text:
# Find the necessary and sufficient conditions on $a$, $b$ so that $ax^2 + b = 0$ has a real solution.

This question is really confusing me, and I'd love some help but not the answer. :D

Is it asking: What values of $a$ and $b$ result in a real solution for the equation $ax^2 + b = 0$? $a = b = 0$ would obviously work, but how does $x$ come into play? There'd be infinitely many solutions if $x$ can vary as well ($a = 1$, $x = 1$, $b = -1$, etc.).

I understand how necessary and sufficient conditions work in general, but how would it apply here? I know it takes the form of "If $p$ then $q$" but I don't see how I could apply that to the question. Is "If $ax^2 + b = 0$ has a real solution, then $a$ and $b =$ ..." it?

-
By the way, if there is a real solution, all (how many?) solutions are real too :) – Tapu Sep 20 '12 at 1:08

I assume the question is "find conditions that are necessary and sufficient to guarantee solutions" rather than "find necessary conditions and also find sufficient sufficient conditions for a solution." If the former is the case, then you're asked for constraints on $a$ and $b$ such that (1) if the conditions are met then $ax^2+b$ is zero for some $x$ and (2) if the conditions are not met then $ax^2+b$ isn't zero for any $x$.

So, when will $ax^2+b$ have some value $x$ for which the expression is zero? Well, as André suggested, try to solve $ax^2+b=0$ "mechanically". By subtracting we have the equivalent equation $ax^2 = -b$ and we'd like to divide by $a$ to get $x^2=-b/a$. Unfortunately, we can't do that if $a=0$, so we need to consider two cases

1. $a=0$
2. $a\ne 0$

In the first case, our equation becomes $0\cdot x^2+b=0$, namely $b=0$ and if $b=0$ (and, of course $a=0$) then any $x$ will satisfy this. In other words, there's a solution (actually infinitely many solutions) if $a=b=0$, as you've already noted.

Now, in case (2) we can safely divide by $a$ to get $x^2=-b/a$. When does this have a solution? We know that $x^2\ge 0$ no matter what $x$ is, so when will our new equation have a solution? You said you don't want the full answer, so I'll denote the answer you discover by $C$, which will be of the form "some condition on $a\text{ and }b$". Once you've done that, your full answer will be:

$ax^2+b=0$ has a solution if and only if

1. $a=b=0$, or
2. Your condition $C$.

These are sufficient, since either guarantees a solution, and necessary, since if neither is true, then there won't be a solution (since we exhausted all possibilities for solutions).

-
I'm a little confused why x^2 = -b/a having a solution is the answer to the problem. And why do we know that x^2 >= 0? Thanks so much for the help, just a little blurp in my knowledge. – Doug Smith Sep 20 '12 at 3:47
For your first question, remember that you're asked to find conditions on $a\text{ and }b$ so that $ax^2+b=0$ will have a solution, so in the case that $a\ne 0$ that equation will have a solution if and only if $x^2=-b/a$ will have a solution. For your second question, note that for any real $x$, we'll always have $x^2\ge 0$. (For example, $1^2=1\ge 0, 3^2=9\ge 0, (-2)^2=4\ge 0$.) Thus, there will be an $x$ that satisfies $x^2=-b/a$ if and only if $-b/a\ge 0$. – Rick Decker Sep 20 '12 at 14:08

Hint: Mechanically "solve" the equation. What could go wrong?

As to what the question is asking, suppose that we are given fixed numbers $a$ and $b$. For example, let $a=-3$ and $b=17$. Does the equation have a real solution $x$? What about if $a=-3$ and $b=-5$? (And don't forget about the possibility that one or both of $a$ and $b$ might be $0$.) Can we come up with simple conditions on $a$ and $b$ so that just glancing at them tells us whether or not there is a solution?

-
A simpler hint might be to note that for real $x$, we should have $x^2\ge0$. – Tapu Sep 20 '12 at 0:54
@Tapu: That is a very useful hint. It may be better at the "write-up" stage. – André Nicolas Sep 20 '12 at 0:55
b^2-4ac > 0 or 0 then you have real solutions. – yiyi Sep 20 '12 at 0:55

If $ax^2 + b=0$, then $x^2=\dfrac{-b}{a}$. The solutions are therefore $x=\pm\sqrt{-b/a}$. For the equation to have a real solution, $-b/a$ must be non-negative. A fraction is non-negative if

• the numerator and denominator are positive, or
• the numerator and denominator are negative, or
• the numerator is $0$ and the denominator is not $0$.
-


text:
# IEEE Transactions on Information Theory

IEEE Transactions on Information Theory publishes papers concerned with the transmission, processing, and utilization of information.

Aims & Scope

## Latest Published Articles

• ### Cache-Aided Interference Channels

Wed Feb 13 00:00:00 EST 2019 Wed Feb 13 00:00:00 EST 2019
• ### Asymptotically Tight Bounds on the Depth of Estimated Context Trees

Sun Dec 23 00:00:00 EST 2018 Sun Dec 23 00:00:00 EST 2018
• ### Rényi Resolvability and Its Applications to the Wiretap Channel

Thu Dec 06 00:00:00 EST 2018 Thu Dec 06 00:00:00 EST 2018
• ### Intrinsic Capacity

Thu Dec 06 00:00:00 EST 2018 Thu Dec 06 00:00:00 EST 2018
• ### Feedback Capacity of Gaussian Channels Revisited

Thu Nov 29 00:00:00 EST 2018 Thu Nov 29 00:00:00 EST 2018
View All Latest Articles

## Popular Articles

• ### Compressed sensing

Mon Apr 03 00:00:00 EDT 2006 Mon Apr 03 00:00:00 EDT 2006
• ### Channel Polarization: A Method for Constructing Capacity-Achieving Codes for Symmetric Binary-Input Memoryless Channels

Tue Jun 16 00:00:00 EDT 2009 Tue Jun 16 00:00:00 EDT 2009
• ### Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information

Mon Jan 23 00:00:00 EST 2006 Mon Jan 23 00:00:00 EST 2006
• ### Speeding Up Distributed Machine Learning Using Codes

Fri Aug 04 00:00:00 EDT 2017 Fri Aug 04 00:00:00 EDT 2017
• ### Channel Coding Rate in the Finite Blocklength Regime

Mon Apr 19 00:00:00 EDT 2010 Mon Apr 19 00:00:00 EDT 2010
View All Popular Papers

## Meet Our Editors

Editor-in-Chief
Alexander Barg

Department of Electrical and Computer Engineering and the Institute for Systems Research, University of Maryland

email: abarg-ittrans@ece.umd.edu

format: plain


text:
# How do I test if a manipulation reduces variance with stan?

Here’s a thought experiment, not too far from what I actually planning to test. Say that I can present sound in 360 degrees surrounding the participant. I then ask the participant a question about the presented sound.

In a classic setup, all sounds would be played equally loud, but I have found that participants are on average better at some positions rather than others. If I staircase this, it seems that on average sounds should be reduced most close to the ears, slightly in front but not at all behind the head. Something like this:

The staircase already seems convincing to me, but I further wish to test this against a classic setup in which all sounds are equally loud - like this:

So each participant would try 2 conditions: classic and balanced, and I believe that the performance of each position should be more comparable in the balanced setup. But is it practically measurable?

My question is how I formally test whether the performance is more “even” in my balanced setup. One option would be to split the data of the two conditions and then run a model that does not know position vs a model knowing the position and compare their performance with loo. Something like:

# Create models
m0 <- stan_lmer(accuracy~-1+(1|participant))
m1a <- stan_lmer(accuracy~-1+factor(position)+(1|participant))

# Compare models
loo_m0 <- loo(m0)
loo_m1a <- loo(m1a)
compare(loo_m0,loo_m1a)

And I would then guess that the “m1a” model for the classic condition would be relatively better than “m0” in the classical condition. Say, it is m1a is 5 standard deviations better than m0 in the classical setup but only 3 standard deviations better in the balanced setup. That implies that the variance for each position was lower in the balanced setup.

This test does not seem very stringent to me (and probably not to a reviewer either), so I wonder if there is a more formal way to test my question?

All feedback would be highly appreciated.

What does “staircase” mean as a verb here? I got confused about what you were actually presenting to subjects and what they were reporting and what those dots in circles meant. What is accuracy? Why is volume on the y axis and position on the x axis? I’m also not sure what’s being balanced in the balanced set up (distance from listener?).

Usually we just directly model the thing we care about and then do posterior inference for that rather than trying to build two models and compare. That is, butild the more complicated model and see if the position makes a difference. Usually we don’t care about comparison with 0, but you could evaluate \mathrm{Pr}[\mathrm{position}[i] > 0]. Or just measure accuracy in both situations, build a hierarchical model to smooth, and compare.


text:
# How to solve Power Density Function?

How to solve this problem:

A radio receiver needs $$1 \textrm{ nW/}\textrm{m}^2$$ of power density function, how far away from a 1-watt point source will it continue to work?

Then $$\frac{P}{A} = S = 1*10^{-9} = \frac{1}{4 \pi r^2}$$ Then $$r = \frac{500}{\sqrt{\pi}} ~= 159m$$
So at a maximum distance of 159 meters your radio receiver can still operate. Note that this is an upper bound obiously. So for any distance $$r \leq 159 m$$ your radio receiver will work.


text:
# Fourth-order tensor tutorial excerpts: Voigt and Mandel representations as well as isotropy topics

I often get asked: when are you going to finish your tensor-analysis book?

Well, I have so many pressures on my time, that this hobby gets pushed to the bottom of the stack.  This said, I find myself often discussing fourth-order tensor operations, the distinction between Voigt and Mandel components, and scalar measures of anisotropy.  To help with such discussions, I am here posting two excerpts from my unpublished tensor-analysis notes.   Enjoy this enthralling topic!

EDIT: you can now cite this topic from a real publication. It is in chapter 26 of my 2018 book: Rotation, Reflection, and Frame change.

The first excerpt from the still unpublished notes, 150606tensorsVoigtMandelExcerpt (for which the cited references may be found in 150606tensorsVoigtMandelExcerptReferences), discusses Voigt and Mandel notation, and introduces some helpful operations on fourth-order tensors.  Here are some highlights taken from this PDF excerpt (all of which are included in the rotation book if you need to cite something):

What we have labeled as the 9×1 “contravariant Voigt array (without factors of 2)” is typically called a “stress-like” array in the composites community, while the 9×1 “covariant Voigt array (with factors of 2)” is called a “strain-like” array.  When these arrays actually represent stress and strain, their last three entries are zero because of symmetry. Likewise, the 9×9 array is, in this context, the elastic stiffness so its last three columns and last three rows are zero because of minor symmetry. Accordingly, in constitutive modeling, you typically see this matrix relationship truncated down to only 6 dimensions. When using computer code to work out components of such tensors, we recommend keeping all 9 dimensions just to serve as a visual cue that you have indeed enforced symmetries properly in your equations.

It is mystifying that the composites community doesn’t seem to even realize that the Voigt representations would be more properly referred to as covariant and contravariant, so please add a comment to this post if you have ever seen any composites articles use this mathematically proper terminology. These are not just matrix equations. There is a tensor basis that goes with Voigt representations, and that basis is a set of mutually orthogonal tensors. The basis is not normalized, so that leads to co/contravariant representations, in which the factors of 2 are metrics.  Whenever you have an orthogonal but not normalized basis, the obvious thing to do is to normalize it! That is what gives the following Mandel form, which you should note has no more of that ugly distinction between contravariant (stress-like) arrays and covariant (strain-like) arrays. Both arrays are treated the same!

In this list, the very last set of basis tensors are the ones that pair with ordinary components of a tensor. For example, the 11 component of an ordinary second-order tensor is paired with a basis tensor whose component matrix is all zeros everywhere except a 1 in the 11 spot.  The 12 component goes with a basis tensor that has all zeros everywhere except for a 1 in the 12 spot, and so on.  The Voigt and Mandel representations merely represent a change of basis so that the first six basis tensors span the manifold of all possible symmetric tensors, while the last three basis tensors span the space of all possible skew tensors.

If you are not convinced that the Mandel representation is the better choice, try comparing it with Voigt for the components of the fourth-order identity tensor. The result is the identity matrix in Mandel form (not so for Voigt). Also, you really need the Mandel form to find eigenvalues and eigentensors.  Ordinary spectral analysis of the Voigt representation is completely meaningless — you need to use Mandel form to get meaningful eigenvalues and eigenmodes of a stiffness tensor.

Page 658 (PDF page 30) of my second excerpt, 150606tensorsFourthOrderOperationsAndMeasureOfIsotropyExcerpt, shows how to find the isotropic (IFOET) part of a fourth-order tensor (which is NOT generally some multiple of the identity), and how to define a scalar measure of anisotropy in the range from zero to one, as determined from the “angle” that the tensor makes with the linear manifold of isotropic tensors.  Doesn’t this sound exciting?  Here is an infographic summary of this process of finding the isotropic part of a fourth-order tensor, as well as setting the scalar measure of anisotropy (equal to 1 minus the scalar measure of isotropy):

The isotropic part of a fourth-order tensor, as well as a scalar measure of anisotropy ranging from 0 to 1.

In this infographic, the acronym IFOET stands for “isotropic fourth-order engineering tensor”  (labeled “ISO” elsewhere in the image).  In the scalar measure of isotropy, the denominator is the L2 norm of the original fourth-order tensor, equal to the square root of the sum of the squares of the tensor’s Mandel components (which is another benefit of Mandel over Voigt because getting the magnitude of a Voigt tensor would require insertions of factors of 2 and 4 — Yuck!). As you can see, the isotropy is 100% if the original tensor is isotropic, and it ranges down to 0% isotropy if the isotropic part of the original tensor is zero.

# F-tables for prescribed deformation

Motion without superimposed rotation

Same deformation with superimposed rotation

When developing constitutive models, it is crucial to run the model under a variety of standard (and some nonstandard) homogeneous deformations. To do this, you must first describe the motion mathematically. As indicated in http://csm.mech.utah.edu/content/wp-content/uploads/2011/03/GoBagDeformation.pdf, a good way to do that is to give the deformation gradient tensor, F. The component matrix [F] contains the deformed edge vectors of an initially unit cube, making this a very easy to way to prescribe deformations.

# Rate of Hencky (logarithmic) strain and similar tasks

The logarithmic (Hencky) strain is evaluated by taking the log of the symmetric stretch tensor in continuum mechanics. Doing so requires transforming to the principal stretch basis, taking logs of the principal stretch eigenvalues, and transforming the result back to the lab basis.  While this procedure is a bit tedious, it certainly is straightforward.

The harder — almost freakishly daunting — question is: how do you get the rate of the logarithmic strain?  This rate must include contributions from both the rate of the stretch eigenvalues and the rate of the stretch eigenvectors, which is difficult to handle when there are repeated eigenvalues causing extra ambiguity of eigenvectors.  Continue reading

# Tutorial: multi-linear regression

The straight line is the linear regression of a function that takes scalars (x-values) as input and returns scalars (y-values) as output. (figure from GANFYD)

You’ve probably seen classical equations for linear regression, which is a procedure that finds the straight line that best fits a set of discrete points $\{(x_1,y_1), (x_2,y_2),...,(x_N,y_N)\}$. You might also be aware that similar formulas exist to find a straight line that is a best (least squares) fit to a continuous function $y(x)$.

The pink parallelogram is the multi-linear regression of a function that takes vectors (gray dots) as input and returns vectors (blue dots) as output

The bottom of this post provides a link to a tutorial on how to generalize the concept of linear regression to fit a function $\vec{y}(\vec{x})$ that takes a vector $\vec{x}$ as input and produces a vector $\vec{y}$ as output. In mechanics, the most common example of this type of function is a mapping function that describes material deformation: the input vector is the initial location of a point on a body, and the output vector is the deformed location of the same point. The image shows a collection of input vectors (initial positions, as grey dots) and a collection of output vectors (deformed locations as blue dots). The affine fit to these descrete data is the pink parallelogram. Continue reading

# Publication: Conjugate stress and strain caveats /w distortion and deformation distinction

The publication, “Caveats concerning conjugate stress and strain measures (click to download)” contains an analytical solution for the stress in a fiber reinforced composite in the limit as the matrix material goes to zero stiffness. Because the solution is exact for arbitrarily large deformations, it is a great test case for verification of anisotropic elasticity codes, and it nicely illustrates several subtle concepts in large-deformation continuum mechanics.

Also see related viewgraphs entitled “The distinction between large distortion and large deformation.”


text:
# What is the relation of basis in linear algebra and basis in topology?

In linear algebra and topology ,it all has the concept basis,but I can not construct the relation of them,could you explain the relation of two basis,such as the basis in linear algebra is special case of basis in topology,and some details about it,thanks a lot.

-
They're conceptually similar in that both of them "generate" something (the vector space and the topology, respectively), neither is a special case of the other or anything like that. –  Alex Becker Apr 6 '12 at 7:13
Both terms rely on the embedded/base language (English), whose cognate ('base') is thus overloaded. –  bgins Apr 6 '12 at 7:18
I will second Alex Becker's comment. Two points: in topology there is little point in seeking a 'minimal' generating set. OTOH in linear algebra this is paramount given that we want to use the basis to introduce coordinates. Also, natural language is not rich enough to capture all the subtleties encountered in math. We just need to make the most of it. –  Jyrki Lahtonen Apr 6 '12 at 7:19
They are unrelated, other than by accident of language. –  copper.hat Apr 6 '12 at 7:32
The only important reason that the worse "base" is used to thoroughly in mathematics is to allow one mastering mathematics say "All your base are belong to us!". –  Asaf Karagila Apr 6 '12 at 12:13

Well, they are two different entities: they are similar in that they are minimal information that give rise to a mathematical object--vector space in linear algebra, a topological space in General topology.

To see that they are different too -- I have adopted the following discussion from Munkres' book on general topology:

Firstly, note the following lemma:

Lemma (LA)

Let $V$ be a vector space and let $\scr B$ be a basis for $V$. Then, every vector $v \in V$ is uniquely written as a linear combination of basis elements.

Lemma (GT)

Let $(X, \scr T)$ be a topological space. Let $\scr B$ be a basis for topology $\scr T$ on $X$. Then, every open set in $\scr T$ can be expressed as a union of basis elements.

The difference arises from the fact that, this expression for the union need not be unique.

Let's look at a situation where this is actually the case:

Consider the three element set, $X=\{a,b,c\}$ and the following subset of $2^X$:

$$B=\{\{a\},\{a,b\}, \{a,b,c\}\}$$

It is not hard to see that $B$ is in fact a basis for some topology, $\scr T$ on $X$. Now, observe that $\scr T$ is actually:

$$\mathscr{T}=\{\varnothing, X, \{a\}, \{a,b\} \}$$

Now see that $$\{a,b,c\}=\{a,b,c\}=\{a,b\} \cup \{a,b,c\}=\{a\}\cup\{a,b\} \cup \{a,b,c\}$$

I'll add two more points from the discussion I had with the user t.b.:

1. Topological bases are a special kind of small collection of open subsets on which you have a nice handle. They help study all open sets. Now, notice that there is no minimality in question nor independence, which however fails to make much sense.

2. The importance of bases in Linear algebra comes from the fact they are minimal spanning sets and maximal linearly independent sets. Notice that there are not immediate analogues of these facts for topological bases.

-
There is a sort of minimality notion for topological bases, in that for a given topological space X there is a unique least possible cardinality of a base of X. (This is known as the weight of X.) But not every base has this cardinality, e.g. the collection of all open sets of X is itself a base. And one base can strictly contain another base, unlike the linear-algebraic notion. –  Brad Apr 6 '12 at 15:09

As previously discussed, the two notions are not perfectly analogous, but here's one way in which they are conceptually similar. If you have a vector space $V$ and a basis $B$ for $V$, and you want to construct some linear-algebraic thing "living on $V$", it is equivalent to constructing something "living on $B$". For example, if you have another vector space $W$, then constructing a linear map $T:V\to W$ is equivalent to choosing a function $T|_B:B\to W,$ as each such function has a unique linear extension.

Similarly, if $X$ is a topological space and $B$ is a base for $X$, some things "living on $X$" can be equivalently defined as "living on $B$". For example, to construct a sheaf of abelian groups on $X$, it is equivalent to define a $B$-sheaf - an assignment of groups to elements of $B$ which satisfies the sheaf axioms "on $B$". (See the definition on Wikipedia.) Any $B$-sheaf extends uniquely to a sheaf on $X$, and conversely any sheaf on $X$ restricts to give a $B$-sheaf. (The same is true for sheaves with values in many other categories.)

In this sense, each notion of basis is "a (possibly) simpler gadget on which to define things".

-


text:
# 50143. AND & OR of Trees

## I'm a slow walker, but I never walk backwards.

We will AND and OR two trees to generate a new tree.

The AND operation is similar to intersection of two sets. If we AND two trees $A$ and $B$ then only the nodes that appear in both $A$ and $B$ will appear in the result tree $C$. The value of a new tree node will be the product of the values from the two corresponding tree nodes from the two trees. An example of ANDing two trees is shown in the following figure.

Similarly we can define the OR operation, which is like the union of two sets. If we OR two trees $A$ and $B$ then a node will appear in the result tress $C$ if the node appears in $A$ or in $B$. The value of the new tree node will be the sum of the values of the two corresponding tree nodes in $A$ and $B$. If a node only appears in $A$ (or in $B$), the the new value will be the value of $A$ (or $B$). An example of OR operation is shown in the following figure.
Write two functions to do tree AND and OR. The input of the function is the pointers to the roots of trees $A$ and $B$, and the function will construct the result tree $C$ and return the root of $C$. You can use malloc() to create new nodes in the function if necessary. The tree node structure and function prototype is defined in the following header file "tree.h".

typedef struct Node{    int val;    struct Node *left, *right;} Node; Node *treeAND(Node *root1, Node *root2);Node *treeOR(Node *root1, Node *root2);

You can use the following main.c to check your functions.

#include <stdio.h>#include <stdlib.h>#include "tree.h" void preorder(Node *root) {    if (root == NULL) return;    printf("%d ", root->val);    preorder(root->left);    preorder(root->right);} void inorder(Node *root) {    if (root == NULL) return;    inorder(root->left);    printf("%d ", root->val);    inorder(root->right);} Node *insertBST(Node *root, int val) {    if (root == NULL) {        root = (Node *) malloc(sizeof(Node));        root->val = val;        root->left = NULL;        root->right = NULL;        return root;    }    if (val < root->val)        root->left = insertBST(root->left, val);    else        root->right = insertBST(root->right, val);    return root;} int main() {    int op, n1, n2, val;    scanf("%d%d%d", &op, &n1, &n2);    Node *tree1 = NULL;    Node *tree2 = NULL;    for (int i = 0; i < n1; ++i) {        scanf("%d", &val);        tree1 = insertBST(tree1, val);    }    for (int i = 0; i < n2; ++i) {        scanf("%d", &val);        tree2 = insertBST(tree2, val);    }     Node *tree = (op) ? treeAND(tree1, tree2) : treeOR(tree1, tree2);     preorder(tree);    printf("\n");    inorder(tree);    printf("\n");     return 0;}

## Input Format of main.c

The input has four lines. The first line is the type of the operation -- $1$ for AND, and $0$ for OR. The second line has two integers $n1$ and $n2$, which are the number of nodes in trees $A$ and $B$. The third line and forth line are the values of the tree nodes in $A$ and $B$. They will be inserted into a binary search tree in the order they appear in the input.

• $1 \lt n1, n2 \lt 100$
• $0 \lt \text{node values} \lt 101$

## Output Format of main.c

The first line is the result of a preorder traversal of the new tree $C$. The second line is the retuls of an inorder traversal of the new tree $C$.

• 50 points: AND opertion
• 50 points: OR opertion

## Sample Input 1

18 84 1 7 3 2 5 6 86 2 1 5 3 4 7 8

## Sample Output 1

24 2 15 6 49 642 6 15 24 49 64

## Sample Input 2

08 84 1 7 3 2 5 6 86 2 1 5 3 4 7 8

## Sample Output 2

10 3 1 8 5 4 14 5 6 161 3 5 4 8 10 5 6 14 16


text:
# [texhax] Unused Labels

Peter Flynn peter at silmaril.ie
Sat Feb 23 21:20:08 CET 2019

On 23/02/2019 16:11, Weil, Clifford wrote:
> Is there a facility that will find all “\label” entries that never
> occur in a “\ref” command? That is, is there a way to eliminate all
> labels that are never cited?

Not as such, but you can do it by forcing all non-commented commands to
the start of a line each, extracting the label and ref commands,
grabbing their arguments into an array, and then testing the labels for
not being in a ref:

grep -v '^%' thesis.tex |\
tr '\\{}' '\012\040\040' |\
grep -E '^(label|ref)' |\
awk '/^label/ {++label[$2]} /^ref/ {++ref[$2]} \
END {for(val in label){if(!(val in ref))print val}}'

This shows me I had rather a lot of over-optimistic labels that I never
used :-)

This should work on any UNIX or GNU Linux system, including Macs; on a
Windows system you can either install the utilities with something like
Cygwin, or rewrite the script into Powershell, which I believe has all
the features needed. But it's subject to the document not having variant
syntax like LaTeX code in literal environments, or changing catcode values.

It's a bit easier in XML (which has the identical concept of an ID
(label) that can only occur once and an IDREF (ref) that can be used
many times, as in HTML), because the syntax is more rigorous and the
parser takes care of all the messy stuff like linebreaks and nesting and
literals; so for a DocBook document, where @linkend is the attribute
which carries IDREF references, you can write an XPath query statement
like:


text:
# How to calculate the maximum height

I am studying radio galaxies and observing the behavior of fluxes at high frequencies and want to calculate the maximum height of the fluxes at where they best correspond(typically at higher frequencies).

My first graph shows

N(<S) vs log(S)

but I want to show instead.

N(>S) vs log(S)

Any idea on how I can do this? I merely switched the axis using a computer code, but I was told I need to compute the max height instead. Graphs here Thanks in advanced!

• Welcome to Mathematics Stack Exchange! Your question needs to provide more information. As written, it does not seem to belong on this site. A quick tour will enhance your experience. Here are helpful tips to write a good question and write a good answer. For equations, please use MathJax. – dantopa Mar 25 at 19:22

It looks from the graphs that $$N(S) = 100$$, can you not compute one from the other?


text:
# What is the incidence angle of a propeller in the terminology of the Wright Brothers and how did they obtain it?

What is the angle of incidence of a propeller in the terminology of the Wright Brothers and how did they obtain it?

In Feb. 1903 (see Screw Test 1903, page 8), using a 0.58 hp engine, the Wright brothers tested a propeller characterized by:

Diameter, D = 8.5 ft

Pitch = 15 deg

Surface, $S = 2\ ft^2$

RPM = 245

They measured what appears to be the static thrust and got:

Thrust = 18.75 lbf

Based on the assumption that the propeller can be modeled by a wing (Pitch = 15 deg, Surface = $2\ ft^2$, Lift = 18.75 lbf) having its entire area concentrated in a single point (called the Center of Pressure, C.P.) traveling at RPM = 245 on a circle with the diameter 0.824 x D, $V_{wing} = RPM \times \pi \times 0.824 \times D$, the Wright brothers evaluated its Drag and from Lift, Drag and Pitch using trigonometric relations they obtained the Normal (perpendicular to the chord) and Axial or Tangential (along the chord) components of Lift + Drag (the resultant force) and some angles related to them.

Now comes something I do not quite understand.

The two brothers calculated a reference force defined as:

Total Normal = $k \times S \times V_{wing}^2$ = 24.8 lbf,

where

$V_{wing} = RPM \times \pi \times 0.824 \times D = 61\ \mathrm{mph}$

$k =$ Smeaton’s coefficient $= 0.0033\ \mathrm{lbf} \cdot \mathrm{ft}^{-2} \cdot \mathrm{mph}^{-2}$

Further, the Wrights divided the Lift by the Total Normal

$\frac{\mathrm{Lift}}{\mathrm{Total\ Normal}} = 0.756$

and from this 0.756 they drew the conclusion that the angle of incidence is 7¼ deg. The question is why? What is this angle of incidence and how can it be calculated?

Update

The correspondence between the percentage 0.756 and the angle of incidence, 7¼, comes likely from a table like this (which is for a slightly different wing curvature):

Percentage of air pressure at various angles of incidence

However, as the propeller did not move, the incidence angle of its equivalent wing appears to be identical with the pitch of the same wing and they are not. Why?

• If a wing (a small helicopter blade) rotates about the axis of a stationary motor, how do you define the pitch and incidence angle? This is the question. It appears that Pitch = angle (chord, plane of rotation), AoA = angle(chord, relative wind) = angle (chord, plane of rotation) as the relative wind speed is in the plane of rotation. It seems that, for the Feb 1903 test, Pitch = AoA also called incidence. – Robert Werner Oct 16 '15 at 9:08
• Possible W. Wright, based on the speed of the air blown back by the propeller, believed he was in the equivalent case of a propeller moving forward, a situation when AoA is different from Pitch and this is the reason he did not see a contradiction when he obtained an angle of incidence different from Pitch. – Robert Werner Oct 16 '15 at 17:21

According to various scholars, the Wright brothers were referring to "angle of attack" when they used the term "incidence." They actually published an article by the same title, so the fact is well-established.

References include the following:

In Wilbur's time, the "angle of incidence" was what we now call the "angle of attack" -- the angle at which the wing of an airplane meets the wind (wright-brothers.org).

and

In the Wright's own words: "The lift is thus balanced against the normal pressure on the resistance surfaces ... Therefore, the lift ... at the given angle of incidence (angle of attack) is to the pressure on a square plane of equal area at 90° as the sine of the (...)

The angle of attack of a propeller is defined in various ways, a summary of which is here.

The angle at which this air (relative wind) strikes the propeller blade is its angle of attack.


text:
# An infinitely differentiable function that is not analytic

In mathematics, smooth functions (also called infinitely differentiable functions) and analytic functions are two very important types of functions. One can easily prove that any analytic function of a real argument is smooth. The converse is not true, with this article constructing a counterexample.

 Contents

## The function

Consider the function

[itex]f(x)=\left\{\begin{matrix}\exp(-1/x^2) & \mbox{if}\ x\neq 0 \\ \\ 0 & \mbox{if}\ x=0 \end{matrix}\right.[itex]

where x is a real number.

## How it is ill-behaved

### f is smooth on R

It can be shown that [itex]f[itex] admits continuous derivatives of any order in all points of [itex]\mathbb{R}[itex], given by

[itex]f^{(n)}(x) = \left\{\begin{matrix} R_n(x)\,f(x) & (x \ne 0) \\ 0 & (x = 0)\end{matrix}\right.,[itex]

where [itex] R_n [itex] is a rational function without pole on [itex]\mathbb{R}^*=\mathbb{R}\setminus\{0\}[itex].

The proof, by induction, is mainly based on the fact that for any integer [itex]m[itex],

[itex]\lim_{x \to 0} x^m e^{-1/x^2} = 0,[itex]

which implies that all [itex]f^{(n)}[itex] are continuous and differentiable in [itex]x = 0[itex], because

[itex]\lim_{x \to 0} \frac{f^{(n)}(x) - f^{(n)}(0)}{x-0} = \lim_{x \to 0} \frac{R_n(x)}{x}\,e^{-1/x^2} = 0[itex].

### f is not analytic

According to what precedes, the Taylor series of [itex]f[itex] is

[itex]\sum_{n=0}^\infty {0\over n!}x^n = 0\neq f(x)[itex]

unless [itex]x = 0[itex]. Consequently [itex]f[itex] is not analytic at 0. This pathology cannot occur with functions of a complex variable rather than of a real variable. Note that although this function has derivatives of all orders over the real line, the analytic continuation of [itex]f[itex] to the complex plane has an essential singularity at the origin, and hence is not even continuous, much less analytic. Indeed, all holomorphic functions are analytic, so that the failure of [itex]f[itex] to be analytic in spite of its being infinitely differentiable is an indication of one of the most dramatic differences between real-variable and complex-variable analysis.

## How this is a good thing...

### ...in negative terms

This example teaches us that functions of a real variable are sometimes ill-behaved in ways to which functions of a complex variable are immune.

### ...in positive terms

One of the most important applications of this function is the construction of so-called mollifiers, which are important in theories of generalized functions, like e.g. Laurent Schwartz's theory of distributions.

The existence of these functions represents one of the main differences between differential geometry and analytic geometry. In terms of sheaf theory, this difference can be stated by saying that the sheaf of differentiable functions on a differentiable manifold is flasque, in contrast with the analytic case.

The function above is generally used to build up partitions of unity on differentiable manifolds.

• Art and Cultures
• Countries of the World (http://www.academickids.com/encyclopedia/index.php/Countries)
• Space and Astronomy


text:
## Matrix repair

When fitting a statistical model we want two things as a minimum:

1. The parameter estimates, e.g. the maximum-likelihood estimates (MLEs), and
2. The estimated variance-covariance matrix, $$\hat V$$, for those estimates.

We can get both from the log-likelihood: the MLEs maximise the value of the log-likelihood function, and an approximation for the covariance matrix comes from inverting the negative information matrix, $$\mathcal J$$, i.e. the matrix of second partial derivatives evaluated at the MLEs. However, the limitations of computer arithmetic can sometimes get in the way, as shown in the information matrix in Figure 1:

Figure 1. Information matrix, $$\mathcal J$$, for a five-parameter model. Only the leading diagonal is shown for clarity.  Source: own calculations.

$\mathcal J=\left(\begin{matrix}-11584.8 & \circ & \circ & \circ & \circ \\ \circ & -10062.7 & \circ & \circ & \circ \\ \circ & \circ & -167.653 & \circ & \circ \\ \circ & \circ & \circ & -7275.98 & \circ \\ \circ & \circ & \circ & \circ & -0.00291038\end{matrix}\right)$

The issue with $$\mathcal J$$ in Figure 1 lies in the bottom right: the pure second partial derivative $$\mathcal J_{5,5}$$ is tiny relative to the pure second partial derivatives for the other parameters. This causes a problem when inverting to estimate the covariance matrix, as shown in Figure 2:

Figure 2. $$\hat V=-\mathcal J^{-1}$$ for the five-parameter model. Only the leading diagonal is shown for clarity.  Source: own calculations.

$-\mathcal J^{-1}=\left(\begin{matrix}-0.000201 & \circ & \circ & \circ & \circ \\ \circ & -0.000181 & \circ & \circ & \circ \\ \circ & \circ & -00642 & \circ & \circ \\ \circ & \circ & \circ & -0.000136 & \circ \\ \circ & \circ & \circ & \circ & -3980.715\end{matrix}\right)$

The leading diagonal of $$\hat V$$ is supposed to be the Cramer-Rao lower bound for the variances of the parameter estimates, i.e. they should all be positive values.  However, the unreliable value for $$\mathcal J_{5,5}$$ has rippled through the entire matrix inversion and ruined the estimates for everything else.  Can anything be done to recover the situation?

One avenue of exploration is to try alternative values for $$\mathcal J_{5,5}$$ and see what the impact is on the leading diagonal of $$\hat V$$.  Table 1 shows the estimated variances for a series of alternative values of $$\mathcal J_{5,5}$$:

Table 1.  Sensitivity of variance estimates to alternative values for $$\mathcal J_{5,5}$$.  Source: own calculations using the R script on the right.

$$\mathcal J_{5,5}$$Var1Var2Var3Var4Var5
-12010 0.01241442 0.01331367 0.08057805 0.01172591 0.009124909
-10010 0.01241442 0.01331367 0.08057805 0.01172591 0.009995005
-8010 0.01241442 0.01331367 0.08057805 0.01172591 0.011173361
-6010 0.01241442 0.01331367 0.08057805 0.01172591 0.012899203
-4010 0.01241442 0.01331367 0.08057805 0.01172591 0.015791667
-2010 0.01241442 0.01331368 0.08057805 0.01172591 0.022305004
-10 0.01241478 0.01331401 0.08057806 0.01172592 0.316277767

Table 1 shows that the variance estimates for the first four parameters are robust to a wide range of alternative values of $$\mathcal J_{5,5}$$. Inspection of the resulting coefficients of variation (not shown) suggests that these are indeed sensible variance estimates. Thus, it is possible to recover something from an information matrix even where one of the values is unreliable.  The R script on the right gives the full details of this example; feel free to adapt it when trying to repair your own matrices!


text:
# Page:A Dynamical Theory of the Electromagnetic Field.pdf/13

Jump to navigation Jump to search
This page has been validated.
471
PROFESSOR CLERK MAXWELL ON THE ELECTROMAGNETIC FIELD.

Mechanical Action between Conductors.

(34) The remaining terms,

 ${\displaystyle {\textstyle {1 \over 2}}{\frac {dL}{dt}}x^{2}+{\frac {dM}{dt}}xy+{\textstyle {1 \over 2}}{\frac {dN}{dt}}y^{2}=W}$ (11)

represent the work done in unit of time arising from the variations of ${\displaystyle L}$, ${\displaystyle M}$, and ${\displaystyle N}$, or, what is the same thing, alterations in the form and position of the conducting circuits ${\displaystyle A}$ and ${\displaystyle B}$.

Now if work is done when a body is moved, it must arise from ordinary mechanical force acting on the body while it is moved. Hence this part of the expression shows that there is a mechanical force urging every part of the conductors themselves in that direction in which ${\displaystyle L}$, ${\displaystyle M}$, and ${\displaystyle N}$ will be most increased.

The existence of the electromagnetic force between conductors carrying currents is therefore a direct consequence of the joint and independent action of each current on the electromagnetic field. If ${\displaystyle A}$ and ${\displaystyle B}$ are allowed to approach a distance ${\displaystyle ds}$, so as to increase ${\displaystyle M}$ from ${\displaystyle M}$ to ${\displaystyle M'}$ while the currents are ${\displaystyle x}$ and ${\displaystyle y}$, then the work done will be

${\displaystyle \left({M'-M}\right)xy,}$

and the force in the direction of ${\displaystyle ds}$ will be

 ${\displaystyle {\frac {dM}{ds}}xy,}$ (12)

and this will be an attraction if ${\displaystyle x}$ and ${\displaystyle y}$ are of the same sign, and if ${\displaystyle M}$ is increased as ${\displaystyle A}$ and ${\displaystyle B}$ approach.

It appears, therefore, that if we admit that the unresisted part of electromotive force goes on as long as it acts, generating a self-persistent state of the current, which we may call (from mechanical analogy) its electromagnetic momentum, and that this momentum depends on circumstances external to the conductor, then both induction of currents and electromagnetic attractions may be proved by mechanical reasoning.

What I have called electromagnetic momentum is the same quantity which is called by Faraday[1] the electrotonic state of the circuit, every change of which involves the action of an electromotive force, just as change of momentum involves the action of mechanical force.

If, therefore, the phenomena described by Faraday in the ninth Series of his Experimental Researches were the only known facts about electric currents, the laws of Ampère relating to the attraction of conductors carrying currents, as well as those of Faraday about the mutual induction of currents, might be deduced by mechanical reasoning.

In order to bring these results within the range of experimental verification, I shall next investigate the case of a single current, of two currents, and of the six currents in the electric balance, so as to enable the experimenter to determine the values of ${\displaystyle L}$, ${\displaystyle M}$, ${\displaystyle N}$.

1. Experimental Researches, Series I. 60, &c


text:
This bipartite network contains person–company shared leadership relations of "the five most representative companies" that are claimed to represent "the small inner ring of South African Finance". Left nodes represent persons and right nodes represent companies. An edge between a person and a company shows that the person had a leadership position in that company.

brunson_south_africa

igraph object


text:
Korean Journal of Veterinary Research 1999;39(5):965-973.
 Effects of vitamin E on prevention and treatment of canine babesiosis Kyoung-kap Lee, Geun-hyoung Kim, Young-jae Lee Department of Veterinary Medicine, College of Agriculture, Cheju National University 바베시아 원충감염 예방과 치료에 대한 비타민 E의 응용 이경갑, 김근형, 이영재 제주대학교 농과대학 수의학과 Abstract This study was carried out to investigate the effects of vitamin E on the prevention and treatment of Babesia gibsoni. Fifteen mongrel dogs, uninfected with Babesia spp, were assigned to three groups according to vitamin E(${alpha}-tocopherol$) concentrations in the RBC. The concentrations in each of the three groups were, respectively : ${alpha}-tocopherol$ in RBC less than $30{mu}g/{mu}l$(Group I), $30{mu}g/{mu}l{sim}60{mu}g/{mu}l$(Group II), more than $60{mu}g/{mu}l$(Group III). Artificial infection was accomplished by injecting $2{ imes}10^7{sim}2{ imes}10^8$ erythrocyte of Babesia gibsoni-infected dog into the cephalic vein. We investigated the clinical signs, vitamin E concentrations in RBC and serum, Vitamin A concentrations in serum, hematological values, white blood cell(WBC) viability and RBC membrane osmotic fragility after infection of Babesia gibsoni for a period of 20 days at 5 day intervals. The results obtained are summarized as follows : 1. After infection by Babesia gibsoni, clinical examination revealed depression, anorexia, pale mucous membranes, dark brown urine and diarrhea in proportion as time went on. After 10 days of infection, one dog each of Groups I, II and III revealed depression and anorexia. Two dogs in Group I and one dog each of Groups II and III showed dark brown urine after 15 days. Diarrhea was observed in one dog in each of the 3 groups after 20 days of infection. 2. After 5 days of infection, two dogs in each of Groups I, II and III showed Babesia gibsoni in RBC of blood smear stained with Giemsa. At the 15th day after infection with Babesia gibsoni, they were observed in all experimental animals. After both 5 days and 10 days of infection, the rate of Babesia gibsoni parasitized RBC(permillage, ‰) was 1‰, and increased as time went on. 3. After 5 days of infection by Babesia gibsoni, Group I, which had the lowest vitamin E concentration, showed significantly decreased RBC and PCV levels(p < 0.01). Group II and group III also showed significantly decreased RBC and PCV levels after 15 days of infection(p < 0.05). Particularly after 10 days of infection, Group I showed lower values in RBC and PCV levels compared to Groups II and III. WBC, RBC, fibrinogen and total protein levels between the groups did not differ during experimental periods. 4. According to the WBC differential counts, the ratios of neutrophil to lymphocyte showed a tendency to be slightly higher in Group III (more than $60{mu}g/{mu}l$) than in Groups I and II. 5. WBC viability did not differ between the groups. 6. RBC membrane osmotic fragility did not differ between the groups. Key Words: Babesia, Vitamin E, Vitamin E on RBC


text:
# zbMATH — the first resource for mathematics

Equivelar toroids with few flag-orbits. (English) Zbl 07308717
Toroids are generalisations of maps on the torus to higher dimensions. To be more specific, an $$(n+1)$$-toroid is a quotient of a tessellation of the n-dimensional Euclidean space with a lattice group.
The classification of regular and chiral toroids of any rank was completed by P. McMullen and E. Schulte [Adv. Math. 117, No. 1, 17–51 (1996; Zbl 0847.52010)]. In particular, they showed that there exist no chiral toroids of rank greater than 3. Furthermore, M. I. Hartley et al. showed for $$n\ge 2$$, the only Euclidean space form that admits chiral tessellations is the $$n$$-torus and proved this is only possible when $$n = 2$$ (see [Can. J. Math. 51, No. 6, 1230–1239 (1999; Zbl 0965.51008)]). In [Discrete Comput. Geom. 48, No. 4, 1110–1136 (2012; Zbl 1263.51016)], I. Hubard et al. studied properties of the automorphism groups of equivelar toroids and classified all equivelar toroids of ranks 3 and 4 into families according to their symmetry types.
The present nice paper gives a classification of equivelar $$(n+1)$$-toroids with at most $$n$$ orbits. In particular, a classification of 2-orbit $$(n+1)$$-toroids is discussed. Symmetry types of families of equivelar $$(n+1)$$-toroids with at most $$n$$ orbits are also determined.
##### MSC:
 51B15 Laguerre geometries 52C22 Tilings in $$n$$ dimensions (aspects of discrete geometry) 51F15 Reflection groups, reflection geometries 51M20 Polyhedra and polytopes; regular figures, division of spaces
##### Keywords:
symmetries of toroids; maps; polytopes; regular tessellation
GAP
Full Text:
##### References:
 [1] Brehm, U.; Kühnel, W., Equivelar maps on the torus, Eur. J. Comb., 29, 8, 1843-1861 (2008) · Zbl 1160.52010 [2] Conder, MDE, Regular maps and hypermaps of Euler characteristic $$-1$$ to $$-200$$, J. Comb. Theory Ser. B, 99, 2, 455-459 (2009) · Zbl 1205.05106 [3] Conder, M.; Dobcsányi, P., Determination of all regular maps of small genus, J. Comb. Theory Ser. B, 81, 2, 224-242 (2001) · Zbl 1028.05045 [4] Coxeter, HSM, Regular Polytopes (1973), New York: Dover, New York · Zbl 0031.06502 [5] Coxeter, H.S.M., Moser, W.O.J.: Generators and Relations for Discrete Groups. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 14. Springer, New York (1972) · Zbl 0239.20040 [6] Cunningham, G.; Del Río-Francos, M.; Hubard, I.; Toledo, M., Symmetry type graphs of polytopes and maniplexes, Ann. Comb., 19, 2, 243-268 (2015) · Zbl 1327.52019 [7] Dixon, J.D., Mortimer, B.: Permutation Groups. Graduate Texts in Mathematics, vol. 163. Springer, New York (1996) · Zbl 0951.20001 [8] Hartley, MI; McMullen, P.; Schulte, E., Symmetric tessellations on Euclidean space-forms, Can. J. Math., 51, 6, 1230-1239 (1999) · Zbl 0965.51008 [9] Hubard, I., Two-orbit polyhedra from groups, Eur. J. Comb., 31, 3, 943-960 (2010) · Zbl 1186.68502 [10] Hubard, I.; Orbanić, A.; Pellicer, D.; Ivić Weiss, A., Symmetries of equivelar 4-toroids, Discrete Comput. Geom., 48, 4, 1110-1136 (2012) · Zbl 1263.51016 [11] Jones, GA; Singerman, D., Theory of maps on orientable surfaces, Proc. Lond. Math. Soc., 37, 2, 273-307 (1978) · Zbl 0391.05024 [12] Jones, G., Singerman, D.: Maps, hypermaps and triangle groups. In: The Grothendieck Theory of Dessins d’Enfants (Luminy 1993). London Mathematical Society Lecture Note Series, vol. 200, pp. 115-145. Cambridge University Press, Cambridge (1994) · Zbl 0833.20045 [13] Matoušek, J.: Lectures on Discrete Geometry. Graduate Texts in Mathematics, vol. 212. Springer, New York (2002) · Zbl 0999.52006 [14] McMullen, P.; Schulte, E., Higher toroidal regular polytopes, Adv. Math., 117, 1, 17-51 (1996) · Zbl 0847.52010 [15] McMullen, P., Schulte, E.: Abstract Regular Polytopes. Encyclopedia of Mathematics and its Applications, vol. 92. Cambridge University Press, Cambridge (2002) · Zbl 1039.52011 [16] Orbanić, A.; Pellicer, D.; Ivić Weiss, A., Map operations and $$k$$-orbit maps, J. Comb. Theory Ser. A, 117, 4, 411-429 (2010) · Zbl 1197.51005 [17] Ratcliffe, J.G.: Foundations of Hyperbolic Manifolds. Graduate Texts in Mathematics, vol. 149. Springer, New York (2006) · Zbl 1106.51009 [18] GAP—Groups, Algorithms, and Programming. Version 4.8.6 (2016). http://www.gap-system.org
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.