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## Playing with 3d printed versions of shapes theorized by Hermann Schwarz

Saw a neat tweet earlier today about 3d printing, math, and engineering:

I recognized some of the shapes in the article as ones that we’d played with before:

The grey shape displayed in the article is a “made thicker for 3d printing” version of the surface $\cos(x) + \cos(y) + \cos(z) = 0.$ I thought it would be fun to print that shape today and use it for a little project with the kids tonight. Here’s the Mathematica code and what the print looks like in the Preform software:

8 hours later the print finished and I asked the boys to describe that shape plus the gyroid. It is always fascinating to hear what kids see in unusual shapes. My younger son went first:

Here’s what my older son had to say (and he’s starting to study trig, so we could go a tiny bit deeper into the math behind the shape I printed today):

Next we watched the video about the shapes made by Rice University:

After watching the video I asked the boys to talk about some of the things they learned:

Of course, mostly they didn’t want to talk about the shapes – they wanted to stand on them! So much for an 8 hour print and 45 min of trying to clean out the supports . . .

Here’s how the standing went:

Definitely a fun project and a fun way to show kids a current application of both theoretical math and 3d printing!

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## Using 3d printing to help explore a few ideas from introductory algebra

Last spring I was playing around with some different 3d printing ideas and found a fun way to explore a common algebra mistake:

Does (x + y)^2 = x^2 + y^2

comparing x^2 + y^2 and (x + y)^2 with 3d printing

Today I decided to revisit that project. We started by looking at the same idea from algebra:

Does $x^2 + y^2 = (x + y)^2$ ?

At first we talked about the two equations using ideas from algebra and arithmetic.

/

Now I asked the boys for their geometric intuition and then showed them the 3d printed graphs of the two functions.

This part ran a little long while my younger son was stuck on a small but important point about the graph $z = (x + y)^2$ – I didn’t want to tell him the answer and it took a couple of minutes for him to work through the idea in his mind.

/

Next I showed them 3d prints of $x^3 + y^3$ and $(x + y)^3$ and asked them to tell me which one was which. It is really neat to hear the reasoning that kids use to go from shapes to equations.

/

For the last part of the project I asked the boys to come up with their own algebra “mistakes” for us to explore. My older son chose to compare the graphs of $\sqrt{x^2 + y^2}$ and $x + y$.

/

My younger son chose the two equations $x^2 - y^2$ and $(x - y)^2$. Changing the + to a – in our first set of equations turns out to have some pretty interesting geometric consequences – “it looks sort of like a saddle” was a fun comment.

One especially interesting idea here was exploring where $x^2 - y^2 = 0$. We used Mathematica’s ContourPlot[] function to explore those two lines because those lines weren’t immediately obvious on the saddle.

/

I’m happy to have had the opportunity to revisit this old project. I think exploring simple algebraic expressions is a fun and sort of unexpected application of 3d printing.

## Playing with sin() and cos()

My older son has just started the trigonometry this week. I know the topic can be a little dry at the beginning, so I wanted to show him more than just unit circle exercises.

Today we looked at a few fun curves that you can make just by playing around with trig functions.

I stared by showing some simple graphs and then we moved to some 3d shapes:

Next we took a cue from an old project inspired by Henry Segerman:

Playing with Shadows inspired by Henry Segerman

Hopefully both the shape and the shadows come through in the filming – I haven’t figured out how to shoot shadows very well yet.

Finally I let the kids play around with the Mathematica code for a bit to create their own shapes. They had a couple of pretty fun ideas.

There was one little issue that came up on my younger son’s plot, unfortunately. I didn’t have enough detail in the plot to multiply the range by 10. That’s why his picture fuzzed out quite a bit. I didn’t see the problem on the fly, though, and wasn’t able to fix it in real time.

I’m excited to help my son learn about trig. Hopefully a few projects like this one will help him see that that there’s more to trig than just triangles!

## Building Paula Beardell Krieg’s cube

Yesterday we studied how to build the pieces of Paula Beardell Krieg’s dissected cube:

3d printing Paula Beardell Krieg’s dissected cube shapes

Today the shapes were done printing and I had the kids talk about them one more time:

After that short conversation I had each kid tweak the code that we used to make the shapes to make a new shape. Here’s what my younger son made:

Here’s what my older son made:

Definitely a fun couple of days with these shapes. Will probably revisit them again in a few months.

## 3D Printing Paula Beardell Krieg’s dissected cube shapes

I’ve been thinking about exposing the boys to math through 3d printing lately. Today I decided to explore making Paula Beardell Krieg’s cube shapes with them. Here’s the exploration the boys did back in March when we first got them:

Even though we’ve played a bit with these shapes before I still thought that thinking through these yellow and pink shapes would be a fun challenge. The project turned out to be a tiny bit harder than I thought it would be, but it still was a nice conversation.

We started by first looking at the three pyramids that can come together to make a cube and continued by looking at what happens when you slice those shapes in half.

In the last video the boys were thinking about trying to describe these shapes by describing the lines that formed the edges. At the beginning of this video I told them that this particular approach was going to be tough since they didn’t know how to write equations of lines in 3 dimensions.

So, I had them continue to search for properties of the shapes that they could describe.

The boys were still struggling to find some ideas about the shape that went beyond the lines on the boundary, but we kept looking.

My older son hit on the idea that the shape was made from “stacking squares on top of each other.” We spent the rest of the video exploring that idea.

Now that we had the idea about stacking squares we went to Mathematica to try to create the shape. It took a few steps to move from the ideas about the squares to generating the code for the shape. We didn’t get all the way there during this video, but we did figure out how to make a cube.

Unfortunately I had to end the video since the camera was about to run out of memory.

While I was getting the videos off the camera the boys worked on how to change the cube shape to the pyramid shape. It was a good challenge for them and they got it. We talked about that shape for a bit and then moved on to the challenge of creating the “pink” and “yellow” shapes that Paula Beardell Krieg created from paper.

We had a little bit of extra time today and it was fun to walk through this challenging problem. I think creating shapes to 3d print is a really fun way to motivate math with kids. Can’t wait to use the printed shapes in a project tomorrow!

## Ten 3D Printing math projects to help students explore math

Yesterday I was able to watch the Global Math Project presentations (well, most of them) via the Facebook Live feed. Hopefully those videos will be preserved here:

The Global Math Project’s Facebook page

One tank that caught my eye was given by Henry Segerman. I’d guess that his work and Laura Taalman’s work account for at least 80% of what I know about exploring math through 3d printing.

As I write this post there are 96 prior posts with the “3D Printing” tag on my blog.  3D Printing is still pretty new, and I think many people around math are only starting to see its use in education. Segerman’s talk made me want to throw together a list of fun projects that we’ve done just in case anyone is looking for a starting point after seeing his talk.

Some of my original thoughts on exploring math through 3d printing can be found in this blog post from March 2014 which features two really neat videos from Brooklyn Tech and Laura Taalman:

Learning from 3D Printing

Here are some sample projects:

(1) James Tanton’s Geometry Problem and 3d printing

Since this blog post was inspired by a talk a James Tanton’s Global Math Project, it seems appropriate to kick it off with a project inspired by Tanton:

Here are some of the shapes we printed as we explored what the shape itself looked like:

and here are the two projects that we’ve done exploring this problem

James Tanton’s geometry problem and 3d printing

Revisiting James Tanton’s Tetrahedron Problem

(2) Hard to highlight just one project that Segerman Inspired, so here’s the first of 2

One of the Segerman’s examples in yesterday’s talk was about bubbles. He showed a few complicated bubble examples but there are simple ones that are amazing, too. Here’s an example showing that the “bubble” formed by dipping a tetrahedron in soap is the same shape as a 4-dimensional shape:

Talking about Henry Segerman’s 5-cell with my 5th grader

(3) A second idea from Segerman – exploring shadows

One of Segerman’s most beautiful creations is on the cover of his book:

It is incredibly fun to have kids explore this shape:

Here’s the project we did after seeing Segerman give a talk last fall:

Playing with Sahdows inspired by Henry Segerman

Here’s a link to all of our project inspired by him:

All of our Henry Segerman-inspired projects

(4) There is also no way to limit Laura Taalman’s work to one example.

Here’s a project where we explored some of here 3d printed knots – one of which was featured in Segerman’s talk yesterday:

Playing with some 3d printed knots

(5) Here’s another project inspred by Taalman – tiling pentagons

Taalman’s 3d printed tiling pentagon designs are one of the most amazing pieces at the intersection of math and education:

We’ve used them for several projects including making cookies!

Here’s that project

Learning about tiling pentagons from Laura Taalman and Evelyn Lamb

and here’s a link to all of our projects inspired by Laura Taalman:

All of our projects inspired by Laura Taalman

(6) Exploring connections between algebra and geometry

3d printing can come in handy for looking at math ideas that previously you could only study on paper or on the computer screen. For example, a common algebra mistake is to think that:

$(x + y)^2 = x^2 + y^2$

Here’s what these two surfaces look like:

Here’s two projects exploring these algebra ideas with the boys:

Comparing x^2 + y^2 and (x + y)^2 with 3d printing/a>

Comparing Sqrt(x^2 + y^2) and Sqrt(x^2) + Sqrt(y^2) with 3d Printing

(7) 3D printing can also be surprisingly useful for studying 2d geometry

We’ve done a few neat projects in this area.

(i) Which triangle has larger area, a 5-5-6 triangle or a 5-5-8 one?

A nice little triangle puzzle

A few follow ups to the triangle puzzle

https://mikesmathpage.wordpress.com/2017/03/01/a-nice-little-triangle-puzzle/

(ii) A neat geometry idea from Patrick Honner

Here’s how we used 3d printing to explore this triange:

Inequalities and Mr. Honner’s Triangles

(iii) A neat geometry problem shared by Tina Cardone

Here’s how I explored this problem with 3d printing

A Cool Geometry Problem Shared by Tina Cardone

which led to a fun and unexpected follow up:

A Follow Up to Our Tina Cardone Geometry Project

(8) 3d printing can be a fun way to review ideas from elementary geometry

In his talk yesterday Segerman mentioned a few prints that his undergraduate students created. As he showed this projects he talked about how the creation process really helps students understand and explore the underlying math.

In the project below, creating the shape of the tile helped me review and explore equations of lines with the boys:

Sharing a Craig Kaplan post with kids

(9) 3d printing can also make abstract math / advanced problems accessible

A few months back I saw this problem shared by Alexander Bogomolny:

Nassim Taleb’s look at the problem on Mathematica made me think that the problem could be shared with kids:

A project for kids inspired by Nassim Taleb and Alexander Bogomolny

After getting some intuition from this problem we extended the problem to 4 dimensions using Taleb’s approach. The prints were really fun to play with and it is amazing to hear kids talk about these shapes that come from 4 dimensions:

Here’s that project:

Extending our Bogomolny / Taleb project from 3 to 4 dimensions

(10) Using 3d printing for calculus and beyond

I’m written a few posts and done a few projects about how to use 3d printing to explore some basic ideas from Calculus.

That collection of posts is here:

Posts about 3d printing and calculus

But 3d printing can help you see even more advanced ideas. Here’s a cube inside of a dodecahedron, for example:

and, of course, many (most!) of examples that Henry Segerman showed in his talk yesterday are perfect for showing how 3d printing can help everyone experience some advanced ideas in mathematics.

I’ll end with the project we did yesterday, which is a delightful example of how 3d printing can help you explore a math idea:

Revisiting the Volume of a Sphere with 3d printing

## Revisiting the volume of a sphere with 3d printing

[Note: 10:30 am on Oct 7th, 2017 – had a hard stop time to get this out the door, so it is published without editing. Will (or might!) edit a bit later]

About two years I found an amazing design by Steve Portz on Thingiverse:

“Archimedes Proof” by Steve Portz on Thingiverse

We did a really fun project using the print back then:

The volume of a sphere via Archimedes

Today we revisited the idea. We began by talking generally about the volume of a cylinder:

The next part of the project was heading down the path to finding the volume of a cone. I thought the right idea would be to talk first about the volume of a pyramid, so I introduced pyramid volume idea through snap cubes.

Also, I knew something was going a little sideways with this one when we were talking this morning, but seeing the video now I see where it was off. The main idea here is the factor of 3 in the division. Ignore the height h that I’m talking about.

Next we looked at some pyramid shapes that we’ve played with in the past. The idea here was to show how three (or 6) pyramids can make a cube. This part was went much better than the prior one 🙂

The ideas here led us to guess at the volume formula for a cone.

Now that we’d talked about the volume formulas for a cone and a cylinder, we could use the 3d print to guess at the volume formala for the sphere.

With all of that prep work behind us, we took a shot at pouring water through the print. It worked nearly perfectly 🙂

I am really happy that Steve Portz designed this amazing 3d print. It makes exploring some elementary ideas in 3d geometry really fun!

## Thinking about a math appreciation class

Steven Strogatz had great series of tweets about math education earlier in the week. These two have stayed in my head since he posted them:

I know that last year Strogatz taught a college level course similar to the one he is describing in the tweets. We even used a couple of his tweets about the course material for some fun Family Math activities. For example:

Here’s a link to that set of projects:

Steven Strogatz’s circle-area exercise part 2 (with a link to part 1)

So, thinking back to projects like those got me thinking about all sorts of other ideas you could explore in an appreciation course. At first my ideas were confined to subjects that are traditionally part of pre-college math programs and were essentially just different ways to show some of the usual topics. Then I switched tracks and thought about how to share mathematical ideas that might not normally be part of a k-12 curriculum. Eventually I tried to see if I could come up with a (maybe) 3 week long exploration on a specific topic.  I chose folding and thought about what sort of ideas could be shared with students.

Below are 9 ideas that came to mind along with 30 second videos showing the idea.

(1) A surprise book making idea shown to me by the mother of a friend of my older son:

(2) Exploring plane geometry through folding

We’ve done many explorations like this one in the last couple of years – folding is an incredibly fun (and surprisingly easy) way for kids to explore ideas in plane geometry without having to calculate:

Our Patty Paper geometry projects

Here’s one introductory example showing how to find the incenter of a triangle:

(3) The Fold and Cut theorem

Eric Demaine’s “fold and cut” theorem is an fantastic bit of advanced (and fairly recent) math to share with kids. Our projects exploring “fold and cut” ideas are here:

OUr Fold and Cut projects

Here’s one fun fold and cut example:

(4) Exploring platonic solids with Laura Taalman’s 3d printed polyhedra nets

You can find Taalman’s post about these hinged polyhedra here:

Laura Taalman’s hinged polyhedra blog post on her Makerhome blog

And if you like the hinged polyhedra, here’s a gif of a dodecahedron folding into a cube!

Which comes from this amazing blog post:

The Golden Section, The Golden Triangle, The Regular Pentagon and the Pentagram, The Dodecahedron

[space filled in with random words to get the formatting in the blog post right 🙂 ]

(5) An amazing cube dissection made by Paula Beardell Krieg

We’ve also done some fun projects with shapes that I wouldn’t have thought to have explored with folded paper. Paula Beardell Krieg’s work with these shapes has been super fun to play with:

Our projects based on Paula Beardell Krieg’s work

(6) And Paula didn’t just stop with one cube 🙂

(7) Two more of Laura Taalman’s prints

Seemingly simple ideas about folding and bending can lead to pretty fantastic mathematical objects! These objects are another great reminder of how 3d printing can be used to make mathematical ideas accessible.

Here’s Taalman’s blog post about the Peano curve:

Laura Taalman’s peano curve 3d print

(8) Getting to some more advanced work from Erik Demaine and Joseph O’Rourke

As hinted at early with the Fold and Cut theorem, some of the mathematical ideas in folding can be extremely deep:

(9) Current research by Laura DeMarco and Kathryn Lindsey

Finally, the Quanta Magazine article linked below references current research involving folding ideas. The article also provides several ways to share the ideas with students.

Quanta Magazine’s article on DeMarco and Lindsey’s work

The two blog posts below show my attempt to understand some of the ideas in the article and share them with kids. The video shows some of the shapes we made while studying the article.

Trying to understand the DeMarco and Lindsey 3d folded fractals

Sharing Laura DeMarco’s and Kathryn Lindsey’s 3d Folded Fractals with kids

So, these are just sort of ideas that popped into my head thinking about one part of a math explorations class. Feels like you could spend three weeks on folding and expose kids to lots of fun ideas that they’d (likely) never seen before.

## Extending our Alexander Bogomolny / Nassim Taleb project from 3 to 4 dimensions

Last week I saw really neat tweet from Alexander Bogomolny:

The discussion about that problem on Twitter led to a really fun project with the boys:

A project for kids inspired by Nassim Taleb and Alexander Bogomolny

That project reminded the boys about a project we did at the beginning of the summer that was inspired by this Kelsey Houston-Edwards video:

Here’s that project:

One more look at the Hypercube

For today’s project I wanted to have the boys focus on the approach that Nassim Taleb used to study the problem posed by Alexander Bogonolny. That approach was to chop the shape into slices to get some insight into the overall shape. Here’s Taleb’s tweet:

So, for today’s project we followed Taleb’s approach to study a 4d space similar to the space in the Bogomolny tweet above. The space is the region in 4d space bounded by:

$|x| + |y| + |z| \leq 1$,

$|x| + |y| + |w| \leq 1$,

$|x| + |w| + |z| \leq 1$, and

$|w| + |y| + |z| \leq 1$,

To start the project we reviewed the shapes from the project inspired by Kelsey Houston-Edwards’s hypercube video. After that we talked about the equations we’d looked at in the project inspired by Alexander Bogomolny’s tweet and the shape we encountered there:

Next we talked a bit about the equations that we’d be studying today and I asked the boys to take a guess at some of the shapes we’d be seeing. We also talked a little bit about absolute value which briefly caused a tiny bit of confusion.

The next part of the project used the computer. First we reviewed Nassim Taleb’s approach to studying the problem posed by Alexander Bogomolny. I think it is really useful for kids to see examples of how people use mathematical ideas to solve problems.

The 2d slicing was a fascinating way to approach the original 3d problem. We’ll use the same idea (though in 3d) to gain some insight on the 4d shape.

One fun thing about this part of the project is that we encountered a few shapes that we’ve never seen before!

Finally, I revealed 3d printed copies of the shapes for the boys to explore. They immediately noticed some similarities with the hypercube project. It was also really interesting to hear them talk about the differences.

At the end, the boys think that the 4d shape we encountered in this project will be the 4d version of the rhombic dodecahedron. We’ve studied that shape before in this project inspired by a Matt Parker video:

Using Matt Parker’s Platonic Solid video with kids

I don’t know if we are looking at a 4d rhombic dodecahedron or not, but I’m glad that the kids think we are 🙂

It amazes me how much much fun math is shared on line these days. I’m happy to have the opportunity to share all of these ideas with my kids!

## Calculating the volume of our rhombic dodecahedron

Yesterday we did a fun project involving a rhombic dodecahedron:

A project for kids inspired by Nassim Taleb and Alexander Bogomolny

At the end of that project we were looking carefully at how you would find the volume of a rhombic dodecahedron in general. Today I wanted to move from the general case to the specific and see if we could calculate the volume of our shapes. This tasked proved to be much more difficult for the boys than I imagined it would be. Definitely a learning experience for me.

Here’s how we got going. Even at the end of the 5 min here the boys are struggling to see how to get started.

So, after the struggle in the first video, we tried to back up and ask a more general question -> how do we find the volume of a cube?

Now we grabbed a ruler and measured the side length of the cube. This task also had a few tricky parts -> do we include the zome balls, for example. But now we were making progress!

Finally we turned to finding the volume of one our our 3d printed rhombic dodecahedrons. We did some measuring and found how many of these shapes it would take to fill our zome shape and how many it would take to fill a 1 meter cube.

So, a harder project than I expected, but still fun. We’ve done so much abstract work over the years and that makes the concrete work a little more difficult (or unusual), I suppose. I’m happy for this struggle, though, since it showed me that we need to do a few more projects like this one.


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# Permutations with identical objects

Note: I tried asking this on math.stackexchange (here), but didn't really receive an answer - so I figured this might be the right place.

How can I find the number of k-permutations of n objects, where there are x types of objects, and r1, r2, r3 ... rx give the number of each type of object?

Example:

I have 20 letters from the alphabet. There are some duplicates - 4 of them are a, 5 of them are b, 8 of them are c, and 3 are d. How many unique 15-letter permutations can I make?

In the example:

n = 20
k = 15
x = 4
r1 = 4, r2 = 5, r3 = 8, r4 = 3

Furthermore, if there isn't a straightforward solution: how efficiently can this problem be solved?

-
Next time you re-ask a question, please include the URL of the question on the other site so that people do not have to search around. math.stackexchange.com/questions/2372/… –  Tsuyoshi Ito Aug 31 '10 at 3:11
@Tsuyoshi Ito: Sure, edited. –  Cam Aug 31 '10 at 3:18

Here's a sketch of an idea. Consider first the problem of computing the number of permutations of $k$ elements given the number of elements $n_i$ in each class. This is easy: it works out to $$\frac{k!}{\Pi_i n_i !}$$ because in any fixed string, you can permute the elements in a class without changing the string.
Now consider the problem of writing down the different partitions of $k$ into the sum of $x$ labelled numbers. Merely finding the sum itself can be done by determining the coefficient of $y^k$ in the polynomial $$\Pi_i \frac{y^{r_i+1}-1}{y-1}$$
But this is not enough, since you need to weight each such partition by a different number. However, we know that the contribution of $y^i$ must be $1/i!$, so the overall number we are looking for is $k!$ times the coefficient of $y^k$ in the polynomial $$\Pi_i \sum_{j=0}^{r_i} \frac{y^j}{j!}$$

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# Maxima & Minima

1. Jan 29, 2005

### footprints

Find the coordinates of a stationary point on the curve $$y= \frac{16x^3 + 4x^2 + 1}{2x^2}$$ And determine the nature of this points.

How I find the coordinates? I know the second part.

2. Jan 29, 2005

### \$id

You have to find the dy/dx of that equation and find points where that is 0

Hint: use the quotient rule to differentiate that

3. Jan 29, 2005

### arildno

You say you know about the second part.
We are therefore, I hope, in agreement that the x-values for stationary points are found by the equation y'(x)=0.
Let us denote a particular solution of this equation by X.
But then, the corresponding y-value for a stationary point on the curve (for which the x-value is X) is simply y(X).

4. Jan 29, 2005

### maverick280857

Assuming that you mean this by a stationary point, set the first derivative equal to zero.

EDIT--Oops...when I was replying to this post I didn't see others' responses. Anyway you have much better ones now. Cheers.

5. Jan 29, 2005

### footprints

How do you get the derivitive? When I let me derivitive equal to 0, I keep getting weird answers :grumpy:

6. Jan 29, 2005

### dextercioby

What do you mean by "weird"??Should they be "lovely"???

On the other hand i don't see a really "nice" cubic...
BTW,because "x=0" is not in the domain of the function,u can simplify the quartic on the numerator and end up with a cubic...

Daniel.

7. Jan 29, 2005

### footprints

I did that the first time I did it. I got the derivitive $$8 - 4x^{-3}$$
Is that correct?

8. Jan 29, 2005

### dextercioby

Yes,it's correct.Now solve the equation into reals...

Daniel.

9. Jan 29, 2005

### Nylex

What else would he mean?

10. Jan 29, 2005

### footprints

When $$8 - 4x^{-3} = 0$$, I get 0.7937..... (thats why I said it was weird). I didn't anything wrong did I?

11. Jan 29, 2005

### dextercioby

$$x=\frac{1}{\sqrt[3]{2}}$$
,then it is correct.

Daniel.

12. Jan 29, 2005

### footprints

Yeah, I got that. However my books answer is $$(\frac{1}{2}, 8)$$

13. Jan 29, 2005

### arildno

You have NOT calculated the derivative correctly (despite what others have told you):
$$y'(x)=\frac{2x^{2}(3*16x^{2}+8x)-4x(16x^{3}+4x^{2}+1)}{4x^{4}}=\frac{2x}{4x^{4}}*((3*16x^{3}+8x^{2})-(2*16x^{3}+8x^{2}+2))=\frac{16x^{3}-2}{2x^{3}}=8-\frac{1}{x^{3}}$$
Hence, your root is $$X=\frac{1}{2}$$

14. Jan 29, 2005

### arildno

15. Jan 29, 2005

### dextercioby

U're right,Arildno,my mistake...

The function,IIRC is
$$y(x)=8x+4+\frac{1}{2x^{2}}$$

whose derivative is immediate
$$y'(x)=8-x^{-3}$$

Daniel.

16. Jan 29, 2005

### arildno

footprints:
Take this as a typical example of why I HATE the differentiation rule for fractions!
It is the nastiest one, it is so easy to make a mistake.

Daniel has kindly provided you with a rewriting which gives you the correct answer right away.

17. Jan 29, 2005

### footprints

I don't understand. When differentiating $$y(x)=8x+4+\frac{1}{2x^{2}}$$, won't I get $$8 - 4x^{-3}$$?

$$\frac{dy}{dx}= 1*8x^{1-1}+ (-2)(2x^{-2-1})$$
$$\frac{dy}{dx}= 8 - 4x^{-3}$$
Whats wrong with that?
Maybe you could show me in detail how it is done?

18. Jan 29, 2005

### arildno

This is wrong; it should be:

$$\frac{dy}{dx}= 1*8x^{1-1}+ (-2)(\frac{1}{2}x^{-2-1})$$

19. Jan 29, 2005

### footprints

Oh right!!!! Thanks for the help guys!


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# 1.5 standard deviations within the mean

Asked Feb 11, 2010, 07:47 PM — 3 Answers
All my professor gave me was what proportion of measurements from a normal distribution falls within 1.5 standard deviations of the mean. No mean value was given and I do not understand how to solve the question. He goes on to ask the same thing only for is more than 1.5 above the mean, is more than 1.5 below the mean, and is between 1.5 and 1.8 above the mean. I am very frustrated

 galactus Posts: 2,272, Reputation: 1436 Ultra Member #2 Feb 12, 2010, 12:36 PM
This is a question about Chebychev's Theorem.

The portion of the data set lying within k standard deviations (k>1) of the mean is at least

$1-\frac{1}{k^{2}}$

Just plug in k=1.5.
 elscarta Posts: 116, Reputation: 103 Junior Member #3 Feb 14, 2010, 06:24 AM
Actually this question is about the Normal Distribution function, not about Chebychev's Theorem! You need to have a calculator capable of calculating Normal Distribution values or a Table which lists Normal Distribution values.

The reason that you were not given the mean (and by the way nor were you given the standard deviation) is because any Normal Distribution can be standardized to distribution with a mean of zero and a standard deviation of one.

Given that Z is the Standardized Normal Distribution function then the questions become:

P(-1.5 <Z<1.5), P(Z>1.5), P(Z<-1.5) and P(1.5<Z<1.8)

http://en.wikipedia.org/wiki/Standard_normal_table gives a table of values for P(Z<z)

Since the Normal Distribution is symmetrical the following property is true.

Property 1: P(Z<-z) = P(Z>z)

Also since the Normal Distribution is a probability density function the following also hold true.
Property 2: P(Z<∞) = 1
Property 3: P(Z>z) = 1 – P(Z<z)
Property 4: P(a<Z<b) = P(Z<b) – P(Z<a)

So P(-1.5 <Z<1.5)
= P(Z<1.5) – P(Z<-1.5) (Property 4)
= P(Z<1.5) – P(Z>1.5) (Property 1)
= P(Z<1.5) – (1 – P(Z<1.5)) (Property 3)
= P(Z<1.5) – 1 + P(Z<1.5)
= 2 x P(Z<1.5) – 1
= 2 x 0.9332 -1 (value from table)

= 0.8664

You should be able to do the remainder of the questions.
 galactus Posts: 2,272, Reputation: 1436 Ultra Member #4 Feb 14, 2010, 06:59 AM
Yeah, I reckon you're correct about that. I thought Chebyshev's theorem would have been close enough for this though. Doing the z-score is certainly better. The way you showed, all one has to do is look them up in the z-table. Much easier than fooling with Chebyshev's theorem and more accurate. after all, it says 'at least'.

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# Ratio Test/ Simple English

1. Nov 22, 2004

### karen03grae

Can anyone explain what exactly the ratio test is comparing?

undefinedSome say it is being compared to a geometric series...I love geo. series and I don't see how the test involves them.

Thanx,
Erin

2. Nov 23, 2004

### Galileo

Ok, Im a bit rusty onseries, but let's see if I remember why the ratio test works.

Ratio test:

If $\limsup \left| \frac{a_{n+1}}{a_n}\right| <1$ then the series $\sum a_n$ converges absolutely.
If $\limsup \left| \frac{a_{n+1}}{a_n}\right| >1$ then the series $\sum a_n$ diverges.

Ill just take the case where the limit < 1. And see why it converges absolutely in that case.
Suppose $\limsup \left| \frac{a_{n+1}}{a_n}\right| =\rho<1$
Then for all $\epsilon >0$, there is an N, so that for n>N:
$$\left| \frac{a_{n+1}}{a_n}\right-\rho|<\epsilon$$
or
$$-\epsilon + \rho < \frac{a_{n+1}}{a_n} < \epsilon +\rho$$
Since $\rho$ is smaller than 1, we can choose an epsilon small enough so that $\epsilon +\rho=r<1$. Then if N is big enough:
$$|\frac{a_{n+1}}{a_n} |< r$$
or
$$|a_{n+1}| < |a_n|r$$
and also
$$|a_{n+2}| < |a_{n+1}|r<|a_n|r^2$$
and so on...
$$|a_{n+k}| < |a_n|r^k$$
for n>N.
Therefore:
$$\sum_{n=0}^{\infty}|a_n|\leq \sum_{n=N}^{\infty}|a_n|<\sum_{n=0}^{\infty}|a_N|r^n$$
So the series is smaller than a convergent geometric series.

Last edited: Nov 23, 2004
3. Nov 23, 2004

### mathwonk

same thing in words,

tests of convergence are usually comparisons. i.e. if a series of positive terms is smaller term by term than some convergent series, then it is also convergent.

This is true even if it only holds from some finite point onward.

The best series to compare with is the geometric series. A geometric series is a series in which the ratio of each term divided by the previous term is the same number. if that number is less than one it converges.

hence if a series has the property that the ratios of each term divided by the previous term eventually all become less than some number r with 0<r<1, then that original series is eventually smaller than the geometric series with ratio r, hence also convergent.

the best possible treatment of convergence of series is in richard courants calculus book.

4. Nov 27, 2004

### karen03grae

Thanks...So may I assume that the ratio of successive terms of a power series ,
_
\
/ cn(x-a)^n, near infinity is constant? Because when we do the ratio test, we are basically testing the ratio of successive terms at infinity.
_

5. Nov 28, 2004

### mathwonk

you do not need them to be constant, just for all of them eventually i.e. near infinity) to be less than some number which itself is less than one. the easy case is where these ratios converge to a limit less than one.

bewst npossible example: exponential series: the nth term is

x^n/n! so the ratio of the nth by the n-1st term is x/n. as n goes to infinity this approaches 0, no matter what x is. so it converges no matter what x is.

Last edited: Nov 28, 2004


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# Why does the trace of the Jacobian and its eigenvalues determine equilibrium stability?

I am a high school student studying differential equations! I just cannot understand exactly why the trace of the Jacobian Matrix and its eigenvalues determine equilibrium stability (I encountered the Jacobian when learning about the Lotka-Volterra equations). Could anybody offer an explanation without invoking topology or Lyapunov stability? Thank you.

• "Why does the trace of the Jacobian and its eigenvalues determine equilibrium stability?" In general they don't, so it is impossible to give an answer to your question. – John B Feb 24 '18 at 13:30
• That's interesting! Most sources I have found claim that whether the eigenvalues of the Jacobian at the equilibrium point(s) are purely imaginary, real, or have both real and imaginary parts determines the kind of stability in a system. I certainly might be wrong so could you please point to a source that opposes that? Thank you very much. – George Orf. Feb 24 '18 at 13:35
• The condition you describe in your comment is not the condition you describe in your question (and in the title). Before asking for references, perhaps make up your mind? – Did Feb 24 '18 at 13:54
• Consider this, math.stackexchange.com/questions/546155/…. But e.g. the trace of $\mathrm{diag([-3,1])}$ is $-2$, however unstable, the trace of $\mathrm{diag([-1,-1])}$ is $-2$, and corrsponding system is stable. – Carlos Feb 24 '18 at 14:20
• I sincerely apologize for any potential discrepancy in the title and comment @Did and would certainly be open to editing the title. I am new on StackExchange and open to constructive feedback. My question is possibly better phrased in my comment above. – George Orf. Feb 24 '18 at 14:32

A system is stable if all eigenvalues $\lambda _i$, satisfy $$Re(\lambda _i) < 0.$$ Otherwise the positive real part of eigenvalue will generate an exponential function which diverges and cause the equilibrium to be unstable.

• I see! Could you perhaps elaborate on the generation of a divergent exponential function? Thank you! – George Orf. Feb 24 '18 at 14:25
• Solutions of linearized system are linear combinations of $e^{\lambda _it}$ where $\lambda _i$s are eigenvalues. The linearized system approximates the original system. – Mohammad Riazi-Kermani Feb 24 '18 at 14:42
• @MohammadRiazi-Kermani Thank you for your explanation, sir! Can you recommend any sites/books about the linearization of systems? – George Orf. Feb 24 '18 at 14:49
• The differential equation book by Polking,Boggess, and Arnold published by Pearson is a down to earth undergraduate book which explains systems very well. – Mohammad Riazi-Kermani Feb 24 '18 at 14:57
• The question is not whether the linearized system is un/stable but if the original system is. Instead of arguing about trivialities, you could correct the answer. – Did Feb 24 '18 at 18:34


text: In general, the trinomial of the ax 2 + bx + c is a perfect square if the discriminant is zero; that is, if b 2 -4ac = 0, because in this case it will only have one root and can be expressed in the form a (xd) 2 = (√a (xd)) 2 , where d is the root already mentioned. Step 3: Write -13x as the sum of -3x and -10x: 5x 2 - 3x - 10x + 6 . The argument appears in the middle term. Just 3 easy steps to factoring trinomials. + ?). For example: $$x^2 + y^2 + xy$$ and $$x^2 + 2x + 3xy$$. Here is the form of a quadratic trinomial with argument x:. General - scope of application Our terms and conditions of sale shall apply exclusively to all deliveries - including future deliveries; we do not recognise any terms and conditions of the customer that conflict with or deviate from our terms and conditions of sale unless we have expressly agreed to their validity … Define trinomial. For K-12 kids, teachers and parents. * (? Binomial: It is an expression that has two terms. [See the related section: Solving Quadratic Equations.] It can be variables, constants and mathematical operators. In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial.According to the theorem, it is possible to expand the polynomial (x + y) n into a sum involving terms of the form ax b y c, where the exponents b and c are nonnegative integers with b + c = n, and the coefficient a of each term is a specific positive … ‘We show that the unconditional means derived in this paper are actually the positions of the central nodes in HW's trinomial tree.’ 2 Biology (of a taxonomic name) consisting of three terms where the first is the name of the genus, the second that of the species, and the third that of the subspecies or variety. Consisting of three names or terms, as a taxonomic designation. 3. Step 4: Group the two pairs of terms: (5x 2 - 3x) - (10x + 6) . The expressions $$x^2 + 2x + 3$$, $$5x^4 - 4x^2 +1$$ and $$7y - \sqrt{3} - y^2$$ are trinomial examples. So to get in the … Step 2: Find of two factors of 30 that add up to 13: 3 and 10 . (This is the part where you are moving the other way). Trinomial definition, consisting of or pertaining to three terms. Solution It means that the highest power of the variable cannot be greater than 2. Trinomial: It is an expression that has three terms. See more. You will be given the trinomial and in order to factor the trinomial, you will need to work backwards to find the two binomials. Monomial: It is an expression that has one term. 7th Grade Math Problems 8th Grade Math Practice From Square of a Trinomial to HOME PAGE. Some examples of monomial are $$8, … A general polynomial (of one variable) could have any number of terms: Degree 2 (Quadratic) can have letters a,b,c: ax 2 + bx + c Degree 3 (Cubic) can have letters a,b,c,d: ax 3 + bx 2 + cx + d Notice that every monomial, binomial, and trinomial is also a polynomial. Well, the first term, x 2, is the square of x.The third term, 25, is the square of 5.Multiplying these two, I get 5x.. Multiplying this expression by 2, I get 10x.This is what I'm needing to match, in order for the quadratic to fit the pattern of a perfect-square trinomial. sum or difference of two cubes: For example, 5x 2 − 2x + 3 is a trinomial. New! The first term came from multiplying the first term in each binomial. Leave me a comment in the box below. 5x 2 - 13 x + 6 . Factoring a Trinomial with Leading Coefficient 1. In general, there are three types of polynomials. Factoring-polynomials.com supplies great facts on Trinomial Factoring Calculator, subtracting fractions and rational numbers and other math subject areas. To be a perfect square trinomial, it must factor into two identical binomials, which means the first term and third term must both be perfect squares. We use the words ‘monomial’, ‘binomial’, and ‘trinomial’ when referring to these special polynomials and just call all the rest ‘polynomials’. You have learned that a term is a constant or the product of a constant and one or more variables. In context|algebra|lang=en terms the difference between trinomial and binomial is that trinomial is (algebra) an expression consisting of 3 terms while binomial is (algebra) a quantity expressed as the sum or difference of two terms. Let's look at an example. A trinomial is an equation that consists of three terms. To figure out how we would factor a trinomial of the form \(x^2+bx+c$$, such as $$x^2+5x+6$$ and factor it to $$(x+2)(x+3)$$, let’s start with two general … The term ‘a’ is referred to as the leading coefficient, while ‘c’ is referred to as the absolute term of f (x). The polynomial root is a number where the polynomial becomes zero; in other words, a number that, by replacing it with … They are special members of the family of polynomials and so they have special names. Definition. Ex: x, y, z, 23, etc. Example: Factor the following trinomial using the grouping method. An example is the equation = + studied by Johann Heinrich Lambert in the 18th century. A trinomial equation is a polynomial equation involving three terms. a, b, c are called constants.In this quadratic, 3x 2 + 2x − 1,. the constants are 3, 2, −1. When a = 1, the trinomial becomes x 2 + bx + c and it is easier to factor. Ask a … UnFOILing is a method for factoring a trinomial into two binomials. While factoring trinomials, it will result in two binomials. When it is of the form $$a{x}^{m}$$, where $$a$$ is a constant and $$m$$ is a whole number, it is called a monomial. There are two general formulas for factoring a perfect square trinomial: x^2 + 2xy + y^2 = (x + y)^2, and x^2 - 2xy + y2 = (x - y)^2. In many applications in mathematics, we need to solve an equation involving a trinomial.Factoring is an important part of this process. To factor the trinomial means to start with the product, and end with the factors. A quadratic equation is a polynomial of second degree usually in the form of f(x) = ax 2 + bx + c where a, b, c, ∈ R and a ≠ 0. You need to think about where each of the terms in the trinomial came from. Second-degree equations have two possible solutions: and The graph of a second-degree equation produces a parabola. The above trinomial examples are the examples with one variable only, let's take a few more trinomial examples with multiple variables. The argument is whatever is being squared. In general, for a trinomial of the form ${x}^{2}+bx+c$ you can factor a trinomial with leading coefficient $1$ by finding two numbers, $p$ and $q$ whose product is c, and whose sum is b. Second-degree equations involve at least one variable that is squared, or raised to a power of two. To factor the trinomial means to start with the product, , and end with the factors, . Previously, various methods have been used to provide the proofs for the general terms of these two series. When you multiply two binomials together, you use the FOIL method, multiplying the First, then the Outer, then the Inner, and finally the Last terms of the two binomials into a trinomial. Thus, the formula of square of a trinomial will help us to expand. When these two binomial terms are multiplied, it results in a given trinomial. Some notable trinomials. Math explained in easy language, plus puzzles, games, quizzes, videos and worksheets. All the roots of the general nth degree trinomial admit certain convenient representations in terms of the Lambert and Euler series for the asymmetric and symmetric cases of the trinomial equation, respectively. The “easy” case happens when the value of a is equal to +1 or - 1, that is a = 1 or a = - 1.You don’t need to write the coefficient of 1 before the {x^2} term because it is understood.. Factoring Trinomials. Quadratics in different arguments. adj. trinomial synonyms, trinomial pronunciation, trinomial translation, English dictionary definition of trinomial. If ever you need assistance on rational functions or even inequalities, Factoring-polynomials.com is certainly the ideal place to … Factor the GCF from the middle terms. Factoring Trinomial – Easy Case. Taking n to be any real or complex number, … Comments Have your say about what you just read! Ex: 2x+y, x 2 – x, etc. Solution: Step 1: Find the product ac: (5)(6) = 30 . A quadratic trinomial is a polynomial with three terms and the degree of the trinomial must be 2. Binomial is a see also of trinomial. But when you need to factor a trinomial, you unFOIL by determining […] As adjectives the difference between trinomial and binomial 1. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture. A trinomial is a perfect square trinomial if it can be factored into a binomial multiplied to itself. To factor the trinomial means to start with the product, and end with the factors. x is being squared.x is called the argument. Factors of Quadratic Trinomials of the Type x 2 + bx + c. The Distributive Law is used in reverse to factorise a quadratic trinomial, as illustrated below.. We notice that: 5, the coefficient of x, is the sum of 2 and 3.; 6, the independent term, is the product of 2 and 3. Example 1. Perfect Square Trinomial – Explanation & Examples. Trinomial is a see also of binomial. For example, x 2 +x – 6 is a trinomial. Factor x 2 − 5x − 6. Example 1: Factoring Trinomials. One of the most well-known second-degree equations is the quadratic where a, b, and c are constants and a is not equal 0. + ?) Factoring General Trinomials Lecture Slides are screen-captured images of important points in the lecture. In a perfect square trinomial… This lesson will only show you how to factor when a = 1. All the roots of the general nth degree trinomial admit certain convenient representations in terms of the Lambert and Euler series for the asymmetric and symmetric cases of the trinomial … They are Monomial, Binomial and Trinomial. Consider the quadratic trinomial x^2 + 24x + 144. A trinomial meaning in math is, it is a type of polynomial that contains only three terms. A trinomial is a 3 term polynomial. ax 2 + bx + c.. The general form of a trinomial is ax 2 + bx + c. Your goal in factoring trinomials is to make ax 2 + bx + c equal to (? Trinomial equation. Let's take a look at another example. Identifying Polynomials, Monomials, Binomials and Trinomials. General terms of delivery and payment (terms and conditions): 1. Foil to find the product. To figure out how we would factor a trinomial of the form such as and factor it to let’s start with two general binomials of the form and . A trinomial can take any form such as perfect square trinomial, a difference of squares and so on. The general form of a quadratic trinomial is written as a{x^2} + bx + c where a, b, and c are constants. Step 5: Take out the common factors from each group: It does not mean that a quadratic trinomial always turns into a quadratic equation when we equate it to zero. ) = 30 on trinomial factoring Calculator, subtracting fractions and rational numbers and other math subject.... And Trinomials section: Solving quadratic equations. squared, or raised to power. X^2 + y^2 + xy\ ) and \ ( x^2 + 2x + 3 a. Perfect square trinomial… it can be variables, constants and mathematical operators and worksheets trinomial. Be greater than 2 solve an equation that consists of three terms trinomial: it is easier to factor trinomial... Are the examples with one variable only, let 's take a few more examples! 2: Find the product,, and end with the product, and end with the factors the trinomial... Will help us to expand of -3x and -10x: 5x 2 - 3x ) (! Practice from square of a trinomial equation is a polynomial equation involving a is. Can take any form such as perfect square trinomial, a difference squares... To start with the product, and end with the product of a constant the. 18Th century methods have been used to provide the proofs for the general terms of these two.... With the factors subtracting fractions and rational numbers and other math subject areas a trinomial equation is a polynomial three! More trinomial examples with one variable only, let 's general trinomial meaning a few more trinomial examples with variable! Highest power of the trinomial means to start with the product, and end with the product,... Equations have two possible solutions: and the degree of the variable not. Factoring Calculator, subtracting fractions and rational numbers and other math subject areas 3xy\ ) x... Images of important points in the trinomial means to start with the product of a equation... Of 30 that add up to 13: 3 and 10 this process the product, and with... Trinomial synonyms, trinomial translation, English dictionary definition of trinomial the trinomial came from multiplying first! +X – 6 is a constant or the product of a trinomial to HOME PAGE quadratic trinomial is trinomial. Mean that a term is a trinomial will help us to expand Grade... Must be 2 factoring Calculator, subtracting fractions and rational numbers and other math subject areas x. Equation = + studied general trinomial meaning Johann Heinrich Lambert in the trinomial means to start with the factors constant one... Factoring a trinomial is a constant or the product, and end with the,... Quadratic trinomial always turns into a quadratic trinomial with argument x: facts trinomial... More variables 3 is a trinomial equation is a trinomial to HOME PAGE a polynomial with three terms 3 a! One term let 's take a few more trinomial examples with multiple....: Solving quadratic equations. English dictionary definition of trinomial step 4: Group the pairs! Trinomial will help us to expand squared, or raised to a power of the trinomial means to with! Each binomial a few more trinomial examples are the examples with one variable only, let 's a... Above trinomial examples with one variable that is squared, or raised a! 4: Group the two pairs of terms: ( 5x 2 − +. 18Th century of -3x and -10x: 5x 2 − 2x + 3 is a into! 7Th Grade math problems 8th Grade math problems 8th Grade math problems 8th Grade math problems 8th Grade problems... Subject areas general terms of these two series 18th century math subject areas few... Take any form such as perfect square trinomial… it can be variables, and... Easier to factor the trinomial becomes x 2 + bx + c and it is expression. To general trinomial meaning the proofs for the general terms of these two series 24x 144. With one variable that is squared, or raised to a power of two points! Here is the form of a constant and one or more variables and one or more variables square... It is an expression that has one term a method for factoring a trinomial is an that... + bx + c and it is an expression that has one term to do practice as... Are special members of the terms in the trinomial must be 2 polynomial equation three... + studied by Johann Heinrich Lambert in the … Identifying polynomials, Monomials, binomials and Trinomials = studied! Find the product of a constant or the product,, and end with the.. 5 ) ( 6 ) example, x 2 – x, y, z,,... Way ) to think about where each of the general trinomial meaning of polynomials in a perfect trinomial! In two binomials the graph of a quadratic equation when we equate it to zero each... Explained in easy language, plus puzzles, games, quizzes, videos worksheets. Download and print out these lecture slide images to do practice problems as well as take notes watching! From square of a trinomial equation is a constant and one or more variables easy,. Translation, English dictionary definition of trinomial an important part of this.. X^2 + 24x + 144, quizzes, videos and worksheets - 10x + 6 here is the of... Write -13x as the sum of -3x and -10x: 5x 2 - 3x - 10x +.! As well as take notes while watching the lecture 's take a more... Monomial: it is an important part of this process practice problems as well take. English dictionary definition of trinomial images of important points in the lecture of important points in the … Identifying,! Comments have your say about what you general trinomial meaning read subject areas trinomial always into. When these two binomial terms are multiplied, it will result in two binomials equate it to zero the century. About where each of the trinomial means to start with the factors, or the product,, and with... Of terms: ( 5 ) ( 6 ) greater than 2,! Terms are multiplied, it results in a given trinomial solution Factoring-polynomials.com great... You need to think about where each of the terms in the … Identifying,. Produces a parabola: 3 and 10 to think about where each of the terms in lecture... Terms and the degree of the variable can not be greater than 2 have been used to provide the for! Watching the lecture complex number, … Define trinomial ac: ( 5x 2 - 3x ) (! Types of polynomials an important part of this process term is a trinomial to HOME PAGE becomes x –!, constants and mathematical operators in easy language, plus puzzles, games quizzes! 6 is a method for factoring a trinomial will help us to expand section! Multiplied, it results in a perfect square trinomial… it can be variables, constants and mathematical operators any... ( x^2 + 2x + 3xy\ ) … Identifying polynomials, Monomials, binomials and Trinomials of points... Two series always turns into a quadratic trinomial with argument x:: Find the product ac: ( )... The 18th century two pairs of terms: ( 5x 2 - 3x - 10x 6. Of these two series types of polynomials and so on 6 is a polynomial with terms... Lambert in the trinomial means to start with the product of a trinomial equation is a trinomial help! Get in the trinomial means to start with the product, and end with the product a! Not mean that a term is a constant or the product, and end the... Lecture Slides are screen-captured images of important points in the lecture taxonomic designation 23 etc! More variables, there are three types of polynomials and so on and! Or terms, as a taxonomic designation x 2 – x, etc and worksheets have used! Step 3: Write -13x as the sum of -3x and -10x: 5x 2 - 3x ) (... Of three names or terms, as a taxonomic designation videos and worksheets the pairs... Lesson will only show you how to factor the trinomial means to start with product... Section: Solving quadratic equations. 5x 2 - 3x ) - ( 10x + 6 ) ) (. The sum of -3x and -10x: 5x 2 − 2x + 3xy\ ) with argument x: the where... About where each of the terms in the … Identifying polynomials, Monomials, binomials and.! Trinomial… it can be variables, constants general trinomial meaning mathematical operators the related section: Solving quadratic equations ]! As take notes while watching the lecture multiple variables argument x: … Identifying polynomials, Monomials, and! Equation is a constant or the product,, and end with the product,, and end with factors! A method for factoring a trinomial into two binomials and print out these lecture images!, we need to think about where each of the terms in the lecture to get in 18th... A few more trinomial examples with multiple variables is squared, or to. Are screen-captured images of important points in the trinomial becomes x 2 +x – 6 a... Pronunciation, trinomial pronunciation, trinomial translation, English dictionary definition of trinomial pronunciation, general trinomial meaning pronunciation trinomial! It can be variables, constants and mathematical operators 2 + bx + c and it is easier factor..., subtracting fractions and rational numbers and other math subject areas these two series the century! Related section: Solving quadratic equations., as a taxonomic designation mean...: Find of two factors of 30 that add up to 13 3..., plus puzzles, games, quizzes, videos and worksheets 5 ) 6...


text:
## Sun, Jingxian

Compute Distance To:
 Author ID: sun.jingxian Published as: Sun, Jingxian; Sun, jing Xian; Sun, Jing-xian; Sun, Jing Xian External Links: dblp
 Documents Indexed: 167 Publications since 1983 1 Further Contribution Reviewing Activity: 21 Reviews Co-Authors: 42 Co-Authors with 126 Joint Publications 1,395 Co-Co-Authors
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### Co-Authors

 35 single-authored 20 Cui, Yujun 12 Zhang, Guowei 11 Xu, Xian 10 Li, Hongyu 8 Liu, Xiaoying 8 Zhang, Kemei 7 Liu, Zhaoli 7 Zhang, Xiaoyan 4 Liu, Lishan 4 Sun, Jinli 4 Sun, Yong 4 Wang, Feng 4 Zhao, Zengqin 3 Guo, Dajun 3 Ji, Hongwei 3 O’Regan, Donal 3 Yang, Zhilin 3 Zhang, Jinqing 3 Zhao, Lühuizi 2 Song, Fumin 2 Sun, Li 2 Sun, Yongzheng 2 Zhang, Tie 2 Zhang, Xingqiu 2 Zou, Yumei 1 Ang, Miin Huey 1 Chen, Fangqi 1 Ciu, Yujun 1 de Silva, Clarence Wilfred 1 Hong, Gap-Seon 1 Hu, Shouchuan 1 Li, Lai 1 Li, Youtang 1 Li, Zhilong 1 Li, Zhong Hua 1 Liu, Yansheng 1 Lou, Bendong 1 Lu, Haixia 1 Lu, Huiqin 1 Poo, Aun Neow 1 Qi, Guijie 1 Su, Jun 1 Sun, Fei 1 Sun, Jie 1 Tan, Kok Kiong 1 Wang, Guangwa 1 Wang, Xi 1 Xing, Hui 1 Zheng, Xiongjun 1 Zhou, Yanqiu
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### Serials

 12 Acta Mathematica Sinica 12 Journal of Shandong University. Natural Science Edition 11 Journal of Mathematical Analysis and Applications 11 Journal of Systems Science and Mathematical Sciences 10 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 8 Chinese Annals of Mathematics. Series A 6 Journal of Mathematical Research & Exposition 5 Acta Mathematica Scientia. Series A. (Chinese Edition) 5 Acta Mathematica Sinica. Chinese Series 4 Indian Journal of Pure & Applied Mathematics 4 Journal of Shandong University. Natural Science 3 Applicable Analysis 3 Mathematics in Practice and Theory 3 Annals of Differential Equations 3 Journal of Qufu Normal University. Natural Science 3 Mathematica Applicata 3 Positivity 3 Acta Analysis Functionalis Applicata. AAFA 3 Fixed Point Theory and Applications 2 Computers & Mathematics with Applications 2 Acta Mathematicae Applicatae Sinica 2 Acta Mathematicae Applicatae Sinica. English Series 2 Journal of Jiangxi Normal University. Natural Science Edition 2 Kexue Tongbao 2 Abstract and Applied Analysis 2 Electronic Journal of Qualitative Theory of Differential Equations 2 Journal of Systems Science and Complexity 2 Acta Mathematica Scientia. Series B. (English Edition) 2 Journal of Xuzhou Normal University. Natural Science Edition 2 Boundary Value Problems 2 Chinese Journal of Engineering Mathematics 2 Nonlinear Analysis. Theory, Methods & Applications 1 Journal of Fluid Mechanics 1 Journal of the Mechanics and Physics of Solids 1 Rocky Mountain Journal of Mathematics 1 Applied Mathematics and Computation 1 Journal of Computational and Applied Mathematics 1 Journal of Differential Equations 1 Monatshefte für Mathematik 1 Proceedings of the American Mathematical Society 1 Theoretical Population Biology 1 Topology and its Applications 1 Numerical Mathematics 1 Advances in Mathematics 1 Northeastern Mathematical Journal 1 Mathematical and Computer Modelling 1 Journal of Scientific Computing 1 Journal of Applied Mathematics and Stochastic Analysis 1 Differential and Integral Equations 1 Applied Mathematics 1 Topological Methods in Nonlinear Analysis 1 Calculus of Variations and Partial Differential Equations 1 Numerical Linear Algebra with Applications 1 Discrete and Continuous Dynamical Systems 1 Journal of Heuristics 1 Journal of Vibration and Control 1 Chinese Quarterly Journal of Mathematics 1 Acta Mathematica Sinica. English Series 1 Journal of Sichuan Normal University. Natural Science 1 Communications on Pure and Applied Analysis 1 Fixed Point Theory
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### Fields

 117 Operator theory (47-XX) 66 Ordinary differential equations (34-XX) 25 Integral equations (45-XX) 15 Global analysis, analysis on manifolds (58-XX) 13 Partial differential equations (35-XX) 7 Functional analysis (46-XX) 6 Numerical analysis (65-XX) 5 General topology (54-XX) 3 Calculus of variations and optimal control; optimization (49-XX) 3 Mechanics of deformable solids (74-XX) 2 Linear and multilinear algebra; matrix theory (15-XX) 2 Dynamical systems and ergodic theory (37-XX) 2 Algebraic topology (55-XX) 1 Order, lattices, ordered algebraic structures (06-XX) 1 Fluid mechanics (76-XX) 1 Operations research, mathematical programming (90-XX) 1 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 1 Biology and other natural sciences (92-XX) 1 Systems theory; control (93-XX)

### Citations contained in zbMATH Open

100 Publications have been cited 788 times in 540 Documents Cited by Year
Invariant sets of descending flow in critical point theory with applications to nonlinear differential equations. Zbl 0995.58006
Liu, Zhaoli; Sun, Jingxian
2001
Positive solutions of $$m$$-point boundary value problems. Zbl 1069.34037
Zhang, Guowei; Sun, Jingxian
2004
Nodal solutions for $$m$$-point boundary value problems using bifurcation methods. Zbl 1141.34009
Sun, Jingxian; Xu, Xian; O’Regan, Donal
2008
Nontrivial solutions of singular superlinear Sturm–Liouville problems. Zbl 1100.34019
Sun, Jingxian; Zhang, Guowei
2006
Positive solutions of superlinear semipositone nonlinear boundary value problems. Zbl 1232.34035
Li, Hongyu; Sun, Jingxian
2011
The fixed point theorem of convex-power condensing operator and applications to abstract semilinear evolution equations. Zbl 1124.34342
Sun, Jing Xian; Zhang, Xiao Yan
2005
On sign-changing solution for some three-point boundary value problems. Zbl 1069.34019
Xu, Xian; Sun, Jingxian
2004
A generalization of the cone expansion and compression fixed point theorem and applications. Zbl 1127.47050
Zhang, Guowei; Sun, Jingxian
2007
Computation of topological degree in ordered Banach spaces with lattice structure and its application to superlinear differential equations. Zbl 1177.47065
Sun, Jingxian; Liu, Xiaoying
2008
Nontrivial solutions of singular sublinear Sturm–Liouville problems. Zbl 1111.34023
Sun, Jingxian; Zhang, Guowei
2007
Spectral indicator method for a non-selfadjoint Steklov eigenvalue problem. Zbl 1422.65361
Liu, J.; Sun, J.; Turner, T.
2019
The Schauder condition in the critical point theory. Zbl 0603.47045
Sun, Jingxian
1986
Existence of positive solutions to a non-positive elastic beam equation with both ends fixed. Zbl 1278.34020
Lu, Haixia; Sun, Li; Sun, Jingxian
2012
Multiple positive solutions of singular third-order periodic boundary value problem. Zbl 1068.34020
Sun, Jingxian; Liu, Yansheng
2005
Computation of topological degree for nonlinear operators and applications. Zbl 1169.47043
Sun, Jingxian; Liu, Xiaoying
2008
Positive solutions of sublinear Sturm-Liouville problems with changing sign nonlinearity. Zbl 1197.34040
Li, Hongyu; Sun, Jingxian
2009
A generalization of Guo’s theorem and applications. Zbl 0632.47044
Sun, Jingxian
1987
Iterative method for coupled quasi-solutions of mixed monotone operator equations. Zbl 0763.65041
Sun, Jingxian; Liu, Lishan
1992
Multiple positive fixed points of weakly inward mappings. Zbl 0709.47052
Sun, Yong; Sun, Jingxian
1990
A generalization of Mahadevan’s version of the Krein-Rutman theorem and applications to $$p$$-Laplacian boundary value problems. Zbl 1252.47045
Cui, Yujun; Sun, Jingxian
2012
Global bifurcation and multiple results for Sturm-Liouville problems. Zbl 1223.34030
Cui, Yujun; Sun, Jingxian; Zou, Yumei
2011
Extremal solutions of the initial value problem for integro-differential equations of mixed type in Banach spaces. Zbl 0771.34059
Sun, Jingxian; Zhao, Zengqin
1992
Computation for topological degree and its applications. Zbl 0866.47043
Sun, Jingxian; Liu, Xiaoying
1996
Calculus of variations and super- and sub-solutions in reverse order. Zbl 0810.47059
Sun, Jingxian; Liu, Zhaoli
1994
Positive solutions of boundary value problems for systems of second-order nonlinear ordinary differential equations. Zbl 1158.34317
Yang, Zhilin; Sun, Jingxian
2004
Computation of topological degree of unilaterally asymptotically linear operators and its applications. Zbl 1191.47076
Liu, Xiaoying; Sun, Jingxian
2009
Multiple positive solutions of singular second-order $$m$$-point boundary value problems. Zbl 1094.34017
Zhang, Guowei; Sun, Jingxian
2006
Nontrivial solutions of superlinear Hammerstein integral equations and applications. Zbl 0633.45006
Sun, Jingxian
1986
Recursive integral method with Cayley transformation. Zbl 07031739
Huang, R.; Sun, J.; Yang, C.
2018
Some extensions of the mountain pass lemma. Zbl 0746.47041
Guo, Dajun; Sun, Jingxian; Qi, Guijie
1988
Fixed point theorems for a class of nonlinear operators in Hilbert spaces with lattice structure and application. Zbl 1476.47033
Cui, Yujun; Sun, Jingxian
2013
Fixed point theorems for a class of nonlinear operators in Riesz spaces. Zbl 1301.47069
Sun, Jingxian; Cui, Yujun
2013
Fixed point theorems for a class of nonlinear operators in Hilbert spaces and applications. Zbl 1234.49009
Cui, Yujun; Sun, Jingxian
2011
An elliptic problem with jumping nonlinearities. Zbl 1161.35395
Liu, Zhaoli; Sun, Jingxian
2005
Multiplicity results for three-point boundary value problems with a non-well-ordered upper and lower solution condition. Zbl 1140.34009
Xu, Xian; O’Regan, Donal; Sun, Jingxian
2007
Flow-invariant sets and critical point theory. Zbl 1053.34060
Sun, Jingxian; Hu, Shouchuan
2003
On multiple sign-changing solutions for some second-order integral boundary value problems. Zbl 1211.34020
Zhang, Xingqiu; Sun, Jingxian
2010
Some global generalizations of the Birkhoff-Kellogg theorem and applications. Zbl 0661.47049
Guo, Dajun; Sun, Jingxian
1988
On existence of positive solutions of coupled integral boundary value problems for a nonlinear singular superlinear differential system. Zbl 1340.34087
Ciu, Yujun; Sun, Jingxian
2012
Solutions for an operator equation under the conditions of pairs of paralleled lower and upper solutions. Zbl 1165.47044
Xu, Xian; Sun, Jingxian
2008
Some fixed point theorems of increasing operators. Zbl 0585.47046
Sun, Jingxian; Sun, Yong
1986
Solutions of ordinary differential equations in Banach spaces. Zbl 0719.34107
Sun, Jingxian
1990
Four versus two solutions of semilinear elliptic boundary value problems. Zbl 0996.35017
Liu, Zhaoli; Sun, Jingxian
2002
A property of connected components and its applications. Zbl 1016.54019
Sun, Jingxian; Song, Fumin
2002
Solutions of nonlinear second order impulsive integro-differential equations of mixed type in Banach spaces. Zbl 1052.45010
Zhang, Xiaoyan; Sun, Jingxian
2002
Multiple solutions for nonlinear operators and applications. Zbl 1120.47050
Sun, Jingxian; Cui, Yujun
2007
Evaluation of topological degree and applications. Zbl 0687.47047
Guo, Dajun; Sun, Jingxian
1988
Existence of multiple positive solutions for fourth-order boundary value problems in Banach spaces. Zbl 1281.34026
Cui, Yujun; Sun, Jingxian
2012
A theorem in point set topology. Zbl 0633.54013
Sun, Jingxian
1987
Fixed point theorems of increasing operators and applications to nonlinear integro-differential equations with discontinuous terms. Zbl 0779.47041
Sun, Jingxian; Zhao, Zengqin
1993
Computation of the topological degree and applications to superlinear system of equations. Zbl 0897.47050
Liu, Xiaoying; Sun, Jingxian
1996
Existence of positive solutions for singular second-order $$m$$-point boundary value problems. Zbl 1080.34517
Zhang, Guowei; Sun, Jingxian
2004
The global structure of the set of characteristic vectors for superlinear operators. Zbl 0706.47033
Sun, Jingxian
1988
Positive solutions for second-order three-point boundary value problems in a Banach space. Zbl 1249.34068
Cui, Yujun; Sun, Jingxian
2011
Positive solutions of a fourth-order boundary value problem involving derivatives of all orders. Zbl 1267.34051
Yang, Zhilin; Sun, Jingxian
2012
Positive solutions of singular $$(k, n-k)$$ multi-point boundary value problems. Zbl 1124.34311
Zhang, Guowei; Sun, Jingxian
2006
Some developments in mathematical demography and their application to the People’s Republic of China. Zbl 0499.92018
Song, J.
1982
Existence theorems of multiple critical points and applications. Zbl 1125.35029
Sun, Jingxian; Sun, Jinli
2005
Some new fixed point theorems of increasing operators and applications. Zbl 0738.47052
Sun, Jingxian
1991
Computation of fixed point index for no-cone mappings and its applications. Zbl 1219.47090
Sun, Jingxian; Liu, Xiaoying
2010
Existence of sign-changing solutions for nonlinear operator equations and its applications. Zbl 1078.47503
Zhang, Kemei; Sun, Jingxian
2003
A general principle on ordered sets and its applications to fixed point theory. Zbl 0659.54027
Sun, Jingxian; Sun, Yong
1989
Fixed point theorems of non-continuous increasing operators and applications to nonlinear equations with discontinuous nonlinearities. Zbl 0689.47020
Sun, Jingxian
1988
Fixed point theorems of two-point extension type and applications. Zbl 0795.47035
Sun, Jingxian
1992
Unbounded connected component of the positive solutions set of some semi-positone problems. Zbl 1239.47045
Xu, Xian; Sun, Jingxian
2011
On the number of fixed points of nonlinear operators and applications. Zbl 1131.47308
Sun, Jingxian; Zhang, Kemei
2003
Weakly topological degree for nonlinear operators. I. Zbl 0599.47103
Sun, Jingxian
1984
Computation of topological degree and applications to nonlinear operators. Zbl 0625.47046
Sun, Jingxian
1985
Global generalizations of the Birkhoff-Kellogg theorem on nonlinear operators. Zbl 0633.47038
Sun, Jingxian
1987
Monotone iterative method of integro-differential equations of mixed type in Banach spaces. Zbl 0777.65077
Sun, Jingxian; Liu, Lishan
1993
Existence and uniqueness of solutions for a class of nonlinear operator equations and its applications. Zbl 1280.47061
Zhang, Xiaoyan; Sun, Jingxian
2005
Eigenvalues and eigenvectors of nonlinear operators and applications. Zbl 0893.47039
Sun, Jingxian; Lou, Bendong
1997
Asymptotic bifurcation points of nonlinear operators. Zbl 0952.47049
Yang, Zhilin; Sun, Jingxian
2000
The existence of positive solution for singular boundary value problems of first order differential equation on unbounded domains. Zbl 1181.34029
Zhang, Xingqiu; Sun, Jingxian
2009
The relation between sign-changing solution and positive-negative solutions for nonlinear operator equations and its applications. Zbl 1075.47035
Zhang, Kemei; Sun, Jingxian
2004
Fixed points and generalized fixed points for increasing operators. Zbl 0687.47042
Sun, Jingxian
1989
Generalizations of the general principle on ordered sets in nonlinear functional analysis. Zbl 0705.47049
Sun, Jingxian
1990
Positive solutions of non-positone nonlinear operator equations and their applications. Zbl 1257.47064
Sun, Jingxian; Xu, Xian
2012
Asymptotic bifurcation points, and global bifurcation of nonlinear operators and its applications. Zbl 1227.47039
Liu, Xiaoying; Sun, Jingxian
2012
Eigenvalue criteria for the existence of positive solutions to nonlinear $$(k,n-k)$$ conjugate boundary value problems. Zbl 1122.34308
Zhang, Guowei; Sun, Jingxian
2006
Multiple solutions for superlinear operator equations in Banach spaces and applications. Zbl 1117.47304
Zhang, Kemei; Sun, Jingxian
2005
Existence of multiple fixed points for nonlinear operators and applications. Zbl 1157.47039
Sun, Jingxian; Zhang, Kemei
2008
Positive solutions of singular nonlinear Sturm-Liouville problems. Zbl 1124.34323
Sun, jing Xian; Zhang, Guo Wei
2005
Positive solutions of boundary value problem for a system of nonlinear ordinary differential equations. Zbl 1094.34512
Li, Hongyu; Sun, Jingxian
2005
A theorem on many fixed points for nonlinear operator. Zbl 0738.47053
Sun, Jingxian; Sun, Yong
1990
Analytical solution for motion of an elliptical inclusion in gradient stress field. Zbl 1244.74097
Li, Y.; Li, Z.; Wang, X.; Sun, J.
2010
Solutions of the second order periodic boundary value problem in a Banach space under the weak$$^*$$ topology. Zbl 0774.34048
Song, Fumin; Sun, Jingxian
1993
Decreasing flow invariant sets and fixed points of quasimonotone increasing operators. Zbl 0977.47050
Sun, Jingxian; Xu, Xi’an
2001
Number of invariant sets of descending flow with applications in critical point theory. Zbl 1048.58008
Liu, Zhaoli; Sun, Jingxian
2003
A problem on successive approximations of operator equations. Zbl 0667.65046
Sun, Jingxian
1988
The existence of positive solutions for nonlinear Hammerstein integral equations and applications. Zbl 0699.45005
Sun, Jingxian
1988
Global structure of positive solution sets of nonlinear operator equations. Zbl 1246.47013
Xu, Xian; Sun, Jingxian; O’Regan, D.
2012
Bifurcation points and global structure of positive eigenvalues for cone mappings with no differentiability at the zero point. Zbl 1230.47087
Liu, Xiaoying; Sun, Jingxian
2010
Three solutions theorems for nonlinear operator equations and applications. Zbl 1097.47054
Sun, Jingxian; Xu, Xi’an
2005
Design of vibration controllers for flexible beams using the mechatronic design quotient (MDQ) approach. Zbl 1182.74105
Sun, J.; Poo, A. N.; Ang, M. H.; Hong, G. S.; De Silva, C. W.; Tan, K. K.
2007
Solutions of nonlinear Sturm-Liouville problems in a Banach space. Zbl 1199.34133
Cui, Yujun; Sun, Jingxian
2009
Positive solutions of nonlinear differential equations on a measure chain. Zbl 1199.34098
Li, Hongyu; Sun, Jingxian; Cui, Yujun
2009
Fixed point theorem of domain compression and expansion of convex functional type. Zbl 1174.47047
Zhang, Guowei; Sun, Jingxian; Zhang, Tie
2008
Positive solutions of singular boundary value problems of fourth-order differential equations. Zbl 1212.34054
Cui, Yujun; Sun, Jingxian; Zou, Yumei
2009
Generalizations of fixed-point theorems of Altman and Rothe types. Zbl 1470.47045
Sun, Li; Sun, Jingxian; Wang, Guangwa
2013
Spectral indicator method for a non-selfadjoint Steklov eigenvalue problem. Zbl 1422.65361
Liu, J.; Sun, J.; Turner, T.
2019
Recursive integral method with Cayley transformation. Zbl 07031739
Huang, R.; Sun, J.; Yang, C.
2018
Fixed point theorems for a class of nonlinear operators in Hilbert spaces with lattice structure and application. Zbl 1476.47033
Cui, Yujun; Sun, Jingxian
2013
Fixed point theorems for a class of nonlinear operators in Riesz spaces. Zbl 1301.47069
Sun, Jingxian; Cui, Yujun
2013
Generalizations of fixed-point theorems of Altman and Rothe types. Zbl 1470.47045
Sun, Li; Sun, Jingxian; Wang, Guangwa
2013
Existence of positive solutions to a non-positive elastic beam equation with both ends fixed. Zbl 1278.34020
Lu, Haixia; Sun, Li; Sun, Jingxian
2012
A generalization of Mahadevan’s version of the Krein-Rutman theorem and applications to $$p$$-Laplacian boundary value problems. Zbl 1252.47045
Cui, Yujun; Sun, Jingxian
2012
On existence of positive solutions of coupled integral boundary value problems for a nonlinear singular superlinear differential system. Zbl 1340.34087
Ciu, Yujun; Sun, Jingxian
2012
Existence of multiple positive solutions for fourth-order boundary value problems in Banach spaces. Zbl 1281.34026
Cui, Yujun; Sun, Jingxian
2012
Positive solutions of a fourth-order boundary value problem involving derivatives of all orders. Zbl 1267.34051
Yang, Zhilin; Sun, Jingxian
2012
Positive solutions of non-positone nonlinear operator equations and their applications. Zbl 1257.47064
Sun, Jingxian; Xu, Xian
2012
Asymptotic bifurcation points, and global bifurcation of nonlinear operators and its applications. Zbl 1227.47039
Liu, Xiaoying; Sun, Jingxian
2012
Global structure of positive solution sets of nonlinear operator equations. Zbl 1246.47013
Xu, Xian; Sun, Jingxian; O’Regan, D.
2012
Positive solutions of superlinear semipositone nonlinear boundary value problems. Zbl 1232.34035
Li, Hongyu; Sun, Jingxian
2011
Global bifurcation and multiple results for Sturm-Liouville problems. Zbl 1223.34030
Cui, Yujun; Sun, Jingxian; Zou, Yumei
2011
Fixed point theorems for a class of nonlinear operators in Hilbert spaces and applications. Zbl 1234.49009
Cui, Yujun; Sun, Jingxian
2011
Positive solutions for second-order three-point boundary value problems in a Banach space. Zbl 1249.34068
Cui, Yujun; Sun, Jingxian
2011
Unbounded connected component of the positive solutions set of some semi-positone problems. Zbl 1239.47045
Xu, Xian; Sun, Jingxian
2011
On multiple sign-changing solutions for some second-order integral boundary value problems. Zbl 1211.34020
Zhang, Xingqiu; Sun, Jingxian
2010
Computation of fixed point index for no-cone mappings and its applications. Zbl 1219.47090
Sun, Jingxian; Liu, Xiaoying
2010
Analytical solution for motion of an elliptical inclusion in gradient stress field. Zbl 1244.74097
Li, Y.; Li, Z.; Wang, X.; Sun, J.
2010
Bifurcation points and global structure of positive eigenvalues for cone mappings with no differentiability at the zero point. Zbl 1230.47087
Liu, Xiaoying; Sun, Jingxian
2010
Positive solutions of sublinear Sturm-Liouville problems with changing sign nonlinearity. Zbl 1197.34040
Li, Hongyu; Sun, Jingxian
2009
Computation of topological degree of unilaterally asymptotically linear operators and its applications. Zbl 1191.47076
Liu, Xiaoying; Sun, Jingxian
2009
The existence of positive solution for singular boundary value problems of first order differential equation on unbounded domains. Zbl 1181.34029
Zhang, Xingqiu; Sun, Jingxian
2009
Solutions of nonlinear Sturm-Liouville problems in a Banach space. Zbl 1199.34133
Cui, Yujun; Sun, Jingxian
2009
Positive solutions of nonlinear differential equations on a measure chain. Zbl 1199.34098
Li, Hongyu; Sun, Jingxian; Cui, Yujun
2009
Positive solutions of singular boundary value problems of fourth-order differential equations. Zbl 1212.34054
Cui, Yujun; Sun, Jingxian; Zou, Yumei
2009
Nodal solutions for $$m$$-point boundary value problems using bifurcation methods. Zbl 1141.34009
Sun, Jingxian; Xu, Xian; O’Regan, Donal
2008
Computation of topological degree in ordered Banach spaces with lattice structure and its application to superlinear differential equations. Zbl 1177.47065
Sun, Jingxian; Liu, Xiaoying
2008
Computation of topological degree for nonlinear operators and applications. Zbl 1169.47043
Sun, Jingxian; Liu, Xiaoying
2008
Solutions for an operator equation under the conditions of pairs of paralleled lower and upper solutions. Zbl 1165.47044
Xu, Xian; Sun, Jingxian
2008
Existence of multiple fixed points for nonlinear operators and applications. Zbl 1157.47039
Sun, Jingxian; Zhang, Kemei
2008
Fixed point theorem of domain compression and expansion of convex functional type. Zbl 1174.47047
Zhang, Guowei; Sun, Jingxian; Zhang, Tie
2008
A generalization of the cone expansion and compression fixed point theorem and applications. Zbl 1127.47050
Zhang, Guowei; Sun, Jingxian
2007
Nontrivial solutions of singular sublinear Sturm–Liouville problems. Zbl 1111.34023
Sun, Jingxian; Zhang, Guowei
2007
Multiplicity results for three-point boundary value problems with a non-well-ordered upper and lower solution condition. Zbl 1140.34009
Xu, Xian; O’Regan, Donal; Sun, Jingxian
2007
Multiple solutions for nonlinear operators and applications. Zbl 1120.47050
Sun, Jingxian; Cui, Yujun
2007
Design of vibration controllers for flexible beams using the mechatronic design quotient (MDQ) approach. Zbl 1182.74105
Sun, J.; Poo, A. N.; Ang, M. H.; Hong, G. S.; De Silva, C. W.; Tan, K. K.
2007
Nontrivial solutions of singular superlinear Sturm–Liouville problems. Zbl 1100.34019
Sun, Jingxian; Zhang, Guowei
2006
Multiple positive solutions of singular second-order $$m$$-point boundary value problems. Zbl 1094.34017
Zhang, Guowei; Sun, Jingxian
2006
Positive solutions of singular $$(k, n-k)$$ multi-point boundary value problems. Zbl 1124.34311
Zhang, Guowei; Sun, Jingxian
2006
Eigenvalue criteria for the existence of positive solutions to nonlinear $$(k,n-k)$$ conjugate boundary value problems. Zbl 1122.34308
Zhang, Guowei; Sun, Jingxian
2006
The fixed point theorem of convex-power condensing operator and applications to abstract semilinear evolution equations. Zbl 1124.34342
Sun, Jing Xian; Zhang, Xiao Yan
2005
Multiple positive solutions of singular third-order periodic boundary value problem. Zbl 1068.34020
Sun, Jingxian; Liu, Yansheng
2005
An elliptic problem with jumping nonlinearities. Zbl 1161.35395
Liu, Zhaoli; Sun, Jingxian
2005
Existence theorems of multiple critical points and applications. Zbl 1125.35029
Sun, Jingxian; Sun, Jinli
2005
Existence and uniqueness of solutions for a class of nonlinear operator equations and its applications. Zbl 1280.47061
Zhang, Xiaoyan; Sun, Jingxian
2005
Multiple solutions for superlinear operator equations in Banach spaces and applications. Zbl 1117.47304
Zhang, Kemei; Sun, Jingxian
2005
Positive solutions of singular nonlinear Sturm-Liouville problems. Zbl 1124.34323
Sun, jing Xian; Zhang, Guo Wei
2005
Positive solutions of boundary value problem for a system of nonlinear ordinary differential equations. Zbl 1094.34512
Li, Hongyu; Sun, Jingxian
2005
Three solutions theorems for nonlinear operator equations and applications. Zbl 1097.47054
Sun, Jingxian; Xu, Xi’an
2005
Positive solutions of $$m$$-point boundary value problems. Zbl 1069.34037
Zhang, Guowei; Sun, Jingxian
2004
On sign-changing solution for some three-point boundary value problems. Zbl 1069.34019
Xu, Xian; Sun, Jingxian
2004
Positive solutions of boundary value problems for systems of second-order nonlinear ordinary differential equations. Zbl 1158.34317
Yang, Zhilin; Sun, Jingxian
2004
Existence of positive solutions for singular second-order $$m$$-point boundary value problems. Zbl 1080.34517
Zhang, Guowei; Sun, Jingxian
2004
The relation between sign-changing solution and positive-negative solutions for nonlinear operator equations and its applications. Zbl 1075.47035
Zhang, Kemei; Sun, Jingxian
2004
Flow-invariant sets and critical point theory. Zbl 1053.34060
Sun, Jingxian; Hu, Shouchuan
2003
Existence of sign-changing solutions for nonlinear operator equations and its applications. Zbl 1078.47503
Zhang, Kemei; Sun, Jingxian
2003
On the number of fixed points of nonlinear operators and applications. Zbl 1131.47308
Sun, Jingxian; Zhang, Kemei
2003
Number of invariant sets of descending flow with applications in critical point theory. Zbl 1048.58008
Liu, Zhaoli; Sun, Jingxian
2003
Four versus two solutions of semilinear elliptic boundary value problems. Zbl 0996.35017
Liu, Zhaoli; Sun, Jingxian
2002
A property of connected components and its applications. Zbl 1016.54019
Sun, Jingxian; Song, Fumin
2002
Solutions of nonlinear second order impulsive integro-differential equations of mixed type in Banach spaces. Zbl 1052.45010
Zhang, Xiaoyan; Sun, Jingxian
2002
Invariant sets of descending flow in critical point theory with applications to nonlinear differential equations. Zbl 0995.58006
Liu, Zhaoli; Sun, Jingxian
2001
Decreasing flow invariant sets and fixed points of quasimonotone increasing operators. Zbl 0977.47050
Sun, Jingxian; Xu, Xi’an
2001
Asymptotic bifurcation points of nonlinear operators. Zbl 0952.47049
Yang, Zhilin; Sun, Jingxian
2000
Eigenvalues and eigenvectors of nonlinear operators and applications. Zbl 0893.47039
Sun, Jingxian; Lou, Bendong
1997
Computation for topological degree and its applications. Zbl 0866.47043
Sun, Jingxian; Liu, Xiaoying
1996
Computation of the topological degree and applications to superlinear system of equations. Zbl 0897.47050
Liu, Xiaoying; Sun, Jingxian
1996
Calculus of variations and super- and sub-solutions in reverse order. Zbl 0810.47059
Sun, Jingxian; Liu, Zhaoli
1994
Fixed point theorems of increasing operators and applications to nonlinear integro-differential equations with discontinuous terms. Zbl 0779.47041
Sun, Jingxian; Zhao, Zengqin
1993
Monotone iterative method of integro-differential equations of mixed type in Banach spaces. Zbl 0777.65077
Sun, Jingxian; Liu, Lishan
1993
Solutions of the second order periodic boundary value problem in a Banach space under the weak$$^*$$ topology. Zbl 0774.34048
Song, Fumin; Sun, Jingxian
1993
Iterative method for coupled quasi-solutions of mixed monotone operator equations. Zbl 0763.65041
Sun, Jingxian; Liu, Lishan
1992
Extremal solutions of the initial value problem for integro-differential equations of mixed type in Banach spaces. Zbl 0771.34059
Sun, Jingxian; Zhao, Zengqin
1992
Fixed point theorems of two-point extension type and applications. Zbl 0795.47035
Sun, Jingxian
1992
Some new fixed point theorems of increasing operators and applications. Zbl 0738.47052
Sun, Jingxian
1991
Multiple positive fixed points of weakly inward mappings. Zbl 0709.47052
Sun, Yong; Sun, Jingxian
1990
Solutions of ordinary differential equations in Banach spaces. Zbl 0719.34107
Sun, Jingxian
1990
Generalizations of the general principle on ordered sets in nonlinear functional analysis. Zbl 0705.47049
Sun, Jingxian
1990
A theorem on many fixed points for nonlinear operator. Zbl 0738.47053
Sun, Jingxian; Sun, Yong
1990
A general principle on ordered sets and its applications to fixed point theory. Zbl 0659.54027
Sun, Jingxian; Sun, Yong
1989
Fixed points and generalized fixed points for increasing operators. Zbl 0687.47042
Sun, Jingxian
1989
Some extensions of the mountain pass lemma. Zbl 0746.47041
Guo, Dajun; Sun, Jingxian; Qi, Guijie
1988
Some global generalizations of the Birkhoff-Kellogg theorem and applications. Zbl 0661.47049
Guo, Dajun; Sun, Jingxian
1988
Evaluation of topological degree and applications. Zbl 0687.47047
Guo, Dajun; Sun, Jingxian
1988
The global structure of the set of characteristic vectors for superlinear operators. Zbl 0706.47033
Sun, Jingxian
1988
Fixed point theorems of non-continuous increasing operators and applications to nonlinear equations with discontinuous nonlinearities. Zbl 0689.47020
Sun, Jingxian
1988
A problem on successive approximations of operator equations. Zbl 0667.65046
Sun, Jingxian
1988
The existence of positive solutions for nonlinear Hammerstein integral equations and applications. Zbl 0699.45005
Sun, Jingxian
1988
A generalization of Guo’s theorem and applications. Zbl 0632.47044
Sun, Jingxian
1987
A theorem in point set topology. Zbl 0633.54013
Sun, Jingxian
1987
Global generalizations of the Birkhoff-Kellogg theorem on nonlinear operators. Zbl 0633.47038
Sun, Jingxian
1987
The Schauder condition in the critical point theory. Zbl 0603.47045
Sun, Jingxian
1986
Nontrivial solutions of superlinear Hammerstein integral equations and applications. Zbl 0633.45006
Sun, Jingxian
1986
Some fixed point theorems of increasing operators. Zbl 0585.47046
Sun, Jingxian; Sun, Yong
1986
Computation of topological degree and applications to nonlinear operators. Zbl 0625.47046
Sun, Jingxian
1985
Weakly topological degree for nonlinear operators. I. Zbl 0599.47103
Sun, Jingxian
1984
Some developments in mathematical demography and their application to the People’s Republic of China. Zbl 0499.92018
Song, J.
1982
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text:
### Home > CC1 > Chapter 5 > Lesson 5.3.4 > Problem5-102

5-102.

Find each sum. Homework Help ✎

1. $−3+7$

1. $7+(−8)$

1. $- 6 \frac { 4 } { 9 } + 9 \frac { 7 } { 9 }$

Drawing a tightrope diagram or a number line may make these expressions easier to solve!

The answer to part (a) is 4. Were you able to find the values of parts (b) and (c)?


text:
find the locus of a moving point P in algebraic form which is equidistant from the point (0,4) and the x-axis

chunwai

New member
Hello,

Please explain what is the question mean and how this answer get.
find the locus of a moving point P in algebraic form which is equidistant from the point (0,4) and the x-axis

Harry_the_cat

Senior Member
Draw a diagram and mark in the point (0, 4).

Now dot in some points which are the same distance from (0, 4) as they are from the x-axis (note that the distance between a point and a line is measured perpendicular to the line.

Call any one of these point (x, y).

Write an expression which represents the distance between (x, y) and (0, 4).
Write another expressions which represents the distance between (x, y) and the x-axis.
Equate these wo expressions and rearrange to get y = …..

Give it a go.

BTW the answer will be an equation involving x and y NOT the expression you have stated.

HallsofIvy

Elite Member
Let (x, y) be such a point. The distance to (0, 4) is $$\displaystyle \sqrt{x^2+ (y-4)^2}$$ and the distance to the x-axis is y. To be "equidistant" we must have $$\displaystyle y= \sqrt{x^2+ (y- 4)^2}$$. Start by squaring both sides: $$\displaystyle y^2= x^2+ (y- 4)^2= x^2+ y^2- 8y+ 16$$. The two "$$\displaystyle y^2$$" terms cancel to get $$\displaystyle x^2- 8y+ 16= 0$$, $$\displaystyle 8y= x^2+ 16$$ so $$\displaystyle y= \frac{1}{8}x^2+ 2$$.

Since the question asked about a "point" the answer is $$\displaystyle (x, \frac{1}{8}x^2+ 2)$$.

Harry_the_cat

Senior Member
Since the question asked about a "point" the answer is $$\displaystyle (x, \frac{1}{8}x^2+ 2)$$.
I always thought the locus of a point was given as an equation.

HallsofIvy

Elite Member
A locus is a set of points. Of course, the points can be given as "$$\displaystyle y= \frac{1}{8}x^2+ 2$$" with the understanding that the points are "(x, y)". But "y= f(x)" is a shortcut for "{(x, f(x))}".

Dr.Peterson

Elite Member
So, the answer is really $$\displaystyle \{(x, \frac{1}{8}x^2+ 2): x\in \mathbb{R}\}$$. It's a set of points, not a point.

But, yes, in my experience, the answer is usually given as an equation, meaning "the set of points that satisfy ...". That is, the locus is the graph of the equation.