text:
14,000 hints and counting!

The following comments are owned by whoever posted them. This site is not responsible for what they say.
Use LaTeX for Mathematics
Authored by: googoo on Feb 18, '11 08:53:46AM
It is nice to have access to special characters for occasional use, but LaTeX is the way to go to typeset mathematical equations. There is a great (and free) Mac implementation of LaTeX available at http://www.tug.org/mactex/.

-Mark
Use LaTeX for Mathematics
Authored by: hamarkus on Feb 18, '11 10:34:39AM
I was about to say the same, nothing is easier as this:
$\pi$
Use LaTeX for Mathematics
Authored by: dom on Feb 18, '11 11:35:32PM

uh, I think option-p is easier than $\pi$

Use LaTeX for Mathematics
Authored by: unforeseen:X11 on Feb 18, '11 10:55:14AM

Absolutely agree, it's amazing how many students fiddle with Pages/Word when they could live happy lives using LaTeX.

---
this is not the sig you`re looking for.

Use LaTeX for Mathematics
Authored by: dschneiderch on Feb 18, '11 01:07:52PM

LaTeX is not useful to students if professors expect papers in .docx for editing and journals/conferences expect submissions in .docx

Use LaTeX for Mathematics
Authored by: Jamin on Feb 21, '11 09:41:53AM
An even handier solution than the full-blown LaTeX solution is to use LaTeXiT (standard in the mactex package mentioned above) an awesome little app that lets you not bother with all the details like the LaTeX headers and \begin{document} and all that, but instead just type in an equation and then typeset, and it gives you a beautiful little pdf (or jpeg or tiff or a variety of other formats) that has been cropped right down to your equation. This is how I put every equation and nearly every greek character into my slides.
Use LaTeX for Mathematics
Authored by: smosit on Feb 22, '11 07:25:03AM

agreed. i just started using LaTeXiT a month or so ago for doing my calculus homework.. it's great! no set of keyboard shortcuts is going to create the kind of complex equations that i have to type up. here's link to a wikibooks page:

http://en.wikibooks.org/wiki/LaTeX/Mathematics

it's very helpful for learning the necessary syntax.


text:
# Recent content by KillaKem

1. ### 12 state counter design

Ended up doing the 0-11 counter.
2. ### 12 state counter design

I want to design a synchronous counter with 12 states and synchronous reset by connecting the overflow signal ( CT =15 ) directly to load input ( /M2M1 ) so that I can start the counter at a custom state, what I'm thinking of doing right now is just put flip-flops pre-loaded with my predefined...
3. ### Open Loop gain in Feedback control system

You make it sound like it's true for only H(s) = 1 when it works for all H(s). If G(s) = (s+2)(s+9) and H(s) = 1/(s+6), then OL poles = -9,-6 OL zeros = -2 Root locus of CL system will travel from [ -9 and -6 ] to [ -2 and $\pm$Infinity ], H(s) just changes the path the locus will take...
4. ### Open Loop gain in Feedback control system

Isn't that how closed loop poles travel, how does the root locus of G(s) = (s+2)(s+9) look like? Consider this, lets assume my plot is correct, if i wanted to find the value of K at the point s = -3 in the root locus plot, I would use the formula \[ k = \frac{\prod |s - p_i| }{ \prod | s -...
5. ### Open Loop gain in Feedback control system

I meant to write characteristic Equation ( GH(s) ) = 0 instead of just GH(s) = 0. Lets take a simple example of G = (s+2)/(s+9) , H = 1. Open Loop Pole = -9 Open loop Zero = -2 The root locus of the closed loop equation ( G/(1 + GH) ) will thus be a straight line travelling from -9 to...
6. ### Open Loop gain in Feedback control system

The way I understand it is that in order to solve 1 + KGH(s) = 0, you first solve GH(s) = 0 which will give you the open loop poles, those are the poles were the root locus plot will begin ( ie the poles for K=0 ) Or you can consider the fact that in root locus plots the value of K can be found...
7. ### Open Loop gain in Feedback control system

Consider a standard feedback control system with controller gain K(s) and plant transfer function G(s) with unity negative feedback( ie H(s) = 1 ). The characteristic equation of the system is 1 + KGH(s) = 1 + K(s)G(s). Now when finding the root locus you find the poles and zeros at K=0 ( Open...
8. ### creating two diffrents clocks

I don't really understand what you are trying to do, are you trying to derive one clock from another one? e.g you have a 20MHz clock and you are trying to derive a 5Hz clock from it so you can use the 20Mhz clock to power one component and the 5Hz clock to power another.If this is what you are...
9. ### How to use global parameters in System verilog

I have modules that define a parameter N, I also have an encapsulating module that instantiates all the other modules and it also defines the parameter N, so the problem I'm having is that every time I want to change the value of N I have to go into all the modules one by one and change N's...
10. ### SystemVerilog - Setting an output high in future clock cycle

How do I set an output high for the future clock cycle in system verilog.I would really want this module to only consist of one code block. Eg. I have one input signal A, and two output signals B and C, if A = 0, I want B to go High in the next clock cycle but if A = 1, I want B to go high in...
11. ### Common Emitter Amplifier Design

Yeah, I guess it was a loading problem, I increased the R1 and R2 resistors in the common collector amplifier to values over 100k and Re to 5k and the circuit simulated well.
12. ### Common Emitter Amplifier Design

The output drops at the output of the amp when I connect the buffer to the end of the amplifier in simulation, I have not connected a load to the end of the buffer yet. Ce = 10microF Test_Freq = 1Khz The output drops at the output of the first stage. Vin(AC) = 0.1v peak to peak Vout(Stage...
13. ### Common Emitter Amplifier Design

But I need to pick the Re2 value carefully because it will help me better choose the capacitor Ce properly, I want to make sure I have all my components designed properly.Right now the values I've given together with a Re2 value of 3.9K give me a small signal gain of 6.7 when I simulate them on...
14. ### How Can I Design Common-Emitter Amplifier ?

Isn't the DC gain the one that is - (R1/R2)? The gain Av refers to the AC gain that you get that from small signal analysis, Av = - gm * ( R1 || RL ). Where gm = 40Ic
15. ### Common Emitter Amplifier Design

I know, AC gain = Rc/Re1 = 6 Re1 and Re2 will determine the bias point but I first need to pick a suitable value for Re2, and for that I will need to use KVL going around the output loop, and for that I need to get VCE.

format: plain


text:
# Posts tagged as “LCA”

You are given the root of a binary tree with n nodes. Each node is uniquely assigned a value from 1 to n. You are also given an integer startValue representing the value of the start node s, and a different integer destValue representing the value of the destination node t.

Find the shortest path starting from node s and ending at node t. Generate step-by-step directions of such path as a string consisting of only the uppercase letters 'L''R', and 'U'. Each letter indicates a specific direction:

• 'L' means to go from a node to its left child node.
• 'R' means to go from a node to its right child node.
• 'U' means to go from a node to its parent node.

Return the step-by-step directions of the shortest path from node s to node t.

Example 1:

Input: root = [5,1,2,3,null,6,4], startValue = 3, destValue = 6
Output: "UURL"
Explanation: The shortest path is: 3 → 1 → 5 → 2 → 6.

Example 2:

Input: root = [2,1], startValue = 2, destValue = 1
Output: "L"
Explanation: The shortest path is: 2 → 1.

Constraints:

• The number of nodes in the tree is n.
• 2 <= n <= 105
• 1 <= Node.val <= n
• All the values in the tree are unique.
• 1 <= startValue, destValue <= n
• startValue != destValue

## Solution: Lowest common ancestor

It’s no hard to see that the shortest path is from the start node to the lowest common ancestor (LCA) of (start, end), then to the end node. The key is to find the LCA while finding paths from root to two nodes.

We can use recursion to find/build a path from root to a target node.
The common prefix of these two paths is the path from root to the LCA that we need to remove from the shortest path.
e.g.
root to start “LLRLR”
root to dest “LLLR”
common prefix is “LL”, after removing, it becomes:
LCA to start “RLR”
LCA to dest “LR”
Final path becomes “UUU” + “LR” = “UUULR”

The final step is to replace the L/R with U for the start path since we are moving up and then concatenate with the target path.

Time complexity: O(n)
Space complexity: O(n)

# Problem

Given a binary tree, find the lowest common ancestor (LCA) of two given nodes in the tree.

According to the definition of LCA on Wikipedia: “The lowest common ancestor is defined between two nodes p and q as the lowest node in T that has both p and q as descendants (where we allow a node to be a descendant of itself).”

Given the following binary tree:  root = [3,5,1,6,2,0,8,null,null,7,4]

        _______3______
/              \
___5__          ___1__
/      \        /      \
6      _2       0       8
/  \
7   4

Example 1:

Input: root = [3,5,1,6,2,0,8,null,null,7,4], p = 5, q = 1
Output: 3
Explanation: The LCA of of nodes 5 and 1 is 3.

Example 2:

Input: root = [3,5,1,6,2,0,8,null,null,7,4], p = 5, q = 4
Output: 5
Explanation: The LCA of nodes 5 and 4 is 5, since a node can be a descendant of itself according to the LCA definition.

Note:

• All of the nodes’ values will be unique.
• p and q are different and both values will exist in the binary tree.

# Solution 1: Recursion

Time complexity: O(n)

Space complexity: O(h)

For a given root, recursively call LCA(root.left, p, q) and LCA(root.right, p, q)

if both returns a valid node which means p, q are in different subtrees, then root will be their LCA.

if only one valid node returns, which means p, q are in the same subtree, return that valid node as their LCA.


text:
### Sidebar

#### Release history

pipeline:viper:sp_proj_compare
This version (2020/08/10 23:16) is a draft.
Approvals: 0/1
The Previously approved version (2019/12/11 18:33) is available.

## sp_proj_compare

Compare re-projections : Compare re-projections to class averages.

## Usage

Usage in command line

sp_proj_compare.py stack input_volume outdir --mode=viper --projparams=angles_file --projselect=img_selection_file --prjmethod=interpolation_method --delta=angular_increment --matchshift=shift_range --matchrad=outer_radius --matchstep=ring_step --symmetry=optional_symmetry --refineparams=refinement_params --partselect=substack_select --refineshift=shift_range --outliers=max_angle_diff --refinerad=outer_radius --refinestep=ring_step --alignmethod=2d_alignment_method --display --verbose
Projections of ribosome (EMDB 10028). For each image in the montage, the left is a projection, and the right is the class average.

## Typical usage

sp_proj_compare compares re-projections of a 3D reconstruction to 2D images, typically class averages. There are three methods for this comparison:

1. Use projections angles from VIPER/RVIPER :

sp_proj_compare.py Class2D/best.hdf Initial3D/main001/run002/rotated_volume.hdf Compare --mode viper --projparams Initial3D/main001/run002/rotated_reduced_params.txt --projselect Initial3D/main001/index_keep_images.txt

Note 1: The projection angles are not required if present in the image header. When in doubt, include this file explicitly.

Note 2: RVIPER, if it finds a stable solution, may exclude some images, and thus their projection angles will be excluded from the parameters file. The file containing the list of included images will have a name like main003/index_keep_images.txt.

2. Run a simple iteration of projection-matching:

sp_proj_compare.py Class2D/best.hdf Initial3D/main001/average_volume.hdf Compare/ProjMatch --mode=projmatch --delta 7.5

3. Use the orientation parameters from refinement:

sp_proj_compare.py Class2D/best.hdf Initial3D/main001/average_volume.hdf Compare/Meridien --mode=meridien --refineparams Refine3D/final_params_037.txt --partselect Substack/isac_substack_particle_id_list.txt --outliers=15

The projection angle for a class will be calculated by taking the vector sum of the projection angles of the particles belonging to that class.

The set of classified particles may be different from the set of refined particles – for example when a substack is generated from a selection of good class averages – and thus the particle numbers will not match. The particle-selection file – e.g., Substack/isac_substack_particle_id_list.txt – will allow mapping of the particle number during classification to that during refinement.

Some particles may have been erroneously assigned to a class, as indiciated by a large angular distance between their projection angles and the average. Such particles can be excluded from the calculation of the average angles with the –outliers option. Furthermore, a list of the remaining included particles will be written to the files OUTPUT_DIRECTORY/Byclass/goodpartsclass###.txt.

## Input

#### Main Parameters

stack
Input images stack
Set of 2D images to be compared. (default required string)
input_volume
Input volume
Reconstruction for which re-projections will be computed. In RVIPER, this file is of the form main003/run002/rotated_volume.hdf
This volume must have the same dimensions as the input images. (default required string)
outdir
Output directory
Directory where outputs will be written. (default required string)
--mode
Comparison method
Choices are: viper, projmatch, and meridien. (default viper)
--projparams
VIPER - Projection parameter file
Parameter file containing projection angles. Not required if the projection parameters are stored in the header of the input images, which for ISAC2 is generally not the case. In RVIPER, this file is of the form main003/run002/rotated_reduced_params.txt. This file is not required if the angles have already been written to the header. (default None)
--mode==viper
--projselect
VIPER - Image selection file
Input selection containing list of images to be included from the input stack. For instance, RVIPER, if it finds a stable solution, may exclude some images, and thus their projection angles will be excluded from the parameters file. The file containing the list of included images will have a name like main003/index_keep_images.txt. (default None)
--mode==viper
--delta
ProjMatch - Sampling angle
Angular-sampling for reference projections. (default 7.5)
--mode==projmatch
--symmetry
ProjMatch - Symmetry
To limit angular projections. (default c1)
--mode==projmatch
--refineparams
MERIDIEN - Alignment parameter file
Input refinement parameter file. (default None)
--mode==meridien
--partselect
MERIDIEN - Particle selection file
Input substack selection file if particles removed before refinement (e.g., Substack/isac_substack_particle_id_list.txt). (default None)
--mode==meridien
--outliers
MERIDIEN - Outlier angle
Particles differing from average Euler angle by more than this threshold (in degrees) will be excluded from average calculation, by default keeps all. (default None)
--mode==meridien

--prjmethod
Interpolation method
Valid choices are trilinear, gridding, and nn. (default trilinear)
--display
e2display
Automatically pops up a window with the output montage. (default False)
--verbose
verbose
Writes additional messages to the terminal during execution. (default False)
--fullstack
VIPER - Input particle stack to which VIPER angles will be added. (default None)
--mode==viper
--matchshift
ProjMatch - Maximum shift
Maximum shift to allow during translation alignment, pixels. (default 2)
--mode==projmatch
ProjMatch - Outer radius
Outer alignment radius, defaults to automatically-determined. (default None)
--mode==projmatch
--matchstep
ProjMatch - Radius step size
Alignment radius step size. (default 1)
--mode==projmatch
MERIDIEN - Outer radius
Outer alignment radius, defaults to automatically-determined. (default None)
--mode==meridien
--refineshift
MERIDIEN - Maximum shift
Maximum shift to allow during translation alignment, pixels. (default 2)
--mode==meridien
--refinestep
MERIDIEN - Radius step size
Alignment radius step size. (default 1)
--mode==meridien
--alignmethod
MERIDIEN - Alignment method
Valid choices are apsh and scf. (default apsh)
--mode==meridien

## Output

angles.txt
Text file of projection parameters of input images.
proj.hdf
Stack of re-projections of input 3D reconstruction.
comp-proj-reproj.hdf
Comparison of re-projections and 2D images.

## Description

### Developer Notes

As an input_volume, RVIPER's main003/average_volume.hdf also worked for me.

VIPER (as opposed to RVIPER) includes all class averages, so an image selection file shouldn't be needed.

Tapu Shaikh

### Keywords

Category 1:: APPLICATIONS

### Files

sphire/bin/sp_proj_compare.py

### Maturity

Beta:: Under evaluation and testing. Please let us know if there are any bugs.

### Bugs

2019/06/18 Strange behavior when the volume/image dimension is odd. In the meantime, will pad by 1.

pipeline/viper/sp_proj_compare.txt · Last modified: 2020/08/10 23:16 by shaikh


text:
# zbMATH — the first resource for mathematics

The Lukacs–Olkin–Rubin theorem without invariance of the “quotient”. (English) Zbl 0993.62043
Summary: The Lukacs theorem [E. Lukacs, Ann. Math. Stat. 26, 319-324 (1955; Zbl 0065.11103)] is one of the most brilliant results in the area of characterizations of probability distributions. First, because it gives a deep insight into the nature of independence properties of the gamma distribution; second, because it uses beautiful and non-trivial mathematics. Originally it was proved for probability distributions concentrated on $$(0,\infty)$$. I. Olkin and H. Rubin [ibid. 33, 1272-1280 (1962; Zbl 0111.34202] extended it to matrix variate distributions. Since that time it has been believed that the fundamental reason such an extension is possible is the assumed property of invariance of the distribution of the “quotient” (properly defined for matrices).
The main result of this paper is that the matrix variate Lukacs theorem holds without any invariance assumption for the “quotient”. The argument is based on solutions of some functional equations in matrix variate real functions, which seem to be of independent interest. The proofs use techniques of differential calculus in the cone of positive definite symmetric matrices.

##### MSC:
 62H05 Characterization and structure theory for multivariate probability distributions; copulas 26B12 Calculus of vector functions 62E10 Characterization and structure theory of statistical distributions 26B99 Functions of several variables
Full Text: