text:
NC State Extension Publications

# Introduction

The soil samples that determine lime and fertilizer needs of crops routinely come from the top 4 to 8 inches of soil. The results of soil tests help to optimize the purchase of fertilizer, maximize yield, and minimize environmental impact. However, there are times when deep soil sampling is appropriate or necessary.

Deep sampling (28 to 32 inches into the ground) represents a way to monitor the movement of nutrients below the topmost 8 inches of soil. The nutrient phosphorus (P) is of particular interest because it builds up in the soil if overapplied, and it may leach into groundwater, seep into streams, or bind itself to soil particles and be carried to water bodies when soil is washed from fields. Thus, monitoring, via deep soil sampling, may be required as part of the new Phosphorus Loss Assessment Tool (PLAT), which seeks to preserve water quality.

Growers and consultants normally do not take deep samples, so the collection of samples requires that they exercise extreme care. Soil near the surface usually has higher nutrient levels than soil in the deeper strata; because of this, the sampler must avoid contamination of the deep sample by soil from near the surface.

# Sampling Tools

Tool selection depends on price, availability, quality of sample desired, and expected frequency of use (Table 1).

• Hydraulic soil probes usually are mounted on a tractor or truck. The soilcollecting tubes usually have no open face and come in various diameters. If contamination is a major concern, plastic liners can be inserted into the tubes to provide intact cores that can be crosssectioned for easy acquisition of subsamples for specific depths. Although hydraulic soil probes are easy to operate and very effective in removing cores with little contamination from above soil, they also are very expensive and not readily available to the public. Another potential drawback is the difficulty of accessing sampling sites with vehicles when field conditions are wet.
• Soil sample probes are hand-held, single units 18 to 36 inches in length. The lower portion of the probe (12 inches) is an open-faced soil tube with an approximate diameter of ¾ inch. Some probes have screw-on extensions for more versatile sampling by depth. Agronomists and growers commonly use the shorter, singleunit probes to obtain the routine shallow soil samples used for fertility and nematode analyses.
Probes are usually pushed into the ground by hand, using necessary body weight. Quality samples can be easily obtained when soils are moist. When soils are dry and there is significant resistance, the probe may be hammered into the ground. It may be difficult to take samples in dry, single-grain sands due to their unstable, shifting nature. Soil sample probes are available through mail order from outdoor or forestry suppliers at a modest cost, usually $50 to$100.
• Soil augers with various handle lengths and extensions for deep sampling are also available through mail-order suppliers, but they are more expensive than soil probes. They are generally easy to use except where rocks are abundant. The auger itself is about 12 inches long and comes in various diameters. Its sides may be open or completely closed (bucket augers). Planing augers provide clean excavation of soil from a hole’s bottom.
Soil extracted with augers comes up in cores due to the twisting and cutting action of the auger, so care is essential in verifying the exact depth of sampling. Since several borings are necessary for sampling deeper depths, contamination may occur from soil sloughing into the hole upon inserting the auger or from side-wall contamination when removing an auger with open sides. Sampling dry, sandy, single-grained soils may be difficult with an auger.
• Post-hole diggers consist of two metal blades bridged together with wooden handles about 4 feet long. They are inexpensive and commonly available. Sampling is usually easy except in dry soil conditions or when soil is compacted. There is some potential for contamination due to soil sloughing. Samples, however, will not be in cores, so depths are best determined by inserting a measuring tape into the hole.

 Tool Cost & Accessibility Ease of Sampling Quality of Sample Hydraulic probe ✔ ✔✔✔✔✔ ✔✔✔✔✔ Soil probe with extensions ✔✔✔✔✔ ✔✔✔ ✔✔✔✔ Soil auger ✔✔✔✔ ✔✔✔ ✔✔✔ Post-hole digger ✔✔✔✔✔ ✔✔✔ ✔✔✔ * The higher the number of checkmarks (✔), the higher the rating of the tool.

# Suggested Deep Sampling Technique for PLAT

A site with high P leaching potential, as determined by PLAT, requires a soil sample from a depth of 28 to 32 inches. The composite sample submitted for analysis should be taken from a mixture of at least five cores to ensure that it is representative of the site. To minimize contamination and acquire a quality sample, follow these steps:

1. Using a post-hole digger or auger at least 2 inches in diameter, excavate to 26 inches. Clean as much soil as possible from the bottom of the hole. Use a measuring tape to measure the depth at the center of the hole.
2. Use a standard soil probe at least 36 inches in length to collect a 6- inch soil core from the center of the hole. Measure the top 2 inches and discard. Place the remaining 4-inch core in a clean plastic bucket.
3. Collect at least four additional cores using the same technique.
4. Thoroughly mix the cores into a composite sample. Mixing is extremely important for best characterization of the site.
5. Take a surface sample to further confirm elevated P levels as found in the initial site characterization. Cores should be taken from areas coinciding with deep sampling.
6. Submit samples to a laboratory that is certified by the NC Department of Environment and Natural Resources, Division of Water Infrastructure. Request Mehlich-3 methodology and results as a phosphorus index (P-I) as routinely performed by the NC Department of Agriculture & Consumer Services (NCDA&CS) Agronomic Division, Soil Testing Section.

For more information concerning deep sampling, contact the NCDA&CS Agronomic Division (919-733-2655) or the county offices of any of these agencies: NCDA&CS Division of Soil and Water Conservation, NC Cooperative Extension, or US Department of Agriculture Natural Resources Conservation Service (NRCS). Direct PLAT inquiries to NRCS at 919-873-2100.

# Reference and Acknowledgments

North Carolina Phosphorus Loss Assessment Tool

This publication is a revision of an earlier version. The authors would like to thank R. Reich for his earlier contributions.

# Authors

Assistant Professor & Extension Specialist - Nutrient Management & Animal Waste
Crop & Soil Sciences
Department Extension Leader (Nutrient Mgt and Water Quality)
Crop & Soil Sciences
Section Chief, Soil Testing, Agronomic Division
NC Department of Agriculture & Consumer Services

Publication date: May 6, 2014
AG-439-40

North Carolina State University and North Carolina A&T State University commit themselves to positive action to secure equal opportunity regardless of race, color, creed, national origin, religion, sex, age, veteran status or disability. In addition, the two Universities welcome all persons without regard to sexual orientation.


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# Tag Info

## Hot answers tagged reference-frames

28

Velocity is relative. There is no special reference frame that would be "at rest". But acceleration is not and was never claimed to be. Reference frames in free fall are special and reference frames that are accelerating relative to the ones in free fall contain inertial forces (circular motion involves acceleration towards the centre; the corresponding ...

18

Because in the frame of reference that is co-rotating, the object doesn't move, and therefore it has no kinetic energy in that frame, which is the frame in which most problems involving objects on earth are looked at. Note that kinetic energy is evidently not a frame-invariant quantity, but it is not required to be.

10

Special relativity deals with "inertial" or "non-accelerating" frames. Physics in inertial frames are equivalent independent of their velocity and the velocity of inertial frames are relative. You are free to assume any inertial frame is stationary and all other frames are moving relative to it. Rotating frames are not inertial, they are accelerating ...

9

In general relativity, angular motion actually does have some "relativity" to it as well. When you're in close proximity to a spinning object, you'll actually be dragged along with it. This is known as the Lense-Thirring effect, or just "frame-dragging". The most dramatic example is the ergosphere of a spinning black hole, a region where no object can remain ...

8

On the quantum level, force is not acceleration. The concept of "fictitious force" makes no sense on a QFT level, because forces are interactions between quantum states, not the classical forces you might imagine. Quantum forces are not vector fields in space. The notion of "fictitious force" would mean that, e.g., the strong force is something influencing ...

8

The classical theory of electrodynamics can indeed be written as a geometrical theory in a similar way to general relativity. As it happens there is a question and answer addressing just this, but it's in the Maths SE: Electrodynamics in general spacetime. Classical electrodynamics is an example of a class of theories called classical Yang-Mills gauge ...

6

Here's a simple demonstration: Consider flat space (i.e. Minkowski), viewed in a rotating frame (in e.g. cylindrical coordinates one just replaces $\phi$ by $\phi'=\phi+\omega t$). One can calculate (without too much trouble) that, in these coordinates, a spatial line element can be expressed in terms of the canonical cylindrical coordinates as  ...

5

The exact quantities of kinetic energy (like momentum) depend on your choice of a reference frame. Don't get too worried though; regardless of your choice of a reference frame, you will find that energy (and likewise momentum) is conserved within the reference frame. Therefore, two observers may not agree on the kinetic energy or momentum of an object, but ...

5

General covariance applies only to freely falling observers -- once you invoke non-gravitational forces, like the inward pressure of the wall, the observer is no longer freely falling.

4

is it possible to consider also the other fundamental forces [...] to be fictitious forces like gravity in the framework of general relativity? No, because the equivalence principle only holds for gravity. If we want a final unification of all fundamental forces, hasn't this feature of gravity to become a feature of the other forces as well? The ...

3

Are Lorentz transformations more adequate representations of motion, than the more intuitive velocities? Yes. The non-associativity that bothers you simply arises because there is no group of three dimensional boosts. Confined to one dimension, boosts form a rather lovely one parameter subgroup of the Lorentz group $SO^+(1,3)$. So everything "works", ...

3

Easy way to distinguish between gravity and rotating space station: Throw a ball straight up in the air. If it comes straight down, gravity. If it moves away from you (behind your tangential velocity), it's a rotating space station.

3

The site rules forbid us from giving the answers to homework problems, but this problem illustrates a fundamental issue in relativity so I think it's worth some general comments. Incidentally you may be interested to read the Wikipedia article on the ladder/barn paradox, though in it's efforts to be comprehensive I think the article gets a bit confusing. ...

3

The Coriolis force $\vec F_{\text{coriolis}} = -2m \, \vec \omega \times \vec v$ only depends on velocity. The centrifugal force $\vec F_{\text{centrifugal}} = -m \, \vec \omega \times (\vec \omega \times \vec r)$ only depends on position. Finally, if the object is not rotating uniformly ($\dot {\vec \omega} \ne 0$), then yet another fictitious force comes ...

3

If the occupants of the space station were not aware of its design and could not look out a window then there is no way to tell if it is rotating or they are near a earth size planet that causes the gravity. Orbiting around another space station will causes a sensation of gravity, and it seems you are contradicting yourself. If there is any rotational ...

2

I see what you're trying to ask. Let me try to rephrase it: Does an observer at the bottom of a massive gravity well perceive that the clocks of actors outside of the gravity well move faster? The answer is yes, but the intensity depends on the depth of the gravitational well. For anyplace far away from exotic things like event horizons and perhaps neutron ...

2

The discussion is mostly semantic. They are both calculated relative to a point, in the case of the torque the point has the additional meaning that if you put an axle trough the point, the object will start to rotatte around it if the net torque is not zero. It happens also that the torque will be the same if you chose any other point along the axis ...

2

My guess would be that $\mathbb E^n$ denotes Euclidean space. In addition to having geometric structure (angles and distances) and motions (rotations, translations, reflections) - not all of it terribly useful in the 1-dimensional case - it is an affine space. Affine spaces have no notion of distinguished origin or zero point. We can use a vector space like ...

2

Your first sentence is not true. There is a whole family of freefall frames that are co-incident at any spacetime event. They correspond to different "initial velocities" as they diverge from that event: in geometric language, their origins follow the many different geodesics defined by different tangent vectors of the tangent space at that spacetime event, ...

2

A photon cannot be said to have its own inertial reference frame, because inertial reference are defined to be a family of coordinate systems that satisfy the two fundamental postulates of SR, one of which is that light moves at c in all frames. You could construct a coordinate system where the photon was at rest, but since this coordinate system wouldn't be ...

1

As pointed out you can't travel at the speed of light but you can look at the limits we are tending towards as we approach it. So, if I were to travel in a spacecraft at the speed of light, would I freeze and stop moving? From the perspective of a stationary observer if your spacecraft was traveling at close to the speed of light, time on the ...

1

How to understand non-associative composition of velocities in STR? Special relativity introduces a weirdness about how your axes can be related to other observers' axes: if your axes are aligned with observer A's axes and theirs are aligned with observer B then special relativity (i.e. the Lorentz transformations) say that B's axes will be rotated with ...

1

Tilting an object in space changes its apparent dimension (think of trying to get furniture through a door: the width of an object depends on its orientation). Objects in relative motion are tilted in space and time (or rather, spacetime), and different observers will see things unfold under different perspectives. Personally, I find relativity of ...

1

Have a look at my answer to Time Dilation Effects from simply being on a spinning planet orbiting a star in a rotating galaxy in an expanding universe.. Compared to an observer far from the Earth, time at the Earth's surface runs more slowly by a factor of 0.9999999993. Over a 70 year lifespan this makes a difference of about 1.5 seconds. So the man in ...

1

There are two types of time dilation: dilation caused by being near a large body, and dilation caused by traveling very fast relative to another observer. Relativistic time dilation plays a bigger role for astronauts aboard a space station similar to the ISS. Even though velocity and gravity produce opposite time dilation, in this scenario, time dilation due ...

1

Kinetic energy is the energy associated with motion. Therefore, we can no more measure kinetic energy free of a reference frame than we can say something is moving free of a reference frame. It must always be moving with respect to something. Energy is conserved because the total energy of a system is the same before and after a process in the same frame of ...

1

This frame is exist. You got wrong result because you ignored that this two photon move in the opposite direction. Set that the first photon move along the z axis and the second photon move against z-axis.$\omega_1$ and $\omega_2$ are the frequency of the first and the second photon correspondingly in the reference frame. In new frame shout be $k'_1=-k'_2$ ...

1

Look in Wikipedia http://en.wikipedia.org/wiki/Coriolis_effect. For understanding intuitively the Coriolis force effect, assume an object moving according to a static (inertial) frame of reference, in the plane perpendicular to the rotation axis, and along the radius, In the rotating frame, see the animation in Wikipedia, the Coriolis force imposes an ...

1

I was doing a question about if a train fits in a tunnel. Did the question assignment include a specific consistent definition of what's meant there by "to fit", in the first place? Presumably, in the setup which is typically considered, the ends of the tunnel (say participants $A$ and $B$) are supposed to be at rest to each other, the ends of the ...

Only top voted, non community-wiki answers of a minimum length are eligible

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# Math Revolution Free Diagnostic Test (\$19 worth)

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# GMAT Question of the Day (July 18)

- Jul 18, 02:00 AM   Comments [0]

Math and are points on a circle of radius and is the center of the circle. If the length of arc is 0.8, what is the value of angle $OAB$

# GMAT Question of the Day (July 17)

- Jul 17, 02:00 AM   Comments [0]

Math If and are non-negative integers, what is the units digit of ? (1) (2)   Question Discussion & Explanation Correct Answer - B - (click and drag your mouse to see the answer) Verbal Plants are more efficient at acquiring...

# GMAT Question of the Day (July 14)

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Tighten your grip on core concepts in the days leading up to the GMAT. If my husband and I move something heavy and have to pause along the way, he tells me, “Hold what you’ve got.” Reaching for more instead of focusing on what you currently...

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Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
 Already signed up? New to KöMaL?

# Problem K. 561. (November 2017)

K. 561. A novel was published in three volumes. The page numbering started with 1 in the first volume, and in the second and third volumes it continued where the previous volume ended. The second volume was 50 pages thicker than the first one, and the third volume was 1.5 times as thick as the second volume. The sum of the first page numbers in the three volumes was 893. How many pages long is the entire novel? How many digits were used altogether in numbering the pages?

(6 pont)

Deadline expired on December 11, 2017.

Sorry, the solution is available only in Hungarian. Google translation

Megoldás. Ha $\displaystyle x$ oldalas egy-egy könyv, akkor az első oldalszámok rendre $\displaystyle 1$, $\displaystyle x+1$ és $\displaystyle 2x+51$. Tehát $\displaystyle 3x+53 = 893$, innen rendezéssel az $\displaystyle x=280$-at kapjuk. Tehát az első könyv $\displaystyle 280$, a második $\displaystyle 330$, a harmadik pedig $\displaystyle 1,5\cdot330=495$, így a regény $\displaystyle 1105$ oldalas. A számjegyek száma:

$\displaystyle 9\cdot1+(99-10+1)\cdot2+(999-100+1)\cdot3+(1105-1000+1)\cdot4=$

$\displaystyle =9\cdot1+90\cdot2+900\cdot3+106\cdot4=9+180+2700+424=3313.$

### Statistics:

 144 students sent a solution. 6 points: 75 students. 5 points: 9 students. 4 points: 6 students. 3 points: 44 students. 2 points: 7 students. 1 point: 2 students. 0 point: 1 student.

Problems in Mathematics of KöMaL, November 2017


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# Our Next Trip to Integer Partitions

Published on 18 Sep 2021 by Susam Pal

## Introduction

After 114 meetings and 75 hours of studying together, our analytic number theory book club has finally reached the final chapter of the book Introduction to Analytic Number Theory (Apostol, 1976). We have less than 18 pages to read in order to complete reading this book. Considering that we read about 2-3 pages in every meeting, it appears that we will complete reading this book in another 2 weeks.

Reading this book has been quite a journey! The previous three blog posts on this blog provide an account of how this journey has been. It has been fun, of course. The best part of hosting a book club like this has been the number of extremely smart people I got an opportunity to meet and interact with. The insights and comments on the study material that some of our book club participants shared during the meetings were very helpful. Special thanks to Andrey who goes by the Libera IRC nick halyavin for joining most of our meetings and sharing his explanations on various steps of the proofs we came across in the book.

The meeting log shows that our book club started really small with only 4 participants in the first meeting in March 2021 and then it gradually grew to about 10-12 regular members within a month. Then a few months later, the number of participants began dwindling a little. This happened because some members of the book club had to drop out as they got busy with other personal or professional engagements. However, six months later, we still have about 4-5 regular participants meeting consistently. I think it is pretty good that we have made it this far.

## Unrestricted Partitions

The final chapter on integer partitions is very unlike all the previous 12 chapters. While the previous chapters dealt with multiplicative number theory, this final chapter deals with additive number theory. For example, the first theorem talks about an interesting property of unrestricted partitions. We study the number of ways a positive integer can be expressed as a sum of positive integers. The number of summands is unrestricted, repetition of summands is allowed, and the order of the summands is not taken into account. For example, the number 3 has 3 partitions: 3, 2 + 1, and 1 + 1 + 1. Similarly, the number 4 has 5 partitions: 4, 3 + 1, 2 + 2, 2 + 1 + 1, and 1 + 1 + 1 + 1.

I have always wanted to learn about partitions more deeply, so I am quite happy that this book ends with a chapter on partitions. The subject of partitions is rich with very interesting results obtained by various accomplished mathematicians. In the book, the first theorem about partitions is a very simple one that follows from the geometric representation of partitions. Let us see an illustration first.

How many partitions of 6 are there? There are 11 partitions of 6. They are 6, 5 + 1, 4 + 2, 4 + 1 + 1, 3 + 3, 3 + 2 + 1, 3 + 1 + 1 + 1, 2 + 2 + 2, 2 + 2 + 1 + 1, 2 + 1 + 1 + 1 + 1, and 1 + 1 + 1 + 1 + 1 + 1. Now how many of these partitions are made up of 5 parts? Each summand is called a part. The answer is 2. There are 2 partitions of 6 that are made up of 5 parts. They are 3 + 1 + 1 + 1 and 2 + 2 + 1 + 1. Let us represent both these partitions as arrangements of lattice points. Here is the representation of the partition 3 + 1 + 1 + 1:

• • •
•
•
•

Now if we read this arrangement from left-to-right, column-by-column, we get another partition of 6, i.e., 4 + 1 + 1. Note that the number of parts in 3 + 1 + 1 + 1 (i.e., 4) appears as the largest part in 4 + 1 + 1. Similarly, the number of parts in 4 + 1 + 1 (i.e., 3) appears as the largest part in 3 + 1 + 1 + 1. Let us see one more example of this relationship. Here is the geometric representation of 2 + 2 + 1 + 1:

• •
• •
•
•

Once again, reading this representation from left-to-right, we get 4 + 2, another partition of 6. Once again, we can see that the number of partitions in 2 + 2 + 1 + 1 (i.e., 4) appears as the largest part in 4 + 2, and vice versa. These observations lead to the first theorem in the chapter on partitions:

Theorem 14.1 The number ofpartitions of $$n$$ into $$m$$ parts is equal to the number of partitions of $$n$$ into parts, the largest of which is $$m$$.

That was a brief introduction to the chapter on partitions. In the next two or so weeks, we will dive deeper into the theory of partitions.

## Next Meeting

If this blog post was fun for you, consider joining our next meeting. Our next meeting is on Tue, 21 Sep 2021 at 17:00 UTC. Since we are at the beginning of a new chapter, it is a good time for new participants to join us. It is also a good time for members of the club who have been away for a while to join us back. Since this chapter does not depend much on the previous chapters, new participants should be able to join our reading sessions for this chapter and follow along easily without too much effort.

To join our book club, see our channel details in the home page here. To get the meeting link for the next meeting, visit the analytic number theory book club page.

It is worth mentioning here that lurking is absolutely fine in our meetings. In fact, most members of the club join in and stay silent throughout the meeting. Only a few members talk via audio/video or chat. This is considered absolutely normal in our meetings, so please do not hesitate to join our meetings!


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# An algebra problem by Ankit Nigam

Algebra Level 3

The roots of the equation $${ z }^{ n }={ \left( z+1 \right) }^{ n }$$ (z is any complex number)

×


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# Newton Applications. Help solving equation with sin and cos.

## Homework Statement

At an airport, you pull a 18-kg suitcase across the floor with a strap that is at 45 degrees above the horizontal. Find the normal force and the tension on the strap, given that the suitcase moves with a constant speed and the coefficient of kinetic friction is 0.38.

$$\mu$$k = 0.38
m = 18kg
$$\sum$$F = 0 (because speed is constant)
angle = 45
N = ?
T = ?

## Homework Equations

Fcos($$\theta$$) - Ff = 0
Ff = mg - Fsin($$\theta$$)($$\mu$$k)

## The Attempt at a Solution

Fcos($$\theta$$) - [mg - Fsin($$\theta$$)($$\mu$$k)]
Fcos(42) - [(18)(9.81) - Fsin(42)](0.38) = 0

I'm trying to solve for F but I don't know how to get rid of, solve equations with cos and sin.
Help is appreciated!

## The Attempt at a Solution

#### Attachments

• 231.1 KB Views: 560

Related Introductory Physics Homework Help News on Phys.org
you just need to evaulate cos(42) and sin(42) using your calculator. Be very careful with roundoff error (don't round too much too soon), and make sure your calculator is in degrees mode (if the problem is stated in degrees.. most intro physics problems are though)

so Fcos(45) turns to .707107F?

Yeah, but $$\cos(45^\circ) = \frac{\sqrt{2}}{2}$$, so maybe you can leave it like that until the very end, then you can approximate with decimals. It'll save you the hassle of round-off error. To be honest, it won't make much difference in this case, especially since it looks like you're rounding out to 6 places anyway, but it's not a bad idea in general.

but the short answer is yes.

I solved the equation,

F = 68.763N
then i found normal force
N = 127.957N

I have to find the tension on the strap, but i don't completely understand how to do that. Could someone explain the concept or how to find it?

The question I have is should I add Force and Force of Friction since you have to overcome Force of Friction?
But Force is on an angle, so does that mean i have to find Fx and then add them, and then combine Fx and Fy?

HallsofIvy
Homework Helper
The horizontal component of force must be equal to the friction force so that there is no acceleration or deceleration.

Okay, so how do I find tension on the strap?

Doc Al
Mentor
Okay, so how do I find tension on the strap?
Set up two equations, one for vertical forces and one for horizontal force. You'll have two equations and two unknowns (Normal force and Tension)--which works out just right.

I already know the normal force.
I was trying to figure it out by myself and thought if i did this:
Ty = 68.763sin(45)
Tx = 68.763cos(45) + (127.957) (0.38)
solve and combine x and y components to have the tension? not sure...at all.

Doc Al
Mentor
I was trying to figure it out by myself and thought if i did this:
Ty = 68.763sin(45)
Looks to me like you already know the tension, if you're able to find it's y component. This equation says Ty = T sin(45) = 68.76 sin(45)

The 68.783n is not the tension, its the force of the pull. Or is it the same thing?
If it is the same thing, i have to add the force of friction to the x component of force since thats working against the movement?

Doc Al
Mentor
The 68.783n is not the tension, its the force of the pull. Or is it the same thing?
The force with which the strap is pulled is the tension.
If it is the same thing, i have to add the force of friction to the x component of force since thats working against the movement?
The sum of the forces in the x-direction will equal zero, if that's what you mean.

Okay, thanks Doc Al!
Sorry, my teacher is pretty confusing sometimes.


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# Calculate the electric field in all of space for a non conducting sphere

## Homework Statement

A) Use Gauss's Law to derive the electric field in all space for a non-conducting sphere with volumetric charge distribution ρ=ρ0r3 and radius, R.
B) Repeat when there is a concentric spherical cavity within the non conducting sphere with radius, A.

## Homework Equations

∫E⋅dA=Qenclosed0
ρ=Q/V (charge density)

## The Attempt at a Solution

Im struggling with part B more than A, but since B uses part A's answer (in a way) I will show both my attempts

A) I drew a picture (sphere.jpg attached) and will use words to explain what Im doing.
My Gaussian Surface has radius, A.
I will be finding dQ of a small spherical piece of the sphere, the radius of this small spherical piece is, r, and has thickness, dr.

First find dQ
we know that ρ=Q/V →dQ=ρdV since ρ=ρ0r3 and dV=(4πr2)(dr) lets plug those into our dQ:
dQ=(ρ0r3)(4πr2)(dr)

We can now find our Qenclosed of gaussian surface.
My below integral's limits go from 0→a (a being the radius of my gaussian surface)
Qenclosed=∫dQ=∫ (ρ0r3)(4πr2)(dr)=4πρ0∫r4→once evaluated= (4πρ0a5)/5 (this is our Qenclosed)
Now we can use gauss's law to find electric field
E(4πa2)=(4πρ0a5)/(5ε0)
and finally:
E=(ρa3)/(5ε0)→E(r)=ρr3)/(5ε0)r(hat)
This is the answer if you are inside the sphere at some distance r<R

Now for outside the sphere:

We can go to the Qenclosed and just change the limits, instead of integrating from 0→a, we can integrate from 0→R. If we do this our Qenclosed becomes:

Qenclosed=(4πρ0R5)/5

My new gaussian surface has radius a'

E(4πa'2)=(4πρ0R5)/(5ε0)
solving for E:
E=(ρ0R5)/(5a'2ε0)

Now for part B

Again, new limits for Qenclosed, the limits now are going from A→a
our new Qenclosed=[(4πρ0a5)/(5)]-[(4πρ0A5)/(5)]
and our new E=[(ρ0a3)/(5ε0)]-[(ρ0A5)/(5a2ε0)

and outside our Qenclosed=[(4πρ0R5)/(5)]-[(4πρ0A5)/(5)]
and our new E=[(ρ0R5)/(5a2ε0)]-[(ρ0A5)/(5a2ε0)

Hope that was clear enough, please ask for more steps if Im wrong.
thanks!

#### Attachments

• Sphere.jpg
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BvU
Homework Helper
Hello,
Qenclosed=∫dQ=∫ (ρ0r3)(4πr2)(dr)=4πρ0∫r4→once evaluated= (4πρ0a5)/5
How do you do this ?


text:
## Elementary Algebra

Each piece of fuel line will be $\frac{23}{8}$ feet long.
The piece of fuel line is 5$\frac{3}{4}$ feet long, or $\frac{23}{4}$ feet long. He needs to cut it into 2 equal pieces. So, we divide $\frac{23}{4}$ by 2: $\frac{23}{4}$ $\div$ 2 = $\frac{23}{4}$ $\times$ $\frac{1}{2}$ = $\frac{23}{8}$ Each piece of fuel line will be $\frac{23}{8}$ feet long.


text:
# Why is Frobenius norm related to the inner product of characters?

This is a continuation of my question asked here.

I am going through Normal Subgroup Reconstruction and Quantum Computation Using Group Representations. In Definition 2, the authors start with the following function.

$$f : G \to \mathbb{C}$$

Then the Fourier transform of $f$ at the irrep $\rho$ is defined as the $d_\rho \times d_\rho$ matrix:

$$\hat{f}(\rho) = \sqrt{\frac{d_\rho}{|G|}} \sum_{g \in G} f (g) \rho(g)$$

In the quantum computational settings, the authors identify the superposition $\sum_{g \in G} f_g |g\rangle$ ($|a\rangle$ is a quantum state which is the $a$-th computational basis vector), with the function $f:G\to\mathbb{C}$ defined by $f(g) = f_g$.

In this notation, $\sum_{g \in G} f(g)|g\rangle$ is mapped under the Fourier transform to $\sum_{\rho \in \hat{G}, 1\le i,j \le d_\rho} \hat{f}(\rho)_{i,j} |\rho, i, j\rangle$.

$(\rho, i, j)$ is the label of the $(\rho, i, j)$-th computational basis vector $|\rho, i, j\rangle$. $\hat{f}(\rho)_{i,j}$ is a complex number and the $(i,j)$-th element of the matrix $\hat{f}(\rho)$. $\hat{G}$ is the collection of all irreps of $G$.

Then the authors say as follows.

When the first portion of this triple is measured, we observe $\rho \in \hat{G}$ with probability

$$\sum_{1\le i,j\le d_\rho} |\hat{f}(\rho)_{i,j}|^2 = ||\hat{f}(\rho)||^2\\ = tr ((\hat{f}(\rho))^* \hat{f}(\rho))$$

Here, ($\rho, i, j$) is the triple the authors are referring to. Measuring $\rho$ means measuring the qubits which encode the label $\rho$.

I understand up to this part. Then in Section 3, right after Equation 2, the authors go on saying:

$$||\hat{f}(\rho)||^2 = ||\sqrt{\frac{d_\rho}{|G|}} \sum_{h\in H} \rho(h)||^2$$

I know that $\rho$ may not be an irrep for $H$. But, the matrix elements in $\hat{f}(\rho)$ comes from the direct sum of the matrices $\rho(h)$. I assume the previous equation works because of the Equation (2) in the paper.

I do not understand from here. The equation written by the authors is as follows:

$$||\hat{f}(\rho)||^2 = ||\sqrt{\frac{d_\rho}{|G|}} \sum_{h\in H} \rho(h)||^2\\ = \frac{d_\rho}{|G|} \frac{1}{|H|} |H|^2 \langle \chi_\rho, \chi_{1_H} \rangle_H \\= \frac{|H|}{|G|} d_\rho \langle \chi_\rho, \chi_{1_H} \rangle_H$$

Here, $\chi_{1_H}$ is the character of the trivial representation of the subgroup $H$.

My question:

Why is $||\sum_{h\in H} \rho (h)||^2 = \frac{1}{|H|} |H|^2 \langle \chi_\rho, \chi_{1_H} \rangle_H$? According to the definition of the Frobenius Norm, it should be like $||\sum_{h\in H} \rho (h)||^2 = Tr \left(\left(\sum_{h\in H} \rho (h)\right)\left(\sum_{h\in H} \rho (h)\right)^\dagger\right)$. I do not understand the following relationship.

$$Tr \left(\left(\sum_{h\in H} \rho (h)\right)\left(\sum_{h\in H} \rho (h)\right)^\dagger\right) = \frac{1}{|H|} |H|^2 \langle \chi_\rho, \chi_{1_H} \rangle_H$$

It's important that on the first page the paper states that, without loss of generality, all of their irreducible representations are unitary, so that $\rho(g)^\dagger=\rho(g)^{-1}$ for all $g\in G$.

By the way, the beginning of section $3$ is in error. The sum $\sum_{h\in H}\rho(h)$ is not a projection, because it is not normalized correctly. Instead, $|H|^{-1}\sum_{h\in H}\rho(h)$ is the projection:

$$\left(\frac{1}{|H|}\sum_{h\in H}\rho(h)\right)^2=\frac{1}{|H|^2}\sum_{h_1,h_2\in H} \rho(h_1 h_2)=\frac{1}{|H|}\sum_{h\in H}\rho(h)$$

because $h_1h_2=h$ has $|H|$ solutions $(h_1,h_2)\in H\times H$.

Next, I am not sure why the bottom of the page introduces $\sqrt{d_\rho/|G|}$. According to the definition of the Fourier transform on the previous page, one introduces this constant when transforming from an initially given function $f:G\to\mathbb{C}$ to $\widehat{f}$. However section $3$ begins by defining $\widehat{f}$ to be a certain projection operator (at least it wanted to it seems), so there's no need for that constant.

Thus, let's just talk about the projection map $\widehat{f}(\rho)=|H|^{-1}\sum_{h\in H}\rho(h)$. We have

$$\begin{array}{ll} \|\widehat{f}(\rho)\|^2 & \displaystyle =\left\|\frac{1}{|H|}\sum_{h\in H}\rho(h)\right\|^2 \\ & \displaystyle =\frac{1}{|H|^2}\mathrm{tr}\left(\sum_{h_1\in H}\rho(h_1)\right)\left(\sum_{h_2\in H}\rho(h_2)\right)^\dagger \\ & \displaystyle =\frac{1}{|H|^2}\mathrm{tr}\sum_{h_1,h_2\in H}\rho(h_1h_2^{-1}) \\ & \displaystyle =\frac{1}{|H|}\sum_{h\in H}\chi(h), \end{array}$$

again because $h_1h_2^{-1}=h$ has $|H|$ solutions $(h_1,h_2)\in H\times H$. The above is $\langle \chi,\chi_{1_H}\rangle_H$.

The inner product $\langle \chi,\psi\rangle_H=\frac{1}{|H|}\sum_{h\in H}\chi(h)\overline{\psi(h)}$ on characters can be used to sift for the multiplicities of irreps within a given rep. Suppose that $V$ is a complex representation of $H$, and that $V\cong \bigoplus_{X\in\widehat{H}}X^{\oplus m(X)}$ is $V$'s decomp into irreps $X\in\widehat{H}$ with multiplicities $m(X)$. Then for a fixed irrep $Y$, because $\hom$ distributes over $\oplus$, we have

$$\begin{array}{ll} \hom_H(V,Y) & \displaystyle =\hom_H\left(\sum_{X\in\widehat{H}} X^{\oplus m(X)},Y\right) \\ & \displaystyle \cong\bigoplus_{X\in\widehat{H}} \hom_H(X,Y)^{\oplus m(X)} \\ & \displaystyle \cong \mathbb{C}^{\oplus m(Y)} \end{array}$$

This follows since, by Schur's lemma, $\hom_H(X,Y)\cong\mathbb{C}$ if $X=Y$ and is $0$ otherwise (when $X,Y$ are irreps of $H$). On the other hand, $\hom(V,Y)$, as a representation of $H$, is isomorphic to the tensor product $V^*\otimes Y$ (a special case of tensor-hom adjunction, the linearization of currying), and so $\dim\hom_H(V,Y)\cong\dim(V^*\otimes Y)^H$. But if $Z$ is an $H$-rep, the dimension of $Z^H$ is just the trace of any projection operator $Z\to Z^H$, which we know $|H|^{-1}\sum_{h\in H}\rho_Z(h)$ is, so applying this with $Z=V^*\otimes Y$ yields $m(Y)=|H|^{-1}\sum_{h\in H}\overline{\chi_V(h)}\chi_Y(h)=\langle \chi_Y,\chi_V\rangle$.

For example, suppose $H=\langle\omega\rangle=\{1,\omega,\omega^2\}$ is the group of $3$rd roots of unity. If we write

$$\sum_{h_2\in H}\rho(h_2)=\rho(1)+\rho(\omega)+\rho(\omega^2)$$

and set $h_1=\omega$, then we have

$$\sum_{h_2\in H}\rho(h_1h_2)=\rho(\omega)+\rho(\omega^2)+\rho(1).$$

It's the same sum, only permuted. Therefore,

$$\sum_{h_1\in H}\left(\sum_{h_2\in H}\rho(h_1h_2)\right)=\left(\color{Purple}{\sum_{h_2\in H}\rho(h_2)}\right)+\left(\color{Blue}{\sum_{h_2\in H}\rho(\omega h_2)}\right)+\left(\color{Green}{\sum_{h_2\in H}\rho(\omega^2 h_2)}\right)$$

$$=\begin{array}{ccccc} & \color{Purple}{\rho(1)} & \color{Purple}{+} & \color{Purple}{\rho(\omega)} & \color{Purple}{+} & \color{Purple}{\rho(\omega^2)} \\ + & \color{Blue}{\rho(\omega)} & \color{Blue}{+} & \color{Blue}{\rho(\omega^2)} & \color{Blue}{+} & \color{Blue}{\rho(1)} \\ + & \color{Green}{\rho(\omega^2)} & \color{Green}{+} & \color{Green}{\rho(1)} & \color{Green}{+} & \color{Green}{\rho(\omega)} \end{array} \quad =3\big[\rho(1)+\rho(\omega)+\rho(\omega^2)\big]$$

• @Omar for instance, $(\sum_i a_i)^2=(\sum_i a_i)(\sum_j a_j)=\sum_{i,j}a_ia_j$. same idea. – arctic tern May 3 '16 at 6:07
• @Omar because for all $h\in H$, $\rho(h)$ appears as a summand in $\sum_{h_1,h_2\in H}\rho(h_1h_2)$ a total of $|H|$ times. If it helps, you can think of $x\mapsto h_1x$ as a bijection, hence $$\frac{1}{|H|^2}\sum_{h_1\in H}\left(\sum_{h_2\in H}\rho(h_1h_2)\right)=\frac{1}{|H|^2}\sum_{h_1\in H}\left(\sum_{h_2\in H}\rho(h_2)\right)=\frac{1}{|H|^2}|H|\sum_{h\in H}\rho(h)$$ – arctic tern May 3 '16 at 6:53
• I'll repeat what I said: "for all $h\in H$, $\rho(h)$ appears as a summand in $\sum_{h_1,h_2\in H}\rho(h_1h_2)$ a total of $|H|$ times." This is because we can write any particular $h$ in the form $h_1h_2$ with $h_1,h_2\in H$ a total of $|H|$ many ways. Equivalently, the map $H\times H\to H$ given by $(h_1,h_2)\mapsto h_1h_2$ is an $|H|$-to-$1$ function. I am running out of ways to say the same thing. Also see my previous comment: Do you get why, for any fixed $h_1$, we have $\sum_{h_2\in H}\rho(h_1h_2)=\sum_{h_2\in H}\rho(h_2)$? – arctic tern May 3 '16 at 19:49
• @OmarShehab Because for fixed $h_1$, the map $x\mapsto h_1x$ is a bijection on $H$, and so both sums have exactly the same summands, just possibly in a different order. – arctic tern May 4 '16 at 1:44
• For example, suppose $H$ is the fifth roots of unity generated by a primitive root $\zeta$. If we write $$\sum_{h_2\in H}\rho(h_2)=\rho(1)+\rho(\zeta)+\rho(\zeta^2)+\rho(\zeta^3)+\rho(\zeta^4),$$ and set $h_1=\zeta^2$, then we have $$\sum_{h_2\in H}\rho(h_1h_2)=\rho(\zeta^2)+\rho(\zeta^3)+\rho(\zeta^4)+\rho(1)+\rho(\zeta),$$ which is exactly the same sum, only with its summands permuted. – arctic tern May 4 '16 at 1:51


text:
Publication 1995 Issue No. 4 - April Abstract - Some New Designs of 2-D Array for Matrix Multiplication and Transitive Closure
Some New Designs of 2-D Array for Matrix Multiplication and Transitive Closure
April 1995 (vol. 6 no. 4)
pp. 351-362
 ASCII Text x Jong-Chuang Tsay, Pen-Yuang Chang, "Some New Designs of 2-D Array for Matrix Multiplication and Transitive Closure," IEEE Transactions on Parallel and Distributed Systems, vol. 6, no. 4, pp. 351-362, April, 1995.
 BibTex x @article{ 10.1109/71.372789,author = {Jong-Chuang Tsay and Pen-Yuang Chang},title = {Some New Designs of 2-D Array for Matrix Multiplication and Transitive Closure},journal ={IEEE Transactions on Parallel and Distributed Systems},volume = {6},number = {4},issn = {1045-9219},year = {1995},pages = {351-362},doi = {http://doi.ieeecomputersociety.org/10.1109/71.372789},publisher = {IEEE Computer Society},address = {Los Alamitos, CA, USA},}
 RefWorks Procite/RefMan/Endnote x TY - JOURJO - IEEE Transactions on Parallel and Distributed SystemsTI - Some New Designs of 2-D Array for Matrix Multiplication and Transitive ClosureIS - 4SN - 1045-9219SP351EP362EPD - 351-362A1 - Jong-Chuang Tsay, A1 - Pen-Yuang Chang, PY - 1995VL - 6JA - IEEE Transactions on Parallel and Distributed SystemsER -

Abstract—In this paper, we present some new regular iterative algorithms for matrix multiplication and transitive closure. With these algorithms, by spacetime mapping the 2-D arrays with $2N - 1$ and $\lceil \left(3N - 1\right)/2\rceil$ execution times for matrix multiplication can be obtained. Meanwhile, we can derive a 2-D array with $4N - 2$ execution time for transitive closure based on the sequential Warshall-Floyd algorithm. All these new 2-D arrays for matrix multiplication and transitive closure have the advantages of faster and more regular than other previous designs.

Index Terms—Algorithm mapping, matrix multiplication, mesh array, systolic array, spherical array, transitive closure, VLSI architecture.

[1] H. T. Kung and C. E. Leiserson,“Systolic arrays (for VLSI),”inProc. Sparse Matrix Symp., Soc. Indust. Appli. Math., 1978, pp. 256–282.
[2] S.K. Rao,“Regular iterative algorithms and their implementations on processor arrays,” PhD thesis, Stanford Univ., 1985.
[3] V. Van Dongen and P. Quinton,“Uniformization of linear recurrence equations: A step towards the automatic synthesis of systolic arrays,”inProc. Int. Conf. Systolic Array, 1988, pp. 473–482.
[4] P. Quinton, “Automatic Synthesis of Systolic Arrays from Uniform Recurrent Equations,” Proc. 11th Ann. Int'l Symp. Computer Architecture, pp. 208-214, June 1984.
[5] V. K. Prasanna Kumar and Y. C. Tsai,“Designing linear systolic arrays,”J. Parallel Distribut. Comput., vol. 7, pp. 441–463, 1989.
[6] ——,“Mapping two dimensional systolic arrays to one dimensional arrays and applications,”inProc. Int. Conf. Parallel Processing, 1988, pp. 39–46.
[7] ——,“Synthesizing optimal family of linear systolic arrays for matrix computations,”inProc. Int. Conf. Systolic Array, 1988, pp. 51–60.
[8] P.J. Varman and I.V. Ramakrishnan, "Synthesis of an Optimal Family of Matrix Multiplication Algorithms on Linear Arrays," IEEE Trans. Computers, vol. 35, no. 11, pp. 989-996, Nov. 1986.
[9] ——,“Modular matrix multiplication on a linear array,”IEEE Trans. Comput., vol. C-33, pp. 952–958, Nov. 1984.
[10] S.Y. Kung, VLSI Array Processors. Prentice Hall, 1988.
[11] L. Melkemi and M. Tchuente,“Complex of matrix product on a class of orthogonally connected systolic arrays,”IEEE Trans. Comput., vol. C-36, pp. 615–619, May 1987.
[12] G. J. Li and B. W. Wah,“The design of optimal systolic arrays,”IEEE Trans. Comput., vol. C-34, pp. 66–77, Jan. 1985.
[13] W. A. Porter and J. L. Aravena,“Cylindrical arrays for matrix multiplication,”inProc. 24th Annu. Allerton Conf. Commun., Control Comput., Mar. 1988, pp. 595–602.
[14] S. C. Kak,“Multilayered array computing,”inProc. 1985 Annu. Conf. Inform. Sci. Syst., Mar. 1984, pp. 4.36–4.41.
[15] ——,“A two-layered mesh array for matrix multiplication,”Parallel Comput., vol. 6, pp. 383–385, 1988.
[16] A. Benaini and Y. Robert,“An even faster systolic array for matrix multiplication,”Parallel Comput., vol. 12, pp. 249–254, 1989.
[17] H. V. Jagadish and T. Kailath,“A family of new efficient arrays for matrix multiplication,”IEEE Trans. Comput., vol. C-38, pp. 149–155, Jan. 1989.
[18] S.Y. Kung,S.C. Lo,, and P.S. Lewis,“Optimal systolic design for the transitive closure and the shortest path problems,” IEEE Trans. Computers, vol. 36, pp. 603-614, May 1987.
[19] D.I. Moldovan and J.A.B. Fortes, “Partitioning and Mapping Algorithms into Fixed Size Systolic Arrays,” IEEE Trans. Computers, vol. 35, no. 1, pp.1-12, Jan. 1986.

Citation:
Jong-Chuang Tsay, Pen-Yuang Chang, "Some New Designs of 2-D Array for Matrix Multiplication and Transitive Closure," IEEE Transactions on Parallel and Distributed Systems, vol. 6, no. 4, pp. 351-362, April 1995, doi:10.1109/71.372789


text:
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text:
# continuous onto map from $(0,1)\to (0,1]$

I need to know whether There exists any continuous onto map from $(0,1)\to (0,1]$

could any one give me any hint?

Hint: $(0,1) = (0,\frac 12] \cup [\frac 12, 1)$. Can you map each part onto $(0,1]$?

• hint: $\times 2$ – Dan Rust Jul 28 '13 at 12:36
• Taxi: you have no idea how to send $(0,\frac12]$ continuously to $(0,1]$? Really no idea? – Did Jul 28 '13 at 12:37
• @TaxiDriver Yes, you do. – Git Gud Jul 28 '13 at 12:42
• @TaxiDriver ... and now: can you map $[\frac 12, 1)$ onto $(0,1]$? – martini Jul 28 '13 at 12:42
• @Taxi Driver no, you're joking! – W_D Jul 28 '13 at 12:43

Find a polynomial that:

1. Crosses the x-axis at $x=0$ and $x=1$.
2. Has an absolute maximum of $f(x)=1$.

$$f(x)=-4(x^2-x),x\in(0,1)$$

From The Hint of Martini the Map $f(x)=2x; x\in (0,{1\over 2}]$ and $f(x)=1;x\in [{1\over 2},1)$ will work

$f(x)=|\sin (\pi x)|$ will work


text:
## Feed aggregator

### Regularized maximal fidelity of the generalized Pauli channels. (arXiv:1810.05160v1 [quant-ph])

arXiv.org: Quantum Physics - Mon, 2018-10-15 22:34

We consider the asymptotic regularization of the maximal fidelity for the generalized Pauli channels, which is a problem similar to the classical channel capacity. In particular, we find the formulas for the extremal channel fidelities and the maximal output $\infty$-norm. For wide classes of channels, we show that these quantities are multiplicative. Finally, we find the regularized maximal fidelity for the channels satisfying the time-local master eqeuations.

Categories: Journals, Physics

### Equiangular quantum key distribution in more than two dimensions. (arXiv:1810.05161v1 [quant-ph])

arXiv.org: Quantum Physics - Mon, 2018-10-15 22:34

We extend the spherical code based key distribution protocols to qudits with dimensions 4 and 16 by constructing equiangular frames and their companions. We provide methods for equiangular frames in arbitrary dimensions for Alice to use and the companion frames, that has one antipode to eliminate one of the possibilities, made up of qudits with $N=4,16$ as part of Bob's code. Non-orthogonal bases that form positive operator valued measures can be constructed using the tools of frames (overcomplete bases of a Hilbert space) and here we apply them to key distribution that are robust due to large size of the bases making it hard for eavesdropping. We demonstrate a method to construct a companion frame for an equiangular tight frame for $\complexes^{p-1}$ generated from the discrete Fourier transform, where $p$ is any odd prime. The security analysis is based on the assumption restricting possible attacks to intercept/resend scenario highlighting the advantages of a qudit over qubit-based protocols.

Categories: Journals, Physics

### Generation of non-classical light in a photon-number superposition. (arXiv:1810.05170v1 [quant-ph])

arXiv.org: Quantum Physics - Mon, 2018-10-15 22:34

The ability to generate light in a pure quantum state is essential for advances in optical quantum technologies. However, obtaining quantum states with control in the photon-number has remained elusive. Optical light fields with zero and one photon can be produced by single atoms, but so far it has been limited to generating incoherent mixtures, or coherent superpositions with a very small one-photon term. Here, we report on the on-demand generation of quantum superpositions of zero, one, and even two photons, via pulsed coherent control of a single artificial atom. Driving the system up to full atomic inversion leads to the generation of quantum superpositions of vacuum and one photon, with their relative populations controlled by the driving laser intensity. A stronger driving of the system, with $2\pi$-pulses, results in a coherent superposition of vacuum, one and two photons, with the two-photon term exceeding the one-photon component, a state allowing phase super-resolving interferometry. Our results open new paths for optical quantum technologies with access to the photon-number degree-of-freedom.

Categories: Journals, Physics

### Scalable in operando strain tuning of multiple quantum dots within a photonic waveguide architecture. (arXiv:1810.05195v1 [quant-ph])

arXiv.org: Quantum Physics - Mon, 2018-10-15 22:34

The quest for an integrated quantum optics platform involving solid state waveguides and cavities combined with semiconductor quantum dots has motivated the field of quantum semiconductor research for two decades. Demonstrations of discrete quantum light sources, single photon switches, transistors, and quantum memory-photon interfaces have become very advanced. Yet the fundamental problem that every quantum dot is different prevents integration and scaling beyond a few quantum dots. Here, we address this challenge by patterning strain via local phase transitions to selectively tune individual quantum dots that are embedded in a photonic architecture. The patterning is implemented with in operando micro-laser crystallization of a thin HfO_2 film "sheath" on the surface of a GaAs waveguide. Using this approach, we tune InAs quantum dot emission energies over the full inhomogeneous distribution with a step size down to the homogeneous linewidth and a spatial resolution better than 1 \mum. As a demonstration, we tune three quantum dots to the same energy within a waveguide. This approach is general, and the same principles extend to other solid-state quantum emitters and photonic structures.

Categories: Journals, Physics

### Exchanging two identical particles. (arXiv:1810.05208v1 [quant-ph])

arXiv.org: Quantum Physics - Mon, 2018-10-15 22:34

The phase factor $(-1)^{2s}$ that features in the exchange symmetry for identical spin-$s$ fermions or bosons contributes to but is not identical to the phase factor one may observe when one actually exchanges two such particles. The observable phase contains {\em single-particle} geometric and dynamical phases as well, induced by both spin and spatial exchange transformations. Extending the analysis to (abelian) anyons by incorporating the Aharonov-Bohm effect shows that the topological phase two anyons pick up under an interchange is at bottom a single-particle effect, in contrast to the fermion/boson exchange phase factor $(-1)^{2s}$, which is a two-particle effect.

Categories: Journals, Physics

### Towards Quantum One-Time Memories from Stateless Hardware. (arXiv:1810.05226v1 [quant-ph])

arXiv.org: Quantum Physics - Mon, 2018-10-15 22:34

A central tenet of theoretical cryptography is the study of the minimal assumptions required to implement a given cryptographic primitive. One such primitive is the one-time memory (OTM), introduced by Goldwasser, Kalai, and Rothblum [CRYPTO 2008], which is a classical functionality modeled after a non-interactive 1-out-of-2 oblivious transfer, and which is complete for one-time classical and quantum programs. It is known that secure OTMs do not exist in the standard model in both the classical and quantum settings.

Here, we propose a scheme for using quantum information, together with the assumption of stateless (i.e., reusable) hardware tokens, to build statistically secure OTMs. Via the semidefinite programming-based quantum games framework of Gutoski and Watrous [STOC 2007], we prove security for a malicious receiver, against a linear number of adaptive queries to the token, in the quantum universal composability framework. We prove stand-alone security against a malicious sender, but leave open the question of composable security against a malicious sender, as well as security against a malicious receiver making a polynomial number of adaptive queries. Compared to alternative schemes derived from the literature on quantum money, our scheme is technologically simple since it is of the "prepare-and measure" type. We also show our scheme is "tight" according to two scenarios.

Categories: Journals, Physics

### Delegating Quantum Computation Using Only Hash Functions. (arXiv:1810.05234v1 [quant-ph])

arXiv.org: Quantum Physics - Mon, 2018-10-15 22:34

In this paper, we construct a new scheme for delegating a large circuit family, which we call "C+P circuits". "C+P" circuits are the circuits composed of Toffoli gates and diagonal gates. Our scheme is non-interactive, only requires small quantum resources on the client side, and can be proved secure in the quantum random oracle model, without relying on additional assumptions, for example, the existence of fully homomorphic encryption. In practice the random oracle can be replaced by an appropriate hash functions, for example, SHA-3, AES.

This protocol allows a client to delegate the most expensive part of some quantum algorithms, for example, Shor's algorithm. The previous protocols that are powerful enough to delegate Shor's algorithm require either many rounds of interaction or the existence of FHE. The quantum resources required by the client are fewer than when it runs Shor's algorithm locally.

Different from many previous protocols, our scheme is not based on quantum one time pad, but on a new encoding called "entanglement encoding". We then generalize the garbled circuit to reversible garbled circuit to allow computation on this encoding.

To prove the security of this protocol, we study key dependent message(KDM) security in the quantum random oracle model. Then as a natural generalization, we define and study quantum KDM security. KDM security was not previously studied in quantum settings.

Categories: Journals, Physics

### Anti-Stokes excitation of solid-state quantum emitters for nanoscale thermometry. (arXiv:1810.05265v1 [physics.app-ph])

arXiv.org: Quantum Physics - Mon, 2018-10-15 22:34

Color centers in solids are the fundamental constituents of a plethora of applications such as lasers, light emitting diodes and sensors, as well as the foundation of advanced quantum information and communication technologies. Their photoluminescence properties are usually studied under Stokes excitation, in which the emitted photons are at a lower energy than the excitation ones. In this work, we explore the opposite Anti-Stokes process, where excitation is performed with lower energy photons. We report that the process is sufficiently efficient to excite even a single quantum system, namely the germanium-vacancy center in diamond. Consequently, we leverage the temperature-dependent, phonon-assisted mechanism to realize an all-optical nanoscale thermometry scheme that outperforms any homologous optical method employed to date. Our results frame a promising approach for exploring fundamental light-matter interactions in isolated quantum systems, and harness it towards the realization of practical nanoscale thermometry and sensing.

Categories: Journals, Physics

### Superfast encodings for fermionic quantum simulation. (arXiv:1810.05274v1 [quant-ph])

arXiv.org: Quantum Physics - Mon, 2018-10-15 22:34

Simulation of fermionic many-body systems on a quantum computer requires a suitable encoding of fermionic degrees of freedom into qubits. Here we revisit the Superfast Encoding introduced by Kitaev and one of the authors. This encoding maps a target fermionic Hamiltonian with two-body interactions on a graph of degree $d$ to a qubit simulator Hamiltonian composed of Pauli operators of weight $O(d)$. A system of $m$ fermi modes gets mapped to $n=O(md)$ qubits. We propose Generalized Superfast Encodings (GSE) which require the same number of qubits as the original one but have more favorable properties. First, we describe a GSE such that the corresponding quantum code corrects any single-qubit error provided that the interaction graph has degree $d\ge 6$. In contrast, we prove that the original Superfast Encoding lacks the error correction property for $d\le 6$. Secondly, we describe a GSE that reduces the Pauli weight of the simulator Hamiltonian from $O(d)$ to $O(\log{d})$. The robustness against errors and a simplified structure of the simulator Hamiltonian offered by GSEs can make simulation of fermionic systems within the reach of near-term quantum devices. As an example, we apply the new encoding to the fermionic Hubbard model on a 2D lattice.

Categories: Journals, Physics

### Second Thoughts on the Second Law. (arXiv:1810.05276v1 [quant-ph])

arXiv.org: Quantum Physics - Mon, 2018-10-15 22:34

We speculate whether the second law of thermodynamics has more to do with Turing machines than steam pipes. It states the logical reversibility of reality as a computation, i.e., the fact that no information is forgotten: nature computes with Toffoli-, not NAND gates. On the way there, we correct Landauer's erasure principle by directly linking it to lossless data compression, and we further develop that to a lower bound on the energy consumption and heat dissipation of a general computation.

Categories: Journals, Physics

### A single-world consistent interpretation of quantum mechanics from fundamental time and length uncertainties. (arXiv:1810.05307v1 [quant-ph])

arXiv.org: Quantum Physics - Mon, 2018-10-15 22:34

Within ordinary ---unitary--- quantum mechanics there exist global protocols that allow to verify that no definite event ---an outcome to which a probability can be associated--- occurs. Instead, states that start in a coherent superposition over possible outcomes always remain as a superposition. We show that, when taking into account fundamental errors in measuring length and time intervals, that have been put forward as a consequence of a conjunction of quantum mechanical and general relativity arguments, there are instances in which such global protocols no longer allow to distinguish whether the state is in a superposition or not. All predictions become identical as if one of the outcomes occurs, with probability determined by the state. We use this as a criteria to define events, as put forward in the Montevideo Interpretation of Quantum Mechanics. We analyze in detail the occurrence of events in the paradigmatic case of a particle in a superposition of two different locations. We argue that our approach provides a consistent (C) single-world (S) picture of the universe, thus allowing an economical way out of the limitations imposed by a recent theorem by Frauchiger and Renner showing that having a self-consistent single-world description of the universe is incompatible with quantum theory. In fact, the main observation of this paper may be stated as follows: If quantum mechanics is extended to include gravitational effects to a QG theory, then QG, S, and C are satisfied.

Categories: Journals, Physics

### The rational fragment of the ZX-calculus. (arXiv:1810.05377v1 [quant-ph])

arXiv.org: Quantum Physics - Mon, 2018-10-15 22:34

We introduce here a new axiomatisation of the rational fragment of the ZX-calculus, a diagrammatic language for quantum mechanics. Compared to the previous axiomatisation introduced in [8], our axiomatisation does not use any metarule , but relies instead on a more natural rule, called the cyclotomic supplementarity rule, that was introduced previously in the literature. Our axiomatisation is only complete for diagrams using rational angles , and is not complete in the general case. Using results on diophantine geometry, we characterize precisely which diagram equality involving arbitrary angles are provable in our framework without any new axioms, and we show that our axiomatisation is continuous, in the sense that a diagram equality involving arbitrary angles is provable iff it is a limit of diagram equalities involving rational angles. We use this result to give a complete characterization of all Euler equations that are provable in this axiomatisation.

Categories: Journals, Physics

### Qubit Measurement by Multi-Channel Driving. (arXiv:1810.05465v1 [quant-ph])

arXiv.org: Quantum Physics - Mon, 2018-10-15 22:34

We theoretically propose and experimentally implement a method to measure a qubit by driving it close to the frequency of a dispersively coupled bosonic mode. The separation of the bosonic states corresponding to different qubit states begins essentially immediately at maximum rate, leading to a speedup in the measurement protocol. Also the bosonic mode can be simultaneously driven to optimize measurement speed and fidelity. We experimentally test this measurement protocol using a superconducting qubit coupled to a resonator mode. For a certain measurement time, we observe that the conventional dispersive readout yields above 100 % higher measurement error than our protocol. Finally, we use an additional resonator drive to leave the resonator state to vacuum if the qubit is in the excited state during the measurement protocol. This suggests that the proposed measurement technique may become useful in unconditionally resetting the resonator to a vacuum state after the measurement pulse.

Categories: Journals, Physics

### Solution of Schrodinger equation using Tridiagonal representation approach in nonrelativistic quantum mechanics: a pedagogical approach. (arXiv:1810.05509v1 [quant-ph])

arXiv.org: Quantum Physics - Mon, 2018-10-15 22:34

We present the pedagogical method of Tridiagonal representation approach,an algebraic method for the solution of Schrodinger equation in nonrelativistic quantum mechanics for conventional potential functions. However, we solved a new three parameters potential function recently encountered using it as an example

Categories: Journals, Physics

### Necessary and sufficient condition of separability for D-symmetric diagonal states. (arXiv:1810.05522v1 [quant-ph])

arXiv.org: Quantum Physics - Mon, 2018-10-15 22:34

For multipartite states we consider a notion of D-symmetry. For a system of $N$ qubits it concides with usual permutational symmetry. In case of $N$ qudits ($d\geq 3$) the D-symmetry is stronger than the permutational one. For the space of all D-symmetric vectors in $(\mathbb{C}^d)^{\otimes N}$ we define a basis composed of vectors $\{|R_{N,d;k}\rangle: \,0\leq k\leq N(d-1)\}$. The aim of this paper is to discuss the problem of separability of D-symmetric states which are diagonal in the basis $\{|R_{N,d;k}\rangle\}$. We show that if $N$ is even and $d\geq 2$ is arbitrary then a PPT property is necessary and sufficient condition of separability for D-invariant diagonal states. In this way we generalize results obtained by Yu for qubits. Our strategy is to use some classical mathematical results on a moment problem.

Categories: Journals, Physics

### Edge states in trimer lattices. (arXiv:1810.05566v1 [cond-mat.mes-hall])

arXiv.org: Quantum Physics - Mon, 2018-10-15 22:34

Topological phases of matter have attracted much attention over the years. Motivated by analogy with photonic lattices, here we examine the edge states of a one-dimensional trimer lattice in the phases with and without inversion symmetry protection. In contrast to the Su-Schrieffer-Heeger model, we show that the edge states in the inversion-symmetry broken phase of the trimer model turn out to be chiral, i.e., instead of appearing in pairs localized at opposite edges they can appear at a $\textit{single}$ edge. Interestingly, these chiral edge states remain robust to large amounts of disorder. In addition, we use the Zak phase to characterize the emergence of degenerate edge states in the inversion-symmetric phase of the trimer model. Furthermore, we capture the essentials of the whole family of trimers through a mapping onto the commensurate off-diagonal Aubry-Andr\'{e}-Harper model, which allow us to establish a direct connection between chiral edge modes in the two models, including the calculation of Chern numbers. We thus suggest that the chiral edge modes of the trimer lattice have a topological origin inherited from this effective mapping. Also, we find a nontrivial connection between the topological phase transition point in the trimer lattice and the one in its associated two-dimensional parent system, in agreement with results in the context of Thouless pumping in photonic lattices.

Categories: Journals, Physics

### Thermal Processes and State Achievability. (arXiv:1810.05578v1 [quant-ph])

arXiv.org: Quantum Physics - Mon, 2018-10-15 22:34

A complete characterization of the set of states that can be achieved through Thermal Processes (TP) is given by describing all vertices, edges and facets of the allowed set of states in the language of thermomajorization curves. TPs are linked to transportation matrices, which leads to the existance of extremal TPs that are not required in implemenation of any transition allowed by TPs, for every dimension $d\geq 4$ of the state space. A property of the associated graphs, biplanarity, which differentiates between these extremal TPs and the necessary ones, is identified.

Categories: Journals, Physics

### Applying quantum algorithms to constraint satisfaction problems. (arXiv:1810.05582v1 [quant-ph])

arXiv.org: Quantum Physics - Mon, 2018-10-15 22:34

Quantum algorithms can deliver asymptotic speedups over their classical counterparts. However, there are few cases where a substantial quantum speedup has been worked out in detail for reasonably-sized problems, when compared with the best classical algorithms and taking into account realistic hardware parameters and overheads for fault-tolerance. All known examples of such speedups correspond to problems related to simulation of quantum systems and cryptography. Here we apply general-purpose quantum algorithms for solving constraint satisfaction problems to two families of prototypical NP-complete problems: boolean satisfiability and graph colouring. We consider two quantum approaches: Grover's algorithm and a quantum algorithm for accelerating backtracking algorithms. We compare the performance of optimised versions of these algorithms, when applied to random problem instances, against leading classical algorithms. Even when considering only problem instances that can be solved within one day, we find that there are potentially large quantum speedups available. In the most optimistic parameter regime we consider, this could be a factor of over $10^5$ relative to a classical desktop computer; in the least optimistic regime, the speedup is reduced to a factor of over $10^3$. However, the number of physical qubits used is extremely large, and improved fault-tolerance methods will likely be needed to make these results practical. In particular, the quantum advantage disappears if one includes the cost of the classical processing power required to perform decoding of the surface code using current techniques.

Categories: Journals, Physics

### Thermodynamic length in open quantum systems. (arXiv:1810.05583v1 [quant-ph])

arXiv.org: Quantum Physics - Mon, 2018-10-15 22:34

Optimal finite-time thermodynamic processes can be constructed by introducing a metric on the thermodynamic state space, known as thermodynamic length. Here, we show how to construct a thermodynamic metric for processes involving a small quantum system, whose Hamiltonian can be externally controlled, in contact with a thermal bath. First, we consider that the Hamiltonian of the system is modified by a sequence of quenches, followed by equilibrations with the bath, and a linear-response approach shows that the metric is induced by the Kubo-Mori-Bogoliubov (KMB) inner product. Second, we consider that the Hamiltonian of the system is continuously modified and the action of the bath is captured by a Lindbladian master equation, and find the corresponding thermodynamic metric. The latter approach has the KMB metric as a particular case, and in general enables the design of better finite-time protocols since it takes into account the nature of the dissipative dynamics. These results are illustrated in a paradigmatic quantum open two-level system.

Categories: Journals, Physics

### Entanglement detection by violations of noisy uncertainty relations: A proof of principle. (arXiv:1810.05588v1 [quant-ph])

arXiv.org: Quantum Physics - Mon, 2018-10-15 22:34

It is well-known that the violation of a local uncertainty relation can be used as an indicator for the presence of entanglement. Unfortunately, the practical use of these non-linear witnesses has been limited to few special cases in the past. However, new methods for computing uncertainty bounds became available. Here we report on an experimental implementation of uncertainty-based entanglement witnesses, benchmarked in a regime dominated by strong local noise. We combine the new computational method with a local noise tomography in order to design noise-adapted entanglement witnesses. This proof-of-principle experiment shows that quantum noise can be successfully handled by a fully quantum model in order to enhance entanglement detection efficiencies.

Categories: Journals, Physics


text:
# Find the Class Mark of the Classes 20-30 and 30-40. - Algebra

Find the class mark of the classes 20-30 and 30-40.

#### Solution

"Class mark" = ("Upper limit"+"Lower limit")/2

∴"Class mark of the class" 20-30 =(30+20)/2=50/2=25

"Class Mark of the class"30-40=(40+30)/2=70/2=35

Concept: Basic Ideas of Probability
Is there an error in this question or solution?


text:
# Integral of cos(1/x)

1. Apr 13, 2015

### Emmanuel_Euler

what is the integral of cos(1/x) dx ??

2. Apr 13, 2015

### Delta²

3. Apr 13, 2015

### Emmanuel_Euler

thank for help (delta)

4. Apr 14, 2015

### HallsofIvy

Staff Emeritus
You have posted a number of these "What is the integral of ... ". Do you understand that very few function can be integrated in terms of elementary functions?


text:
LLVM  mainline
InstCombineCasts.cpp File Reference
#include "InstCombineInternal.h"
#include "llvm/Analysis/ConstantFolding.h"
#include "llvm/IR/DataLayout.h"
#include "llvm/IR/PatternMatch.h"
#include "llvm/Analysis/TargetLibraryInfo.h"
Include dependency graph for InstCombineCasts.cpp:

Go to the source code of this file.

## Defines

#define DEBUG_TYPE   "instcombine"

## Functions

static ValuedecomposeSimpleLinearExpr (Value *Val, unsigned &Scale, uint64_t &Offset)
Analyze 'Val', seeing if it is a simple linear expression.
static Instruction::CastOps isEliminableCastPair (const CastInst *CI, unsigned opcode, Type *DstTy, const DataLayout &DL)
This function is a wrapper around CastInst::isEliminableCastPair.
static bool canEvaluateTruncated (Value *V, Type *Ty, InstCombiner &IC, Instruction *CxtI)
Return true if we can evaluate the specified expression tree as type Ty instead of its larger type, and arrive with the same value.
static InstructionfoldVecTruncToExtElt (TruncInst &Trunc, InstCombiner &IC, const DataLayout &DL)
Given a vector that is bitcast to an integer, optionally logically right-shifted, and truncated, convert it to an extractelement.
static bool canEvaluateZExtd (Value *V, Type *Ty, unsigned &BitsToClear, InstCombiner &IC, Instruction *CxtI)
Determine if the specified value can be computed in the specified wider type and produce the same low bits.
static bool canEvaluateSExtd (Value *V, Type *Ty)
Return true if we can take the specified value and return it as type Ty without inserting any new casts and without changing the value of the common low bits.
static ConstantfitsInFPType (ConstantFP *CFP, const fltSemantics &Sem)
Return a Constant* for the specified floating-point constant if it fits in the specified FP type without changing its value.
static ValuelookThroughFPExtensions (Value *V)
If this is a floating-point extension instruction, look through it until we get the source value.
static InstructionoptimizeVectorResize (Value *InVal, VectorType *DestTy, InstCombiner &IC)
This input value (which is known to have vector type) is being zero extended or truncated to the specified vector type.
static bool isMultipleOfTypeSize (unsigned Value, Type *Ty)
static unsigned getTypeSizeIndex (unsigned Value, Type *Ty)
static bool collectInsertionElements (Value *V, unsigned Shift, SmallVectorImpl< Value * > &Elements, Type *VecEltTy, bool isBigEndian)
V is a value which is inserted into a vector of VecEltTy.
static ValueoptimizeIntegerToVectorInsertions (BitCastInst &CI, InstCombiner &IC)
If the input is an 'or' instruction, we may be doing shifts and ors to assemble the elements of the vector manually.
static InstructioncanonicalizeBitCastExtElt (BitCastInst &BitCast, InstCombiner &IC, const DataLayout &DL)
Canonicalize scalar bitcasts of extracted elements into a bitcast of the vector followed by extract element.

## Define Documentation

 #define DEBUG_TYPE   "instcombine"

Definition at line 22 of file InstCombineCasts.cpp.

## Function Documentation

 static bool canEvaluateSExtd ( Value * V, Type * Ty )  [static]

Return true if we can take the specified value and return it as type Ty without inserting any new casts and without changing the value of the common low bits.

This is used by code that tries to promote integer operations to a wider types will allow us to eliminate the extension.

This function works on both vectors and scalars.

Definition at line 1060 of file InstCombineCasts.cpp.

Referenced by llvm::InstCombiner::visitSExt().

 static bool canEvaluateTruncated ( Value * V, Type * Ty, InstCombiner & IC, Instruction * CxtI )  [static]

Return true if we can evaluate the specified expression tree as type Ty instead of its larger type, and arrive with the same value.

This is used by code that tries to eliminate truncates.

Ty will always be a type smaller than V. We should return true if trunc(V) can be computed by computing V in the smaller type. If V is an instruction, then trunc(inst(x,y)) can be computed as inst(trunc(x),trunc(y)), which only makes sense if x and y can be efficiently truncated.

This function works on both vectors and scalars.

Definition at line 331 of file InstCombineCasts.cpp.

Referenced by llvm::InstCombiner::visitTrunc().

 static bool canEvaluateZExtd ( Value * V, Type * Ty, unsigned & BitsToClear, InstCombiner & IC, Instruction * CxtI )  [static]

Determine if the specified value can be computed in the specified wider type and produce the same low bits.

If not, return false.

If this function returns true, it can also return a non-zero number of bits (in BitsToClear) which indicates that the value it computes is correct for the zero extend, but that the additional BitsToClear bits need to be zero'd out. For example, to promote something like:

B = trunc i64 A to i32 C = lshr i32 B, 8 E = zext i32 C to i64

CanEvaluateZExtd for the 'lshr' will return true, and BitsToClear will be set to 8 to indicate that the promoted value needs to have bits 24-31 cleared in addition to bits 32-63. Since an 'and' will be generated to clear the top bits anyway, doing this has no extra cost.

This function works on both vectors and scalars.

Definition at line 724 of file InstCombineCasts.cpp.

Referenced by llvm::InstCombiner::visitZExt().

 static Instruction* canonicalizeBitCastExtElt ( BitCastInst & BitCast, InstCombiner & IC, const DataLayout & DL )  [static]

Canonicalize scalar bitcasts of extracted elements into a bitcast of the vector followed by extract element.

The backend tends to handle bitcasts of vectors better than bitcasts of scalars because vector registers are usually not type-specific like scalar integer or scalar floating-point.

Definition at line 1768 of file InstCombineCasts.cpp.

Referenced by llvm::InstCombiner::visitBitCast().

 static bool collectInsertionElements ( Value * V, unsigned Shift, SmallVectorImpl< Value * > & Elements, Type * VecEltTy, bool isBigEndian )  [static]

V is a value which is inserted into a vector of VecEltTy.

Look through the value to see if we can decompose it into insertions into the vector. See the example in the comment for OptimizeIntegerToVectorInsertions for the pattern this handles. The type of V is always a non-zero multiple of VecEltTy's size. Shift is the number of bits between the lsb of V and the lsb of the vector.

This returns false if the pattern can't be matched or true if it can, filling in Elements with the elements found here.

Definition at line 1630 of file InstCombineCasts.cpp.

Referenced by optimizeIntegerToVectorInsertions().

 static Value* decomposeSimpleLinearExpr ( Value * Val, unsigned & Scale, uint64_t & Offset )  [static]

Analyze 'Val', seeing if it is a simple linear expression.

If so, decompose it, returning some value X, such that Val is X*Scale+Offset.

Definition at line 28 of file InstCombineCasts.cpp.

 static Constant* fitsInFPType ( ConstantFP * CFP, const fltSemantics & Sem )  [static]

Return a Constant* for the specified floating-point constant if it fits in the specified FP type without changing its value.

Definition at line 1218 of file InstCombineCasts.cpp.

Referenced by lookThroughFPExtensions().

 static Instruction* foldVecTruncToExtElt ( TruncInst & Trunc, InstCombiner & IC, const DataLayout & DL )  [static]

Given a vector that is bitcast to an integer, optionally logically right-shifted, and truncated, convert it to an extractelement.

Example (big endian): trunc (lshr (bitcast <4 x i32> X to i128), 32) to i32 ---> extractelement <4 x i32> X, 1

Definition at line 439 of file InstCombineCasts.cpp.

Referenced by llvm::InstCombiner::visitTrunc().

 static unsigned getTypeSizeIndex ( unsigned Value, Type * Ty )  [static]

Definition at line 1616 of file InstCombineCasts.cpp.

References llvm::Type::getPrimitiveSizeInBits().

Referenced by collectInsertionElements().

 static Instruction::CastOps isEliminableCastPair ( const CastInst * CI, unsigned opcode, Type * DstTy, const DataLayout & DL )  [static]

This function is a wrapper around CastInst::isEliminableCastPair.

It simply extracts arguments and returns what that function returns.

Parameters:
 CI First cast instruction opcode Opcode for the second cast DstTy Target type for the second cast

Definition at line 234 of file InstCombineCasts.cpp.

Referenced by llvm::InstCombiner::commonCastTransforms(), and foldConstantCastPair().

 static bool isMultipleOfTypeSize ( unsigned Value, Type * Ty )  [static]

Definition at line 1612 of file InstCombineCasts.cpp.

References llvm::Type::getPrimitiveSizeInBits().

Referenced by collectInsertionElements().

 static Value* lookThroughFPExtensions ( Value * V )  [static]

If this is a floating-point extension instruction, look through it until we get the source value.

Definition at line 1229 of file InstCombineCasts.cpp.

Referenced by llvm::InstCombiner::visitFPTrunc().

 static Value* optimizeIntegerToVectorInsertions ( BitCastInst & CI, InstCombiner & IC )  [static]

If the input is an 'or' instruction, we may be doing shifts and ors to assemble the elements of the vector manually.

Try to rip the code out and replace it with insertelements. This is to optimize code like this:

tmp37 = bitcast float inc to i32 tmp38 = zext i32 tmp37 to i64 tmp31 = bitcast float inc5 to i32 tmp32 = zext i32 tmp31 to i64 tmp33 = shl i64 tmp32, 32 ins35 = or i64 tmp33, tmp38 tmp43 = bitcast i64 ins35 to <2 x float>

Into two insertelements that do "buildvector{%inc, %inc5}".

Definition at line 1739 of file InstCombineCasts.cpp.

Referenced by llvm::InstCombiner::visitBitCast().

 static Instruction* optimizeVectorResize ( Value * InVal, VectorType * DestTy, InstCombiner & IC )  [static]

This input value (which is known to have vector type) is being zero extended or truncated to the specified vector type.

Try to replace it with a shuffle (and vector/vector bitcast) if possible.

The source and destination vector types may have different element types.

Definition at line 1560 of file InstCombineCasts.cpp.

Referenced by llvm::InstCombiner::visitBitCast().


text:
4

# Question Which one 0f / the 9 following 'buffer ! statements Is an 'True? aqucous = The solutlon , common composed C 'Ion In a '0( (o = bull...

## Question

###### Question Which one 0f / the 9 following 'buffer ! statements Is an 'True? aqucous = The solutlon , common composed C 'Ion In a '0( (o = buller 'weak acids; soluuon buller E does = Tesists [ not / pH have change any eliea ( on lhe t nculrali/ana - Ulssechuon d Ud *e bufler {edded acida und batcs Oetnt docs nol change ! pH when ' lurge ' AnUnls 0/ 0 Wiong buffer can absorb = ecuor 4to1t unlimilcd bas Drdukd amount of ncid or base .QUESTION 5Which of the

Question Which one 0f / the 9 following 'buffer ! statements Is an 'True? aqucous = The solutlon , common composed C 'Ion In a '0( (o = buller 'weak acids; soluuon buller E does = Tesists [ not / pH have change any eliea ( on lhe t nculrali/ana - Ulssechuon d Ud *e bufler {edded acida und batcs Oetnt docs nol change ! pH when ' lurge ' AnUnls 0/ 0 Wiong buffer can absorb = ecuor 4to1t unlimilcd bas Drdukd amount of ncid or base . QUESTION 5 Which of the following is the cOnect cqualion relating Q to Kip lor w cquilibnam [cacpon Q < Ksp Q > Ksp Q = Ksp

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##### Time lett 0 55.590con5x-Y+2z=5 The linear system 2x-2y+432 =[0 has infinitely many solutions 3x - 3y+62 =15Manedcut dcelct 57o
Time lett 0 55.59 0con5 x-Y+2z=5 The linear system 2x-2y+432 =[0 has infinitely many solutions 3x - 3y+62 =15 Manedcut d celct 57o...
##### The combustion of ethanol (CzHsOH) is shown below: CzHsOH 3 02 (g) 2 CO2 (g) HzO() LH = -1367.4 kJThe ethanol can also react with oxygen to form acetic acid_ The reaction is:CzHsoh 02 (9) CH;COOH HzO()The heats of formation of CH3COOH, COz and Hzo are listed below: 2 Hz (g) 02 (g) CH3CooH(l) AH 484 kl{mol Cts) 02 (g) - cO2 (g LH 393.5 kllmol H2(g) 1/2 02 (g) ~ HzO (} H = -285.8 kJmol H2(g} 1/2 0z2 (g) Hz0 {g) 4H = -241.8 kl/molUse the above information to answer the next two questions_Determine
The combustion of ethanol (CzHsOH) is shown below: CzHsOH 3 02 (g) 2 CO2 (g) HzO() LH = -1367.4 kJ The ethanol can also react with oxygen to form acetic acid_ The reaction is: CzHsoh 02 (9) CH;COOH HzO() The heats of formation of CH3COOH, COz and Hzo are listed below: 2 Hz (g) 02 (g) CH3CooH(l) AH 4...
##### Lt_6g (R) b2 to_space_Sall_teal polynomial 25 bale 43 @t D bz te diftountiation eucatet Bn Vzlz(R) what t minimal ~pold fol_D 2 9marKs is
lt_6g (R) b2 to_space_Sall_teal polynomial 25 bale 43 @t D bz te diftountiation eucatet Bn Vzlz(R) what t minimal ~pold fol_D 2 9marKs is...
##### Using the half-reaction method, show how you can balancethe reaction of ethanol with chromic acid to produce acetic acidand chromium dioxide.
Using the half-reaction method, show how you can balance the reaction of ethanol with chromic acid to produce acetic acid and chromium dioxide....


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# How do I find the inverse function of a polynomial with $x^5$?

I've been stumped on this problem for hours and cannot figure out how to do it from tons of tutorials.

Please note: This is an intro to calculus, so we haven't learned derivatives or anything too complex.

Here's the question:

Let $f(x) = x^5 + x + 7$. Find the value of the inverse function at a point. $f^{-1}(1035) =$___?

I tried setting $f(x)$ as $y$.. and solving for $x$. Clearly that doesn't help lol. I've tried many different approaches and cannot figure out the answer. I used wolframalpha, my textbook, notes, examples, and tons of Google searches and nothing makes sense. Can someone please help? Thanks!!

• Guess and check! There is no nice formula for solving a fifth degree equation (that's a theorem). Your function is increasing, so there is a unique solution to $f(x)=1035$. If you fool around, you will find it very fast. – André Nicolas Sep 10 '13 at 4:20
• Oh no, André edited his comment as I wrote my answer, and now they're the same. All credit to André, the comment FGITW. – davidlowryduda Sep 10 '13 at 4:24

$$1035-7=1028=1024+4=4^5+4$$ Therefore $f^{-1}(1035)=4$.

HINT(s)

1. $f$ is an increasing function.
2. Since $f$ is increasing, you will be able to modify your guesses to close in on the answer quickly.
• So it's literally a guess and check type ordeal? – Cozen Sep 10 '13 at 4:24
• @Justin: Why not? – davidlowryduda Sep 10 '13 at 4:25
• Since $f$ is rapidly increasing, this will converge quickly. Take $x_0=1035^{\frac 15}$, ignoring the other terms. Then $x_i=(x_{i-1}^5-x_{i-1}-7)^{\frac 15}$ – Ross Millikan Sep 10 '13 at 4:32

In general, polynomials won't have an inverse. This one happens to have one, but it's not fun to express, as far as I know.

Since you only need to find the inverse at a particular number, not any $y$, just plug it in and rearrange until something looks nice: $x^5 + x + 7 = 1035$ means $x(x^4 + 1) = 1028$. The factors of $1028$ are a good place to start.

• Ah, that's a nice one too. – davidlowryduda Sep 10 '13 at 4:26


text:
Compile a function with undetermined number of variables

I have been wrapping my head about a question it may look simple to you. I want compile a function that depends on a number of undetermined variables as,

Compile[{{x, _Real}, {y, _Real}, {z, _Real}, {w,_Real},...}, Evaluate[f(x,y,z,w,..,)],
RuntimeAttributes -> {Listable}, RuntimeOptions -> "Speed",
Parallelization -> True]

Where {x,y,z,...} can be as long as needed. I have tried the simple exercise of doing,

var={x,y,z}
list=Table[{var[[i]], _Re},{i,Length@var}];
Compile[list, Evaluate[f(x,y,z,w,..,)], RuntimeAttributes -> {Listable}, RuntimeOptions -> "Speed", Parallelization -> True]

but it does not seem to work since _Re needs to be symbolic? (I am asking but I do not know).

UPDATE:

I found a solution thanks to: Evaluating arguments of module (inside compile)

This is what it worked for me:

Compile[Evaluate@({#, _Real} & /@ var), Evaluate[f(x,y,z,w,..,)], RuntimeAttributes -> {Listable}, RuntimeOptions -> "Speed", Parallelization -> True]

• Why not just have your compiled function accept a list of reals? – J. M.'s technical difficulties May 11 at 10:17
• Hi J.M. thanks for replying. I am not sure I understand what you mean. Are you asking to pass to compile a list of reals instead of a function? If this is the case, this does not work for me since the list is huge and I have to pass it many times. – Xisco May 11 at 15:13

I think this is what @JM was suggesting:

cfun = Compile[{{list, _Real, 1}}, Sin[list]];

In other words, define your function to take a rank-1 tensor (i.e. a list) of reals; you do not need to specify the length beforehand! Then, just evaluate whatever you need on that list. For instance, here are two examples with lists of different lenghts:

cfun[Range[10000]]
(* {0.841471, 0.909297, 0.14112, -0.756802, ..., 0.992973, 0.636087, -0.305614}*)

cfun[Range[3]]
(* {0.841471, 0.909297, 0.14112} *)


text:
Math Help - Differential Equations

1. Differential Equations

Recall that a set of functions is linearly independent if their Wronskian is not identically 0. Calculate the Wronskian of the set to show that they are linearly independent. Write the differential equation for which they are solutions.

The set is:

$e^{\alpha x}, \ xe^{\alpha x}$ <---- I believe that these are not linearly independent, is that the case?

I can do the wronskian... What does it mean to write the differential equation? I'm not sure how to proceed.

2. I would do the Wronskian only to check: Let $a,b \in \mathbb{R}$ such that $ae^{\alpha x} +bxe^{\alpha x} \equiv 0$ then evaluating in $x=0$ gives $a=0$ and in $x=1$ gives $b=0$ so they are l.i.

As for the second question here's what I would do:

Clearly $g' - \alpha g =0$ is satisfied by $e^{ \alpha x}$. The second one is trickier let $a(x), b(x) \in C^0(\mathbb{R} )$ such that $b(x)f'(x)+a(x)f(x) \equiv 0$ then after some calculations we get $a(x)= \frac{ -b(x)(1+ \alpha x) }{ x}$ now $a(x) \notin C^0(\mathbb{R} )$ but we take $c(x)=x$ and $d(x)= 1+ \alpha x$ and we get $c(x)f'(x)+d(x)f(x) \equiv 0$ (this results from taking $b \equiv 1$, substituting $a,b$ in the equation and then multiplying by $x$ the whole thing). And with this equations we have a representation of $V:=span\{ e^{\alpha x}, xe^{\alpha x} \} \subseteq C^1(\mathbb{R} )$ and what I think you're asked is to try and find a second order (linear) dif. eq. such that $V$ $\cap C^2(\mathbb{R} )$ is the set of solutions of said equation.

3. I appreciate that, my understanding in the subject is now a lot better because of that explanation, but you are right, the question "Write the differential equations for which they are solutions." means to find one differential equation that both of those satisfy, and it would surely be the one that has the solution set $V\cap C^2(\mathbb{R})$.

Can you get me started on finding that equation? I'll work on it and see what you think.

By the way, can you see what I've done wrong in the wronskian? I can't seem to get it to be identically zero...

$W = \left| \begin{array}{cc}
e^{\alpha x} & xe^{\alpha x} \\
\alpha e^{\alpha x} & (e^{\alpha x} + \alpha x e^{\alpha x}) \end{array} \right|$

$W = e^{\alpha x}(e^{\alpha x} + \alpha x e^{\alpha x}) - \alpha x e^{2\alpha x} = e^{2\alpha x}$