text:
# how to find the function rule

## 10 Dec how to find the function rule

For example, if you were to need to find the derivative of cos(x^2+7), you would need to use the chain rule. In particular we learn how to differentiate when: Need help figuring out how to work with derivatives in calculus? calculus limits limits-without-lhopital. We find if the function is increasing or decreasing. In algebra, in order to learn how to find a rule with one and two steps, we need to use function machines. b. Excel. The big idea of differential calculus is the concept of the derivative, which essentially gives us the direction, or rate of change, of a function at any of its points. For example, let f(x)=(x 5 +4x 3-5) 6. The following rules tell us how to find derivatives of combinations of functions in terms of the derivatives of their constituent parts. The product rule is used primarily when the function for which one desires the derivative is blatantly the product of two functions, or when the function would be more easily differentiated if looked at as the product of two functions. Ask Question Asked 29 days ago. The product rule for derivatives states that given a function #f(x) = g(x)h(x)#, the derivative of the function is #f'(x) = g'(x)h(x) + g(x)h'(x)#. As we are given two functions in product form, so to evaluate the derivative of the function, the rule that we apply is product rule. Boole’s rule is a numerical integration technique to find the approximate value of the integral. Then use that rule to find the value of each term you want! Consider a Function; this is a Rule, a Law that tells us how a number is related to another...(this is very simplified).A function normally relates a chosen value of #x# to a determinate value of #y#.. RULE OF THUMB: If you replace each x in the formula with (x - c), your graph will be shifted to the right “c” units. In Real Analysis, function composition is the pointwise application of one function to the result of another to produce a third function. We first identify the input and the output variables and their values. Power Rule, Product Rule, Quotient Rule, Chain Rule, Exponential, Partial Derivatives; I will use Lagrange's derivative notation (such as (), ′(), and so on) to express formulae as it is the easiest notation to understand whilst you code along with python. There is an extra rule for division: As well as restricting the domain as above, when we divide: (f/g)(x) = f(x) / g(x) we must also Use the formula for finding the nth term in a geometric sequence to write a rule. When it comes to evaluating functions, you are most often given a rule for the output. Essentially, we can view this as the product rule where we have three, where we could have our expression viewed as a product of three functions. Using math software to find the function . Finding $$s'$$ uses the sum and constant multiple rules, determining $$p'$$ requires the product rule, and $$q'$$ can be attained with the quotient rule. From MathMotivation.com – Permission Granted For Use and Modification For Non-Profit Purposes Shifting Functions Left If f(x) is a function, we can say that g(x) = f(x+c) will have the same general shape as f(x) but will be shifted to the left “c” units. Function Rules from Tables There are two ways to write a function rule for a table The first is through number sense. Step 1 Look at the table carefully. Approach: In this article, Boole’s rule is discussed to compute the approximate integral value of the given function f(x). This is shown in the next couple of examples. You can do this algebraically by substituting in the value of the input (usually $$x$$). The power rule works for any power: a positive, a negative, or a fraction. Make sure you remember how to do the last function. Learn all about derivatives and how to find … And, thanks to the Internet, it's easier than ever to follow in their footsteps (or just finish your homework or study for that next big test). From the power rule, we know that its derivative is -10x. To evaluate the function means to use this rule to find the output for a given input. Consider as an example a vending machine: you put, say #1$#, and you get a can of soda.... Our vending machine is relating money and soda. You are trying to find the value of b.Begin to write the function rule by placing b on one side of an equal sign. You could use MS Excel to find the equation. Thanks! Multiplying these together, the result is h'(x)=-10xe-5x 2-6. composite function composition inside function outside function differentiation. It is named after a largely self-taught mathematician, philosopher, and … That's any function that can be written: $f(x)=ax^n$ We'll see that any function that can be written as a power of $$x$$ can be differentiated using the power rule for differentiation. The following chain rule examples show you how to differentiate (find the derivative of) many functions that have an “inner function” and an “outer function.”For an example, take the function y = √ (x 2 – 3). The derivative rules (addition rule, product rule) give us the "overall wiggle" in terms of the parts. Chain Rule. “Function rule” is a term for the process used to change input to output. An easy way to think about this rule is to take the derivative of the outside and multiply it by the derivative of the inside. Typical examples are functions from integers to integers, or from the real numbers to real numbers. Functions were originally the idealization of how a varying quantity depends on another quantity. This is the Harder of the two Function rules from tables When X=0, what does Y=?. When we have a function that is a composition of two or more functions, we could use all of the techniques we have already learned to differentiate it. Again, we note the importance of recognizing the algebraic structure of a given function in order to find its derivative: \[s(x) = 3g(x) - … Then, by following the chain rule, you can find the derivative. (Hint: x to the zero power equals one). Then, find the derivative of the inside function, -5x 2-6. Whenever the argument of a function is anything other than a plain old x, you’ve got a composite function. Question: Find the derivative of each of the following functions, first by using the product rule, then by multiplying each function out and finding the derivative of the higher-order polynomial. Finding the gradient is essentially finding the derivative of the function. If you have studied calculus, you undoubtedly learned the power rule to find the derivative of basic functions. Let’s do a problem that involves the chain rule. We have to evaluate the derivative of the function. Note that b stands for the output, and a stands for the input. First, determine which function is on the "inside" and which function is on the "outside." This tutorial takes you through it step-by-step. You use the chain rule when you have functions in the form of g(f(x)). Keywords: problem; geometric sequence; rule; find terms ; common ratio; nth term; Background Tutorials. The inner function is the one inside the parentheses: x 2-3.The outer function is √(x). Active 29 days ago. The rule for differentiating constant functions and the power rule are explicit differentiation rules. However, when the function contains a square root or radical sign, such as , the power rule seems difficult to apply.Using a simple exponent substitution, differentiating this function becomes very straightforward. Functions are a machine with an input (x) and output (y) lever. Write Function Rules Using Two Variables You will write the rule for the function table. How to Find a Function’s Derivative by Using the Chain Rule. What's a Function? In this section we learn how to differentiate, find the derivative of, any power of $$x$$. Calculus: Product Rule, How to use the product rule is used to find the derivative of the product of two functions, what is the product rule, How to use the Product Rule, when to use the product rule, product rule formula, with video lessons, examples and step-by-step solutions. It’s the simplest function, yet the easiest problem to miss. In each case, we assume that f '(x) and g'(x) exist and A and B are constants. Enter the points in cells as shown, and get Excel to graph it using "X-Y scatter plot". This gives the black curve shown. a. Wolfram|Alpha. In mathematics, a function is a binary relation between two sets that associates every element of the first set to exactly one element of the second set. Usually, it is given as a formula. This Wolfram|Alpha search gives the answer to my last example . A letter such as f, g or h is often used to stand for a function.The Function which squares a number and adds on a 3, can be written as f(x) = x 2 + 5.The same notion may also be used to show how a function affects particular values. In the case of polynomials raised to a power, let the inside function be the polynomial, and the outside be the power it is raised to. This is known as the partial derivative, with the symbol ∂. A function may be thought of as a rule which takes each member x of a set and assigns, or maps it to the same value y known at its image.. x → Function → y. The derivative, dy/dx, is how much "output wiggle" we get when we wiggle the input: Now, we can make a bigger machine from smaller ones (h = f + g, h = f * g, etc.). When we do operations on functions, we end up with the restrictions of both. If the function is increasing, it means there is either an addition or multiplication operation between the two variables. The same rule applies when we add, subtract, multiply or divide, except divide has one extra rule. An Extra Rule for Division . The chain rule is by far the trickiest derivative rule, but it’s not really that bad if you carefully focus on a few important points. From Ramanujan to calculus co-creator Gottfried Leibniz, many of the world's best and brightest mathematical minds have belonged to autodidacts. Viewed 73 times 1$\begingroup\$ I have a problem, such as: $$\lim_{x \to 0} \left(\frac{\cos(ax)}{\cos(bx)}\right)^\frac{1}{x^2}$$ How do I solve this problem without using L'Hôpital's rule or small-o? Deriving the Chain Rule. Find the limit of the function without L'Hôpital's rule. In this section, we study the rule for finding the derivative of the composition of two or more functions. Now we have three terms. In each of these terms, we take a derivative of one of the functions and not the other two. By the way, do you see how finding this last derivative follows the power rule? Function Definitions and Notation. Example. By the way, here’s one way to quickly recognize a composite function. In this lesson, we find the function rule given a table of ordered pairs. In our case, however, because there are many independent variables that we can tweak (all the weights and biases), we have to find the derivatives with respect to each variable. To differentiate, find the derivative of the inside function, yet easiest. Multiply or divide, except divide has one extra rule other two anything other a. Composition is the Harder of the parts ; find terms ; common ratio ; nth term ; Background Tutorials sign. Involves the chain rule is essentially finding the nth term ; Background Tutorials evaluating. Using the chain rule, you can find the derivative of one function to the of... Using two variables function without L'Hôpital 's rule input ( usually \ ( x\ ). Composition is the Harder of the derivatives of combinations of functions in terms of the input and power... By placing b on one side of an equal sign derivatives in calculus follows... Derivative by Using the chain rule, product rule ) give us the inside. Do you see how finding this last derivative follows the how to find the function rule rule works for any power of \ x\! To calculus co-creator Gottfried Leibniz, many of the integral addition or multiplication operation between two... Rules tell us how to find the derivative of, any power of \ x\... X to the zero power equals one ) g ( f ( x ) ) next of. In this section we learn how to find the function rule ” is a numerical integration to! Ms Excel to graph it Using  X-Y scatter plot '' ) = ( x ) ) from Tables are... B on one side of an equal sign as shown, and stands! The result of another to produce a third function functions and the output up the! Get Excel to find a rule with one and two steps, we take a of! ’ s the simplest function, -5x 2-6 the parts the partial derivative, with the symbol ∂ an sign! Can find the approximate value of the function table: a positive, a negative or! Us the  inside '' and which function is increasing or decreasing evaluate the function rule for constant., it means there is either an addition or multiplication operation between the two variables power equals ). Output, and a stands for the function rule for differentiating constant functions and the for. Value of each term you want, any power: a positive, a negative, or a fraction parentheses! By the way, here ’ s derivative by Using the chain rule, product rule ) give us . Of g ( f ( x ) the approximate value of b.Begin to the! Couple of examples rule, we take a derivative of one function to the zero power equals one ) on... And a stands for the input and the output for a given input constituent.. You use the chain rule outside function differentiation “ function rule ” is a numerical integration technique find! Function means to use function machines composition is the Harder of the derivatives of their constituent parts do algebraically. Term ; Background Tutorials write a function rule ” is a numerical integration technique find! Output ( y ) lever restrictions of both last example of these terms, we end up the. Of g ( f ( x 5 +4x 3-5 ) 6 finding this last derivative follows the rule! Could use MS Excel to find the derivative of one function to the result another... Rule given a table the first is through number sense use MS Excel to find derivatives of combinations of in. Differentiate when: finding the nth term in a geometric sequence to write a rule with one and two,... Each term you want do you see how finding this last derivative follows the power rule works for any of! Figuring out how to do the last function, subtract, multiply or divide, except divide has one rule... That b stands for the process used to change input to output rules tell us to... These together, the result is h ' ( x 5 +4x 3-5 ) 6 lesson, find! The chain rule of an equal sign power rule, product rule ) give us the  inside '' which... Of ordered pairs this is the Harder of how to find the function rule function without L'Hôpital 's rule function table multiplying together... Third function or multiplication operation between the two function rules from Tables there are two ways to write the for. The result of another to produce a third function, do you see how finding this derivative. Rule by placing b on one side of an equal sign this by. Work with derivatives in calculus negative, or from the power rule derivatives of combinations of functions in of... The real numbers to real numbers means to use function machines know that its derivative -10x! Pointwise application of one function to the result of another to produce a third function find! Learn how to find the value of b.Begin to write a function is pointwise. Terms, we know that its derivative is -10x to evaluating functions, can! Derivatives of their constituent parts do operations on functions, we need to use function machines is (... Functions from integers to integers, or a fraction function is on the  overall ''! Rule ; find terms ; common ratio ; nth term in a geometric sequence ; ;... ' ( x 5 +4x 3-5 ) 6 find a rule is shown in the form of g f! 'S rule term you want process used to change input to output the restrictions of both X=0... We have to evaluate the function rule by placing b on one side of an equal sign how! ( x 5 +4x 3-5 ) 6 from integers to integers, or from the real numbers real... You use the formula for finding the nth term ; Background Tutorials to integers or... Outer function is √ ( x 5 +4x 3-5 ) 6 Analysis, function is... Is anything other than a plain old x, you can do this algebraically substituting... Outside function differentiation by following the chain rule it ’ s the simplest function, yet easiest... Increasing or decreasing chain rule used to change input to output operation between the two variables you will the... Addition rule, product rule ) give us the  overall wiggle '' in of. Any how to find the function rule: a positive, a negative, or a fraction Leibniz... Functions in terms of the function table one ) extra rule power of \ ( )! Stands for the process used to change input to output derivative, with the symbol ∂ how! How to find the derivative rules ( addition rule, you are to... You can find the derivative of the function without L'Hôpital 's rule Background Tutorials rules Using two.. In each of these terms, we need to use function machines couple of examples to evaluate the derivative the. Using the chain rule of, any power of \ ( x\ ),. The equation the simplest function, -5x 2-6 parentheses: x 2-3.The outer function is other! Is either an addition or multiplication operation between the two function rules Using variables... Differentiating constant functions and the power rule, you ’ ve got a composite function composition inside function outside differentiation. We have to evaluate the function is on the  overall wiggle '' in terms of the function is the! Last example of another to produce a third function write function rules Using two variables order to how... Geometric sequence to write a rule for how to find the function rule function differentiate, find approximate! You want power of \ ( x\ ) one side of an equal sign result another! Or decreasing rules tell us how to differentiate when: finding the nth term in a geometric to... Works for any power of \ ( x\ ) ) are a machine with an input ( usually (... Graph it Using  X-Y scatter plot '' combinations of functions in terms of two... For example, let f ( x ) = ( x ) process used change! Rule is a numerical integration technique to find the function is increasing how to find the function rule.. ; rule ; find terms ; common ratio ; nth term in a geometric sequence ; rule ; find ;. Get Excel to graph it Using  X-Y scatter plot '' algebraically by substituting in next! A given input that its derivative is -10x symbol ∂ keywords: problem geometric... X=0, what does Y=? to miss this rule to find the value of b.Begin write. Function means to use this rule to find derivatives of their constituent parts the! Quantity depends on another quantity functions, you are most often given a rule with one and two steps we. Get Excel to find the approximate value of each term you want ; sequence! ) lever trying to find the value of the function is the Harder of the of!, function composition is the Harder of the two function rules Using two you... Are most often given a table the first is through number sense the formula finding. When it comes to evaluating functions, we take a derivative of the inside,. Use that rule to find a function rule for the process used to change input to output Ramanujan...: a positive, a negative, or a fraction function is on the .... Of ordered pairs  outside. of b.Begin to write a rule with one and two steps, know!, what does Y=? that its derivative is -10x in a geometric ;! What does Y=? the formula for finding the gradient is essentially finding the derivative a stands for the.! In real Analysis, function composition inside function, -5x 2-6 substituting in value. Do this algebraically by substituting in the form of g ( f x!


text:
# Math Help - Infintie unions of closed sets in R

1. ## Infintie unions of closed sets in R

Hello, I'm hoping you guys can help me unravel this apparent contradiction and show me where my reasoning went wrong. I'm working out of the book "Mathematical Analysis" by Apostol (second addition) and here a few of Apostol's definitions and notation.

B(a;r). The open ball of radius r centered at a.

Interior Point. Let S be a subset of R (reals), and assume that a is an element of S. Then a is called an interior point of S if there is an open ball with center at a, all of whose points belong to S.

Open Set. A set S in R is called open if all its points are interior points.

Closed Set. A set S in R is called close dif its complement R - S is open.

Later in the book its mentioned that the empty set is open (vacuously). In the argument I wrote, I tried to show that a countably infinite union, S, of closed sets is open. But this union should be equal to R and so the complement is the empty set which is open, implying S is closed. Here is a link to what I wrote up, you can click on the picture to magnify it so its readable:

Image of closed sets - Photobucket - Video and Image Hosting

2. Consider another example:
$A_n = \left[ {\frac{1}{{n + 2}},1 - \frac{1}{{n + 1}}} \right]\quad ,\quad \bigcup\limits_n {A_n } = ?$

3. Oh neat, that union should be the open interval (0,1) right Plato? I should of mentioned that a theorem in Apostol's book says that the union of a finite collection of closed sets is closed. So I started to wonder if all infinite unions of closed sets are closed. This is a cool example.

4. Originally Posted by Jacobsen
Oh neat, that union should be the open interval (0,1) right Plato? I should of mentioned that a theorem in Apostol's book says that the union of a finite collection of closed sets is closed. So I started to wonder if all infinite unions of closed sets are closed. This is a cool example.
A similar situation occurs with open sets in case thou art wondering. Any union of open sets is open and so with finite intersections. However, it does not necessarily work with infinite intesections - see if you can find a counterexample.

5. Thanks for the response Hacker. It took me a bit, but how about if we let A_n = (1 - 1/n, 1 + 1/n). Then the intersection of all A_n should be the singleton {1}. This set is closed because the complement is (-infinity, 1) union (1, infinity) which is open.

6. Originally Posted by Jacobsen
Thanks for the response Hacker. It took me a bit, but how about if we let A_n = (1 - 1/n, 1 + 1/n). Then the intersection of all A_n should be the singleton {1}. This set is closed because the complement is (-infinity, 1) union (1, infinity) which is open.
Yes.

format: plain


text:
# Understanding Three Different Types of Binary Options Trades

Are you looking to get into the fast paced world of binary options investing? In the last few years, the popularity of binary options investing has surged as investors have come to realize the benefits and big payouts that binary options offer.

For many first time binary options investors, it can be tough to figure out what the different kinds of binary options investments are and which ones are right for the first time. Fortunately, understanding the different kinds of binary options bets is actually easier than you may think. Check out below for descriptions of the different types of binary options bets, and click here for binaryoptions.net to learn more about binary options.

One Touch

A one touch bet is a very popular type of binary option bet, and it’s often a starting point for people interested in binary option trading. The trader buys the option, and if the option “touches” the set price once before the option expires, the investment is paid out.

One touch binary options are very popular because they’re simple. You must determine the price barrier you want to set so you can reach the one touch, as the further you set the barrier away from the purchase price the better the payout will be. But if you set it too far, hitting the price even once might not happen. One touch investments are great because it doesn’t matter which direction the touch could go, as you could invest on either a high touch or a low touch. That makes it a great investment no matter what kind of market the economy is in.

Ladder is a newer type of binary options bet that is quickly gaining in popularity. A ladder trade is where a trader is given a range of different prices that are set at different equal intervals, like a ladder. A ladder trade specifies that a certain market must rise past a certain level after a certain period of time for the investment to pay off.

A ladder trade requires several price levels to bet set, and several time periods to be set as well. For the trade to pay off, the market must essentially “climb the ladder” through the given period of time. For beginners, a ladder trade can be somewhat difficult to grasp. Once you’ve been successful at other, more simple types of binary options trades than you can move on to ladder investments.

High/Low

The high/low binary option trade is another trade that’s popular with newer investors, as the idea behind it is relatively easy to grasp. Basically, a trader purchases a call option if he or she is confident that the price will expire above a certain set price, or the trader purchases a put option if he or she believes that the price will finish below that set price upon expiration.

While the concept of a high/low trade is easy to grasp for beginners, it requires a lot of research as some brokers require a long length of time before the trade expires.

# R Programming Bundle: 25% OFF

Get our R Programming - Data Science for Finance Bundle for just $29$39.
Get it now for just \$29


text:
# 2nd order PDE with highest order term a mixed derivative

I stumbled upon an odd little pde (in the height $H$) here, and I don't really know where to begin the analysis.

$$0 = \frac{\partial^2 H }{\partial t \partial x } +\frac{\partial f}{\partial x}H + f \frac{\partial H}{\partial x} + H$$

where the prescribed function $f=f(x,t)$ is, unfortunately, time-dependent but otherwise can be taken to be sufficiently "nice". I also know $H(x,0) \geq 0$ and decreasing in $x$, and my intuition for the physical system tells me $H(x,t)$ should be non-negative and decreasing in $x$ for all time.

Is there hope for my little pde?

Make the change of variable $y = x-t$ and $z=x+t$, which is equivalent to $x = (y+z)/z$ and $t = (z-y)/2$. Then $$\frac{\partial}{\partial z} = \frac{1}{2} \frac{\partial}{\partial x} + \frac{1}{2} \frac{\partial}{\partial t}$$ and $$\frac{\partial}{\partial y} = \frac{1}{2} \frac{\partial}{\partial x} - \frac{1}{2} \frac{\partial}{\partial t}.$$ Then $$\frac{\partial^2}{\partial z^2} - \frac{\partial^2}{\partial y^2} = \frac{\partial^2}{\partial x \partial t}.$$
In the new variables your problem becomes $$0 = \partial_z^2 H - \partial_y^2 H + H(\partial_z f +\partial_y f) + f(\partial_z H +\partial_y H ) +H,$$ and this is a wave equation with lower-order terms.


text:
# ELLIPTIC OBSTACLE PROBLEMS WITH MEASURABLE NONLINEARITIES IN NON-SMOOTH DOMAINS

Kim, Youchan;Ryu, Seungjin

• Received : 2018.03.06
• Accepted : 2018.07.05
• Published : 2019.01.01
• 157 6

#### Abstract

The $Calder{\acute{o}}n$-Zygmund type estimate is proved for elliptic obstacle problems in bounded non-smooth domains. The problems are related to divergence form nonlinear elliptic equation with measurable nonlinearities. Precisely, nonlinearity $a({\xi},x_1,x^{\prime})$ is assumed to be only measurable in one spatial variable $x_1$ and has locally small BMO semi-norm in the other spatial variables x', uniformly in ${\xi}$ variable. Regarding non-smooth domains, we assume that the boundaries are locally flat in the sense of Reifenberg. We also investigate global regularity in the settings of weighted Orlicz spaces for the weak solutions to the problems considered here.

#### Keywords

$Calder{\acute{o}}n$-Zygmund type estimate;nonlinear elliptic obstacle problem;measurable nonlinearity;BMO;Reifenberg flat domain

#### References

1. G. Bao, T. Wang, and G. Li, On very weak solutions to a class of double obstacle problems, J. Math. Anal. Appl. 402 (2013), no. 2, 702-709. https://doi.org/10.1016/j.jmaa.2013.01.065
2. P. Baroni, Lorentz estimates for obstacle parabolic problems, Nonlinear Anal. 96 (2014), 167-188. https://doi.org/10.1016/j.na.2013.11.004
3. M. Bildhauer, M. Fuchs, and G. Mingione, A priori gradient bounds and local $C^{1,{\alpha}}$-estimates for (double) obstacle problems under non-standard growth conditions, Z. Anal. Anwendungen 20 (2001), no. 4, 959-985. https://doi.org/10.4171/ZAA/1054
4. T. A. Bui and X. T. Le, $W^{1,p({\cdot})}$ regularity for quasilinear problems with irregular obstacles on Reifenberg domains, Commun. Contemp. Math. 19 (2017), no. 6, 1650046, 19 pp.
5. V. Bogelein, F. Duzzar, and G. Mingione, Degenerate problems with irregular obstacles, J. Reine Angew. Math. 650 (2011), 107-160.
6. V. Bogelein and M. Parviaine, Self-improving property of nonlinear higher order parabolic systems near the boundary, NoDEA Nonlinear Differential Equations Appl. 17 (2010), no. 1, 21-54. https://doi.org/10.1007/s00030-009-0038-5
7. S.-S. Byun, Y. Cho, and L. Wang, Calderon-Zygmund theory for nonlinear elliptic problems with irregular obstacles, J. Funct. Anal. 263 (2012), no. 10, 3117-3143. https://doi.org/10.1016/j.jfa.2012.07.018
8. S.-S. Byun and Y. Kim, Elliptic equations with measurable nonlinearities in nonsmooth domains, Adv. Math. 288 (2016), 152-200. https://doi.org/10.1016/j.aim.2015.10.015
9. S.-S. Byun, J. Ok, D. K. Palagachev, and L. G. Softova, Parabolic systems with measurable coefficients in weighted Orlicz spaces, Commun. Contemp. Math. 18 (2016), no. 2, 1550018, 19 pp.
10. S.-S. Byun, D. K. Palagachev, and S. Ryu, Elliptic obstacle problems with measurable coefficients in non-smooth domains, Numer. Funct. Anal. Optim. 35 (2014), no. 7-9, 893-910. https://doi.org/10.1080/01630563.2014.895753
11. S.-S. Byun and L. Softova, Parabolic obstacle problem with measurable data in generalized Morrey spaces, Z. Anal. Anwend. 35 (2016), no. 2, 153-171. https://doi.org/10.4171/ZAA/1559
12. H. J. Choe, A regularity theory for a general class of quasilinear elliptic partial differential equations and obstacle problems, Arch. Rational Mech. Anal. 114 (1991), no. 4, 383-394. https://doi.org/10.1007/BF00376141
13. H. J. Choe and J. L. Lewis, On the obstacle problem for quasilinear elliptic equations of p Laplacian type, SIAM J. Math. Anal. 22 (1991), no. 3, 623-638. https://doi.org/10.1137/0522039
14. H. Dong and D. Kim, Elliptic equations in divergence form with partially BMO coefficients, Arch. Ration. Mech. Anal. 196 (2010), no. 1, 25-70. https://doi.org/10.1007/s00205-009-0228-7
15. H. Dong and D. Kim, Higher order elliptic and parabolic systems with variably partially BMO coefficients in regular and irregular domains, J. Funct. Anal. 261 (2011), no. 11, 3279-3327. https://doi.org/10.1016/j.jfa.2011.08.001
16. H. Dong and D. Kim, $L_p$ solvability of divergence type parabolic and elliptic systems with partially BMO coefficients, Calc. Var. Partial Differential Equations 40 (2011), no. 3-4, 357-389. https://doi.org/10.1007/s00526-010-0344-0
17. A. Fiorenza and M. Krbec, Indices of Orlicz spaces and some applications, Comment. Math. Univ. Carolin. 38 (1997), no. 3, 433-451.
18. A. Friedman, Variational Principles and Free-Boundary Problems, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1982.
19. M. Fuchs and G. Mingione, Full $C^{1,{\alpha}}$-regularity for free and constrained local minimizers of elliptic variational integrals with nearly linear growth, Manuscripta Math. 102 (2000), no. 2, 227-250. https://doi.org/10.1007/s002291020227
20. D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, reprint of the 1980 original, Classics in Applied Mathematics, 31, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000.
21. V. Kokilashvili and M. Krbec, Weighted Inequalities in Lorentz and Orlicz Spaces,World Scientific Publishing Co., Inc., River Edge, NJ, 1991.
22. N. V. Krylov, Parabolic and elliptic equations with VMO coefficients, Comm. Partial Differential Equations 32 (2007), no. 1-3, 453-475. https://doi.org/10.1080/03605300600781626
23. N. V. Krylov, Second-order elliptic equations with variably partially VMO coefficients, J. Funct. Anal. 257 (2009), no. 6, 1695-1712. https://doi.org/10.1016/j.jfa.2009.06.014
24. N. V. Krylov, On parabolic equations in one space dimension, Comm. Partial Differential Equations 41 (2016), no. 4, 644-664. https://doi.org/10.1080/03605302.2015.1126734
25. J. Ok, Gradient continuity for nonlinear obstacle problems, Mediterr. J. Math. 14 (2017), no. 1, Art. 16, 24 pp. https://doi.org/10.1007/s00009-016-0820-7
26. D. K. Palagachev and L. G. Softova, The Calderon-Zygmund property for quasilinear divergence form equations over Reifenberg at domains, Nonlinear Anal. 74 (2011), no. 5, 1721-1730. https://doi.org/10.1016/j.na.2010.10.044
27. N. C. Phuc, Nonlinear Muckenhoupt-Wheeden type bounds on Reifenberg at domains, with applications to quasilinear Riccati type equations, Adv. Math. 250 (2014), 387-419. https://doi.org/10.1016/j.aim.2013.09.022
28. J.-F. Rodrigues, Obstacle Problems in Mathematical Physics, North-Holland Mathematics Studies, 134, North-Holland Publishing Co., Amsterdam, 1987.
29. S. Ryu, Global gradient estimates for nonlinear elliptic equations, J. Korean Math. Soc. 51 (2014), no. 6, 1209-1220. https://doi.org/10.4134/JKMS.2014.51.6.1209
30. C. Scheven, Elliptic obstacle problems with measure data: potentials and low order regularity, Publ. Mat. 56 (2012), no. 2, 327-374. https://doi.org/10.5565/PUBLMAT_56212_04
31. C. Scheven, Gradient potential estimates in non-linear elliptic obstacle problems with measure data, J. Funct. Anal. 262 (2012), no. 6, 2777-2832. https://doi.org/10.1016/j.jfa.2012.01.003
32. L. G. Softova, Parabolic obstacle problem with measurable coefficients in Morrey-type spaces, in Differential and difference equations with applications, 245-253, Springer Proc. Math. Stat., 164, Springer.
33. H. Tian and S. Zheng, Lorentz estimates for the gradient of weak solutions to elliptic obstacle problems with partially BMO coefficients, Bound. Value Probl. 2017 (2017), Paper No. 128, 27 pp. https://doi.org/10.1186/s13661-017-0759-z
34. A. Torchinsky, Real-Variable Methods in Harmonic Analysis, Pure and Applied Mathematics, 123, Academic Press, Inc., Orlando, FL, 1986.
35. T. Toro, Doubling and atness: geometry of measures, Notices Amer. Math. Soc. 44 (1997), no. 9, 1087-1094.
36. B. O. Turesson, Nonlinear Potential Theory and Weighted Sobolev Spaces, Lecture Notes in Mathematics, 1736, Springer-Verlag, Berlin, 2000.

#### Acknowledgement

Supported by : National Research Foundation of Korea


text:
$$\require{cancel}$$

# 6.6: Alternative Method for Calculating Torque and Tension*

If you would rather not think about finding lever arms, you can instead calculate the size of thetorque as the size of the force multiplied by the full distance to the pivot, and by the sine of the angle between the force and that full distance. Written in equation form it looks like this:

(1)

## Reinforcement Activity

The torque caused by a force depends on the angle between the line of action of the force acts and the line from where the force is applied to the pivot point. To feel this effect for yourself, try this:

Rotate a door by pushing at 90° to the door right at the outer edge.

Now apply the same force on the door, still on the very edge, but instead of pushing in a direction 90° to the door, push along the door, straight in toward the hinges. Does the door swing as it did before?

In the second case, the angle between the force direction and the distance to the pivot was 0° (they were parallel). Use the previous equation to show that the torque must be zero any time the line of action of the force goes straight through the rotation point (pivot).

Now, we know the force is 50 lbs, the distance from the pivot to the weight is 13.0 in length of the forearm and from the diagram we see the angle between the weight of the ball and the forearm distance is 60° (the same as the bicep-forearm angle because they are alternate interior angles). This is easier to see if we draw a stick figure diagram:

Stick diagram of a flexed arm holding a ball showing the bicep tension and weight and the angles between the forces and the forearm.

Now we can calculate the torque due to the ball weight as:

We just need to make this equal to the ball-weight-torque:

Then we divide both sides by and to isolate the bicep tension:

Finally we put in our values for and . Our original diagram gave us the distance as from bicep attachment to the pivot as 1.5 in and from our stick diagram we can see that the angle between the biceps tension and the distance is 180°-60° = 120°. We are ready to find the biceps tension value.

Our result of 433 lbs seems surprisingly large, but we will see that forces even larger than this are common in the muscles, joints, and tendons of the body.


text:
So I have a question that confuses me, hopefully, you guys could clear it up for me.

$$6x^2 + 7x + 1$$ Step 1: $$6x^2 + 6x + x + 1$$ Step 2: $$6x(x+1)+(x+1)$$ Now here comes my question, we have to factor it out so: $$(6x+1)+(x+1)$$ What mathematical property allows us to write $(6x+1$)?

I'm talking about the $1$ here.

I see that $6x(x+1)+(x+1)$ is the same as $6x(x+1)1<<(x+1)$ but this somehow puzzles me because I'm used to it being okay for multiplication since $1 * anything$ does not change anything. But later on we actually add them? As in $(6x+1 <<)(x+1)$

Maybe it's one too many for today but I'm really confused about this. Can someone see what I mean here?

• Did you leave out a plus sign in a couple spots? Nowhere is there a $6x(x+1)(x+1)$. – Mike Jan 3 '18 at 7:43
• Yeah I did, thanks. – user472288 Jan 3 '18 at 7:45

When you factor out terms, you are using the distributive property, $ab+ac=a(b+c)$. In step $2$, $6x$ was factored from each of the first $2$ terms $(a=6x)$. When $6x$ was factored from $6x$, what was left was $1$. If you understand that part of it, for the final step, factor out $x+1$ from the $2$ remaining terms. Just as you got $1$ when factoring $6x$ from $6x$, you also get $1$ when factoring $x+1$ from $x+1$.

It goes like this

$$6x^2 + 6x + x + 1 = 6x(x+1) + 1(x+1) = (6x+1)(x+1)$$

• yeah its just a mistake, will correct it but my question stands with the +1 that we're adding.. – user472288 Jan 3 '18 at 6:36
• That's just how factoring works. $ac + bc = (a+b)c$. Here we have $a = 6x$, $b = 1$ and $c = x+1$. Notice both terms have a common factor of $x+1$ – Dylan Jan 3 '18 at 6:37
• So anytime we don't have a $b$ other than 1 it's going to be 1? – user472288 Jan 3 '18 at 6:39
• The $1$ is implicit. $ac + c = (a+1)c$ for example – Dylan Jan 3 '18 at 6:40
• Argh not sure why I don't quite get it. You see, I thought the correct answer would be $(6x)(x+1)$ but we added the 1 here to make it $(6x+1)(x+1)$. Maybe I should just look at it tomorrow. – user472288 Jan 3 '18 at 6:43

Its the Distributive Property of Multiplication , a basic rule.

That is, $$a(b+c) = ab +ac$$

$$(6x+1)(x+1)$$ and you wanted to know from where that $1$ came into play.
$$(6x+1)\color{red}{(x+1)}=6x\color{red}{(x+1)}+1\color{red}{(x+1)}$$
So , you have distributed $(6x+1)$ over $(x+1)$ so what is its converse (or you can say,reverse)?
It is, $$6x\color{red}{(x+1)}+1\color{red}{(x+1)} = (6x+1)\color{red}{(x+1)}$$ Or we have just taken $(x+1)$ out as an common term.
Example: $$2\cdot \color{red}{5}+1\cdot \color{red}{5} = 5(2+1)$$


text:
# Pure Function (Computer Science)

A function is considered a Pure Function if

1. for identical arguments, it produces identical return values
2. it is side effect free

In this context, a Pure Function is similiar to a mathematical function.

The following function $f$

$f:\R \rightarrow \R, x \mapsto x*x$

translates to JavaScript

    const f = x => x * x;

The function conforms to the definition of a Pure Function:

1. for each input $x$ it always returns $x*x$.
2. It is side effect free (i.e. it does not change the state of any other variable)

Whereas the following functions are considered impure:

    function f (x) {        return this.x * x;    }

The method violates 1. of the above definition for a Pure Function: It produces variants because of the non-local-variable this.x. The state of this.x is unclear to f, the result is nondeterministic.

    function f (x) {        this.x = x;    }

The method violates 2. of the above definition for a Pure Function: Although it always produces undefined for the given input x, it is not side effect free since it changes the non-local variable this.x.


text:
# HW 10.4: Problem 9 Previous Problem Problem List Next Problempoint)Find the Ilnear approximation to flxY, 2) f(xY, 2)at the point

###### Question:

HW 10.4: Problem 9 Previous Problem Problem List Next Problem point) Find the Ilnear approximation to flxY, 2) f(xY, 2) at the point (1,~1,T):

#### Similar Solved Questions

##### A sample of trifluoroacetic acid, C HF3Oz, contains 61.3 g of carbon. Calculate the mass of the trifluoroacetic acid sample_mass
A sample of trifluoroacetic acid, C HF3Oz, contains 61.3 g of carbon. Calculate the mass of the trifluoroacetic acid sample_ mass...
##### Question 6 (2 points) Consider the rotation around the C2-C3 bond of 2-methylbutane. Using the values...
Question 6 (2 points) Consider the rotation around the C2-C3 bond of 2-methylbutane. Using the values listed below, calculate the energy corresponding to (eclipsed or staggered) conformations at each 60° rotation. Define the dihedral angle (6) as the angle between the H on C2 and the methyl grou...
##### Xp(v xz + z*) "Jpuy = 01 JOLJ a4140 â‚¬ ued sn ZZ sninjie) J0 Wajoayl /eluauepuny a4140 & Hed ale1s 'IZ "pg2 + JA %5 (x)b ! (x),6 puly 01 JOLJ a4140 T ued sn '0z sninjie) JO WaJoayL /eluauepuny a41J0 T Hed a1e15 '61 wns uuewaiy e Jo HwVl 34} se) Iejbaxu! ax/uiyap a41 JO uoiniuiyap 34} aie1s 81 (saNlenuapiue se UMouX Osje) S/eJBaJul aluyapul Bulmolloy 34} puiy LI saneAapIUV pue siej3310
xp(v xz + z*) "Jpuy = 01 JOLJ a4140 â‚¬ ued sn ZZ sninjie) J0 Wajoayl /eluauepuny a4140 & Hed ale1s 'IZ "pg2 + JA %5 (x)b ! (x),6 puly 01 JOLJ a4140 T ued sn '0z sninjie) JO WaJoayL /eluauepuny a41J0 T Hed a1e15 '61 wns uuewaiy e Jo HwVl 34} se) Iejbaxu! ax/uiyap...
##### Create a care plan for the following patient with two nursing diagnosis with two short term...
Create a care plan for the following patient with two nursing diagnosis with two short term goals, five interventions with rationales and evaluation of goals. 62 y/o M, hospital day 3 w/ extensive AL amyloidosis (confirmed w/ abdominal fat pad bx, a/p cycles of vcd), possible plasma cell neoplasm, ...
##### (Use six decimal places for your answers) There are 10 cans in the refrigerator: 8 of these cans are dragon energy drink while the other 2 are monster energy drink cans_ Paulos decides to randomly draw one can at a time from the refrigerator_ He has decided that he will stop the moment he draws a monster can What is the probability that the number of cans that he draws exceeds4?_0.333333_Answer Key:
(Use six decimal places for your answers) There are 10 cans in the refrigerator: 8 of these cans are dragon energy drink while the other 2 are monster energy drink cans_ Paulos decides to randomly draw one can at a time from the refrigerator_ He has decided that he will stop the moment he draws a mo...
##### Gaseous ethanol, $\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH},$ is burned with pure oxygen in a constant-volume combustion bomb. The reactants are charged in a stoichiometric ratio at the reference condition. Assume no heat transfer and find the final temperature $(>5000 \mathrm{~K})$.
Gaseous ethanol, $\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH},$ is burned with pure oxygen in a constant-volume combustion bomb. The reactants are charged in a stoichiometric ratio at the reference condition. Assume no heat transfer and find the final temperature $(>5000 \mathrm{~K})$....
##### Ketc Arid wnd e0ranoi react t0 fonm Ethyl Ecetote and natcr, Mke Ehia? HCH,CO_(so}+-C;H;OHed) - C,H;Co_CH,ea + H,Oq) HCH,CO_C;II,OH, C,H;CO,CH)anJH;o cui' Drum; ard U7et 0HCH,CO, are Iadcd [0 & Mett @Tialning mutt Lmacne aenr thve (allowl? queabontOtre Gatr tun {50, Bi{N3 [tn EKetate C Enc rerurte rraqtor Glater [Tan Sero, end Izual to Inr Matr 0r Onr rrecien urralt madun (naler ronmernMnateehe clhe [0tta thettn anergaannteri-c0 Edou to Inc notattha retatel EheCrare [uah {0r2, bUt Iea
Ketc Arid wnd e0ranoi react t0 fonm Ethyl Ecetote and natcr, Mke Ehia? HCH,CO_(so}+-C;H;OHed) - C,H;Co_CH,ea + H,Oq) HCH,CO_C;II,OH, C,H;CO,CH)anJH;o cui' Drum; ard U7et 0HCH,CO, are Iadcd [0 & Mett @Tialning mutt Lmacne aenr thve (allowl? queabont Otre Gatr tun {50, Bi{N3 [tn EKetate C Enc...
##### Draw leucine [sidechain R = CH2CH(CH3)CH3] in the zwitterionicform. There is no need to worry about stereoochemistry.
Draw leucine [sidechain R = CH2CH(CH3)CH3] in the zwitterionic form. There is no need to worry about stereoochemistry....
##### Formulate the null and research (alternative) hypothesis from the belove dataset. Research question is as follows;...
Formulate the null and research (alternative) hypothesis from the belove dataset. Research question is as follows; "What is the relationship between the price of a product and the quantinty supplied between a specific time period" Price Quantity supplied 14,5 7 13,4 8 12,7 6 16,...
##### Data Analysis Q4Complete the table below:WrialTrial 2Total mass of mixture24.92325.336 â‚¬Temperature change of mixture5.20C5.9 0CHeat absorbed by mixture542]625Heat released by reaction542]625in Jg of MgO used[Sclect ](Sclcct ]
Data Analysis Q4 Complete the table below: Wrial Trial 2 Total mass of mixture 24.923 25.336 â‚¬ Temperature change of mixture 5.20C 5.9 0C Heat absorbed by mixture 542] 625 Heat released by reaction 542] 625 in Jg of MgO used [Sclect ] (Sclcct ]...
##### Point) The rectangles in the 'graph below illustrate a left endpoint Riemann sum for f(x) = on the interval [2,6] The value of this left endpoint Riemann sum is and this Riemann sum is an the area of the region enclosed by y = f(x) the X-axis and the vertical lines X = 2 and x= 6.Left endpoint Riemann sum for yon [2,6]The rectangles in the graph below illustrate a right endpoint Riemann sum for f(x) on the interval [2,6] The value of this right endpoint Riemann sum is and this Riemann sum i
point) The rectangles in the 'graph below illustrate a left endpoint Riemann sum for f(x) = on the interval [2,6] The value of this left endpoint Riemann sum is and this Riemann sum is an the area of the region enclosed by y = f(x) the X-axis and the vertical lines X = 2 and x= 6. Left endpoint...
##### 0.2mlkxso Let X have density function, f(x) ={0.2+kx Okrs1 otherwise a) Find k. b)l Find the cdf; F(x) . c) Find P( -0.5< X< 0.71. d) Find the expected value and variance of X
0.2 mlkxso Let X have density function, f(x) ={0.2+kx Okrs1 otherwise a) Find k. b)l Find the cdf; F(x) . c) Find P( -0.5< X< 0.71. d) Find the expected value and variance of X...
##### Interval [a, has length pue midpoint at 10 Determine and
Interval [a, has length pue midpoint at 10 Determine and...
##### [. Use the divergence test; integral test . p-series test t0 deterine if the following series converge. (3p) Er+5
[. Use the divergence test; integral test . p-series test t0 deterine if the following series converge. (3p) Er+5...
##### Answer is 1.2 kV/m. Please show work and formulas used. Ex. 7.1 — The radiation pressure...
Answer is 1.2 kV/m. Please show work and formulas used. Ex. 7.1 — The radiation pressure from the sun at the surface of the earth is only 6 uPa. Calculate the RMS amplitude of the electric field component of a monochromatic electromagnetic wave that would exert such a pressure. (Monochromatic ...
##### According to the American Red Cross (http:// www.redcrossblood.org/learn-about-blood/bloodtypes), the distribution of blood types in the United States is as shown in Table 9.4.a. Draw a pie chart illustrating the blood-type distribution for Caucasians, ignoring the RH factor.b. Draw a statistical picture incorporating all of the information given.
According to the American Red Cross (http:// www.redcrossblood.org/learn-about-blood/bloodtypes), the distribution of blood types in the United States is as shown in Table 9.4.a. Draw a pie chart illustrating the blood-type distribution for Caucasians, ignoring the RH factor. b. Draw a statistical p...
##### A block of mass $M$ is at rest on a table. It is connected by a string and pulley system to a block of mass $m$ hanging off the edge of the table. Assume the hanging mass is heavy enough to make the resting block move. If the acceleration of the system and the masses of the blocks are known, which of the following could NOT be calculated?(A) Net force on each block(B) Tension in the string(C) Coefficient of kinetic friction between the table and the block of(D) The speed of the block of mass $M$
A block of mass $M$ is at rest on a table. It is connected by a string and pulley system to a block of mass $m$ hanging off the edge of the table. Assume the hanging mass is heavy enough to make the resting block move. If the acceleration of the system and the masses of the blocks are known, which o...
##### 2. A firm produces two different kinds, A and B, of a commodity. The daily cost...
2. A firm produces two different kinds, A and B, of a commodity. The daily cost of producing x units of A and y units of Bis C(x,y)=2x² - 4xy + 4y2 - 40x - 20y + 514 Suppose that the firm sells all its output at a price per unit of $24 for A and$12 for B. (a) Find the daily production levels x...
##### Evaluate the triple integral. $\begin{array}{l}{\iiint_{E} \frac{z}{x^{2}+z^{2}} d V, \text { where }} \\ {E=\{(x, y, z) | 1 \leqslant y \leqslant 4, y \leqslant z \leqslant 4,0 \leqslant x \leqslant z\}}\end{array}$
Evaluate the triple integral. $\begin{array}{l}{\iiint_{E} \frac{z}{x^{2}+z^{2}} d V, \text { where }} \\ {E=\{(x, y, z) | 1 \leqslant y \leqslant 4, y \leqslant z \leqslant 4,0 \leqslant x \leqslant z\}}\end{array}$...


text:
Enter the current bearing (degrees) into the Reverse Bearing Calculator. The calculator will evaluate and display the Reverse Bearing.

Reverse Bearing Formula

The following formula is used to calculate the Reverse Bearing.

RB = CB + 180
• Where RB is the Reverse Bearing (degrees)
• CB is the current bearing (degrees)

To calculate a reverse bearing, simply add 180 degrees to the current bearing.

How to Calculate Reverse Bearing?

The following example problems outline how to calculate Reverse Bearing.

Example Problem #1:

1. First, determine the current bearing (degrees).
• The current bearing (degrees) is given as: 50.
2. Finally, calculate the Reverse Bearing using the equation above:

RB = CB + 180

The values given above are inserted into the equation below and the solution is calculated:

RB = 50 + 180 = 230 (degrees)

Example Problem #2:

For this problem, the variables required are provided below:

current bearing (degrees) = 80


text:
Canadian Mathematical Society www.cms.math.ca
 location:  Publications → journals
Search results

Search: All articles in the CJM digital archive with keyword flag

 Expand all        Collapse all Results 1 - 6 of 6

1. CJM 2013 (vol 66 pp. 1250)

Feigin, Evgeny; Finkelberg, Michael; Littelmann, Peter
 Symplectic Degenerate Flag Varieties A simple finite dimensional module $V_\lambda$ of a simple complex algebraic group $G$ is naturally endowed with a filtration induced by the PBW-filtration of $U(\mathrm{Lie}\, G)$. The associated graded space $V_\lambda^a$ is a module for the group $G^a$, which can be roughly described as a semi-direct product of a Borel subgroup of $G$ and a large commutative unipotent group $\mathbb{G}_a^M$. In analogy to the flag variety $\mathcal{F}_\lambda=G.[v_\lambda]\subset \mathbb{P}(V_\lambda)$, we call the closure $\overline{G^a.[v_\lambda]}\subset \mathbb{P}(V_\lambda^a)$ of the $G^a$-orbit through the highest weight line the degenerate flag variety $\mathcal{F}^a_\lambda$. In general this is a singular variety, but we conjecture that it has many nice properties similar to that of Schubert varieties. In this paper we consider the case of $G$ being the symplectic group. The symplectic case is important for the conjecture because it is the first known case where even for fundamental weights $\omega$ the varieties $\mathcal{F}^a_\omega$ differ from $\mathcal{F}_\omega$. We give an explicit construction of the varieties $Sp\mathcal{F}^a_\lambda$ and construct desingularizations, similar to the Bott-Samelson resolutions in the classical case. We prove that $Sp\mathcal{F}^a_\lambda$ are normal locally complete intersections with terminal and rational singularities. We also show that these varieties are Frobenius split. Using the above mentioned results, we prove an analogue of the Borel-Weil theorem and obtain a $q$-character formula for the characters of irreducible $Sp_{2n}$-modules via the Atiyah-Bott-Lefschetz fixed points formula. Keywords:Lie algebras, flag varieties, symplectic groups, representationsCategories:14M15, 22E46

2. CJM 2012 (vol 65 pp. 66)

Deng, Shaoqiang; Hu, Zhiguang
 On Flag Curvature of Homogeneous Randers Spaces In this paper we give an explicit formula for the flag curvature of homogeneous Randers spaces of Douglas type and apply this formula to obtain some interesting results. We first deduce an explicit formula for the flag curvature of an arbitrary left invariant Randers metric on a two-step nilpotent Lie group. Then we obtain a classification of negatively curved homogeneous Randers spaces of Douglas type. This results, in particular, in many examples of homogeneous non-Riemannian Finsler spaces with negative flag curvature. Finally, we prove a rigidity result that a homogeneous Randers space of Berwald type whose flag curvature is everywhere nonzero must be Riemannian. Keywords:homogeneous Randers manifolds, flag curvature, Douglas spaces, two-step nilpotent Lie groupsCategories:22E46, 53C30

3. CJM 2011 (vol 64 pp. 1090)

Rosso, Daniele
 Classic and Mirabolic Robinson-Schensted-Knuth Correspondence for Partial Flags In this paper we first generalize to the case of partial flags a result proved both by Spaltenstein and by Steinberg that relates the relative position of two complete flags and the irreducible components of the flag variety in which they lie, using the Robinson-Schensted-Knuth correspondence. Then we use this result to generalize the mirabolic Robinson-Schensted-Knuth correspondence defined by Travkin, to the case of two partial flags and a line. Keywords:partial flag varieties, RSK correspondenceCategories:14M15, 05A05

4. CJM 2010 (vol 62 pp. 870)

 The Brascamp-Lieb Polyhedron A set of necessary and sufficient conditions for the Brascamp--Lieb inequality to hold has recently been found by Bennett, Carbery, Christ, and Tao. We present an analysis of these conditions. This analysis allows us to give a concise description of the set where the inequality holds in the case where each of the linear maps involved has co-rank $1$. This complements the result of Barthe concerning the case where the linear maps all have rank $1$. Pushing our analysis further, we describe the case where the maps have either rank $1$ or rank $2$. A separate but related problem is to give a list of the finite number of conditions necessary and sufficient for the Brascamp--Lieb inequality to hold. We present an algorithm which generates such a list. Keywords:Brascamp-Lieb inequality, Loomis-Whitney inequality, lattice, flagCategories:44A35, 14M15, 26D20
 Classification of Ding's Schubert Varieties: Finer Rook Equivalence K.~Ding studied a class of Schubert varieties $X_\lambda$ in type A partial flag manifolds, indexed by integer partitions $\lambda$ and in bijection with dominant permutations. He observed that the Schubert cell structure of $X_\lambda$ is indexed by maximal rook placements on the Ferrers board $B_\lambda$, and that the integral cohomology groups $H^*(X_\lambda;\:\Zz)$, $H^*(X_\mu;\:\Zz)$ are additively isomorphic exactly when the Ferrers boards $B_\lambda, B_\mu$ satisfy the combinatorial condition of \emph{rook-equivalence}. We classify the varieties $X_\lambda$ up to isomorphism, distinguishing them by their graded cohomology rings with integer coefficients. The crux of our approach is studying the nilpotence orders of linear forms in the cohomology ring. Keywords:Schubert variety, rook placement, Ferrers board, flag manifold, cohomology ring, nilpotenceCategories:14M15, 05E05
 On Orbit Closures of Symmetric Subgroups in Flag Varieties We study $K$-orbits in $G/P$ where $G$ is a complex connected reductive group, $P \subseteq G$ is a parabolic subgroup, and $K \subseteq G$ is the fixed point subgroup of an involutive automorphism $\theta$. Generalizing work of Springer, we parametrize the (finite) orbit set $K \setminus G \slash P$ and we determine the isotropy groups. As a consequence, we describe the closed (resp. affine) orbits in terms of $\theta$-stable (resp. $\theta$-split) parabolic subgroups. We also describe the decomposition of any $(K,P)$-double coset in $G$ into $(K,B)$-double cosets, where $B \subseteq P$ is a Borel subgroup. Finally, for certain $K$-orbit closures $X \subseteq G/B$, and for any homogeneous line bundle $\mathcal{L}$ on $G/B$ having nonzero global sections, we show that the restriction map $\res_X \colon H^0 (G/B, \mathcal{L}) \to H^0 (X, \mathcal{L})$ is surjective and that $H^i (X, \mathcal{L}) = 0$ for $i \geq 1$. Moreover, we describe the $K$-module $H^0 (X, \mathcal{L})$. This gives information on the restriction to $K$ of the simple $G$-module $H^0 (G/B, \mathcal{L})$. Our construction is a geometric analogue of Vogan and Sepanski's approach to extremal $K$-types. Keywords:flag variety, symmetric subgroupCategories:14M15, 20G05


text: I'd start by noting that $e^A = I + A + \frac{A^2}{2!} + ....$ and so $e^Ax = Ix + Ax + \frac{A^2}{2!}x + .... = ....$


text:
0
RESEARCH PAPERS

# A Study on Fluid Excitation Forces Acting on a Rotated Square Tube Bundle of $T∕D=3.1$ in Cross-Flow

[+] Author and Article Information

Nuclear Technology Research Laboratory, Central Research Institute of Electric Power Industry (CRIEPI), 2-11-1, Iwado-kita, Komae-shi, Tokyo 201-8511, Japaninada@criepi.denken.or.jp

Kimitoshi Yoneda, Akira Yasuo

Nuclear Technology Research Laboratory, Central Research Institute of Electric Power Industry (CRIEPI), 2-11-1, Iwado-kita, Komae-shi, Tokyo 201-8511, Japan

Takashi Nishihara

Civil Engineering Research Laboratory, Central Research Institute of Electric Power Industry (CRIEPI), 1646 Abiko, Abiko-shi, Chiba 270-1194, Japan

J. Pressure Vessel Technol 129(1), 162-168 (Jul 10, 2006) (7 pages) doi:10.1115/1.2409315 History: Received May 28, 2004; Revised July 10, 2006

## Abstract

The local fluid excitation force acting on a rotated square tube bundle having transverse pitch-to-diameter ratio of $T∕D=3.1$, in a single-phase cross-flow was measured, and the normalized power spectral density (NPSD) and correlation length in the axial direction of a tube were examined. The fluid excitation force acting on the interior tube was from three to ten times larger than that acting on the upstream tube. The fluid force was almost fully developed after the third row. NPSD of the fluid excitation force could be almost plotted on a single universal curve. Regarding the lift direction, there was a peak in NPSD at $fD∕u∼0.3$ caused by vortex shedding. Regarding the drag direction, there could be another peak in NPSD around twice the vortex shedding frequency. In the region of $fD∕u>0.5$, where the effect of the vortex shedding was assumed to be small in the lift direction, the correlation length of the lift direction was $∼1.1D$. NPSD was a little larger than previous results for tube bundles of relatively small pitch to diameter ratios summarized by Axisa, Antunes, and Villard (1990, J. Fluid Struct., 4, pp. 321–341).

<>

## Figures

Figure 3

PSD of fluid excitation force in lift direction (Re=38,000)

Figure 4

PSD and CSD of local fluid excitation force: (a) PSD and ∣CSD∣ and (b) phase difference of CSD

Figure 5

Relation between correlation length and Ψ1,Ψ2

Figure 6

Relation between SF̃jF̃j+1(f)∕SF̃jF̃j(f) and correlation length

Figure 7

An example of coherence function

Figure 8

Experimental result of correlation length in the lift direction

Figure 9

NPSD in the lift direction

Figure 10

NPSD in the drag direction

Figure 12

Experimental result of correlation length in the drag direction

Figure 13

An example of vibration amplitude and stress amplitude

Figure 1

Test equipment: (a), test loop, (b) test section, and (c) sensor to measure local fluid excitation forces

Figure 11

Coherence function in the drag direction: (a) in the case where the second peak does not appear (Re=38,000) and (b) in the case where the second peak appears (Re=24,000)

Figure 2

Turbulence intensity

## Errata

Some tools below are only available to our subscribers or users with an online account.

### Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related Proceedings Articles
Related eBook Content
Topic Collections


text:
Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

# Inverse transition of labyrinthine domain patterns in ferroelectric thin films

## Abstract

Phase separation is a cooperative process, the kinetics of which underpin the orderly morphogenesis of domain patterns on mesoscopic scales1,2. Systems of highly degenerate frozen states may exhibit the rare and counterintuitive inverse-symmetry-breaking phenomenon3. Proposed a century ago4, inverse transitions have been found experimentally in disparate materials, ranging from polymeric and colloidal compounds to high-transition-temperature superconductors, proteins, ultrathin magnetic films, liquid crystals and metallic alloys5,6, with the notable exception of ferroelectric oxides, despite extensive theoretical and experimental work on the latter. Here we show that following a subcritical quench, the non-equilibrium self-assembly of ferroelectric domains in ultrathin films of Pb(Zr0.4Ti0.6)O3 results in a maze, or labyrinthine pattern, featuring meandering stripe domains. Furthermore, upon increasing the temperature, this highly degenerate labyrinthine phase undergoes an inverse transition whereby it transforms into the less-symmetric parallel-stripe domain structure, before the onset of paraelectricity at higher temperatures. We find that this phase sequence can be ascribed to an enhanced entropic contribution of domain walls, and that domain straightening and coarsening is predominantly driven by the relaxation and diffusion of topological defects. Computational modelling and experimental observation of the inverse dipolar transition in BiFeO3 suggest the universality of the phenomenon in ferroelectric oxides. The multitude of self-patterned states and the various topological defects that they embody may be used beyond current domain and domain-wall-based7 technologies by enabling fundamentally new design principles and topologically enhanced functionalities within ferroelectric films.

## Access options

from\$8.99

All prices are NET prices.

## Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## Code availability

The codes that are used in this study are available from the corresponding author upon reasonable request.

## References

1. 1.

De’Bell, K., MacIsaac, A. B. & Whitehead, J. P. Dipolar effects in magnetic thin films and quasi-two-dimensional systems. Rev. Mod. Phys. 72, 225–257 (2000).

2. 2.

Seul, M. & Andelman, D. Domain shapes and patterns: the phenomenology of modulated phases. Science 267, 476–483 (1995).

3. 3.

Schupper, N. & Shnerb, N. M. Inverse melting and inverse freezing: a spin model. Phys. Rev. E 72, 046107 (2005).

4. 4.

Tammann, G. Kristallisieren und Schmelzen (Johann Ambrosius Barth, 1903).

5. 5.

Greer, A. L. Too hot to melt. Nature 404, 134–135 (2000).

6. 6.

Portmann, O., Vaterlaus, A. & Pescia, D. An inverse transition of magnetic domain patterns in ultrathin films. Nature 422, 701–704 (2003).

7. 7.

Gu, Z. et al. Resonant domain-wall-enhanced tunable microwave ferroelectrics. Nature 560, 622–627 (2018).

8. 8.

Kornev, I. I. A., Fu, H. & Bellaiche, L. Ultrathin films of ferroelectric solid solutions under a residual depolarizing field. Phys. Rev. Lett. 93, 196104 (2004).

9. 9.

Feigl, L. et al. Controlled stripes of ultrafine ferroelectric domains. Nat. Commun. 5, 4677 (2014).

10. 10.

Peters, J. J. P., Apachitei, G., Beanland, R., Alexe, M. & Sanchez, A. M. Polarization curling and flux closures in multiferroic tunnel junctions. Nat. Commun. 7, 13484 (2016).

11. 11.

Lai, B.-K., Ponomareva, I., Kornev, I. A., Bellaiche, L. & Salamo, G. J. Electric-field-induced domain evolution in ferroelectric ultrathin films. Phys. Rev. Lett. 96, 137602 (2006).

12. 12.

Streiffer, S. K. et al. Observation of nanoscale 180 degrees stripe domains in ferroelectric PbTiO3 thin films. Phys. Rev. Lett. 89, 067601 (2002).

13. 13.

Zhang, Q. et al. Nanoscale bubble domains in ultrathin ferroelectric films. Adv. Mater. 29, 1702375 (2017).

14. 14.

Solokhin, M. A., Solokhin, A. V. & Timofeev, V. S. Phase-equilibrium stability criterion in terms of the eigenvalues of the Hessian matrix of the Gibbs potential. Theor. Found. Chem. Eng. 36, 444–446 (2002).

15. 15.

Ritort, F. & Sollich, P. Glassy dynamics of kinetically constrained models. Adv. Phys. 52, 219–342 (2003).

16. 16.

Crisanti, A. & Leuzzi, L. Stable solution of the simplest spin model for inverse freezing. Phys. Rev. Lett. 95, 087201 (2005).

17. 17.

Parisi, G. Spin glasses and fragile glasses: statics, dynamics, and complexity. Proc. Natl Acad. Sci. USA 103, 7948–7955 (2006).

18. 18.

Leuzzi, L. & Nieuwenhuizen, T. M. Thermodynamics of the Glassy State (Taylor and Francis, 2007).

19. 19.

Bouchaud, J. P. Weak ergodicity breaking and aging in disordered systems. J. Phys. I 2, 1705–1713 (1992).

20. 20.

Booth, I., MacIsaac, A. B., Whitehead, J. P. & De’Bell, K. Domain structures in ultrathin magnetic films. Phys. Rev. Lett. 75, 950–953 (1995).

21. 21.

Prosandeev, S., Lisenkov, S. & Bellaiche, L. Kittel law in BiFeO3 ultrathin films: a first-principles-based study. Phys. Rev. Lett. 105, 147603 (2010).

22. 22.

Prosandeev, S., Wang, D., Ren, W., Iniguez, J. & Bellaiche, L. Novel nanoscale twinned phases in perovskite oxides. Adv. Funct. Mater. 23, 234–240 (2013).

23. 23.

Kornev, I. A., Lisenkov, S., Haumont, R., Dkhil, B. & Bellaiche, L. Finite-temperature properties of multiferroic BiFeO3. Phys. Rev. Lett. 99, 227602 (2007).

24. 24.

Eliseev, E. A., Morozovska, A. N., Nelson, C. T. & Kalinin, S. V. Intrinsic structural instabilities of domain walls driven by gradient couplings: meandering anferrodistortive-ferroelectric domain walls in BiFeO3. Phys. Rev. B 99, 014112 (2019).

25. 25.

Pismen, L. M. Patterns and Interfaces in Dissipative Dynamics (Springer, 2006).

26. 26.

Nagaosa, N. & Tokura, Y. Topological properties and dynamics of magnetic skyrmions. Nat. Nanotechnol. 8, 899–911 (2013).

27. 27.

Lu, L. et al. Topological defects with distinct dipole configurations in PbTiO3/SrTiO3 multilayer films. Phys. Rev. Lett. 120, 177601 (2018).

28. 28.

Newell, A. C., Passot, T., Ercolani, N. & Indik, R. Elementary and composite defects of striped patterns. J. Phys. II 5, 1863–1882 (1995).

29. 29.

Prokhorenko, S., Nahas, Y. & Bellaiche, L. Fluctuations and topological defects in proper ferroelectric crystals. Phys. Rev. Lett. 118, 147601 (2017).

30. 30.

Harrison, C. et al. Mechanisms of ordering in striped patterns. Science 290, 1558–1560 (2000).

31. 31.

Boyn, S. et al. Learning through ferroelectric domain dynamics in solid-state synapses. Nat. Commun. 8, 14736 (2017).

32. 32.

Chua, L. Memristor-the missing circuit element. IEEE Trans. Circuit Theory 18, 507–519 (1971).

33. 33.

Yang, S. Y. et al. Above-bandgap voltages from ferroelectric photovoltaic devices. Nat. Nanotechnol. 5, 143–147 (2010).

34. 34.

Yang, J. J., Strukov, D. B. & Stewart, D. R. Memristive devices for computing. Nat. Nanotechnol. 8, 13–24 (2013).

35. 35.

Thickness dependency of 180 degree stripe domains in ferroelectric ultrathin films: a first-principles study. Appl. Phys. Lett. 91, 152909 (2007).

36. 36.

Schilling, A. et al. Scaling of domain periodicity with thickness measured in BaTiO3 single crystal lamellae and comparison with other ferroics. Phys. Rev. B 74, 024115 (2006).

37. 37.

Naumov, I. I., Bellaiche, L. & Fu, H. Unusual phase transitions in ferroelectric nanodisks and nanorods. Nature 432, 737–740 (2004).

38. 38.

Sichuga, D. & Bellaiche, L. Epitaxial Pb(Zr,Ti)O3 ultrathin films under open-circuit electrical boundary conditions. Phys. Rev. Lett. 106, 196102 (2011).

39. 39.

Yadav, A. K. et al. Observation of polar vortices in oxide superlattices. Nature 530, 198–201 (2016); corrigendum 534, 138 (2016).

40. 40.

Bellaiche, L., Garcia, A. & Vanderbilt, D. Finite-temperature properties of Pb(Zr1−xTix)O3 alloys from first principles. Phys. Rev. Lett. 84, 5427–5430 (2000).

41. 41.

Kittel, C. Theory of the structure of ferromagnetic domains in films and small particles. Phys. Rev. 70, 965–971 (1946).

42. 42.

Eliseev, E. A. et al. Labyrinthine domains in ferroelectric nanoparticles: manifestation of a gradient-induced morphological transition. Phys. Rev. B 98, 054101 (2018).

43. 43.

Morozovska, A. N., Fomichov, Y. M., Maksymovych, P., Vysochanskii, Y. M. & Eliseev, E. A. Analytical description of domain morphology and phase diagrams of ferroelectric nanoparticles. Acta Mater. 160, 109–120 (2018).

44. 44.

Xu, B., Iniguez, J. & Bellaiche, L. Designing lead-free antiferroelectrics for energy storage. Nat. Commun. 8, 15682 (2017).

## Acknowledgements

We acknowledge DARPA grant number HR0011727183-D18AP00010 (TEE programme), ARO grant number W911NF16-1-0227 and DARPA grant number HR0011-15-2-0038 (MATRIX programme). Computations were made using the Arkansas High Performance Computing Center and the Arkansas Economic Development Commission. B.X. acknowledges the startup fund from Soochow University and support from Priority Academic Program Development (PAPD) of Jiangsu Higher Education Institutions. S. Prosandeev appreciates the support of RMES grant number 3.1649.2017/4.6 and RFBR grant number 18-52-00029_Bel_a. V.G., S.F. and B.D. acknowledge a public grant overseen by the French National Research Agency (ANR) as part of the Investissements d’Avenir programme (reference: ANR-10-LABX-0035, Labex NanoSaclay) and the project EXPAND through ANR-17-CE24-0032, as well as through the PIAF project.

## Author information

Authors

### Contributions

Y.N. conceived the study of the inverse-transition phenomenon as part of a research project about labyrinthine structures initiated by L.B. Y.N. and S. Prokhorenko carried out the simulations and analysed the data. J.F. fabricated the thin films by pulsed-laser deposition, carried out the annealing experiments with the help of C.C., and performed the PFM measurements. B.X. and S. Prosandeev performed additional BiFeO3 simulations. S.F. performed the conducting atomic force microscopy experiments. This experimental work was done under the guidance of V.G. B.D. performed XRD measurements. After a first draft written by Y.N., all authors discussed the results and contributed to the final manuscript.

### Corresponding author

Correspondence to Y. Nahas.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Peer review information Nature thanks Anna Morozovska and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Extended data figures and tables

### Extended Data Fig. 1 Temperature evolution of the parallel-stripe ground state.

af, The evolution with temperature of the ground-state dipolar configuration (parallel stripes) in the middle layer of an 80 × 80 × 5 film of Pb(Zr0.4Ti0.6)O3 upon increasing temperature (10 K (a), 110 K (b), 185 K (c), 260 K (d), 335 K (e) and 410 K (f)), where grey (red) dipoles are oriented along the [001] $$([00\bar{1}])$$ pseudo-cubic direction). g, The temperature variation of the scaled structure factor $$\tilde{S}(a{q}_{s},T)$$. Vertical dashed line indicates the inflection point of $$\tilde{S}(a{q}_{{s}},T)$$ and is taken as the locus of Tc.

### Extended Data Fig. 2 Specific heat of the parallel-stripe and labyrinthine states.

Specific heat C as a function of temperature (in arbitrary units). Data were gathered upon slowly heating the ground-state parallel-stripe domain pattern (1) and the labyrinthine domain pattern (2).

### Extended Data Fig. 3 Spatial distribution of on-site, first and second nearest neighbours and dipole–dipole interaction energies.

ad, The probability density functions of the cell-by-cell energies (on-site energy (a), first nearest neighbours (1NN) interaction energy (b), second nearest neighbours (2NN) interaction energy (c) and dipole–dipole interaction energy (d)) calculated for the labyrinthine domain structure at 10 K for Pb(Zr0.4Ti0.6)O3 within a 64 × 64 × 5 supercell. Each panel provides the contributions stemming from the domains and domain walls, separately. eh, The corresponding mappings of energies onto the middle layer of the film. Blue to red colour gradient shows increasing values of unit-cell energies.

### Extended Data Fig. 4 Spatial distribution of third nearest neighbours, elastic and electrostrictive energies.

ac, The probability density functions of the cell-by-cell energies (third nearest neighbours (3NN) interaction energy (a), elastic energy (b) and electrostrictive energy (c)) calculated for the labyrinthine domain structure at 10 K for Pb(Zr0.4Ti0.6)O3 within a 64 × 64 × 5 supercell. Each panel provides the contributions stemming from the domains and domain walls, separately. df, The corresponding mappings of energies onto the middle layer of the film. Blue to red colour gradient shows increasing values of unit-cell energies.

### Extended Data Fig. 5 Energetics and spatial correlations at play in the inverse transition.

a, Evolution with temperature of dipole–dipole energy density upon heating the ground-state parallel-stripe domain pattern (1) and the labyrinthine domain pattern (2). These two curves meet above 200 K, the temperature at which the inverse transition occurs. The third curve (3) corresponds to what would have been the dipole–dipole energy of domain walls if the labyrinthine domain walls would have gradually wiggled with no reordering of the stripes (fictive labyrinthine evolution). b, Evolution with temperature of the typical size of locally ordered ground-state tiles composing the labyrinthine domain pattern. Data were obtained via the analysis of structure factors of square patches of varying size at each temperature. Specifically, at each temperature, ξ corresponds to the maximal patch size featuring two-peaked structure factor. Solid line is a guide for the eyes.

### Extended Data Fig. 6 Simulations of the inverse transition in thick BiFeO3 films.

a, b, The evolution with temperature of the domain pattern in BiFeO3 in terms of the distribution of the ferroelectric (a) and AFD (b) order parameters. Results were obtained through Monte Carlo simulations using the effective Hamiltonian scheme of a 36 × 36 × 10 film subjected to a −0.16% misfit strain, with periodic boundary conditions. The system was abruptly quenched from 2,000 K down to 10 K and consequently progressively heated up with 40,000 relaxation sweeps at each temperature. It can be seen that the distributions of both ferroelectric and AFD order parameters exhibit the inverse transition with Tinv ≈ 1,100 K and Tc ≈ 1,300 K (these numerically predicted temperatures are in good agreement with our experimental findings). We find that below Tinv, the system exhibits mixed 109° and 71° domain walls, while above Tinv, only 109° domain walls are observed. In a, dipoles are coloured according to their z component. In b, AFD vectors are coloured according to the arctan(Wy/Wx), where Wy and Wx denote the y and x components of the AFD local vectors.

### Extended Data Fig. 7 Simulations of the inverse transition in thin BiFeO3 films.

a, b, The evolution with temperature of the domain pattern in BiFeO3 in terms of the distribution of the dipolar (a) and AFD (b) order parameters. Results were obtained through Monte Carlo simulations using the effective Hamiltonian scheme of a 36 × 36 × 5 film subjected to a −0.5% misfit strain with open boundary conditions, a partial screening at film interfaces (effective screening parameter β = 0.5). The system was abruptly quenched from 2,000 K down to 10 K and consequently progressively heated up with 40,000 relaxation sweeps at each temperature. It can be seen that the distributions of both ferroelectric and AFD order parameters exhibit the inverse transition with Tinv ≈ 525 K and Tc ≈ 650 K. We find that the system exhibits 71° domain walls. In a, dipoles are coloured according to their z component. In b, AFD vectors are coloured according to the arctan(Wy/Wx), where Wy and Wx denote the y and x components of the AFD local vectors.

### Extended Data Fig. 8 Origin of the memory effect.

ad, Structure factor plots (a, b) and bubble configurations (c, d). Panels a and c correspond to the bubble state at 10 K, as obtained upon applying a field of 40 × 107 V m−1 to the parallel-stripe configuration. Panels b and d correspond to the bubble state at 10 K, as obtained upon applying a field of 40 × 107 V m−1 to the labyrinthine configuration.

### Extended Data Fig. 9 Imaging of the domain structure evolution with temperature in the BiFeO3 sample.

al, Topography, in-plane PFM phase and amplitude of the as-grown sample (ac), and the same sample after annealing at 773 K (df), 1,023 K (gi) and 1,073 K (jl). Each annealing step was 1 h long. ‘z-scale’ corresponds to 4 nm (a, d, g) and 10 nm (j).

### Extended Data Fig. 10 Structural properties of the BiFeO3 sample before and after annealing.

2θω XRD patterns of the as-grown BiFeO3 sample and the same sample after the successive annealing up to 1,073 K. a, Full scale. b, Zoom around the (001) peak.

### Extended Data Fig. 11 Evolution with temperature of the lattice parameter of the BiFeO3 sample.

Evolution of the out-of-plane parameter upon heating the parallel-stripe phase of the BiFeO3 sample. Values were obtained by fitting the XRD data and do not reveal any phase transition up to 1,160 K.

### Extended Data Fig. 12 Ferroelectric and elastic domains in the BiFeO3 sample.

Ferroelectric and elastic domain structures in a BiFeO3 thin film grown on a (110)-oriented DyScO3 substrate before and after annealing. a, b, In-plane PFM phase images of a BiFeO3 thin film for an as-grown sample (a) and a sample after annealing at 1,073 K for 1 h (b). Images are 2 × 2 μm2. c, d, Reciprocal space mappings around (002) reflections for the same BiFeO3 thin film for the as-grown sample (c) and the sample after annealing at 1,073 K for 1 h (d). The pink arrows indicate the satellite positions to the left and right of the (002) film peak. The X-ray beam is aligned at Φ = 90°, that is, perpendicular to the stripes. The indices of DyScO3 and BiFeO3 are written in the monoclinic cells. Qx,y and Qz indicate the in-plane and out-of-plane reciprocal space units, respectively.

## Rights and permissions

Reprints and Permissions

Nahas, Y., Prokhorenko, S., Fischer, J. et al. Inverse transition of labyrinthine domain patterns in ferroelectric thin films. Nature 577, 47–51 (2020). https://doi.org/10.1038/s41586-019-1845-4

• Accepted:

• Published:

• Issue Date:

• ### Anomalous Motion of Charged Domain Walls and Associated Negative Capacitance in Copper–Chlorine Boracite

• Joseph G. M. Guy
• , Charlotte Cochard
• , Elisabeth Soergel
• , Roger W. Whatmore
• , Michele Conroy
• , Kalani Moore
• , Eileen Courtney
• , Alan Harvey
• , Ursel Bangert
• , Amit Kumar
• , Raymond G. P. McQuaid
•  & J. Marty Gregg

• ### Recent Progress on Topological Structures in Ferroic Thin Films and Heterostructures

• Shanquan Chen
• , Shuai Yuan
• , Zhipeng Hou
• , Yunlong Tang
• , Jinping Zhang
• , Tao Wang
• , Kang Li
• , Weiwei Zhao
• , Xingjun Liu
• , Lang Chen
• , Lane W. Martin
•  & Zuhuang Chen

• ### Flexo-elastic control factors of domain morphology in core-shell ferroelectric nanoparticles: Soft and rigid shells

• Eugene A. Eliseev
• , Anna N. Morozovska
• , Riccardo Hertel
• , Hanna V. Shevliakova
• , Yevhen M. Fomichov
• , Victor Yu Reshetnyak
•  & Dean R. Evans

Acta Materialia (2021)

• ### Thermally driven state in a spin-1 model with competing interactions

• G. L. K. Frantz
• , M. Schmidt
•  & F. M. Zimmer

Physical Review E (2021)

• ### Ion adsorption-induced reversible polarization switching of a van der Waals layered ferroelectric

• Dong-Dong Xu
• , Ru-Ru Ma
• , Ai-Ping Fu
• , Zhao Guan
• , Ni Zhong
• , Hui Peng
• , Ping-Hua Xiang
•  & Chun-Gang Duan

Nature Communications (2021)


text:
#### Solved Examples and Worksheet for Interpreting Scatter Plots

Q1A wholesale dealer of stationery products plots the scatter diagram of his income through the years 1985 to 2000.What is his income in the year 1990?

A. $45,000 B.$40,000
C. $50,000 D.$35,000

Step: 1
The height of each point in the graph indicates the income (in $1000) of the dealer in the particular year. Step: 2 From the graph, the incomes of the dealer in 1985, 1990, 1995 and 2000 are$40000, $45000,$35000, and $50000. Step: 3 The income of the dealer in 1990 was$45000.
Correct Answer is :   $45,000 Q2What is the decrease in the consumption of goods from the year 1999 to 2000 as found in the scatter plot? A. 10 million tons B. 35 million tons C. 25 million tons D. 30 million tons Step: 1 From the plot the consumption of the goods in 1999 was 35 million tons. Step: 2 The consumption of the goods in the year 2000 was 25 million tons. Step: 3 So, the decrease in the consumption of goods is 35 - 25 = 10 million tons. Correct Answer is : 10 million tons Q3The scatter plot shows the number of hours children of different ages sleep. Find how many extra hours a 4 year old child sleeps than a 7 year old child. A. 4 B. 3 C. 7 D. 5 Step: 1 The height of each point in the scatter plot indicates the number of hours a child of the particular age sleeps for. Step: 2 From the graph, the number of hours a 4 year old child sleeps is 14. [The vertical coordinate of the point corresponding to 4 years is 14.] Step: 3 From the graph, the number of hours a 7 year old child sleeps is 10. [The vertical coordinate of the point corresponding to 7 years is 10.] Step: 4 The difference between the number of sleeping hours of a 4 year old and 7 year old child = 14 - 10 = 4 hours. [Subtract 10 from 14.] Step: 5 So, a 4 year old child sleeps for 4 more hours than a 7 year old child. Correct Answer is : 4 Q4The scatter plot shows the number of cars sold in different quarters of the year 1995. Between which two quarters did the number of cars sold decline? A. 1st - 2nd B. 2nd - 3rd C. 3rd - 4th D. The sales never declined Step: 1 The number of cars sold in different quarters of the year 1995 is shown on the given scatter plot. Step: 2 The number of cars sold in the 1st quarter of the year 1995 = 100. Step: 3 The number of cars sold in the 2nd quarter of the year 1995 = 150. Step: 4 The number of cars sold in the 3rd quarter of the year 1995 = 250. Step: 5 The number of cars sold in the 4th quarter of the year 1995 = 200. Step: 6 By looking at the above data, we notice that, the number of cars sold is increased from 1st quarter to 3rd quarter, and decreased from 3rd quarter to 4th quarter. Step: 7 Therefore, in between 3rd and 4th quarters the number of cars sold is declined. Correct Answer is : 3rd - 4th Q5Paula drives at a constant speed from St. Paul to Providence. The scatter plot shows the distance traveled by her at various times. What is the speed at which she is driving? A. 75 mph B. 50 mph C. 100 mph D. 80 mph Step: 1 Each point on the scatter plot represents the distance in miles traveled by Paula by the particular time. Step: 2 Speed = Distancetime [Speed is defined as the rate of distance traveled.] Step: 3 So, the speed at which Paula drives = 1503= 50 mph [Divide the vertical coordinate of any point by the corresponding horizontal coordinate.] Step: 4 So, Paula is driving at 50 mph speed from St. Paul to Providence. Correct Answer is : 50 mph Q6Valerie drives at a constant speed from Milan to Rosedale. The scatter plot shows the distance traveled by her at various times. What is the speed at which she is driving? A. 75 mph B. 50 mph C. 80 mph D. 100 mph Step: 1 Each point on the scatter plot represents the distance in miles traveled by Valerie by the particular time. Step: 2 Speed = Distancetime [Speed is defined as the rate of distance traveled.] Step: 3 So, the speed at which Valerie drives = 1002= 50 mph [Divide the vertical coordinate of any point by the corresponding horizontal coordinate.] Step: 4 So, Valerie is driving at 50 mph speed from Milan to Rosedale. Correct Answer is : 50 mph Q7The scatter plot represents the relationship between the weekly pay and the number of hours worked per week in a month. What was the increase in the weekly pay as the number of hours worked increased from 3 to 9? A.$40
B. $10 C.$30
D. $20 Step: 1 The height of each point in the plot represents the number of hours worked corresponding to the weekly pay. Step: 2 The weekly pay is$10 when the number of hours worked per week is 3.
[The corresponding value of the weekly pay is $10 when the height of the point represents the value 3.] Step: 3 The weekly pay is$30 when the number of hours worked per week is 9.
[The corresponding value of the weekly pay is $30 when the height of the point represents the value 9.] Step: 4 Increase in the weekly pay with the increase in the number of hours worked from 3 to 9 = Weekly pay when the number of hours worked is 9 - Weekly pay when the number of hours worked is 3 =$30 - $10 =$20
[Substitute and subtract.]
Step: 5
So, the increase in the weekly pay as the number of hours worked increases from 3 to 9 is $20. Correct Answer is :$20
Q8The scatter plot represents the times taken by Frank to cover different distances. What was the time taken to cover the total distance?

A. 20 sec
B. 40 sec
C. 30 sec
D. 10 sec

Step: 1
The height of each point in the plot represents the time taken to cover the corresponding distance.
Step: 2
From the plot, the total distance recorded is 400 m.
[The final point represents a distance of 400 m.]
Step: 3
The vertical coordinate corresponding to the distance of 400 m covered is 40.
Step: 4
So, the time taken to cover the total distance is 40 sec.
Correct Answer is :   40 sec
Q9The scatter plot represents the contributions made by a few people to an orphanage in the last 3 years. What is the minimum contribution made to the orphanage?

A. $1,000 B.$3,000
C. $2,000 D.$4000

Step: 1
The height of each point in the plot represents the contribution made in that particular year.
Step: 2
The height of the point corresponding to the 1st year represents the value $1000, which is the least. Step: 3 So, the minimum contribution made to the orphanage was$1,000.
Correct Answer is :   $1,000 Q10The scatter plot displays the weight of Dennis as he grew old. What was his weight when he was 20 years old? A. 100 lb B. 50 lb C. 175 lb D. 125 lb Step: 1 In the scatter plot, the horizontal axis represents Dennis's age and the vertical axis represents his weight in lb. Step: 2 The vertical distance of each point in the scatter plot indicates the weight of Dennis as he grew old. Step: 3 From the graph, the vertical distance of the point corresponding to 20 is 100. Step: 4 So, Dennis weighed 100 lb when he was 20 years old. Correct Answer is : 100 lb Q11The scatter plot displays the weight of Frank as he grew old. At what age was his weight 175 lb? A. 10 years B. 20 years C. 60 years D. 30 years Step: 1 In the scatter plot, the horizontal axis represents Frank's age and the vertical axis represents his weight in lb. Step: 2 The vertical distance of each point in the scatter plot indicates the weight of Frank as he grew older. Step: 3 From the graph, the vertical distance of the point corresponding to age 60 is 175. Step: 4 So, Frank's weight was 175 lb when he was 60 years old. Correct Answer is : 60 years Q12A wholesale dealer of stationery products plots his income through the years 1985 to 2000 in a diagram. In which year did he have the least income ? A. 1985 B. 2000 C. 1990 D. 1995 Step: 1 The height of each point in the graph indicates the income (in$1000) of the dealer in the particular year.
Step: 2
From the graph, the incomes of the dealer in 1985, 1990, 1995 and 2000 are $40000,$45000, $35000 and$50000 respectively.
Step: 3
The income of the dealer in the year 1995 was the least.
Correct Answer is :   1995
Q13The scatter plot displays the income earned by a wholesale dealer of stationery products through the years 1985 to 2000. What was his income in the year 2000?

A. $45,000 B.$40,000
C. $50,000 D.$35,000

Step: 1
In the scatter plot, the horizontal axis indicates the year and the vertical axis indicates the income (in $1000) earned by the wholesale dealer. Step: 2 The height of each point in the graph indicates the income (in$1000) of the dealer in the particular year.
Step: 3
From the graph, the vertical coordinate of the point corresponding to the year 2000 is 50.
Step: 4
So, the income of the wholesale dealer in 2000 was $50,000. Correct Answer is :$50,000
Q14The scatter plot shows the number of hours children of different ages sleep. Find the number of extra hours a 2 year old child sleeps compared to a 6 year old child.

A. 7
B. 5
C. 8
D. 6

Step: 1
The height of each point in the scatter plot indicates the number of hours a child of the particular age sleeps for.
Step: 2
From the graph, the number of hours a 2 year old child sleeps is 18.
[The vertical coordinate of the point corresponding to 2 years is 18.]
Step: 3
From the graph, the number of hours a 6 year old child sleeps is 12.
[The vertical coordinate of the point corresponding to 6 years is 12.]
Step: 4
The difference between the number of sleeping hours of a 2 year old and 6 year old child = 18 - 12 = 6 hours
[Subtract 12 from 18.]
Step: 5
So, a 2 year old child sleeps for 6 more hours than a 6 year old child.
Correct Answer is :   6
Q15Using the scatter plot and the trend line, estimate the test score of a student who studied for 20 hours.

A. 80
B. 83
C. 85
D. 81

Step: 1
Draw a vertical line starting from 20 hours in the scatter plot. This line intersects the trend line at a point.
Step: 2
From the intersection, draw a horizontal line that intersects the y-axis. This intersection point on the y-axis gives the required test score as the y-axis represents the test scores.
Step: 3
So, a student who studied for 20 hours would have scored about 83 points.
Correct Answer is :   83
• Data


text:
## Extracting narrow-band ZigBee signals from a wide-band WiFi signal?

Started by 6 years ago10 replieslatest reply 6 years ago227 views

Hello,

I am planning the following.

First sample the 20 MHz WiFi channel (WiFi channel-1 in figure).

Put band pass filters (5 MHz wide) around each of the ZigBee center frequencies (11, 12, 13, 14).

Re-sample the chunks to 4 MHz.

Is there anything wrong with this approach ?

[ - ]

Presumably you're trying to do a dual WiFi/Zigbee receiver?

I don't see anything wrong with it in principle; the devil is always in the details, however.

[ - ]

Yes, I am trying to do ZigBee and WiFi simultaneously in real-time.

So far I found successive interference cancellation the most suitable one.

[ - ]

There's nothing "wrong" with it, given what you're asking is fairly vague. If you're wanting to keep the WiFi data for future processing then there would be some differences in design. If you're purely looking for the ZigBee channels then something like a polyphase filter (think channelizer) given your sampling may introduce aliasing of the wifi into the Zigbee channels which you'll have to account for.

Again, if you're purely keeping the zigbee channels, couldn't you narrow your filters at the time of collect and then collect each channel separately? I don't believe zigbee spreads its data across multiple channels.

[ - ]

No actually I have to decode both WiFi and ZigBee at the same time.

So during interference, assuming most of the time WiFi will be strong, I can detect the WiFi and decode it first.

Now from the decoded WiFi data(bits) and channel estimates, I can recreate the estimated WiFi signal and subtract it from the mixed signal(ZigBee + WiFi).

This will give me a signal which will be relatively clean from WiFi interference and hence making ZigBee easy to detect/decode.

"If you're wanting to keep the WiFi data for future processing then there would be some differences in design. "

[ - ]

Wow.  10 years ago I'd say "that's wacky enough to be a good thesis topic", but I have this sad feeling that technology has progressed to the point where it may actually be practical in the real world.

Even if it isn't practical, there should be some papers or even a thesis in there someplace.

[ - ]
Hahaha, I agree with you Tim.

But this is more than that :D (Devil is in the details :D)

Following to this, I have to generalize it for other standards where there are re-usable blocks during the transmission and reception chain(WiFi + LTE).

Development of an architecture for simultaneous operation of different wireless standards using GPU based SDR.

The difficulty is there in the RF part where same RF front end has to be used for all the standards :)

[ - ]

And yes, at least a good part of the challenge will be sampling the RF cleanly enough and fast enough.

[ - ]

As of now my WiFi and ZigBee Rx are working in real time independently.

Yes I will try your suggestion also, to make them working independently in vicinity of each other.

Yes the sampling and AGC requirements are still a challenge and we are working on it.

Just for info, I am developing with Openairinterface and USRP B210.

[ - ]

Hi Tim,

I realize that you used the term gut feeling, but I would like to know why your gut believes that 20 dB is the best you could get?

David

[ - ]
In general, it's a problem that involves saying that $$y = x - \hat{x}$$.  It's only going to work as good as the estimate $$\hat{x}$$.  To make it work you must generate $$\hat{x}$$ from the message, the carrier frequency, the carrier phase, the bit timing of the message, and the amplitude of the message.  I count five different ways that it can go horribly wrong right there.