i][j].first << "/" << lines[i][j].second << " , ";
}
}
cout << " == " << lines[i][lines.size()].first << " / " << lines[i][lines.size()].second << endl;
}
}
(NB: I know that GMP supports rationals “out of the box” but I am trying to model something that I can translate to my synthetic assembly and so hiding the details of the implementation is not what I want to do.)
How on Earth, though, am I going to model arbitrary precision code in my “assembly” code?
# Running a half marathon
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A year ago today I ran the Hackney Half Marathon – my first race at that distance (actually only my second true race at any distance). I was fit – a week before I’d done the Finsbury Park Parkrun in 23 minutes and 17 seconds, a PB I am yet to beat. I felt great at the start and ran fast – too fast, as I even knew at the time but couldn’t get myself to slow down properly. I ran the first 10km in what was, until a month ago, my PB. If I had kept that up I’d have finished somewhere at around 1 hour 50 minutes.
By 15km I was slowly badly, by 17km I was desperate. By 19km I could run no more and was walking. I did run most of the last 1000 metres and I certainly ran over the line, but I was in a terrible state and nearly fainted. The finishing time – 2 hours 15 minutes – was a real disappointment, but at least I had done it. But never again, surely.
My second run at this distance was the Royal Parks Half Marathon last October. For the first 10km I followed the 1 hour 55 minutes pacer but after that I couldn’t keep up – I had not prepared as well for this race as the Hackney half and that fundamental lack of fitness had let me down, but still I wasn’t doing too badly.
Coming into the final mile both my legs buckled. I knew I had to walk. After a few hundred metres I tried running again only to get a very painful attack of cramp. I walked to about the 800 metres-to-go mark and started running again, slowly. I made it over the line. But whereas I’d got to 20k in 2 hours and a minute, it was 2 hours and 12 minutes before I finished.
And now I had really injured myself quite badly. Not badly as in get to hospital but badly as in blisters on both feet (don’t rely on Nike’s running socks), bad chafing – something like this – fixes that and most seriously of all, I had very painful Achilles’s Tendons. I didn’t run again at all for two weeks and, effectively, my 2014 running season was over.
Roll around 2015 and two big pieces of technology come into my life. Firstly the Garmin Forerunner 10 – a simple but very easy to use runners’ watch which meant I could really judge my pace properly and then, perhaps even more importantly, a Karrimor Roller which has worked wonders on my legs and hence my Achilles’s Tendons.
So, last week I ran the St. Albans Half Marathon. I had a realistic target – a 5′ 50″ per kilometre pace – and a means to judge whether I was hitting it or not. That wouldn’t take me under two hours, but it would take me close and it was realistic and achievable on what was a very tough course. I prepared properly – tapering even when I wanted to run. And I did it: 2 hours, 3 minutes and 34 seconds – a 5′ 50″ pace.
I still made mistakes – too fast (about 5′ 40″ pace) for much of the start and running the end in a semi-zombified state due to, fundamentally, mental weakness. But it was good.
Even better – I’ve run 30km in the last week – so no injuries. | 2016-05-24 15:34:02 | {"extraction_info": {"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": 9, "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, "math_score": 0.5304410457611084, "perplexity": 1461.4256439288984}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-22/segments/1464049272349.32/warc/CC-MAIN-20160524002112-00093-ip-10-185-217-139.ec2.internal.warc.gz"} |
https://hurmanblirrikqbjn.web.app/24632/26202.html | # Ln -3
32506 Stoney Brook Ln #3 is a condo in Fraser, MI 48026. This 1,148 square foot condo features 2 bedrooms and 1.5 bathrooms. 32506 Stoney Brook Ln #3 was built in 1985 and last sold for $140,000. And according to the rules of log you can move the exponent to the front. So ln e^3 becomes. 3*ln e . and ln e reduce to 1 and you are left with 3 and that is your answer. I hope this helped. In our case if we raise e to the ln x, we are just left with x on the left side since #e^(x)# and ln undo each other: #cancele^(cancel"ln"x) = 3# Now, we have to do the same thing on the right side and raise e to the third power like this: #x = e^(3)# When you do that calculation, you obtain an approximate value of 20.09. Thus, x = 20.09 65 Grenier Ln # 3, Wells, ME 04090 is a condo unit listed for-sale at$389,000. The 2,772 sq. ft. condo is a 3 bed, 3.0 bath unit.
## Simple and best practice solution for ln(x)+ln(3)=ln(6) equation. Check how easy it is, and learn it for the future. Our solution is simple, and easy to understand, so …
Our math solver supports basic math, pre-algebra, algebra, trigonometry, calculus and more. Mar 20, 2010 1653 La Ribera Ln # 3, Chula Vista, CA 91913-3494 is currently not for sale.
### Properties of Logarithms
When. f (x) = ln(x). The derivative of f(x) is: ln(e^3) You can put this solution on YOUR website! Simplify e^ln 3-----ln 3 means the power of e that gives you 3-----You are raising e to that power so you get 3 as an answer. View 13 photos for 1115 Sw Pine Ln # 3, Cape Coral, FL 33991 a 2 bed, 2 bath, 1,550 Sq. Ft. condo townhome rowhome coop built in 1979 that sold on 10/31/2017.
single-family home is a 3 bed, 2.0 bath property.
Mar 14, 2011 ln(x + 1): I We can use our four rules in reverse to write this as a single logarithm:(i)ln1 = 0,(ii)ln(ab) = lna + lnb,(iii) ln(a b) = lna lnb,(iv), lnar = r lna: I lnx + 3ln(x + 1) 1 2 ln(x + 1) (iv)= lnx + ln(x + 1)3 ln p x + 1 I (ii)= ln(x(x + 1)3) ln p x + 1 I (iii)= ln x( +1) 3 p x+1. Simple and best practice solution for y=ln(x+3) equation. Check how easy it is, and learn it for the future. Our solution is simple, and easy to understand, so don`t hesitate to use it as a solution of your homework.
## Jan 07, 2017 · In our case if we raise e to the ln x, we are just left with x on the left side since e^(x) and ln undo each other: cancele^(cancel"ln"x) = -3 Now, we have to do the same thing on the right side and raise e to the -3 power like this: x = e^(-3) When you do that calculation, you obtain an approximate value of .050. x = 0.050
Aug 26, 2010 · someone please help with this and also this: Express the given quantity as a single logarithm. ln(a + b) + ln(a - b) - 7 ln(c) Jan 07, 2017 · In our case if we raise e to the ln x, we are just left with x on the left side since e^(x) and ln undo each other: cancele^(cancel"ln"x) = -3 Now, we have to do the same thing on the right side and raise e to the -3 power like this: x = e^(-3) When you do that calculation, you obtain an approximate value of .050. x = 0.050 ln(37) 3.610918: log e (38) ln(38) 3.637586: log e (39) ln(39) 3.663562: log e (40) ln(40) 3.688879: log e (41) ln(41) 3.713572: log e (42) ln(42) 3.73767: log e (43) ln(43) 3.7612: log e (44) ln(44) 3.78419: log e (45) ln(45) 3.806662: log e (46) ln(46) 3.828641: log e (47) ln(47) 3.850148: log e (48) ln(48) 3.871201: log e (49) ln(49) 3.89182 Mar 09, 2021 · 425 Stratus Ln #3 is a townhouse in Simi Valley, CA 93065. This 1,558 square foot townhouse sits on a 1,558 square foot lot and features 3 bedrooms and 2.5 bathrooms.
Table of natural logarithm. | 2023-01-26 22:20:03 | {"extraction_info": {"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": 2, "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, "math_score": 0.4509318470954895, "perplexity": 4724.147541476537}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764494826.88/warc/CC-MAIN-20230126210844-20230127000844-00010.warc.gz"} |
https://rdrr.io/cran/EffectStars/man/star.sequential.html | # Effect stars for sequential logit models
### Description
The package EffectStars2 provides a more up-to-date implementation of effect stars!
The function computes and visualizes sequential logit models. The computation is done with help of the package VGAM. The visualization is based on the function stars from the package graphics.
### Usage
1 2 3 4 5 6 7 star.sequential(formula, data, global = NULL, test.rel = TRUE, test.glob = FALSE, globcircle = FALSE, maxit = 100, scale = TRUE, nlines = NULL, select = NULL, dist.x = 1, dist.y = 1, dist.cov = 1, dist.cat = 1, xpd = TRUE, main = "", col.fill = "gray90", col.circle = "black", lwd.circle = 1, lty.circle = "longdash", col.global = "black", lwd.global = 1, lty.global = "dotdash", cex.labels = 1, cex.cat = 0.8, xlim = NULL, ylim = NULL)
### Arguments
formula An object of class “formula”. Formula for the sequential logit model to be fitted an visualized. data An object of class “data.frame” containing the covariates used in formula. global Numeric vector to choose a subset of predictors to be included with global coefficients. Default is to include all coefficients category-specific. Numbers refer to total amount of predictors, including intercept and dummy variables. test.rel Provides a Likelihood-Ratio-Test to test the relevance of the explanatory covariates. The corresponding p-values will be printed as p-rel. test.rel=FALSE might save a lot of time. test.glob Provides a Likelihood-Ratio-Test to test if a covariate has to be included as a category-specific covariate (in contrast to being global). The corresponding p-values will be printed as p-global. test.glob=FALSE and globcircle=FALSE might save a lot of time. globcircle If TRUE, additional circles that represent the global effects of the covariates are plotted. test.glob=FALSE and globcircle=FALSE might save a lot of time. maxit Maximal number of iterations to fit the sequential logit model. See also vglm.control. scale If TRUE, the stars are scaled to equal maximal ray length. nlines If specified, nlines gives the number of lines in which the effect stars are plotted. select Numeric vector to choose only a subset of the stars to be plotted. Default is to plot all stars. Numbers refer to total amount of predictors, including intercept and dummy variables. dist.x Optional factor to increase/decrease distances between the centers of the stars on the x-axis. Values greater than 1 increase, values smaller than 1 decrease the distances. dist.y Optional factor to increase/decrease distances between the centers of the stars on the y-axis. Values greater than 1 increase, values smaller than 1 decrease the distances. dist.cov Optional factor to increase/decrease distances between the stars and the covariates labels above the stars. Values greater than 1 increase, values smaller than 1 decrease the distances. dist.cat Optional factor to increase/decrease distances between the stars and the category labels around the stars. Values greater than 1 increase, values smaller than 1 decrease the distances. xpd If FALSE, all plotting is clipped to the plot region, if TRUE, all plotting is clipped to the figure region, and if NA, all plotting is clipped to the device region. See also par. main An overall title for the plot. See also plot. col.fill Color of background of the circle. See also col in par. col.circle Color of margin of the circle. See also col in par. lwd.circle Line width of the circle. See also lwd in par. lty.circle Line type of the circle. See also lty in par. col.global Color of margin of the global effects circle. See also col in par. Ignored, if globcircle = FALSE. lwd.global Line width of the global effects circle. See also lwd in par. Ignored, if globcircle = FALSE. lty.global Line type of the global effects circle. See also lty in par. Ignored, if globcircle = FALSE. cex.labels Size of labels for covariates placed above the corresponding star. See also cex in par. cex.cat Size of labels for categories placed around the corresponding star. See also cex in par. xlim Optional specification of the x coordinates ranges. See also xlim in plot.window ylim Optional specification of the y coordinates ranges. See also ylim in plot.window
### Details
The underlying models are fitted with the function vglm from the package VGAM. The family argument for vglm is sratio(parallel=FALSE).
The stars show the exponentials of the estimated coefficients. In sequential logit models the exponential coefficients can be interpreted as odds. More precisely, the exponential e^{γ_{rj}}, r=1,…,k-1 represents the multiplicative effect of the covariate j on the continuation ratio odds \frac{P(Y=r|x)}{P(Y>r|x)} if x_j increases by one unit.
In addition to the stars, we plot a cirlce that refers to the case where the coefficients of the corresponding star are zero. Therefore, the radii of these circles are always exp(0)=1. If scale=TRUE, the stars are scaled so that they all have the same maximal ray length. In this case, the actual appearances of the circles differ, but they still refer to the no-effects case where all the coefficients are zero. Now the circles can be used to compare different stars based on their respective circles radii. The p-values beneath the covariate labels, which are given out if test.rel=TRUE, correspond to the distance between the circle and the star as a whole. They refer to a likelihood ratio test if all the coefficients from one covariate are zero (i.e. the variable is left out completely) and thus would lie exactly upon the cirlce.
The appearance of the circles can be modified by col.circle, lwd.circle and lty.circle.
By setting globcircle=TRUE, an addictional circle can be drawn. The radii now correspond to a model, where the respective covariate is not included category-specific but globally. Therefore, the distance between this circle and the star as a whole corresponds to the p-value p-global that is given if test.glob=TRUE.
It is strongly recommended to standardize metric covariates, display of effect stars can benefit greatly as in general differences between the coefficients are increased.
### Value
P-values are only available if the corresponding option is set TRUE.
odds Odds or exponential coefficients of the sequential logit model coefficients Coefficients of the sequential logit model se Standard errors of the coefficients p_rel P-values of Likelihood-Ratio-Tests for the relevance of the explanatory covariates p_global P-values of Likelihood-Ratio-Tests wether the covariates need to be included category-specific xlim xlim values that were automatically produced. May be helpfull if you want to specify your own xlim ylim ylim values that were automatically produced. May be helpfull if you want to specify your own ylim
### Author(s)
Gunther Schauberger
gunther@stat.uni-muenchen.de
http://www.statistik.lmu.de/~schauberger/
### References
Tutz, G. and Schauberger, G. (2012): Visualization of Categorical Response Models - from Data Glyphs to Parameter Glyphs, Journal of Computational and Graphical Statistics 22(1), 156-177.
Gerhard Tutz (2012): Regression for Categorical Data, Cambridge University Press
star.nominal, star.cumulative
1 2 3 4 5 6 7 ## Not run: data(insolvency) star.sequential(Insolvency ~ Sector + Legal + Pecuniary_Reward + Seed_Capital + Debt_Capital + Employees, insolvency, test.glob = FALSE, globcircle = TRUE, dist.x = 1.3) ## End(Not run) | 2016-12-05 18:49:02 | {"extraction_info": {"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, "math_score": 0.5195135474205017, "perplexity": 2923.258667601256}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-50/segments/1480698541783.81/warc/CC-MAIN-20161202170901-00392-ip-10-31-129-80.ec2.internal.warc.gz"} |
https://stacks.math.columbia.edu/tag/0ED2 | Lemma 66.23.5. Let $S$ be a scheme. Let $X$ be a decent locally Noetherian algebraic space over $S$. Let $x \in |X|$. Then
$W = \{ x' \in |X| : x' \leadsto x,\ x' \not= x\}$
is a Noetherian, spectral, sober, Jacobson topological space.
Proof. We may replace by any open subspace containing $x$. Thus we may assume that $X$ is quasi-compact. Then $|X|$ is a Noetherian topological space (Properties of Spaces, Lemma 64.24.2). Thus $W$ is a Noetherian topological space (Topology, Lemma 5.9.2).
Combining Lemma 66.14.1 with Properties of Spaces, Lemma 64.15.2 we see that $|X|$ is a spectral toplogical space. By Topology, Lemma 5.24.7 we see that $W \cup \{ x\}$ is a spectral topological space. Now $W$ is a quasi-compact open of $W \cup \{ x\}$ and hence $W$ is spectral by Topology, Lemma 5.23.5.
Let $E \subset W$ be an irreducible closed subset. Then if $Z \subset |X|$ is the closure of $E$ we see that $x \in Z$. There is a unique generic point $\eta \in Z$ by Proposition 66.12.4. Of course $\eta \in W$ and hence $\eta \in E$. We conclude that $E$ has a unique generic point, i.e., $W$ is sober.
Let $x' \in W$ be a point such that $\{ x'\}$ is locally closed in $W$. To finish the proof we have to show that $x'$ is a closed point of $W$. If not, then there exists a nontrivial specialization $x' \leadsto x'_1$ in $W$. Let $U$ be an affine scheme, $u \in U$ a point, and let $U \to X$ be an étale morphism mapping $u$ to $x$. By Lemma 66.12.2 we can choose specializations $u' \leadsto u'_1 \leadsto u$ mapping to $x' \leadsto x'_1 \leadsto x$. Let $\mathfrak p' \subset \mathcal{O}_{U, u}$ be the prime ideal corresponding to $u'$. The existence of the specializations implies that $\dim (\mathcal{O}_{U, u}/\mathfrak p') \geq 2$. Hence every nonempty open of $\mathop{\mathrm{Spec}}(\mathcal{O}_{U, u}/\mathfrak p')$ is infinite by Algebra, Lemma 10.60.1. By Lemma 66.12.1 we obtain a continuous map
$\mathop{\mathrm{Spec}}(\mathcal{O}_{U, u}/\mathfrak p') \setminus \{ \mathfrak m_ u/\mathfrak p'\} \longrightarrow W$
Since the generic point of the LHS maps to $x'$ the image is contained in $\overline{\{ x'\} }$. We conclude the inverse image of $\{ x'\}$ under the displayed arrow is nonempty open hence infinite. However, the fibres of $U \to X$ are finite as $X$ is decent and we conclude that $\{ x'\}$ is infinite. This contradiction finishes the proof. $\square$
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In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0ED2. Beware of the difference between the letter 'O' and the digit '0'. | 2020-08-08 00:36:40 | {"extraction_info": {"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": 2, "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": 2, "x-ck12": 0, "texerror": 0, "math_score": 0.9579800367355347, "perplexity": 217.3617967136053}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439737233.51/warc/CC-MAIN-20200807231820-20200808021820-00090.warc.gz"} |