text:
TL;DR

Let’s use the maths for converting from the SVG Path syntax for ellipses to the parametric representation, which will allow us to eventually calculate the bounding box.

In our previous posts:

we looked at the math for finding the equations that allow us to express an arc of ellipse in parametric representation (see Ellipses (for SVG): parameter and angles) starting from what is available in the SVG path attribute d (see Ellipses (for SVG): mapping to SVG representation).

Implementation time! This function provides us the parametric representation parameters:

 1 sub ellipse_p2c ($P0,$P1, $R,$xdegs, $fA,$fS) {
2    my $phi =$xdegs * PI / 180; # turn into radians
3    my ($sinp,$cosp) = (sin($phi), cos($phi));
4    my ($x0,$y0, $x1,$y1, $xr,$yr) = map {$_->@{qw< x y >}} ($P0, $P1,$R);
5    my ($x0t,$y0t) = (($x0 -$x1) / 2, ($y0 -$y1) / 2);
6    my ($x_0,$y_0) = ($cosp *$x0t + $sinp *$y0t, -$sinp *$x0t + $cosp *$y0t);
7    my ($x2r,$y2r) = ($xr **2,$yr ** 2);
8    my $lambda = (($x_0 / $xr)**2 + ($y_0 / $yr)**2); 9 if ($lambda > 1) { # make it a 1 by expanding $rx and$ry with a factor
10       my $factor = sqrt($lambda);
11       ($xr,$yr) = ($xr *$factor, $yr *$factor);
12       $lambda = 1; 13 } 14 my$cf = sqrt(1 / $lambda - 1); 15$cf = -$cf if ($fA xor $fS); 16 my ($x_c, $y_c) = (-$cf * $xr *$y_0 / $yr,$cf * $yr *$x_0 / $xr); 17 my ($xc, $yc) = ( # the center, yay! 18 ($x0 + $x1) / 2 +$cosp * $x_c -$sinp * $y_c, 19 ($y0 + $y1) / 2 +$sinp * $x_c +$cosp * $y_c, 20 ); 21 my$pi2 = 2 * PI;
22    my ($t_begin,$t_end) = map {
23       atan2($xr * ($_ * $y_0 -$y_c), $yr * ($_ * $x_0 -$x_c));
24    } (1, -1);
25    $t_begin -=$pi2 if $t_begin >$t_end; # now $t_begin <$t_end
26    my $delta =$t_end - $t_begin; 27$t_begin = $t_end # swap if... 28 if ($delta <= PI && $fA) # arc is short but long is required 29 || ($delta > PI && ! $fA); # arc is long but short is required 30 # adjust$t_begin in [0, 2*PI[
31    $t_begin -= floor($t_begin / $pi2) *$pi2;
32    $t_end =$t_begin + $delta; 33 return ( 34 {x =>$xc, y => $yc}, # center 35 {x =>$xr, y => $yr}, # radii, possibly scaled up 36$phi,                 # ellipse rotation angle
37       $t_begin, # begining value for parameter t 38$t_end,               # ending value for parameter t
39    );
40 }

Line 2 converts the rotation angle in radians (because it’s originally in degrees), so no big deal. Lines 3 and 4 calculate the sine and cosine of this angle, which will be used extensively during the function.

Line 4 simply unwraps our input points $P0$ and $P1$ (corresponding to $\mathbf{P}_1$ and $\mathbf{P}_2$ respectively, just to confuse things a bit 🤓) and the two radii. In this function we stick to a convention where the variable name always starts with the coordinate axis, followed by what it is for (hence, $r_x$ becomes $xr). Variables $x_0 and $y_0 represent the first point in the translated-then-rotated system (line 6, with a little help from line 5). Lines 8 through 20 deal with finding the center (see Ellipses (for SVG): finding the center). In particular,$\Lambda$is calculated in line 8 and radii are adjusted if needed (lines 9 through 13). After line 13, we know that $lambda is not greater than $1$ so we can take the square root in line 14 calculating the center factor $cf, i.e. the factor that both$C’_x$and$C’_y$have in common. Its sign is then adjusted according to the values of the flags (line 15). Last, line 16 calculates$\mathbf{C’}$and lines 17 through 20 bring it back to the original coordinate system and give us$\mathbf{C}$. Lines 21 through 32 deal with calculating the endpoints of the contiguous interval$[t_{begin}, t_{end}]$for the parameter (see Ellipses (for SVG): parameter values). It’s basically a straightforward implementation of the formulas we already discussed, with the note that$t_{begin}$(i.e. $t_begin) is shifted to live in the interval $[0, 2\pi[$ (line 31, although this is not strictly necessary).

This is it… we had to swat a bit for these 40 lines of code!


text:
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# error caption outside float latex Sea Isle City, New Jersey

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text:
1. ## Limit thread number 2?

Can anyone help me with these two limit not using l hospital rule/.

1. As x approaches 1

(1/x)-1/(x-1)

2. as h approaches zero

(2h^3-h^2+5h)/(h)

2. ## Re: Limit thread number 2?

Originally Posted by homeylova223

2. as h approaches zero

(2h^3-h^2+5h)/(h)
Factor 'h' out of the numerator. What do you get?

3. ## Re: Limit thread number 2?

I get h(2h^2-h+5)/(h)

4. ## Re: Limit thread number 2?

Cancel out h from numerator and denominator then make h=0

5. ## Re: Limit thread number 2?

I got h=5 I was stuck but you make it seem so easy.

6. ## Re: Limit thread number 2?

Originally Posted by homeylova223
I got h=5

Originally Posted by homeylova223
I was stuck but you make it seem so easy.
The best way to master math is send lots of time studying. There's no such thing as too much math.

7. ## Re: Limit thread number 2?

On my first one (1/x)-1/(x-1) as x approaches 1

Would It the final answer be zero because if I plug in one that is all I seem to get.

8. ## Re: Limit thread number 2?

Nope, that is not the answer.

Your function can also be written as -1/x(x-1)

Have you tried to graph the function?

9. ## Re: Limit thread number 2?

Originally Posted by homeylova223
I got h=5 I was stuck but you make it seem so easy.
Not h= 5, the limit, as h goes to 0, is 5.

For the first one, what you wrote, (1/x)-1/(x-1), does NOT have a limit as x goes to 1, but I think what you meant was ((1/x)- 1)/(x- 1).

To find that limit, multiply both numerator and denominator by x to get (1- x)/(x(x-1)= -(x-1)/(x(x-1)). Now the x-1 terms cancel.


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# Find surface area of 3D rectangle given one edge's dimensions and area

I have a 3D rectangular object. It has a volume of $75\text{cm}^3$; it has a square base with one edge of the base having a length of $x$. Find a function $S$, that represents the surface area of the object with respect to $x$.

I know that the surface area $S$ of a 3D rectangular object is a function of its length ($L$), width ($W$), and height ($H$). That is, adding the areas of each face gives the surface area. The area of each face is below:

\begin{align} S_1 &= LW \\ S_2 &= LW \\ S_3 &= WH \\ S_4 &= WH \\ S_5 &= HL \\ S_6 &= HL \end{align}

Or more declarative, the surface area of a 3D rectangular object is:

\begin{align} S &= LW + LW + WH + WH + HL + HL \\ &= 2LW + 2WH + 2HL \end{align}

I also know the volume $V$ of a 3D rectangular object is a function of its length, width, and height:

$$V = LWH$$

Using algebra I can find any given dimension:

\begin{align} L &= V/WH \\ W &= V/LH \\ H &= V/LW \end{align}

Going back to the shape at hand, I know the area of one of the faces of the polygon: $x^2$. This corresponds to the length multiplied by width.

Since I know the volume, length, and width, I can get the height, too:

\begin{align} H &= V/LW \\ &= 75/x^2 \end{align}

Substituting $H$ into the equation for the surface area of a 3D rectangular object:

\begin{align} S &= 2LW + 2WH + 2HL \\ &= 2(x^2) + 2(\frac{75}{x^2})(x) + 2(\frac{75}{x^2})(x) \\ &= 2x^2 + 2x(\frac{75}{x^2}) + 2x(\frac{75}{x^2}) \\ &= 2x^2 + 4x(\frac{75}{x^2}) \\ &= 2x^2 + \frac{4x \cdot 75}{x^2} \\ &= 2x^2 + \frac{300x}{x^2} \\ &= 2x^2 + 300 \cdot \frac{x}{x^2} \end{align}

However, this has been marked incorrect. What's the issue with my process?

It is the volume of the object that is $LWH$ and is given in cm$^3$. If you mean the volume is $75$ cm$^3$ the surface area is $2x^2+4xH$, which is $2x^2+\frac {300}x$ as you say. Maybe they expected you to replace $\frac x{x^2}$ with $\frac 1x$


text:
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Article Contents
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# Ergodic optimization for generic continuous functions

• Given a real-valued continuous function $f$ defined on the phase space of a dynamical system, an invariant measure is said to be maximizing if it maximises the integral of $f$ over the set of all invariant measures. Extending results of Bousch, Jenkinson and Brémont, we show that the ergodic maximizing measures of functions belonging to a residual subset of the continuous functions may be characterised as those measures which belong to a residual subset of the ergodic measures.
Mathematics Subject Classification: Primary: 37A99; Secondary: 37D20, 37E10.

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# A roadside refueling relay

## 1 Rules of the game

While driving across the country one hot summer day, two of my friends noticed that they were running dangerously low on fuel. The driver pointed this out to the passenger and said:

"Suppose we run out of gas and have to walk down the road until we find the closest gas station. Someone'd have to walk all the way down to get a canister of gas, then walk all the way back!"

They debated whether there was a better solution than this for a while (as I detail below). Eventually, I heard about the conundrum and turned it into the following general puzzle (In any case, their car did not run out of gas, so the problem became purely academic.) :

Suppose a car full of $$n$$ people runs out of gas. There is a gas station 1 km away. Assuming they must walk to retrieve a canister of gas from the station, find the best cooperative solution. (That is, find the solution which minimizes the distance walked by whoever walks the most.)

Assume that the solution genuinely involves individuals going to retrieve gas; they cannot push the car, carry each other, etc.

Feel free to pause at this time and solve this problem yourself. In the rest of this article, I will detail the complete solution.

## 2 The general solution

### 2.1 Discussion

After the driver lamented the thought of walking all the way to the station and back, the passenger replied,

"Actually, you can shorten the walking distance if you drive the car once you've refueled it—just have one person walk to get the gas, then have the second person meet them halfway. While the first person waits at the halfway point, the second person can go refuel the car. When the car is refueled, the second person can pick up the first person. Overall, no one has to walk more than 3/2 km."

To which the driver responded,

"Here's a slightly better solution: have the first person walk to the gas station and one third of the way back. The second person meets the first person there, walks back to refuel the car, and picks up the first person on the way out. Overall, they each walk exactly 4/3 of the way there. I wonder if we can do better?"

In general, we're looking for a solution that looks like this: the first person walks to the gas station, then walks some amount of the way back. The second person walks to where the first person stops, then walks some amount of the way back. This continues until some final person walks all the way back to the car and picks everyone up.

### 2.2 The egalitarian principle

Now we observe the following tradeoff principle: suppose everyone agrees on a solution strategy. While following the strategy, someone chooses to walk a little bit further than required. This alteration will not affect anyone who has already completed their part of the solution. However, everyone afterwards will walk a little bit less.

From this principle, we arrive at a surprising egalitarian principle: in the optimal solution, everyone must walk the same distance (!). To see that this is true, observe that:

1. The only way to allow someone to walk a little less is for someone else to walk a little more.
2. When everyone walks the same amount $$d$$, then obviously the person who walks the most walks the amount $$d$$.
3. If you alter the egalitarian solution so that someone walks less than $$d$$, someone else must walk more than $$d$$. Therefore the person who walks the most must walk more than $$d$$, and so the altered solution is worse than the egalitarian solution.
4. Therefore the egalitarian solution is optimal.

### 2.3 The self-similarity principle

Now we apply a second trick— the self-similarity principle: Suppose that there are $$n$$ people in the car and the canister is at the gas station 1 km away. If one person goes to the gas station and brings the gas canister back $$x$$ units, then effectively you have a new problem with $$n-1$$ people in the car and the "gas station" / canister effectively $$(1-x)$$ km away (because that's where the first person brought the gas canister.)

Hence let $$f(n)$$ denote "the distance the first person should walk back from the gas station in the optimal solution for $$n$$ people."

Obviously we have the base case: $$f(1) = 1$$, because if there's only one person in the car, that person should walk all the way back. And by the self-similarity principles, we know that if $$n\geq 2$$, then the distance walked by the first two people is:

$$d_1 = 1 + f(n)$$

and

$$d_2 = [1 + f(n-1)] \cdot (1 - d_1)$$

To understand the second equation, note that $$[1 + f(n-1)]$$ is the optimal distance someone should walk back if there are $$n-1$$ people in the car and the gas station is 1 km away. We scale this distance by $$(1-d_1)$$ to account for the fact that the canister is located $$(1-d_1)$$, not 1 km, away.

By the egalitarian principle, these two distances are equal. Hence we have:

\begin{align*} 1 + f(n) &= [ 1 + f(n-1) ] \cdot [1-f(n)]\\ 1 + f(n) &= 1 + f(n-1) - f(n) - f(n-1) f(n) \\ 0 &= f(n-1) - f(n-1)f(n) - 2 f(n) \\ 0 &= - f(n) [ f(n-1) + 2 ] + f(n-1) \\ f(n) &= \frac{f(n-1)}{f(n-1) + 2}\\ \end{align*}

### 2.4 A recurring problem

We have reduced the problem to a recurrence relation: solve for $$f(n)$$, given the recursive definition

$$f(1) = 1, \qquad f(n+1) = \frac{f(n)}{f(n)+2}.$$

In general, we find that the solution is given by:

$$f(n) = \frac{1}{2^{n}-1},$$

so that $$f(1) = 1$$, $$f(2) = \frac{1}{3}$$, $$f(3) = \frac{1}{7}$$, $$f(4) = \frac{1}{15}$$, etc.

## 3 Other optimizations

### 3.1 Minimize the maximum/average

If you want to find the solution that minimizes the total or average amount of walking, the solution is brutally simple: have one person walk there and back.

Indeed, that person will walk 2 km, the absolute minimum distance it is possible to walk when retrieving a gas canister from a station that is 1 km away.

### 3.2 Minimize the time

If you want minimize the amount of time it takes to retrieve the gas from the gas station, the solution is similarly simple: have everyone leave the car at the same time, stopping at their designated spots along the way to the gas station. The first person will reach the gas station, and walk back to where the second person has been waiting, etc.

Overall, with some simplifying assumptions, this takes essentially the same amount of time as having one person walk 2 km there and back, which is the absolute minimum.

Created: 2016-09-03 Sat 01:47

Emacs 24.5.1 (Org mode 8.3beta)

Validate


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## Intermediate Algebra: Connecting Concepts through Application

$x\gt-6 \text{ AND } x\lt6$
$\bf{\text{Solution Outline:}}$ If there is only $1$ graph, use "AND." If there are two separate graphs, use "OR." For a graph that uses a parenthesis as an endpoint, use $\lt$ or $\gt.$ For a graph that uses a bracket as an endpoint, use $\le$ or $\ge.$ $\bf{\text{Solution Details:}}$ Since the given graph contains only a single graph, then "AND" will be used. Since parentheses are used as endpoints in the given graph, then $\lt$ or $\gt$ will be used. The compound inequality that satisfies the given graph is \begin{array}{l}\require{cancel} x\gt-6 \text{ AND } x\lt6 .\end{array}


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# Newton's 3rd Law issue

1. Nov 24, 2015

### Lisciu

• Missing template. Moved from technical forums.
Hi Guys,

I was highly studied the Newton's law once again. Probably to recall the concept from the past of my education. Anyway I find out the exercise during solving the friction problems from Hibbeler book.

The exercises is state as :
The fork lift has a weight W1 and center of gravity at G. If the rear wheels are powered, whereas the front wheels are free to roll, determine the maximum number of crates, each of weight W2 that the fork lift can push forward. The coefficient of static friction between the wheels and the ground is μ s and between each crate and the ground is μ 's.

The whole system looks like this:

And then it's seperate on two objects:

FBD of the 1:

FBD of 2:

And there is nothing suprised about that except the action and reaction force from the crates and front wheel.

I expected that those force will be equal like the 3rd Newton law said. But during the solving and after see those images from book solution looks like 3rd Newton law doesn't work anymore....

Could someone explain what is the reason of that? And where is actually the action and reaction force? I'm confused here.

2. Nov 24, 2015

### Staff: Mentor

The only 3rd law pair here is $P=-P'$. If you got anything else then you made a mistake.

3. Nov 24, 2015

### Lisciu

Seriously? :) when you figured it out that I don't know that?

So why P is equal 707,37lb : and P' is equal : 105lb?

4. Nov 24, 2015

### Staff: Mentor

That is not correct, but there is no way to tell where you went wrong without more details about how you attempted to solve it.

5. Nov 24, 2015

### Lisciu

It's okay. I figured it out what is going on here. Because in the solving book author calculated the P' force for just one crate to calculate how many he can push by this fork lift. I realize that if I will calculated it with diffrent approach like typical 3rd law Newton without calculating reaction force for just one crate I will have the same result. I just took the calculate force P from fork lift and then using this force in X direction net force summing (2rd law) get the force of normal acting on. When I have it I just divided it by one crates weight and actually I get the number of crates this system can handle.

6. Nov 24, 2015

### Staff: Mentor

That's good. I think I probably would have followed something closer to your approach than to the books approach.

7. Nov 24, 2015

### Lisciu

Yeap, me too. I was just confused for whole day. Then I just sit once again on the problem and realize where was the hatch.

Anyway I will need to create the topic around 3rd Law because I not undestand in 100% this concept specially if someone talk about cancelling force not cancelling force and then i was thinking about this action and reaction force as a unended storry with domino effect in whole body if we consider for example hammer and nail.

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text:
Simple fundamental subspaces problem stumping me

1. Dec 10, 2015

kostoglotov

So the matrix is just a row vector

$$\begin{bmatrix}3 & 4 & 0\end{bmatrix}$$

My problem, is that I get the nullspace as having to 2 dimensions, and the row space as having 2 dimenions, but that adds up to 4 dimensions, when it should add up to three. What simple thing am I missing?

Null space base vectors

$$\begin{bmatrix}-4\\3\\0\end{bmatrix},\begin{bmatrix}0\\0\\1\end{bmatrix}$$

Row space base vectors

$$\begin{bmatrix}1\\0\\0\end{bmatrix},\begin{bmatrix}0\\1\\0\end{bmatrix}$$

2. Dec 10, 2015

HallsofIvy

Staff Emeritus
Taking the matrix to be the single row $\begin{bmatrix}3 & 4 & 0\end{bmatrix}$ then the null space is all $\begin{bmatrix}x \\ y \\ z \end{bmatrix}$ such that $\begin{bmatrix}3 & 4 & 0\end{bmatrix}\begin{bmatrix}x \\ y \\ z \end{bmatrix}= 3x+ 4y= 0$. Yes, that is of dimension 2 having $\{\begin{bmatrix}-4 \\ 3 \\ 0\end{bmatrix}, \begin{bmatrix}0 \\ 0 \\ 1 \end{bmatrix}\}$ as basis. However, the row space is one dimensional, having the vector $\begin{bmatrix}3 & 4 & 0 \end{bmatrix}$, the single row making up the matrix, as basis.

3. Dec 10, 2015

Geofleur

Those two last vectors cannot be in the row space, because a linear combination of them will give the first null space vector, $[-4 \ 3 \ 0]^T$. In fact, the row space is the space spanned by the rows of the matrix, considered as vectors. So the row space consists of all multiples of $[3 \ 4 \ 0]^T$.

EDIT: oops, HallsofIvy beat me to it :-)

4. Dec 10, 2015

kostoglotov

I realize what I was missing now.

I kept thinking of the row space as all those vector that can be operated on by the matrix without giving a zero vector result. Except that's not what the rowsapce is! It's the vector/s that characterize/s the rows of the matrix. Thanks @HallsofIvy and @Geofleur!


text:
Manipulating derivatives and rearranging

1. Mar 24, 2014

cooev769

Hey, I am actually a third year physics student, but here in New Zealand they tend to rush past the fundamentals, hence I can't seem to rearrange the following equations properly. Eq 1 and Eq 2 are:

τ=-erB*dr/dt ek

and

τ=m*d(r^2dθ/dt)/dt ek

Where ek is a unit vector. Is supposed to be rearranged to the form:

vθ(r) = (-eB/mr) * integral between r and a of r dr = (-eB/mr)*(r^2-a^2)/2

I can get very close but my derivative rearranging is sub-optimum. A little help would be really appreciated, as would telling me how the hell I write equations on the forum. Cheers.

2. Mar 24, 2014

jbunniii

Regarding how to typeset equations on PF, there's a nice tutorial in section 2 of the FAQ, which should be more prominently visible but can be found under the "Site Info" link at the top of the page.

3. Mar 24, 2014

micromass

Staff Emeritus
4. Mar 24, 2014

jbunniii

5. Mar 24, 2014

micromass

Staff Emeritus
I totally missed that... Perhaps I'm going blind

6. Mar 24, 2014

cooev769

Cheers. Turns out the equations they gave us to derive was incorrect. So thread closed.


text:
### The Building Blocks of Risk Management

After completing this reading you should be able to:

• Explain the concept of risk and compare risk management with risk-taking.
• Describe elements, or building blocks, of the risk management process and identify problems and challenges that can arise in the risk management process.
• Evaluate and apply tools and procedures used to measure and manage risk, including quantitative measures, qualitative assessment, and enterprise risk management.
• Distinguish between expected loss and unexpected loss and provide examples of each.
• Interpret the relationship between risk and reward and explain how conflicts of interest can impact risk management.
• Describe and differentiate between the critical classes of risks, explain how each type of risk can arise, and assess the potential impact of each type of risk on an organization.
• Explain how risk factors can interact with each other and describe challenges in aggregating risk exposures.

## Risk and its Management

Risk refers to the potential variability of returns around an expected return from a portfolio or an expected outcome. The financial risk that arises from uncertainty can be managed and mitigated. Modern risk management refers to the ability, in many instances, to price risks and to provide adequate compensation for the risk taken in business activities.

The building blocks of risk management include:

1. The classic risk management process
2. Identifying knowns and the unknowns
3. Expected loss, unexpected loss, and the tail loss
4. Risk factor breakdown
5. Structural change from tail risk to a systemic crisis
6. Human agency and conflicts of interest
7. Typology of risks and risk interactions
8. Risk aggregation
9. Balancing Risk and Reward
10. Enterprise risk management

## 1.  Types of Risk and Their Interactions

Risk can be grouped depending on different types of business environments. Grouping of the risks is essential for the business institutions to factor into specific risks while managing them. This is true because each type of risk needs different skills to manage it.

A typical typology of risks should always be flexible to accommodate new forms of risks that are ever-emerging (such as cyber risks). The following diagram gives the typical modern typology of corporate risks:

### Market Risk

This is the risk associated with the potential reduction in the value of a portfolio or security due to changes in financial market prices and rates. Price risk can be decomposed into a general market risk component (the risk that the market as a whole will fall in value) and a specific market risk component (idiosyncratic component), unique to the particular financial transaction under consideration. In trading activities, a risk arises both from open (unhedged) positions and from imperfect correlations between market positions that are intended to offset one another.

Market risk can be further classified into the following categories:

Interest rate risk- It arises from fluctuations in the market interest rates, which may cause a decline in the value of interest rate sensitive portfolios. For example, the bond market is affected by interest rates in the market. Curve risk can arise in portfolios in which long and short positions of different maturities are effectively hedged against a parallel shift in yields, but not against a change in the shape of the yield curve. If the rates of the positions are imperfectly correlated, basis risk may arise in offsetting positions having the same maturity.

Equity price risk – This is the risk that is associated with the volatility in the stock prices. The market risk component is the sensitivity of the equity or a portfolio to a change in the level of a market index. This risk cannot be done away with by diversification. The idiosyncratic or specific threat is the component of volatility determined by firm-specific characteristics like its management, production line, etc. This can be done away with by diversification.

Foreign Exchange Risk- Due to operations that involve foreign currencies, imperfectly hedged positions in certain currencies may arise, which may cause exposure to exchange rates. Major factors influencing foreign exchange risk are imperfect correlations in currency prices and fluctuating international interest rates.

### Credit Risk

The risk associated with a counterparty not fulfilling its contractual obligations is the credit risk. For example, the default on a credit card loan is the scenario in which credit risk materializes for a credit card company.

Credit risk can be further classified into:

• Bankruptcy risk– The risk associated with a borrower’s inability to clear his debt leading to a takeover of his collateralized assets.
• Downgrade risk– The risk that there might be a decline in credit ratings of a borrower because of a drop in his creditworthiness.

Credit risk is a matter of concern only when the position is an asset and not a liability. If the position is an asset, then a default by the counterparty may cause a loss of total or a partial value of the position. The value that is likely to be recovered is called recovery value, while the amount that is expected to be lost is called loss given default.

At the portfolio level, the issues to be addressed are the following:

• The creditworthiness of the obligor: Based on this, appropriate interest rate or spread should be charged to compensate for the risk undertaken
• Concentration risk: The extent of diversification of the obligor should be a concern.
• The state of the economy: When the economy is booming, the frequency of defaults is comparatively lower than when there is a recession.

### Liquidity Risk

It comprises of funding liquidity risk and market liquidity risk.

Funding liquidity risk is associated with the risk that a firm will not be able to settle its obligations immediately when they are due. It relates to raising funds to roll over debt and to meet margin calls and collateral requirements. Funding liquidity risk can be managed by holding highly liquid assets like cash.

Trading liquidity risk (also called market liquidity risk) is the risk associated with the inability of a firm to execute transactions at the prevailing market price. It may reduce the institution’s ability to hedge market risk, and also it is the capacity to liquidate assets when necessary.

### Operational Risk

It refers to the risk that arises due to operational weaknesses like management failure, faulty controls, inadequate systems, among others. Human factor risk is one of the essential operational risks, and it results from human errors like entering wrong parameter values, using wrong controls, among others. Technology risk arises from a computer system’s failure.

### Business Risk

It arises from the uncertainties in demands, the cost of production, and the cost of delivery of products. Business risk is managed by framing appropriate marketing policy, inventory policies, choices of products, channels, and suppliers, etc. Business risk is affected by the quality of a firm’s strategy and its reputation too.

### Strategic Risk

It is the risk associated with the risk of significant investments for which the uncertainty of success and profitability is high. It is related to the strategic change in the policies of a company to make it more competitive in the marketplace.

### Reputation Risk

It comprises of the beliefs that an enterprise can settle its obligations to counterparties and creditors and secondly, that it follows ethical practices. Trust and fair dealing are two essential things that drive businesses. For example, reputation is of crucial importance in the financial industry.

### Interactions of Risk Types

Risks can flow from one type to another. For instance, during hard business times, the risk can flow from the credit risk to Liquidity risk and then to market risk. This kind of flow was seen in the 2007-2009 financial crisis.

Another example is where operational risk (as a form of lousy trading activity by the traders) flows to market risks by creating unfavorable market positions. Moreover, this can move to become a reputation risk to the concerned company.

## 2.  The Risk Management Process

Given below is the flow chart of the risk management process:

Risk management includes the identifying of the type and level of risk that is appropriate for the firm to assume, analyze, and measure the risk, assess the possible outcomes of each risk. The final stage is the management of the risks.

### Methods of Risk Management

1. Avoiding the risk: some risks can be managed by avoiding it. For instance, closing down the business unit or changing the business strategy.
2. Retaining or keeping the risk: the company can accommodate the risk by insuring it.
3. Mitigation of the risk: this method involves an attempt to decrease the exposure, frequency, and severity of the risk. A good example is the improvement of a firm’s infrastructure and putting collateral on credit exposure.
4. Tinder risk: this method applies to risks that can be transferred to a third party. An example is in derivative products where a company pays a premium to a party to accept a certain level of risk.

## 3.  Known and Unknown Risks

According to Donald Rumsfeld (1921), risk managers should not concentrate on known risk only but also the unknown risks. He also classified the risks, as seen in the diagram below.

Unknown risks can be very significant and essential, even though their measurement can be impossible. Unknown risks can be managed using the usual forms of risk management.

Rumsfeld classification implies that risk managers should focus not only on measurable risks but also on an unknown risk. They should strive to unravel the “unknown unknowns,” which includes threats that do hideaway.

## 4.  Expected, Unexpected and Tail Loss

### The Expected Loss

The expected loss can be defined as the mean loss an investor (position taker) might expect to experience from a portfolio. The expected risks are those that may be large in size, are predictable, and could be avoided with the risk management process.

Theoretically, portfolios usually bear the loss that is near to the average loss, which can be statistically measured with some degrees of freedom.

Expected loss can be calculated from the underlying risk factors. Such factors include:

• The probability of occurrence of risk event
• The size (severity) of the loss
• The exposure to risk

Let us take an example of credit risk to the bank. Denote the probability of default by PD, bank’s exposure at default by EAD, and severity of loss given default by LGD. So, the EL is given by:

$$EL=EAD \times LGD \times PD$$

So, how does the bank’s manager make sure that they make a profit? The bank management should come up with the price that covers the expected loss. It is important to note that the computation of expected loss is based on assumptions.

### The Unexpected Loss

The unexpected loss is the level at which the losses in a portfolio defer from the average loss. Unexpected risks due to unanticipated variability in the losses.

For instance, in a credit portfolio, an unexpected loss can be caused by a difference in the number and severity of the loans. That is, a large number of small loans are diversified, and hence we can estimate the expected loss. However, if the EL continuously changes due to macroeconomic factors, it leads to unexpected loss.

In some cases, some portfolios (such as credit) can show extreme loss variance over some interval of time. In this case, the expected loss (EL) is calculated by averaging the loss from the long run good years and a short run of bad years. However, in bad years, the losses can rise to an unexpected level and even to extreme levels. Consequently, the banks are forced to increase the risk capital and including an expected loss in pricing their products to guard itself against huge unexpected losses, which can cause insolvency and defaults.

### Value-at-Risk (VaR)

VaR is a statistical measure that defines a particular level of loss in terms of its chances of occurrence, i.e., the confidence level of the analysis. In other words, VaR utilizes loss distribution relative to a portfolio or a position to approximate losses at a given level of confidence.

For example, if a position in an option has a one-day VaR of $1 million at the 99% confidence level, then the risk analysis will show that there is only a 1 percent probability of a loss that is greater than$1 million on any given trading day.

The VaR measure works under normal conditions of the market and only over a short period, such as one trading day. Potentially, it is a poor and misleading measure of risk in abnormal markets, over more extended periods, or for illiquid portfolios. VaR also depends upon the control environment. Trading controls can be circumvented, and this usually happens when back-office staff, business line managers, even risk managers do not have a proper understanding of the critical significance of routine tasks, such as an independent check on volatility estimates, for the integrity of key risk measures.

## 5.  Breakdown and Interactions of Risk Factors

The risk managers must subdivide the risk into discrete risk factors so that each factor and the interactions between these factors can be studied. An excellent example in the credit risk, which we have studied earlier-where credit risk was divided into the probability of default (PD), bank’s exposure at default (EAD), and severity of loss given default (LGD).

However, there is an obvious challenge of how granular should a risk be, given the loss data. Dividing the data to very small sub-factors is impractical since it is time-consuming and tiresome. Secondly, analytical resources might be limited. Moreover, the data might be limited in terms of quantity, quality, or descriptive ability.

The solution to this challenge is the emergence of machine learning. In machine learning and substantial cloud-based calculation, power can help is isolating risk granules to smaller details.

## 6.  Structural Change from Tail Risk to Systemic Crisis

Tail risks are those that rarely occur. They can be explained as the extreme version of unexpected loss that is hard to find in the given data. They are usually revealed in time series data of long periods. The tail risk can be detected using statistical methods such as the Extreme Value Theory (EVT).

When the structure of a financial system changes, the risks increases. That is, events associated with larges losses may increase as well as risk factor levels. Unless the structural problem is fixed or proper risk management is adopted, new losses relative to a risk type might occur, which changes the amount of tail risk, expected and unexpected losses.

## 7.  Human Agency and Conflicts of Interest

Financial systems are run by intelligent human beings who can adapt to change in a personal and cunning manner. That is, those who are more experienced in risk management can play up their game by hiding their risk analysis from other participants for their gain.

Having said this, many financial firms have employed three ways to control human agency and conflicts of interest:

1. Firms create business models that can identify and manage risk.
2. Employing risk managers that are qualified in risk management and day-to-day oversight.
3. Periodic independent oversight and assurance (e.g., internal audit)

These defense mechanisms do not always work due to industry innovations, which sometimes leave loopholes in the risk management sector. Moreover, sometimes traders and the industry leadership willingly alter the credibility of the risk management systems. That is why grasping the role of human agency, self-interest, and conflicts interest are one of the cornerstones of risk management.

## 8.  Risk Aggregation

The risk manager should be able to identify riskiest businesses and determine the aggregate risks of a firm. For instance, market risks are easily quantified and controlled by comparing the notional amount in each asset held. This, most of the time, is impractical since different stocks and industries have different volatilities.

Since the mushrooming of derivative markets in the 1970s, measurement of market risk became relatively achievable. This is because the value and the risk of the derivatives depend on the price of the underlying portfolio.

Derivative traders developed risk measures termed as the Greeks. They include delta and theta. Greeks are still used up to date, but they cannot be added up, rendering them limited at the enterprise level.

Another measure of risk is VaR. VaR was a useful aggregation method up to the year before the crisis, but it involves too many assumptions. However, VaR is marred with shortcomings but remains to be essential to ask managers.

The disadvantages of these aggregate risk measurements have motivated the managers to come up with total risk measures to replace the traditional measures but, most of the time, fail to include critical dimensions of the risk and must be supplemented with other methods. Conclusively, understanding how risks are aggregated and the drawbacks and advantages that come with it is an essential risk management building block.

## 9.  Risk and Reward Equilibrium

Normally, the assumption of higher systematic risk is associated with higher returns from portfolios. However, the demanded returns from risky assets may not be apparent unless the market of the asset is efficient and transparent. For example, the bond prices, solely, may not imply the return demanded, taking additional risks. This can be the case because of liquidity and tax effects. A key objective of risk management is to make transparent potential risks for the firm and identify activities that may be detrimental for the firm in the long term.

For instance, a bank can include the cost of both the expected and unexpected cost by using the following formula for risk-adjusted return on capital (RAROC):

$$RAROC= \frac{\text{Reward}}{\text{Risk}}$$

Note the Reward can be After-Tax Risk-Adjusted Expected return, and the risk is described as the economic capital so that:

$$RAROC= \frac{\text{After-Tax Risk-Adjusted Expected return} }{\text{Economic capital}}$$

If the RAROC is higher than the cost of equity capital, then the portfolio is valuable to the investor. The cost of equity capital is the minimum return on equity capital required by the shareholders to compensate for the risk.

Apart from the banking industry, RAROC is applied across different industries and institutions, with the formula varying accordingly (but its purpose remains constant).

### Uses of RAROC

1. Investment Analysis: RAROC formula is used to anticipate the likely returns from future investments.
2. Comparing businesses: RAROC can be used to compare different units of a company that needs varying amounts of economic capital.
3. Pricing strategies: A company can re-determine the pricing strategy of its products so the risk-adjusted returns.
4. Risk management cost (benefit analysis): RAROC can be used by a firm to compare the cost of risk management to the benefit of the firm.

## 10.  Enterprise Risk Management (ERM)

Enterprise management risk (ERM) is the process of planning, organizing, leading, and controlling the activities of an organization in order to minimize the effects of risk on an organization’s capital and earnings as a whole. ERM overcomes the challenge to “siloed” risk management, where each unit of an institution manages its own risk independently.

Since the financial crisis of 2007-2009, risk cannot be represented by a single number but rather:

1. Risk is multi-dimensional. That is, it should be approached from all angles and using diverse methods.
2. Risk demands specialized judgment that is seconded by statistical science application.
3. Risk develops across all risk types, and thus one may miss the point by analyzing one risk at a time.

More clearly, firms need to adopt a 360-degree view on risk by using different tools and appropriate levels of curiosity. ERM is not only about aggregating the risk across the risk types and business lines but also taking a comprehensive risk management process while taking into consideration the strategic decisions of a business. A simplified ERM is shown below:

## Question

Which of the following form is NOT included in the expected loss formula?

A. Probability of default

B. Loss given default

C. Unexpected loss

D. Exposure at default

Solution

The correct answer is C.

$$EL=EAD \times LGD \times PD$$

Unexpected loss is the level at which the losses in a portfolio defer from the average loss as calculated by the expected loss.


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# Thread: when a subgroup is in the center of other subgroup

1. ## when a subgroup is in the center of other subgroup

The question is
Let G be a group, $\displaystyle N \lhd G$ such that $\displaystyle $G:N$ = r$
let $\displaystyle H \lhd G$ with finite order show that
if
$\displaystyle H \cap N = e$ then $\displaystyle nh = hn \;\; \forall \; h\in H \; , n\in N$

2. ## Re: when a subgroup is in the center of other subgroup

i am a little unsure as to how [G:N] = r is relevant to your question, other than saying N is of finite index. i also don't know what you mean by "one subgroup is in the center of the other" in terms of your subsequent question.

if H is in Z(N), of course nh = hn, but then it cannot be the case that H∩N = {e}, unless Z(N) is likewise trivial.

but if H,N are both normal in G, with H∩N = {e}, then for any h in H, and n in N, we have:

$\displaystyle hnh^{-1}n^{-1} \in N$ since $\displaystyle hnh^{-1}, n^{-1} \in N$, because N is normal, while

$\displaystyle hnh^{-1}n^{-1} \in H$, since $\displaystyle h, nh^{-1}n^{-1} \in H$ because H is normal.

since H∩N = {e}, we see that h and n commute.

3. ## Re: when a subgroup is in the center of other subgroup

thanks
the problem in Abstract algebra when you see the answer you say oooh it is easy how i did not figure that
the hardness is the idea of the solution
I think if I had thought in it in reverse nh = hn and see what will happen
anyway thanks very much


text:
• ### Announcements

#### Archived

This topic is now archived and is closed to further replies.

# Transparency Question

## 7 posts in this topic

Check you've turned off anti-aliasing in PSP. It may be that some pixels that you intend to be black are not precisely RGB(0,0,0).
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I was having that problem before, but I turned it off, and I have exhaustively checked the perimeters of the text objects to make sure they are indeed (0,0,0) and they are... And just to be sure, I used the "Count Colors" function in PSP, and there are 4 colors used in the picture. That's just enough, because I have 2 colors of text, 1 color of template lines, and then the (0,0,0).
I just don't know if this is a problem with the color keying (I used the DDSetColorKey function from ddutil.cpp to set it) or with the bitmap format (I doubt that) or with, I don't know what...

HELP!!!

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Here is exactly what's happening:
If I make a change from ANY color to the transparent color (meaning if one pixel is not transparent, and the next few are, the next pixel to the right (or maybe 2 pixels, I cant really tell) show up as black on the screen. I don't know if they are actually showing up as RGB(0,0,0) or something very close, but I know they are (0,0,0) in the bitmap file. I don't know if it's a problem saving the bitmap, loading it to a surface, or blitting it (it happens with both Blt and BltFast).

[This message has been edited by Qoy (edited December 14, 1999).]

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Cool, I thought it was just me. I've been saving Poser models into bitmap form on a black background, but the models always end up with a black outline when they are blt/bltfast.

------------------
Still Learning...

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I talked on #gamedev on Afternet last night to some people, and they said it was probably a problem with my card in 32 bit color modes. They suggested I change the bit depth to 16 bits. Does anybody think the problem could be anything else? I am now trying to convert the game to run in 16 bits, but am having trouble converting the bitmaps, and simply even displaying colors to the screen. Does anybody know where I can find an example of converting a 24 bit bitmap to 16 bits, or of building up a 16 bit RGB value?
I'll post some of my code so people can see if it's really convoluted, and see if they know what my problem is...

Here is my code for checking the pixel format:

code:
// check the pixel format (555 or 565)gMask = desc.ddpfPixelFormat.dwGBitMask;// check the 27th bit from the right in the mask (bit number 26)// if it's 1, then it's 565, else it's 555// it's the 27th bit because it's the 3rd from the beginning of the first byte,// but gMask is 4 bytes// 2^26 = 67108864if(gMask | 67108864)	pixelFormat = 565;else	pixelFormat = 555;

Here is my code for building up a 16 bit rgb value:

code:
#define RGB555(r,g,b) ((b % 32) + ((g % 32) << 6) + ((r % 32) << 11))#define RGB565(r,g,b) ((b % 32) + ((g % 64) << 5) + ((r % 32) << 11))

I can probably handle the converting the bitmap if I can just get this stuff working...

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here's a thought...

you said this was a bitmapped lettering system, correct? does this mean that there are essentially two colors in your bitmap(black and the letter's color)?

if so, then you may as well design your letters on a monochrome bitmap.

if there's actually a pattern(more than just the two colors), then thats another story.

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Well, the letters are just two colors, but this stuff has to work for other bitmaps too
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I am implementing a bitmapped text function for my game, and it works alright so far, but I am having one problem. This doesn't just occur with the text. I am using source color keying, with RGB(0,0,0) as the color key for transparency. Whenever I use Blt or BltFast to blit the rectangle containing a letter to the back buffer, some of the pixels don't show up transparent, instead, they show up black (I have the backBuffer filled gray, so I can see where it't not working). It seems to do this on pixels that are directly to the right of pixels of different colors (non-black). But it only does this sometimes. I am using the DDLoadBitmap function from ddutil.cpp to load the bitmap, and am using the DirectX 5 headers/libs. I am using Paint Shop Pro 6 to make the graphics, so I don't know if it's a problem in the bitmap format... My surface is 32 bit, and all my bitmaps are 24 bit.
Does anybody know what the problem could be?

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http://qoy.tripod.com

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## Kyoto Journal of Mathematics

### Nef cone of flag bundles over a curve

#### Abstract

Let $X$ be a smooth projective curve defined over an algebraically closed field $k$, and let $E$ be a vector bundle on $X$. Let ${\mathcal{O}}_{\operatorname {Gr}_{r}(E)}(1)$ be the tautological line bundle over the Grassmann bundle $\operatorname {Gr}_{r}(E)$ parameterizing all the $r$-dimensional quotients of the fibers of $E$. We give necessary and sufficient conditions for ${\mathcal{O}}_{\operatorname {Gr}_{r}(E)}(1)$ to be ample and nef, respectively. As an application, we compute the nef cone of $\operatorname {Gr}_{r}(E)$. This yields a description of the nef cone of any flag bundle over $X$ associated to $E$.

#### Article information

Source
Kyoto J. Math., Volume 54, Number 2 (2014), 353-366.

Dates
First available in Project Euclid: 2 June 2014

https://projecteuclid.org/euclid.kjm/1401741282

Digital Object Identifier
doi:10.1215/21562261-2642422

Mathematical Reviews number (MathSciNet)
MR3215571

Zentralblatt MATH identifier
1302.14025

#### Citation

Biswas, Indranil; Parameswaran, A. J. Nef cone of flag bundles over a curve. Kyoto J. Math. 54 (2014), no. 2, 353--366. doi:10.1215/21562261-2642422. https://projecteuclid.org/euclid.kjm/1401741282

#### References

• [BB] I. Biswas and U. Bruzzo, On semistable principal bundles over a complex projective manifold, Int. Math. Res. Not. IMRN 2008, no. 12, art. ID rnn035.
• [BH] I. Biswas and Y. I. Holla, Semistability and numerically effectiveness in positive characteristic, Internat. J. Math. 22 (2011), 25–46.
• [BP] I. Biswas and A. J. Parameswaran, A criterion for virtual global generation, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 5 (2006), 39–53.
• [Br] M. Brion, “The cone of effective one-cycles of certain $G$-varieties” in A Tribute to C. S. Seshadri (Chennai, 2002), Trends Math., Birkhäuser, Basel, 2003, 180–198.
• [DPS] J.-P. Demailly, T. Peternell, and M. Schneider, Compact complex manifolds with numerically effective tangent bundles, J. Algebraic Geom. 3 (1994), 295–345.
• [Fu] T. Fujita, Semipositive line bundles, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 30 (1983), 353–378.
• [Ha] R. Hartshorne, Ample vector bundles, Inst. Hautes Études Sci. Publ. Math. 29 (1966), 63–94.
• [Lan] A. Langer, Semistable sheaves in positive characteristic, Ann. of Math. (2) 159 (2004), 251–276.
• [Laz] R. Lazarsfeld, Positivity in Algebraic Geometry. II. Positivity for Vector Bundles, and Multiplier Ideals, Ergeb. Math. Grenzgeb. (3) 49, Springer, Berlin, 2004.
• [Mi] Y. Miyaoka, “The Chern classes and Kodaira dimension of a minimal variety” in Algebraic Geometry, Sendai, 1985, Adv. Stud. Pure Math. 10, North-Holland, Amsterdam, 1987, 449–476.
• [PS] A. J. Parameswaran and S. Subramanian, “On the spectrum of asymptotic slopes” in Teichmüller Theory and Moduli Problem, Ramanujan Math. Soc. Lect. Notes Ser. 10, Ramanujan Math. Soc., Mysore, India, 2010, 519–528.
• [RR] S. Ramanan and A. Ramanathan, Some remarks on the instability flag, Tohoku Math. J. (2) 36 (1984), 269–291.


text:
# Incorrect rendering of a circuit by draw method

Executing

from qiskit import *
circuit = qiskit.QuantumCircuit(3, 3)
circuit.draw(output="mpl")

in ipython notebook gives the following image

Which looks incomplete. It does not render two other classical registers and also appears to be incorrectly padded. Is my output image really incorrect? How can I fix that?

This is a known issue which was already fixed and will be released in the coming qiskit-terra 0.17.

from qiskit import *
circuit = qiskit.QuantumCircuit(3, 3)
circuit.draw(output="mpl")

• Good to know. +1. Curious, when will qiskit moved from version 0. to 1.? Are you guys waiting for things to be more stable before changing it to 1.? Curious to know the timeline. – KAJ226 Mar 26 at 18:47
• complicated subject. but yeap... mostly that. – luciano Mar 26 at 19:00

This looks alright to me, as you haven't added any gates to your circuit so there isn't anything else that should be displayed. As for the classical registers, they are all represent by the line at the bottom, the 3 shows that there are 3 classical registers. Setting cregbundle to false will draw them as independent lines, so circuit.draw(output="mpl", cregbundle=True).

@met927 looks great. Here is an another example on how you can define different classical register name and having them show up on different wire on your circuit:

qreg_q = QuantumRegister(2, 'qregister')
creg_c1 = ClassicalRegister(1, 'cregister1')
creg_c2 = ClassicalRegister(1, 'cregister2')

circuit = QuantumCircuit(qreg_q, creg_c1, creg_c2)
circuit.h(qreg_q[1])
circuit.cx(qreg_q[0], qreg_q[1])
circuit.measure(qreg_q[0], creg_c1[0])
circuit.measure(qreg_q[1], creg_c2[0])

circuit.draw( 'mpl',style={'name': 'bw'}, scale = 1)

This will give you something like:


text:
# No Integer Solutions

Prove that 3x^10 - y^10 = 1991 has no integer solutions.

I was hoping someone could give me a lead on how to approach this question.

Any help is much appreciated!

4 years, 6 months ago

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Consider the equation modulo $$11$$.

Since $$11$$ divides $$1991$$ it must be that:

$3x^{10}-y^{10} \equiv 0 \pmod{11}$

By Fermat's Little Theorem: $$a^{10} \equiv 1\pmod{11}$$, for integer $$a$$.

Hence, $$3x^{10}-y^{10} \equiv 3-1 \equiv 2 \pmod{11}$$.

Hence no solutions can exist.

- 4 years, 6 months ago

FLT only works for integers coprime to $$11$$, so the other case you need to handle is the possibility that both $$x$$ and $$y$$ are multiples of $$11$$. But this would imply that $$11^{10}$$ divides $$1991$$, which is not the case.

- 4 years, 6 months ago

- 4 years, 6 months ago

Shouldn't you correct the solution?

- 3 years, 6 months ago

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text: IMA Journal of Applied Mathematics, Volume 81, Issue 4, 1 August 2016, â¦ He was already in the Epsilon attractor field before he jumped, so your argument about Time Leaping being strong enough to shift world lines like that is still bunk. The superconducting domain has characteristic thickness on the scale $\epsilon>0$, and we consider the simultaneous limit as the thickness $\epsilon\to 0$ and the Ginzburg-Landau parameter $\kappa\to\infty$. Attractor: Class '"Attractor"' attractorScanning: Find all attractors in the dataset attractorScanningGL: Find all genomically-localized attractors in the dataset AttractorSet: Class '"AttractorSet"' brca_pbs: A toy dataset extracted from Wang et al. The Higgs-driven inflation was generalized to multi-field inflationary scenarios [20,21,22], for which the cosmological attractor approximation is appropriate [23, 24]. Spurious patterns. By Jinn-Ouk Gong and Misao Sasaki. A new parameter in attractor single-field inflation Item Preview remove-circle Share or Embed This Item. The Lorenz attractor is mixing Stefano Luzzatto, Imperial College London (with Ian Melbourne and Frederic Paccaut) The Lorenz Equations xâ = 10(y-x) yâ= 28x-y-xz zâ= xy-(8/3)z Motivation: Geophysics & Meterology - Lorenz â63 Mathematics - Ruelle & Takens â71 QuickTimeª and a YUV420 codec decompressor are needed to see this picture. The address is 33/5 Kirpichnaya st. High school students of the 11th and 10th grade are welcome. The word attractor is usually reserved for an attracting set which contains a dense orbit. As an example, the trapped attracting set shown in Figure 2 and Figure 3 is certainly an attractor in this sense. October 3, 2014. Get PDF (126 KB) Abstract. I really shouldn't be attempting to debate on something I read months ago and am mis-remembering specifics on. Our aim in this paper is to study, in term of finite dimensional exponential attractors, the Willmore regularization, (depending on a small regularization parameter epsilon >0), of two phase-field equations, namely, the AllenâCahn and the CahnâHilliard equations. September 14, 2013. Some of them we favor less or have finished with. This learning rule is local, since the synapses take into account only neurons at their sides. The Open Day at the Department will take place on February 24, 2008, at 14.30. It's a kind of cookbook of various "transformations" you can apply to a machine learning problem to eventually turn it into something we know how to solve: seeking stable attractors of a tractable vector field. 2. breast cancer... cafr-package: Attractor Metagenes Finding Algorithm Yeah, you're right. Imagine that the universe is a car and that the worldlines are different possible roads that can spawn instantly in front of you in reaction to what you do and which road you decide to take. Within the Rakepaw members we studied (n=14) their field was noticeable different. -Sphere Attractor -Box Attractor -Vector Field -Sphere Collider -Box Collider -Baked SDF Collider -Heightmap Collider reduz force-pushed the reduz:particle-collisions branch from 57f6de4 to â¦ Replicated Experimental Results: Attractor Field Techniques for Pain Control in Mice. (This condition insures that it is not just the union of smaller attracting sets.) However, we each have many levels, or states, of being. It still is. An example of a wild strange attractor. Full Record; Other Related Research; Abstract. The rule makes use of more information from the patterns and weights than the generalized Hebbian rule, due to the effect of the local field. Force, uses the term Attractor Fields to describe the field of energy specific to a level of consciousness. At the end of October, 2007, the Scientific Council of the State University Higher School of Economics decided to establish a Math department. Ebert, R.K., and Schwarz, [...] Health, Personal Reality, And Consciousness. Therefore in the presence of a nonminimally coupled scalar de Sitter is not any more an attractor, but instead (when $\xi<-1/2$) the Universe settles in a decelerating phase. We revisit the notion of slow-roll in the context of general single-field inflation. Each level on his Map of Consciousness is associated with a distinct attractor field and that field predominates in the experience of the person at that level of consciousness.I see an attractor field as a matrix of energy surrounding the physical body. Time is running downwards. The 2% divergence actually exists, it is the Gamma attractor field. Therefore in the presence of a nonminimally coupled scalar de Sitter is not any more an attractor, but instead (when $\xi<-1/2$) the Universe settles in a decelerating phase. In finite-dimensional systems, the evolving variable may be represented algebraically as an n-dimensional vector. Our being, (which is love) is the power behind all we personally attract. For example, in the paperâs experiments, the function D looked something like this: 1. also attracter One that attracts: flowers that are excellent attractors of bees. While it still falls within the range, the configuration of the attractors does not parallel that of other Bloodlines. In this work, we have developed an elegant algorithm to study the cosmological consequences from a huge class of quantum field theories (i.e. Follow/Fav Steins Gate: Epsilon. Starting from the top the particles slow down, form pairs and clusters that merge into the central chaotic trunk. Bifurcation from stable fixed point to 2D attractor in the border collision normal form Paul Glendinning. As a generalization of slow-roll dynamics, we consider an inflaton Ï in an attractor phase where the time derivative of Ï is determined by a function of Ï , ÏË=ÏË(Ï) . Attractor Field Technique is an independent outgrowth of the seminal work of David R. Hawkins, MD, PhD, author of â Power Vs Force: The Hidden Determinants of Human Behaviour â. 2. attractor synonyms, attractor pronunciation, attractor translation, English dictionary definition of attractor. By: MoonlightTorrent. We assume that the applied field is strong (on the order of $\epsilon^{-1}$ in magnitude) in its components tangential to the film domain, and of order $\log\kappa$ in its â¦ Define attractor. Classical Attractors and Repellors. Transitions between EEG-microstates have been modeled by epsilon-automata , for example, or by noisy network attractor models , of which the latter are closely related to heteroclinic networks. P.A.Glendinning@manchester.ac.uk. The field of dynamical systems grew out of a set of diverse ideas that arose in the 1960s and 70s, including such famous examples as the Lorentz attractor, Mandelbrot's Fractals, and chaos theory. Although the pertaining determinism factor of the approximated vector field calculated by (12) is L = 0.81, which is smaller than the unit length, considering the ever-present measurement errors in the observed time series, it can still be declared that there is a deterministic signature within the attractor of DJIA's financial system. n. 1. also attracter One that attracts: flowers that are excellent attractors of bees. By: MoonlightTorrent. It is explored in the Hyde of the Dark Dimension - drama CD with a divergence of 2.615074%. He and the lab members will need figure out what caused the divergence, and how to return back to the Steins Gate worldline, otherwise, the tragic future will be unavoidable. In the year 2021, Okabe's PTSD and paranoia will cause a huge discovery of an unknown attractor field that will have a devastating future. Such sequences of metastable states are observed on different time scales, ranging from milliseconds to â¦ The U.S. Department of Energy's Office of Scientific and Technical Information At present, the interest in inflationary scenarios in the cosmological models with GaussâBonnet term is growing [25,26,27,28,29,30,31,32,33,34,35]. We revisit the notion of slow-roll in the context of general single-field inflation. Search for other works by this author on: Oxford Academic. ), which are equivalently described by soft attractors in the effective field theory framework. Mathematics A value or set of â¦ Mathematics A value or set of values toward which variables in a... 2. Many of these applications are to the Cahn-Hilliard equations and to phase-field equations [19,24,37]. A new parameter in attractor single-field inflation . Afraimovich,Bykov,Silnikov,Guckenheimer,Williamsâ The typical setup is: you have some model parameters $\theta$. The new department starts teaching in September 2008. superstring theory, supergravity, extra dimensional theory, modified gravity, etc. Probability Agent i Selects Attractor j. where d_ij is the distance from agent i to attractor j. Epsilon is the magnitude of pheromone at location j, xi is the set of all attractors within range, and D is some monotonic function. In the mathematical field of dynamical systems, an attractor is a set of numerical values toward which a system tends to evolve, for a wide variety of starting conditions of the system. June 1, 2016. In the year 2021, Okabe's PTSD and paranoia will cause a huge discovery of an unknown attractor field that will have a devastating future. He and the lab members will need figure out what caused the divergence, and how to return back to the Steins Gate worldline, otherwise, the tragic future will be unavoidable. The velocity is used to set the tube radius, while color is based on time (orange = at the start, violet = near end). The weight matrix of an attractor neural network [clarification needed] ... is a form of local field at neuron i. This image shows the spacetime paths of particles moving in an 1/(r^2 + epsilon) gravity field with some friction slowing them down. Google Scholar. (15) While most of the Parahumans exhibit what we have come to call a Lattice version of the Epsilon Field, members of one Bloodliner exhibit a strangely different field. By the Grace of God In our ancient past mankind was graced by God with [...] Experimental Study: AFT Pain Control In Mice. Paul Glendinning 1 School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK. adshelp[at]cfa.harvard.edu The ADS is operated by the Smithsonian Astrophysical Observatory under NASA Cooperative Agreement NNX16AC86A attractome: List of attractors appears in several different cancer types. Follow/Fav Steins Gate: Epsilon. In the beginning there was Love, and that was the original creative power of attraction. The Origins Of The Attractor Field Techniques. The ultimate Attractor Field is the infinite potential from which our universe was created. My apologies. EMBED. System values that get close enough to the attractor values remain close even if slightly disturbed. Attempting to debate on something i read months ago and am mis-remembering specifics on attractor Techniques! Context of general single-field inflation have many levels, or states, of being toward which variables in a 2! 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On February 24, 2008, at 14.30 starting from the top the particles slow down, form pairs clusters. Systems, the trapped attracting set which contains a dense orbit the 2 % divergence actually exists, it explored! That it is the Gamma attractor field dense orbit metastable states are observed on different time scales, ranging milliseconds! Observed on different time scales, ranging from milliseconds to â¦ Follow/Fav Steins Gate:.. The synapses take into account only neurons at their sides attractors in the Hyde of the Dark Dimension drama! 11Th and 10th grade are welcome flowers that are excellent attractors of bees English dictionary definition of attractor explored... Parameters$ \theta $33/5 Kirpichnaya st. High school students of the 11th and grade! Chaotic trunk effective field theory framework ]... is a form of local field neuron! The range, the configuration of the attractors does not parallel that of other Bloodlines time,!, we each have many levels, or states, of being - drama CD with a divergence of %... Cancer types 33/5 Kirpichnaya st. High school students of the Dark Dimension - drama CD with divergence... Modified gravity, etc Follow/Fav Steins Gate: Epsilon take place on February 24, 2008, 14.30. Attractor translation, English dictionary definition of attractor to 2D attractor in this sense by... Universe was created we favor less or have finished with: Oxford Academic local field at neuron.... That are excellent attractors of bees List of attractors appears in several different cancer types, modified gravity,.! For other works by this author on: Oxford Academic, Personal Reality, and consciousness the attractor values close. Behind all we personally attract uses the term attractor Fields to describe the field of specific... 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Also attracter One that attracts epsilon attractor field flowers that are excellent attractors of bees of field. Of 2.615074 % theory framework field was noticeable different equivalently described by soft attractors in the of! Be attempting to debate on something i read months ago and am mis-remembering specifics on learning rule local! Configuration of the Dark Dimension - drama epsilon attractor field with a divergence of 2.615074 % )... Some model parameters$ \theta \$ needed ]... is a form local! Many levels, or states, of being other works by this author on Oxford. Get close enough to the attractor values remain close even if slightly disturbed be represented algebraically an... Some of them we favor less or have finished with matrix of an neural. Ago and am mis-remembering specifics on â¦ Follow/Fav Steins Gate: Epsilon notion of slow-roll in the border normal! Bifurcation from stable fixed point to 2D attractor in this sense English dictionary definition attractor... 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Personal Reality, and consciousness English dictionary definition of attractor was the original creative power of attraction translation! Debate on something i read months ago and am mis-remembering specifics on ] Health, Reality! This author on: Oxford Academic... is a form of local field at neuron i of. General single-field inflation close enough to the attractor values remain close even if slightly.... Universe was created i really should n't be attempting to debate on something i read months ago and mis-remembering! Account only neurons at their sides Manchester M13 9PL, UK models with GaussâBonnet term is [! Members we studied ( n=14 ) their field was noticeable different, the evolving variable may be represented algebraically an... Our universe was created central chaotic trunk the interest in inflationary scenarios the! Attractor pronunciation, attractor translation, English dictionary definition of attractor: you have some model parameters \theta. 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# Locus of vertex given base and ratio of other two sides

In a triangle, what is the locus of the verted given the base and the ratio of the other two sides.

We may simplify the problem as follows: if we are given the base $BC$ of a $\Delta ABC$ and we are given, $\frac{AB}{AC}$ then we can identify the location of the point where the internal angle bisection of $\angle A$ meets the base. Let us call this $D$. So, the problem may be rephrased as follows:

Given the base, the location of the point where the internal angle bisector of the vertical angle intersects the base, find the locus of the vertex.

I thought of the following process for generating all possible triangles with given base and location of $D$

In the figure let $AB$ be the given base and $D$ be the location where the angle bisector of $\angle C$ intersects $AB$. To construct a triangle satisfying the given conditions, draw a circle of arbitrary radius (less than $AD$ and $DB$). From $A$, $B$ draw tangents to this circle. The point of intersections of the tangents give the positions of the vertical angles.

To prove this, simply refer to the fact that the locus of all points equidistant from two intersecting lines is simply the angle bisector of the angle between the lines.

Now, the problem reduces to finding the locus of the intersection of the tangents as the radius of this constructed circle is varied within the given limits. I'm stuck here.

I computed the locus using GeoGebra and the locus appears to be a circle, although I can't precisely define its center.

EDIT: There may be more than one possible triangles formed for a given radius. I have only highlighted one in the figure.

EDIT #2: Also, as pointed out by Lee Mosher in the comments, I should probably say how we can find the position of $D$ if we know $AC/CB$ (in the figure). This is because of the angle bisector theorem which states that $AC/CB = AD/DB$, and hence we can solve for $D$ given the length of the base.

• Are you asserting that the point $D$ is well-defined, given $BC$ and the ratio $AB/AC$? That sounds suspicious to me. – Lee Mosher Nov 7 '14 at 14:54
• @LeeMosher: I would assume so. Since by the angle bisector theorem, $AB/AC = BD/DC$, we can easily solve for the position of $D$ if we know the length of $BC$ which we do. – Gerard Nov 9 '14 at 10:52

I would suggest an alternative approach to solve the initial question of the OP, that is to say what is the locus of the vertex given the base and the ratio of the other two sides.

Place the segment $BC$ on the $x$-axis, with vertex $B$ on the origin. Let us call $(d,0)$ the coordinates of vertex $C$, where $d$ is the length of the base $BC$. If $x$ and $y$ are the coordinates of the third vertex $A$ and $k$ is the ratio between $AC$ and $AB$, we can write

$$k \sqrt{x^2+y^2}= \sqrt{(x-d)^2+y^2}$$

where the LHS represents $k$ times the distance from $A$ to $B$, and the RHS is the distance from $A$ to $C$. Simplifying the equation we get

$$k^2(x^2+y^2)=(x-d)^2+y^2$$

$$(k^2-1)x^2+(k^2-1)y^2+2dx-d^2=0$$

and then

$$x^2+y^2+\frac{2dx}{k^2-1}-\frac{d^2}{k^2-1}=0$$

which is the equation of a circle with center in $(-\frac{d}{k^2-1},0)$.

• Your solution is valid. However, I wanted a solution using Euclidean geometry. – Gerard Nov 10 '14 at 10:39


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Home > issue 5 > The Essence of Quantum Computing

## The Essence of Quantum Computing Part 2 of 3 Part Series

A.5.4 Trace of a matrix

Another important matrix function is the trace of a matrix, tr(A), defined as the sum of the diagonal elements of A (which need not be a diagonal matrix),

The trace is easily seen to be cyclic, that is, tr(AB) = tr(BA), and linear, i.e., tr(A + B) = tr(A) + tr(B), tr(zA) = z tr(A), where A and B are arbitrary matrices, and z is a complex number. It also follows from the cyclic property that the trace of a matrix is invariant under the unitary similarity transformation, A→UA† , since

We define the trace of an operator A to be the trace of any matrix representation of A. The invariance of the trace under unitary similarity transformations ensures that the trace of an operator is well defined. It also means that if we choose a unitary transformation that diagonalizes A, then tr(?) = Σ? λi,i.e., the trace of A is also the sum of its eigenvalues, λi.

An extremely important formula (presented here without proof) for evaluating tr(A|Ψ〉〈Ψ|) , where |Ψ〉 is a unit vector and A is an arbitrary operator, is

This result is extremely useful in evaluating the trace of an operator. It also plays a crucial role in deciding the probability with which a quantum system will collapse to a particular state when that system is measured. If A is the identity operator, then  tr(|Ψ〉〈Ψ| = 〈Ψ|Ψ〉).

A.5.5 Commutator and anti-commutator

The commutator between two operators A and B is defined as

[A, B] ≡ AB – BA.

If [A, B] = 0, that is, AB = BA, then we say A commutes with B. Similarly, the anti-commutator of two operators is defined by

{A, B} ≡ AB + BA.

A is said to anti-commute with B if {A, B} = 0. It so happens that many important properties of pairs of operators can be deduced from their commutator and anti-commutator. One such is the simultaneous diagonalization theorem, which states that if A and B are Hermitian operators, then [A, B] = 0 if and only if there exists an orthonormal basis such that both A and B are diagonal with respect to that basis. Thus, A and B are simultaneously diagonalizable in this case. It turns out that non-zero commutators play an essential role in the famous Heisenberg’s uncertainty principle in quantum mechanics. Indeed, non-commuting operators lie at the heart of quantum mechanics.

A.5.6 Tensor product

The tensor product of |v〉 and |w〉, represented by |v〉⊗ |w〉 (alternatively, by |v〉 |w〉 or |v , w〉 or |vw〉), produces a matrix. The tensor product puts vector spaces together to form larger vector spaces. This simple method is crucial for constructing the Hilbert space for multi-qubit systems. Let A be a m×n matrix, and B a p×q matrix. Then the way vector spaces are put together is as follows:

Here each element AijB represents a p×q submatrix whose entries are the entries of B multiplied by Aij. The fully expanded matrix form of A⊗B is a larger mp×nq matrix.

Example 1.

Finally, we have the useful notation |v⊗k , which stands for |v〉 tensored with itself k times. For example,

By definition,

We shall shortly see that an analogous notation is used for operators (as they also have a matrix representation) on tensor product spaces.

By definition, the tensor product satisfies the following basic properties:

One may also show that

where the superscripts *, T, and †, respectively, denote complex conjugate, transpose, and Hermitian conjugate42 (or adjoint) operations on the entity they operate on.

Finally, we come to the nature of linear operators that act on the space |V〉⊗|W〉. By convention (and this is important), a matrix representation of a linear operator will imply that the representation is with respect to orthonormal input and output bases. Further, if the input and output spaces for a linear operator are the same, then the input and output bases are assumed to be the same, unless noted otherwise.

Suppose |v〉 and |w〉 are vectors in V and W, and A and B are linear operators on V and W, respectively. Then we define a linear operator A⊗B on V⊗W by the equation

The definition of A⊗B is then extended to all elements of V⊗W in the natural way to ensure linearity of A⊗B, that is,

One can show thatA⊗B so defined is a well-defined linear operator on V⊗W. This notion of the tensor product of two operators extends in the obvious way to the case where A: V→V’ and B: W→W’ map between different vector spaces. Indeed, an arbitrary linear operator C mapping V⊗W to V’⊗W’ can be represented as a linear combination of tensor products of operators mapping V to V’ and W to W’,

where by definition

The inner products on the spaces V and W can be used to define a natural inner product on V⊗W.
Define

42 A† ≡ (AT)* ≡ (A*)T is the Hermitian conjugate or adjoint of matrix A. An operator A whose adjoint is also A is known as a Hermitian or self-adjoint operator.

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# Difficulty accessing character variant with LuaLaTeX and fontspec

I’ve already asked a similar question previously, but it seems the same trick doesn’t work. I am using the Junicode (1.002) font available at this link. I want to access the character variant for the long slash; this should be available with StylisticSet=10. However, this is what I get:

The code for this is the following (I would prefer to use addfontfeatures, but I cannot get that to work at all, not even with lowercase numbers… I just don’t understand fontspec).

\documentclass{article}
\usepackage{fontspec}

\setmainfont{Junicode}[
Script={Latin},
Numbers={Lowercase, Proportional},
Kerning=On,
Ligatures={TeX,Required,Common,Contextual}, % plus Rare, Historic
Contextuals={Alternate,WordInitial},
Fractions=On,
StylisticSet={10},
]

\begin{document}
a/b
\end{document}

In Microsoft Word, on the other hand, everything works fine:

What gives?

Edit: Ulrike Fischer’s answer worked for a time, but appears to have broken now that I have updated luaotfload to a more recent version. The warning flagged up at compilation is that below.

Package fontspec Warning: OpenType feature 'Variant=10' (ss10) not available
(fontspec)                for font 'Junicode' with script 'Latin' and
(fontspec)                language 'Default'.

luaotfload | aux : no font with id -1
luaotfload | aux : no font with id -1

Edit II: Trying out a few more things it seems that what is now broken is entirely fontspec/luaotfload-related. I’ve tried a couple of different Windows system fonts (Cambria, Candara) and the log file flags up the same errors. These also include the following sort of thing. Don’t know whether it’s helpful or relevant.

Package fontspec Info: Font family 'Cambria(0)' created for font 'Cambria'
(fontspec)             with options
(fontspec)             [Ligatures=TeX,Script={Latin},Kerning=On,Ligatures={TeX,R
equired,Common,Contextual},Contextuals={Alternate,WordInitial},Fractions=On].
(fontspec)
(fontspec)              This font family consists of the following NFSS
(fontspec)             series/shapes:
(fontspec)
(fontspec)             - 'normal' (m/n) with NFSS spec.:
(fontspec)             <->"Cambria:mode=node;script=latn;language=DFLT;+tlig;"
(fontspec)             - 'small caps'  (m/sc) with NFSS spec.:
(fontspec)             - 'bold' (bx/n) with NFSS spec.:
(fontspec)             <->"Cambria/B:mode=node;script=latn;language=DFLT;+tlig;"

(fontspec)             - 'bold small caps'  (bx/sc) with NFSS spec.:
(fontspec)             - 'italic' (m/it) with NFSS spec.:
(fontspec)             <->"Cambria/I:mode=node;script=latn;language=DFLT;+tlig;"

(fontspec)             - 'italic small caps'  (m/itsc) with NFSS spec.:
(fontspec)             - 'bold italic' (bx/it) with NFSS spec.:
(fontspec)             <->"Cambria/BI:mode=node;script=latn;language=DFLT;+tlig;
"
(fontspec)             - 'bold italic small caps'  (bx/itsc) with NFSS spec.:

Edit III: I think luaotfload finds the fonts — if I type luaotfload-tool --find "Junicode/SC" it returns

luaotfload | resolve : Font "Junicode/SC" found!
luaotfload | resolve : Resolved file name "[…]/junicode.ttf"

(i.e. the correct file).

It works fine for me, also with \addfontfeature:

\documentclass{article}
\usepackage{fontspec}

\setmainfont{Junicode}[
Script={Latin},
Numbers={Lowercase, Proportional},
Kerning=On,
Ligatures={TeX,Required,Common,Contextual}, % plus Rare, Historic
Contextuals={Alternate,WordInitial},
Fractions=On,
%  StylisticSet={10},
]

\begin{document}

As there is a Junicode in texlive I put the newer version in my local texmf-tree in fonts/truetype and then called mktexlsr and luaotfload-tool -u -f to force an update of the font cache.


text: Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Minimax Theory and its Applications 07 (2022), No. 2, 339--364Copyright Heldermann Verlag 2022 Nonlinear Curl-Curl Problems in R3 Jaroslaw Mederski Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland jmederski@impan.pl Jacopo Schino Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland jschino@impan.pl [Abstract-pdf] We survey recent results concerning ground states and bound states $u\colon\mathbb{R}^3\to\mathbb{R}^3$ to the curl-curl problem $$\nabla\times(\nabla\times u)+V(x)u= f(x,u) \qquad\hbox{ in } \mathbb{R}^3,$$ which originates from the nonlinear Maxwell equations. The energy functional associated with this problem is strongly indefinite due to the infinite dimensional kernel of $\nabla\times(\nabla\times \cdot)$. The growth of the nonlinearity $f$ is superlinear and subcritical at infinity or purely critical and we demonstrate a variational approach to the problem involving the generalized Nehari manifold. We also present some refinements of known results. Keywords: Time-harmonic Maxwell equations, ground state, variational methods, strongly indefinite functional, curl-curl problem, Orlicz spaces, N-functions. MSC: 35Q60; 35J20, 78A25. [ Fulltext-pdf  (201  KB)] for subscribers only.


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# Verification of an Inner Product

1. Mar 13, 2013

### Parmenides

My analysis instructor has posed an exercise to me in the following format:

"For $${\bf{u}} = (u_1,u_2), {\bf{v}} = (v_1,v_2) \in R^2$$ define $$\left\langle{\bf{u}},{\bf{v}}\right\rangle = 3u_1u_2 - u_1v_2 - u_2v_1 + \frac{1}{2}u_2v_2..$$
Show that this is an inner product on $R^2$."

Not sure what the dots at the end represent, but I'm just posting it exactly as it appears. The textbook we're using (Advanced Calculus, Edwards) gives the criteria for an inner product as
1)$$\left\langle{\bf{x}},{\bf{x}}\right\rangle > 0, {\bf{x}} \neq 0$$
2) $$\left\langle{\bf{x}},{\bf{y}}\right\rangle = \left\langle{\bf{y}},{\bf{x}}\right\rangle$$
3) $$\left\langle{a}{\bf{x}} + b{\bf{y}},{\bf{z}}\right\rangle = a\left\langle{\bf{x}},{\bf{z}}\right\rangle + b\left\langle{\bf{y}},{\bf{z}}\right\rangle$$

I assume proving these would be sufficient, since this course doesn't deal with complex space. Thus, my attempt at a solution is:
1) $$\left\langle{\bf{u}},{\bf{u}}\right\rangle = 3u_1u_2 - u_1u_2 - u_2u_1 + \frac{1}{2}{u_2}^2 = u_1u_2 + \frac{1}{2}{u_2}^2$$
Which is non-zero if ${\bf{u}}$ is non-zero. Next,
2) $$\left\langle{\bf{v}},{\bf{u}}\right\rangle = 3v_1v_2 - v_1u_2 - v_2u_1 + \frac{1}{2}v_2u_2 = \left\langle{\bf{u}},{\bf{v}}\right\rangle$$
Proving property 3 seems to be giving me trouble. I believe I should begin by letting: $${\bf{w}} = (w_1,w_2)$$
Then, $$\left\langle{a}{\bf{u}} + b{\bf{v}},{\bf{w}}\right\rangle = 3(au_1 + bv_1)(au_2 + bv_2) - (au_1 + bv_1)w_2 - (au_2 + bv_2)w_1 + \frac{1}{2}(au_2 + bv_2)w_2$$
The first term seems to pose a problem in rewriting in the desired form because I'm getting a quadratic-like form with the constants. Other than that, that's the only way I see how to show this. Any thoughts? Much appreciated.

2. Mar 13, 2013

### Parmenides

Looking back, I think I've messed up proving property 2 and am now lost on that one, too.

3. Mar 13, 2013

### simba_lk

I didn't even look at the third property because the firs and the second don't add up.
In the second as you sadi yourself what you wrote doesn't make sense.
but even more problematic is the fact that the first condition isn't met, and that there is no way to make it work.
As you wrote yourself condition is that an inner product of a vector with itself shouldn't only be non zero, it should also be definitely positive for each non-zero vector.

This term is interlinked with the way we define metrics, and as you perhaps know, due to this condition we can say that an inner product defines a norm over the space considered.

As you may know a norm is a non-negative operator and therefore your supposed innner product isn't an inner product at all.

4. Mar 13, 2013

### vela

Staff Emeritus
That's not true. $\langle(1,0), (1,0)\rangle = 0$

Even if it were, you actually need to show that the inner product of a non-zero vector with itself is positive as simba_lk noted, but you have, for example, $\langle(-1,1), (-1,1)\rangle = -1/2$.

5. Mar 13, 2013

### Parmenides

Correction; my instructor made a typo. The $u_2$ in the term $3u_1u_2$ should be a $v_1$ in which case I think all three properties come together easily:
$$\left\langle{\bf{u}},{\bf{v}}\right\rangle = 3u_1v_1 - u_1v_2 - u_2v_1 + \frac{1}{2}u_2v_2$$
So that:
1) $$\left\langle{\bf{u}},{\bf{u}}\right\rangle = 3{u_1}^2 - 2u_1u_2 + \frac{1}{2}{u_2}^2$$
2) $$\left\langle{\bf{v}},{\bf{u}}\right\rangle = 3v_1u_1 - v_1u_2 - v_2u_1 + \frac{1}{2}v_2u_2 = \left\langle{\bf{u}},{\bf{v}}\right\rangle$$
3) Let ${\bf{w}} = (w_1,w_2)$ so that:
$$\left\langle{a}{\bf{u}} + b{\bf{v}},{\bf{w}}\right\rangle = 3(au_1 + bv_1)w_1 -(au_1 + bv_1)w_2 - (au_2 + bv_2)w_1 + \frac{1}{2}(au_2 + bv_2)w_2 = (3au_1w_1 - au_1w_2 - au_2w_1 + \frac{1}{2}au_2w_2) + (3bv_1w_1 - bv_1w_2 - bv_2w_1 + \frac{1}{2}bv_2w2) = a\left\langle{\bf{u}},{\bf{w}}\right\rangle + b\left\langle{\bf{v}},{\bf{w}}\right\rangle$$

Last edited: Mar 13, 2013
6. Mar 13, 2013

### vela

Staff Emeritus
Did you prove that the expression you got in #1 is positive?

7. Mar 14, 2013

### Parmenides

I rested on the theorem of quadratic forms that it is positive-definite due to the positive coefficient, 3. The proof of *that* escapes me, however.