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https://mathoverflow.net/questions/115465/lattice-in-a-certain-lie-group/115468 | 1,600,449,166,000,000,000 | text/html | crawl-data/CC-MAIN-2020-40/segments/1600400188049.8/warc/CC-MAIN-20200918155203-20200918185203-00405.warc.gz | 514,174,239 | 29,371 | # Lattice in a certain Lie group
Let $G_n$ be the Lie group consisting of $n \times n$ upper triangular matrices of determinant $1$ with real entries. In other words, $$G_n = \{\text{\left(\begin{matrix} a_{11} & a_{12} & a_{13} & \cdots & a_{1n} \\ 0 & a_{22} & a_{23} & \cdots & a_{2n} \\ 0 & 0 & a_{33} & \cdots & a_{3n} \\ \vdots & \vdots & \vdots & & \vdots \\ 0 & 0 & 0 & \cdots & a_{nn} \end{matrix}\right) | a_{ij} \in \mathbb{R} and a_{11} \cdots a_{nn} = 1}\}.$$ Does $G_n$ contain a lattice (i.e. a discrete subgroup of finite covolume)? The obvious thing to try is to take the subgroup consisting of matrices with integer entries, but that does not work (though it would work if we were working with strictly upper triangular matrices).
• If a locally compact group $G$ contains a lattice, then $G$ has to be unimodular and you $G_n$ is not. – Misha Dec 5 '12 at 4:41
• @Misha : Can you give me a reference for the fact that locally compact groups that contain lattices have to be unimodular? – Edward Cooper Dec 5 '12 at 4:50
• See e.g. Corollary 1 on page 3 of Venkataramana's notes on lattices: math.lsu.edu/~pdani/conferences/goa2010/SpeakerNotes/… – Misha Dec 5 '12 at 5:08
• To Misha's remarks I'd add the comment that Example 3 (on page 2 of the lecture notes he links from Tata) illustrates for $n=2$ why this type of solvable group isn't unimodular. I'm not sure how to sort out those solvable Lie groups which are unimodular, but an old theorem of Mostow shows that any lattice must then be arithmetic. (Venkatarama gives a few of the standard references.) – Jim Humphreys Dec 5 '12 at 17:05
• The proof that the group of upper triangular matrices $U_n$ is not unimodular could be found for instance in Bump's book "Automorphic forms and representations", page 426. – Misha Dec 5 '12 at 22:25
@Edward, here is a short proof which I wrote awhile ago for my notes on group theory.
Lemma. If a 2nd countable locally compact group $G$ contains a lattice $\Gamma$ then $G$ is unimodular.
Proof. For arbitrary $g \in G$ consider the push-forward $\nu=R_g(\mu)$ of the (left) Haar measure $\mu$ on $G$; here $R_g$ is the right multiplication by $g$: $$\nu(E)= \mu(Eg).$$ Then $\nu$ is also a left Haar measure on $G$. By the uniqueness of Haar measure, $\nu= c\mu$ for some constant $c > 0$. The lattice $\Gamma \le G$ has a fundamental domain $D\subset G$, i.e., a measurable subset of $G$ such that $$\bigcup_{\gamma\in \Gamma} \gamma D= G, \quad \mu(\gamma D\cap D)=0, \quad \forall \gamma\in \Gamma \setminus 1.$$ (Proof of existence of such domain uses 2nd countable assumption.) In particular, $0<\mu(D)<\infty$, since $\Gamma$ is a lattice. Then $Dg$ is again a fundamental domain for $\Gamma$ and, thus, $\mu(D)=\mu(Dg)$. Hence, $\mu(D) = \mu(Dg) = c\nu(D)$. It follows that $c=1$. Thus, $\mu$ is also a right Haar measure. qed
Another proof could be found in Chapter I of Raghunathan's book "Discrete subgroups of Lie groups".
• A reference is also in Deitmar-Echterhoff "Principles of Harmonic Analysis" Chapter 9 somewhere, not sure though if they refer only to uniform lattices. Nice proof +1. – Marc Palm Dec 5 '12 at 9:37
• It's not clear why two fundamental domains have the same measure. – Guntram Dec 5 '12 at 18:21
• @Guntram: Because the measure of a fundamental domain is the same as the measure of the quotient $\Gamma\backslash G$. – Misha Dec 5 '12 at 20:47
• If $G$ is locally compact and abelian, and $D$ is a lattice in $G$, do you know if one can prove the existence of a measurable FD without the second countability assumption?.. Should I ask this as a question, or is it OK if I ask here? – Nick S Feb 15 '14 at 20:48
• @NickS: I think you should ask a separate question. I do not know an off hand answer and will not be able to think about this for a week. – Misha Feb 15 '14 at 20:57 | 1,190 | 3,837 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.109375 | 3 | CC-MAIN-2020-40 | latest | en | 0.863038 |
https://quantdare.com/foreseeing-future-viterbi-algorithm/ | 1,713,299,782,000,000,000 | text/html | crawl-data/CC-MAIN-2024-18/segments/1712296817106.73/warc/CC-MAIN-20240416191221-20240416221221-00169.warc.gz | 423,458,021 | 21,132 | Unlock the Power of Quantitative Strategies: Explore Our Cutting-Edge Website Today!
Foreseeing the future: a user’s guide
Jose Leiva
06/09/2017
2
Everybody would like to see the future. If you’re a portfolio manager, you’d definitely love to see the future.
Many posts here on QuantDare deal with the challenge of predicting the future (with Prophet, Random Forests, Lasso, etc). This time, we talk about something different: imagine we are able to predict the future exactly. Now what? How could we exploit this priceless information? As we will see, the Viterbi algorithm gives us an answer.
A simple case
Let’s suppose you have 4 € to invest in either of two assets and you have seen in the crystal ball that the returns for the next three days are: (+20%,+20%,+20%) for the first asset and (+40%,+40%,-60%) for the second. How would you manage your money in the next days? You would probably (and correctly) invest your 4 € in asset 2 today, keep your position tomorrow, and then sell all your stocks of asset 2 and buy asset 1 the next day. This way, you would make a profit of +135%, ignoring transaction costs.
That was easy. Now imagine you are a portfolio manager and have 4 billion € to invest in the same assets. In this case, applying the former sequence of actions might not be possible. Sell/buy orders at such scale can have a huge, undesirable impact on the market, incurring enormous costs. Also, additional constraints often exist on the maximum turnover of a portfolio. For these reasons, we might only be allowed to trade a fraction of our positions in a given day.
To illustrate the consequences, imagine now that you are only allowed to sell/buy 25% of your portfolio each day, and you start with a 50% exposure to each of the assets. How would you manage your money now? If your answer is “I’d sell as much as possible of the worst asset to buy as much as possible of the best one in each period”, then… you’d be wrong!
Let’s examine why this is not optimal. Following this myopic strategy, we would move 25% of our portfolio to asset 2 today, then another 25% tomorrow, and finally 25% back to asset 1 on the third day. Graphically, we can represent the portfolio as a set of 4 “boxes” (each containing 25%) in orange (asset 1) or blue (asset 2), with the shown returns:
The accumulated return of the portfolio is +13.4%.
Now consider this alternative sequence of actions: we preserve the 50/50 composition today, then we add 25% of the portfolio to asset 1 tomorrow and another 25% the following day.
The cumulative return of the portfolio is now +95%.
What happened here? The most intuitive explanation is that in the first case, our exposure to asset 2 was too large previous to its crash, while in the second case we have no more asset 2 when the crash arrives. In other words: in the first case, we chose a wrong sequence of decisions despite being able to see the future, because we made each decision day by day (i.e. myopically).
This example illustrates how we should be prepared to exploit our clairvoyant superpowers; otherwise, we might feel like the Greatest American Hero, a hilarious character who was unable to properly use his Superman-like suit because he lost the user’s manual in the first episode.
So, how can we identify the optimal trajectory in the space of all the possible active portfolio managements?
The quick answer is “explore them all, and choose the most profitable”, but is this computationally tractable? Imagine that instead of a 3-day period, we aim at managing the portfolio during a 90-day interval. How many alternative trajectories are there? Each day we have at least two actions at our disposal (either selling one of the assets to buy the other or do nothing), so the number of trajectories is at least $$2^{90}$$. That’s higher than the estimated number of atoms in the Universe!
The Viterbi algorithm
Don’t panic! There’s actually no need to explore all the trajectories. Instead, we can make use of the Viterbi algorithm. It’s widely used on hidden Markov models and digital communications to determine the most likely sequence of hidden states or symbols, respectively. When applied to our hindsight portfolio management problem, it makes use of a very intuitive simplification to avoid the exponential explosion of alternative trajectories: for a given portfolio composition at the n-th day, there’s no need to store all the alternative trajectories that lead to that state, but only the most profitable one. Recursively applying this simplification leads to a procedure with polynomial complexity in both the number of states (portfolio compositions) and period length.
Let us apply the algorithm to our example at hand. In the first day, there are three alternative actions to take, each one leading to the returns shown by the arrows:
In the second day, we have the following tree of possible actions. Each arrow shows the accumulated return from the beginning:
The discarded trajectories are shown as dashed lines. We only have to keep track of the solid lines, which show the best trajectory entering each state. Finally, in the third day we have the following trajectories:
As we can see, the best trajectory is the one which yields a 95% return, as stated before.
An example with real data
Finally, we apply the Viterbi algorithm to an example with real data. It’s Christmas 2015, and an angel comes to shows us the future as in Frank Capra’s movie It’s a Wonderful Life. Specifically, the angel shows us the evolution of five assets along the forthcoming year: Euro Stoxx 50, S&P 500, Nikkei 225, an Emergent Markets index and cash. Their normalized prices are shown in the Figure:
Let us suppose we’re running a portfolio with these five assets, and we are only allowed to trade 5% of the portfolio each day (think of twenty boxes of five colors). In the following figures, we show the portfolio compositions that result from the application of both the myopic and the Viterbi approaches. We have assumed a transaction cost of 10 basis points, which is taken into account by the algorithm to evaluate the net return of each trajectory.
The next Figure shows the returns obtained by each strategy, including the best asset and an equally weighted portfolio, for comparison.
As we can see, the Viterbi approach strongly outperforms the myopic one. This is even the case when the myopic is consistently better than the best asset in the portfolio.
Conclusion
What’s the best you can do if you can see the future? We have shown that the answer if not obvious if we are subject to operational constraints. The Viterbi algorithm gives us an answer without the need of exploring the whole set of alternative trajectories. This might not be very useful since most of us cannot see the future. But it’s so much fun! | 1,494 | 6,822 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.75 | 3 | CC-MAIN-2024-18 | latest | en | 0.953777 |
https://forum.azimuthproject.org/plugin/viewcomment/19900 | 1,656,695,763,000,000,000 | text/html | crawl-data/CC-MAIN-2022-27/segments/1656103943339.53/warc/CC-MAIN-20220701155803-20220701185803-00783.warc.gz | 322,790,565 | 1,933 | **Puzzle 168**
So the 2 morphisms are :
$$\alpha_{a',b'}(g \circ h \circ F(f))$$
$$G(g) \circ \alpha_{a,b}(h) \circ f$$
Now check if these two are equal using the fact that \$$\alpha_{a,b}\$$ is like taking \$$G\$$ and \$$G(F(h)) = h \$$ :
$$\alpha_{a',b'}(g \circ h \circ F(f))$$
$$= G(g \circ h \circ F(f))$$
$$= G(g) \circ G(h) \circ f$$
$$= G(g) \circ \alpha_{a,b}(h) \circ f$$ | 162 | 385 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.3125 | 3 | CC-MAIN-2022-27 | latest | en | 0.556357 |
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... figure, together with four right angles, are equal to twice as many right angles as the figure has be divided into as many triangles as the figure has sides, by drawing straight lines from a point F within the figure to each of its angles.
Euclid's Elements of Geometry: The Six First Books. To which are Added ... - Σελίδα 374
των Euclid, Rev. John Allen - 1822 - 494 σελίδες
Πλήρης προβολή - Σχετικά με αυτό το βιβλίο
## The Young Surveyor's Guide: Or, A New Introduction to the Whole Art of ...
Edward Laurence - 1716 - 375 σελίδες
...external < a, is equal to t wo right Angles (by tbe^tb /)confequently all the internal and external Angles are equal to twice as many right Angles as the Figure has fides. But all its internal Angles are equal to twiee as many right Angles eicept 4 as it has fides...
## The Young Gentleman's Arithmetick, and Geometry: Containing Such Elements of ...
Edward Wells - 1723 - 294 σελίδες
...the Sum of all the Angles in all the Tri~ angles, into which the Figure is divided, will together be equal to twice as ma-ny right Angles, as the Figure has Sides. But the Angles about P, the inward Point of each Figure, wherein all the Triangles concur, are (by...
## The Elements of Euclid: The Errors, by which Theon, Or Others, Have Long Ago ...
Robert Simson - 1762 - 466 σελίδες
...gether with four right angles. Therefore all the angles of the figure^ together with four right angles, are equal to twice as many right angles as the figure has fides. C o R. 2 . All the exterior angles of any rectilineal figure are together equal to four right...
## Euclid's Elements of Geometry: The First Six, the Eleventh and Twelfth Books
Euclid, Edmund Stone - 1765 - 464 σελίδες
...taken together) therefore .all the angles of a right-lined figure, together with four right angles, are equal to twice as many right angles as the figure has fides. And taking away four right angles from each, there will remain all the angles of the figure...
## A Royal Road to Geometry: Or, an Easy and Familiar Introduction to the ...
Thomas Malton - 1774 - 440 σελίδες
...by the Sides. ie equal to four Right Angles. And, all the internal Angles of any Right-lined Figure are equal to twice as many Right Angles as the Figure has Sides, wanting four, (Th. i. i0. i.) confequently, the external Angles being equal to thofe four (Th. 2. of...
## The Elements of Euclid, Viz: The Errors, by which Theon, Or Others, Have ...
Robert Simson - 1775 - 520 σελίδες
...together with four right angles. Therefore all the angles of the figure, together with four right angles, are equal to twice as many right angles as the figure has fides. CoR. 2. All the exterior angles of any re&ilineal figure, are together equal to four right angles....
## The First Six Books: Together with the Eleventh and Twelfth
Euclid - 1781 - 520 σελίδες
...together with four right angles. Thprefpre all the angles of the figure, together with four right angles, are equal to twice as many right angles as the figure has fides. CoR. 2. All the exterior angles of any rectilineal figure, arc together equal to four right...
## A Complete Treatise on Practical Mathematics: Including the Nature and Use ...
John McGregor (teacher of mathematics.) - 1792 - 431 σελίδες
...ft, I. 32. Euclid. All the anterior angles of any reoilineal figure, together with four right angles, are equal to twice as many right angles as the figure has fides. Hence the following rule. RULÉ. From double thé number of fides f übt vail: 4, and the remainder...
## Elements of Geometry: Containing the First Six Books of Euclid, with Two ...
John Playfair - 1795 - 400 σελίδες
...&c. Q^ED CoR. i. All the interior angles of any reftilineal figure, together with four right angles, are equal to twice as many right angles as the figure has fides. For any reftilineal figure ABCDE can be divided into as many triangles as the figure has lides,...
## The Elements of Euclid: Viz. the First Six Books, with the Eleventh and ...
Alexander Ingram - 1799 - 351 σελίδες
...&c. Q^ED CoR. i. All the interior angles of any rectilineal figure, together with four right angles, are equal to twice as many right angles as the figure has fides. For For any reftilineal figure ABCDE can be divided into as many triangles as the figure has... | 1,153 | 4,399 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.875 | 3 | CC-MAIN-2020-50 | latest | en | 0.815808 |
https://testbook.com/learn/digital-electronics-representation-of-boolean-functions/ | 1,669,596,588,000,000,000 | text/html | crawl-data/CC-MAIN-2022-49/segments/1669446710462.59/warc/CC-MAIN-20221128002256-20221128032256-00646.warc.gz | 604,653,609 | 43,517 | # Representation of Boolean Functions: SOP Form, POS Form, Truth Table, Venn Diagram and K-maps
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A boolean function is defined by an algebraic expression consisting of binary variables, constants such as 0 and 1, and the logic operation symbols. Whereas a variable in a boolean function is defined as a variable or a symbol which is generally an alphabet that depicts the logical quantities such as 0 or 1.
For a given set of values of the binary variables concerned, the boolean function can hold a value of 0 or 1. For example, the boolean function x+y’z is defined in terms of three binary variables(x,y, z), where x,y,z can take either 0 or 1 only.
A Boolean function can be converted from a Boolean expression into a circuit diagram formed of logic gates connected in a particular structure. With this article on the Representation of Boolean Functions, you will learn about the various ways of representing a Boolean Function through SOP and POS form, truth table form, Venn Diagram form and the K-map.
## Representation of Function
A Boolean function representation can be done in canonical(in this type of representation all the terms contain all the variables either in complementary or uncomplimentary form) form and minimal(in this type of representation minimum number of literals are used) form. Every boolean function can be expressed by an algebraic expression, or in terms of a Truth Table.
$$F\left(A,\ \ B,\ C,\ \ D\right)\ \ \ =\ \ \ \ \ A+\overline{BC}+ACD$$
$$\left(boolean\ function\right)\ \ \ \left(boolean\ \exp ression\right)$$
### Minterms and Maxterms
There are two methods in which we can place the Boolean function. These are the minterm canonical form and maxterm canonical form. The term “literals” refers to a binary variable either in complemented or in uncomplemented form.
#### Minterm
The product of all literals, either in complemented or uncomplemented form, is known as minterm. In the minterm, each uncomplemented term is assigned ‘1’, and each complemented term is assigned ‘0’.
Example- 100= AB’C’
#### Maxterm
The sum of all the literals, either in complemented or uncomplemented form, is known as maxterm. In maxterm, each uncomplemented term is assigned by ‘0’ and each complemented term is assigned by ‘1’.
Example: – 101= A’+B+C’
### Sum of Product (SOP) Form
In SOP or Sum of Product form two or more ANDed(i.e multiplied) variables are ORed with two or more such terms and used to describe outputs with logic ‘1’ combination. In SOP form each product term is identified as a minterm.
$$SOP\ form\ notation:$$
$$If\ a\ function\ is\ given\ as;\ f\left(A,\ B,\ C\right)=\sum_{ }^{ }m\left(3,5,6,7\right)$$
$$Y=m_3+m_5+m_6+m_7$$
$$Y=\overline{A}BC+A\overline{B}C+AB\overline{C}+ABC$$
The SOP form is also called the “disjunctive normal form”. The SOP form is widely used to develop the truth table and timing diagram of data.
### Product of Sum (POS) Form
In Product of Sum(POS) form two or more ORed(i.e added) variables are ANDed with two or more such terms together and applied to represent outputs with logic ‘0’ combination. In POS form every single term is understood as the maxterm.
$$POS\ form\ notation:$$
$$If\ a\ function\ is\ given\ as;\ f\left(A,\ B,\ C\right)=\Pi M\left(0,1,2,4\right)$$
$$Y=M_0\ M_1\ M_2\ M_4$$
$$Y=\left(A+B+C\right)\left(A+B+\overline{C}\right)\left(A+\overline{B}+C\right)\left(\overline{A}+B+C\right)$$
The POS form is also called the “conjunctive normal form”. The standard SOP or POS from each term of the expression contains all the variables of the function either in complemented or uncomplemented form. This type of representation is also called canonical SOP or POS form.
### Truth Table Form
Apart from the algebraic expression, the Boolean function can also be represented in terms of the truth table. In truth table representation, we represent all the possible combinations of inputs and their respective outputs. We can also transform the switching equations into truth tables.
Below is the truth table representation for three input combinations.
A B C Y 0 0 0 0 0 0 1 1 0 1 0 2 0 1 1 3 1 0 0 4 1 0 1 5 1 1 0 6 1 0 1 7
Let us consider an example for better understanding:
A B Y 0 0 0 0 1 1 1 0 1 1 1 0
$$Find\ the\ \exp ression\ for\ the\ above\ truth\ table\ in\ SOP\ form$$
$$The\ \exp ression\ is\ as\ follows:$$
$$Y=\overline{A}B+A\overline{B}$$
Also, check out the Binary Codes, here.
### Venn Diagram Form
A boolean algebra can also be represented by a Venn Diagram as shown below:
In the Venn Diagram, the AND operation is considered as an intersection and the OR operation is considered as a union. In the above image, A depicts AND operation, B depicts OR operation and C depicts NOT operation.
Let us consider an example regarding the Venn diagram.
$$The\ shaded\ region\ includes;$$
$$\ Y=AB\overline{C}+A\overline{B}C+\overline{A}BC+ABC$$
$$Y=AB\left(C+\overline{C}\right)+AC\left(B+\overline{B}\right)+BC\left(A+\overline{A}\right)$$
$$Y=AB+AC+BC$$
A boolean expression can also be represented in statement form, where the details are given in the form of statements.
### K-map Form
The karnaugh Map or K-map is a graphical method that provides a planned approach for simplifying and operating the Boolean expressions or to transform a truth table to its analogous logic circuit in a simplistic manner. In this process, the information available in the form of a truth table or SOP (Sum of Product) form or POS (Product of Sum) form is expressed on the K-map.
The below image shows the two inputs, three inputs and four inputs K-map in SOP and POS form.
Will will study K-map in detail in a separate article.
Also, check out other topics of Digital Electronics, here.
We hope that the above article on the Representation of Boolean Functions is helpful for your understanding and exam preparations. Stay tuned to the Testbook app for more updates on related topics from Digital Electronics, and various such subjects. Also, reach out to the test series available to examine your knowledge regarding several exams.
## Representation of Boolean Functions FAQs
Q.1 What is a Boolean function, state an example?
Ans.1 A boolean function is represented by an algebraic expression consisting of binary variables, constants such as 0 and 1, and the logic operation symbols. An example of a Boolean function is this, f(a,b,c) = a b – c. These functions are implemented with the logic gates.
Q.2 What are two forms of Boolean expression?
Ans.2 Two canonical forms of any Boolean function are a “sum of minterms” and a “product of maxterms.” The terms Sum of Products or SOP and Product of sum or POS are widely used for the canonical representation of boolean expressions.
Q.3 What does a canonical representation imply?
Ans.3 In canonical representation, all the terms of the expression contain all the variables either in complementary or uncomplimentary form.
Q.4 What are SOP and POS?
Ans.4 SOP and POS are two ways of expressing Boolean expressions. The main difference between SOP and POS is that SOP is a way of representing a Boolean expression using minterms or maxterms.
Q.5 How do you simplify Boolean functions?
Ans.5 A given boolean expression can be simplified by minimizing it into an equivalent expression by applying Boolean identities, laws and theorems. | 1,867 | 7,342 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4 | 4 | CC-MAIN-2022-49 | latest | en | 0.87323 |
https://socratic.org/questions/57d3098211ef6b56eaa55f2d#307999 | 1,725,825,870,000,000,000 | text/html | crawl-data/CC-MAIN-2024-38/segments/1725700651017.43/warc/CC-MAIN-20240908181815-20240908211815-00834.warc.gz | 512,826,559 | 6,176 | # Question 55f2d
Sep 9, 2016
${\lim}_{x \rightarrow 2} \frac{{x}^{3} - 3 {x}^{2} + 4}{{x}^{3} - {x}^{2} - 8 x + 12} = \textcolor{g r e e n}{\frac{3}{5}}$
#### Explanation:
The problem is that both:
$\textcolor{w h i t e}{\text{XXX}} {x}^{3} - 3 {x}^{2} + 4$ and
$\textcolor{w h i t e}{\text{XXX}} {x}^{3} - {x}^{2} - 8 x + 12$
are equal to $0$ if $x = 2$
...but this means that both of these expressions are divisible by $\left(x - 2\right)$
and using either synthetic or long division we can get:
$\textcolor{w h i t e}{\text{XXX")(x^3-3x^2+4)/(x^3-x^2-8x+12)=(cancel(""(x-2))(x^2-x-2))/(cancel(} \left(x - 2\right)} \left({x}^{2} + x - 6\right)$
Unfortunately both
$\textcolor{w h i t e}{\text{XXX}} {x}^{2} - x - 2$ and
$\textcolor{w h i t e}{\text{XXX}} {x}^{2} + x - 6$
are also both equal to $0$ if $x = 2$
...but (again) this means that both of these expressions are divisible by $\left(x - 2\right)$
and we can get
color(white)("XXX")(x^2-x-2)/(x^2+x-6)=(cancel(""(x-2))(x+1))/(cancel(""(x-2))(x+3))#
So we have
$\textcolor{w h i t e}{\text{XXX}} {\lim}_{x \rightarrow 2} \frac{{x}^{3} - 3 {x}^{2} + 4}{{x}^{3} - {x}^{2} - 8 x + 12} = {\lim}_{x \rightarrow 2} \frac{x + 1}{x + 3}$
$\textcolor{w h i t e}{\text{XXXXXXXXXXXXXXXXX}} = \frac{2 + 1}{2 + 3} = \frac{3}{5}$
Sep 9, 2016
I found: $\frac{3}{5}$
Try this: | 598 | 1,333 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 16, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.46875 | 4 | CC-MAIN-2024-38 | latest | en | 0.613964 |
http://masteringolympiadmathematics.blogspot.com/2016/02/analysis-quiz-19-multiple-choice-test.html | 1,532,229,011,000,000,000 | text/html | crawl-data/CC-MAIN-2018-30/segments/1531676593004.92/warc/CC-MAIN-20180722022235-20180722042235-00031.warc.gz | 240,719,948 | 38,765 | ## Wednesday, February 3, 2016
### Analysis Quiz 19: Multiple-Choice Test (Improve Analytical Skill)
Question 1: If you're asked to simplify [MATH]\left(1+\frac{\sqrt{2}+\sqrt{3}+\sqrt{4}}{\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}+4}\right)^2[/MATH], do you think by turning the $1$ as [MATH]\left(\frac{\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}+4}{\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}+4}\right)[/MATH] is feasible in order to simplify the expression?
A. Yes.
B. No.
Yes, of course, since that could be the good strategy to start working on the problem.
Question 2: If you answered yes to question 1, then what would you expect how the expression [MATH]\frac{(\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}+4)+(\sqrt{2}+\sqrt{3}+\sqrt{4})}{\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}+4}[/MATH] would look like?
A. [MATH]\frac{a(\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}+4)}{(\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}+4)}[/MATH], with [MATH]a[/MATH] an irrational number.
B. [MATH]\frac{a(\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}+4)}{(\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}+4)}[/MATH], with [MATH]a[/MATH], with [MATH]a\in \Bbb{N}[/MATH].
C. [MATH]\frac{a(\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}+4)}{(\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}+4)}[/MATH], with [MATH]a[/MATH], with [MATH]a\gt 0[/MATH].
D. [MATH]\frac{a(\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}+4)}{(\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}+4)}[/MATH], with [MATH]a[/MATH], with [MATH]a\lt 0[/MATH].
We can rule out the option D at once since we know the given target expression [MATH]1+\frac{\sqrt{2}+\sqrt{3}+\sqrt{4}}{\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}+4}[/MATH] must be a positive term.
The chance that $a$ will be an irrational number or a natural number is 50-50, and we wouldn't know it unless we work on it. And the option C that speculates $a\gt 0$ must be correct.
Therefore, we have to check the first three options as the answer for this question.
Question 3: How are you going to simplify [MATH](\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}+4)+(\sqrt{2}+\sqrt{3}+\sqrt{4})[/MATH]?
A. First collect [MATH]\sqrt{2}+\sqrt{3}+\sqrt{2}+\sqrt{3}[/MATH] and then look for the common factor of the whole expression to see if we can further factorize it.
B. After collecting $\sqrt{2}+\sqrt{3}+\sqrt{2}+\sqrt{3}$as $2(\sqrt{2}+\sqrt{3})$, and turning the sum into $2(\sqrt{2}+\sqrt{3})+\sqrt{4}+\sqrt{6}+\sqrt{8}+4$, there is nothing left to simplify.
We will of course first collect [MATH]\sqrt{2}+\sqrt{3}+\sqrt{2}+\sqrt{3}[/MATH] soo that we get:
[MATH](\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}+4)+(\sqrt{2}+\sqrt{3}+\sqrt{4})[/MATH]
[MATH]=2(\sqrt{2}+\sqrt{3})+\sqrt{6}+\sqrt{8}+4+\sqrt{4}[/MATH]
[MATH]=2\sqrt{2}+2\sqrt{3}+\sqrt{2}\sqrt{3}+\sqrt{4}\sqrt{2}+4+\sqrt{4}[/MATH](*)
It's tempting now to jump into conclusion that the expression (*) above couldn't be simplified further as there is no common factor. But wait a minute, note that $\sqrt{2}$ exists in
[MATH]=2\color{red}\sqrt{2}\color{black}+2\sqrt{3}+\color{red}\sqrt{2}\color{black}\sqrt{3}+\sqrt{4}\color{red}\sqrt{2}\color{black}+4+\sqrt{4}[/MATH]
for those red terms, so what we want to go is we want to turn $2$ as $\sqrt{2}\sqrt{2}$, that yields:
[MATH]2\color{red}\sqrt{2}\color{black}+2\sqrt{3}+\color{red}\sqrt{2}\color{black}\sqrt{3}+\sqrt{4}\color{red}\sqrt{2}\color{black}+4+\sqrt{4}[/MATH]
[MATH]=2\color{red}\sqrt{2}\color{black}+\color{red}\sqrt{2}\sqrt{2}\color{black}\sqrt{3}+\color{red}\sqrt{2}\color{black}\sqrt{3}+\sqrt{4}\color{red}\sqrt{2}\color{black}+\sqrt{16}+\sqrt{2}\sqrt{2}[/MATH]
[MATH]=2\color{red}\sqrt{2}\color{black}+\color{red}\sqrt{2}\sqrt{2}\color{black}\sqrt{3}+\color{red}\sqrt{2}\color{black}\sqrt{3}+\sqrt{4}\color{red}\sqrt{2}\color{black}+\color{red}\sqrt{2}\color{black}\sqrt{8}+\color{red}\sqrt{2}\color{black}\sqrt{2}[/MATH]
[MATH]=\color{red}\sqrt{2}\color{yellow}\bbox[5px,purple]{(2+\sqrt{2}\sqrt{3}+\sqrt{3}+\sqrt{4}+\sqrt{8}+\sqrt{2})}[/MATH]
Now, what we ought to bear in mind is, the second factor above, i.e. [MATH]\color{yellow}\bbox[5px,purple]{(2+\sqrt{2}\sqrt{3}+\sqrt{3}+\sqrt{4}+\sqrt{8}+\sqrt{2})}[/MATH] most probably could be rewritten to take the form
[MATH]\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}+4[/MATH]
Let's see:
[MATH]\color{yellow}\bbox[5px,purple]{(2+\sqrt{2}\sqrt{3}+\sqrt{3}+\sqrt{4}+\sqrt{8}+\sqrt{2})}[/MATH]
[MATH]=2+\sqrt{2\cdot 3}+\sqrt{3}+\sqrt{4}+\sqrt{8}+\sqrt{2}[/MATH]
[MATH]=2+\sqrt{6}+\sqrt{3}+2+\sqrt{8}+\sqrt{2}[/MATH]
[MATH]=4+\sqrt{6}+\sqrt{3}+\sqrt{8}+\sqrt{2}[/MATH]
[MATH]\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}+4[/MATH]
[MATH]\therefore (\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}+4)+(\sqrt{2}+\sqrt{3}+\sqrt{4})=\sqrt{2}(\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}+4)[/MATH]
Question 4: Do you think you could simplify the fraction [MATH]\frac{\sqrt{2}+\sqrt{3}+\sqrt{4}}{\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}+4}[/MATH]?
A. Yes, by rationalizing the denominator [MATH]\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}+4[/MATH].
B. Yes, by not rationalizing the denominator [MATH]\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}+4[/MATH].
Yes, we could definitely simplify the fraction [MATH]\frac{\sqrt{2}+\sqrt{3}+\sqrt{4}}{\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}+4}[/MATH], and given the denominator consists of more than 2 terms, it's a bit of a stretch to try it from the perspective of rationalizing it.
Plus, we might want to suspect if the denominator could be factored nicely where one of the factors is equivalent to the expression of the numerator.
Question 5: How you would begin to rationalize the denominator [MATH]\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}+4[/MATH]?
A. By multiplying it with [MATH]\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}-4[/MATH].
B. By multiplying it with [MATH]\sqrt{2}+\sqrt{3}+\sqrt{6}-\sqrt{8}-4[/MATH].
C. By multiplying it with [MATH]\sqrt{2}+\sqrt{3}-\sqrt{6}-\sqrt{8}-4[/MATH].
D. By multiplying it with [MATH]\sqrt{2}-\sqrt{3}-\sqrt{6}-\sqrt{8}-4[/MATH].
E. Neither of the above.
As we've already mentioned in the discussion for question 4, we suspected the denominator could be factored into $(\sqrt{2}+\sqrt{3}+\sqrt{4})(P(x))$, therefore, we don't want to attempt it from rationalizing the denominator approach.
The answer for this question is hence E.
Question 6: If [MATH]\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}+4=P(x)(\sqrt{2}+\sqrt{3}+\sqrt{4})[/MATH], what would you do to find $P(x)$?
A. By working with [MATH]\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}+4[/MATH] and try to factor it.
B. By doing the long division.
The simplest way to determine if [MATH]\sqrt{2}+\sqrt{3}+\sqrt{4}[/MATH] is a factor of [MATH]\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}+4[/MATH] is through the long division method:
[MATH]\begin{array}{r}1+\sqrt{2}\hspace{33px}\\\sqrt{2}+\sqrt{3}+\sqrt{4}\enclose{longdiv}{\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}+4} \\ -\underline{\left(\sqrt{2}+\sqrt{3}+\sqrt{4}\right)} \hspace{60px} \\ 2+\sqrt{6}+\sqrt{8}\hspace{33px}\\-\underline{\left(2+\sqrt{6}+\sqrt{8}\right)} \hspace{20px} \end{array}[/MATH]
Therefore, we can safely say [MATH]\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}+4=(1+\sqrt{2})(\sqrt{2}+\sqrt{3}+\sqrt{4})[/MATH].
Question 7: What is the simplified answer for [MATH]\left(1+\frac{\sqrt{2}+\sqrt{3}+\sqrt{4}}{\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}+4}\right)^2[/MATH]?
A. $1+\sqrt{2}$
B. $\sqrt{2}$
C. $2$
[MATH]\begin{align*}\left(1+\frac{\sqrt{2}+\sqrt{3}+\sqrt{4}}{\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}+4}\right)^2&=\left(1+\frac{\sqrt{2}+\sqrt{3}+\sqrt{4}}{(1+\sqrt{2})(\sqrt{2}+\sqrt{3}+\sqrt{4})}\right)^2\\&=\left(1+\frac{\cancel{\sqrt{2}+\sqrt{3}+\sqrt{4}}}{(1+\sqrt{2})(\cancel{\sqrt{2}+\sqrt{3}+\sqrt{4}})}\right)^2\\&=\left(1+\frac{1}{1+\sqrt{2}}\right)^2\\&=\left(1+\frac{(1-\sqrt{2})}{(1+\sqrt{2})(1-\sqrt{2})}\right)^2\\&=\left(1+\frac{(1-\sqrt{2})}{-1}\right)^2\\&=(1-1+\sqrt{2})^2\\&=2\end{align*}[/MATH]
Question 8: Can you think of any other method(s) that is/are different than the previously discussed two methods(turning $1$ as [MATH]\frac{\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}+4}{\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}+4}[/MATH] or to simplify [MATH]\frac{\sqrt{2}+\sqrt{3}+\sqrt{4}}{\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}+4}[/MATH]) to simplify the given square?
No, those two methods (first by turning $1$ as [MATH]\frac{\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}+4}{\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}+4}[/MATH] and proceed from there and the second by simplifying the fraction [MATH]\frac{\sqrt{2}+\sqrt{3}+\sqrt{4}}{\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}+4}[/MATH]) are the only methods one could use to simplify the expression. | 3,466 | 8,414 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.796875 | 4 | CC-MAIN-2018-30 | latest | en | 0.677766 |
http://www.madsci.org/posts/archives/2001-04/988231102.Ph.r.html | 1,534,588,751,000,000,000 | text/html | crawl-data/CC-MAIN-2018-34/segments/1534221213540.33/warc/CC-MAIN-20180818095337-20180818115337-00484.warc.gz | 538,032,439 | 3,302 | ### Re: my question is regarding question 982969482.Ph
Date: Wed Apr 25 14:33:19 2001
Posted By: Benn Tannenbaum, Post-doc/Fellow, Physics and Astronomy, University of California, Los Angeles
Area of science: Physics
ID: 987793769.Ph
Message:
```
Dear Jamie,
I'm not familiar with the original question/answer, so I'll have to work
First, let's clear up a misunderstanding. There is no such thing as
negative energy. Energy is what is known as a positive definite quantity--
that means it can only have positive values. Other positive definite
quantities include length, mass, volume. What would negative mass mean?
For example, let's consider the equation E = m c^2. Can you imagine what
negative mass is? I certainly can't! It's got to be positive. Also, any
number times itself is positive (c^2) [except for imaginary numbers, but c
is quite real!], so that means that E has to be positive as well.
Next, what's the difference between an electron and a positron? A positron
is the anti-matter version of an electron. Matter/anti-matter pairs differ
in only one way. They have the same mass, the same spin, the same
everything, except for charge. Matter/anti-matter pairs have the same
magnitude of charge, but opposite in sign. Thus, electrons have charge -1,
and positrons have charge of +1.
What is the difference between real and virtual? This is very subtle. The
short answer is that virtual particles are terminated in Feynman diagrams.
Pretty meaningless, eh? What that really means is that virtual particles
are created and destroyed in a very very short period of time. Real
particles are created and last for some macroscopic period of time. That is
the only difference.
To make the virtual electrons and positrons real, the electric field must
be large enough so the energy density of the field is equal to or greater
than the equivalent mass of the electron/positron pair. The mass of an
electron is (in my favorite units, megaelectron Volts) 0.511 MeV/c^2. That
means the energy of the electric field must be greater than twice that
(since there is an electron and a positron to make) or 1.022 MeV. The
electric field seperates them, since they have opposite charge, and things
with opposite charges move in opposite directions in an electric field.
Once the electron and positron are far enough apart, the will continue to
exist until they meet their anti-matter counter part and annihilate. As you
might imagine, the electron lives longer than the positron simply because
the universe is made of matter-- there are more electrons for the positron
to interact with than there are positrons for the electron to interact
with.
Finally, as for the Casimir effect, I'm not sure I understand your
question. The Casimir effect is simply that there will be a tiny force
applied to two conductive plates when they are close together. Once they
are far apart, the vacuum is unable to muster enough strength to apply any
force....
I hope this helps! If you want futher information, I recommend Introduction
to Elementary Particles, by Griffiths.
```
Current Queue | Current Queue for Physics | Physics archives | 725 | 3,127 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.96875 | 3 | CC-MAIN-2018-34 | latest | en | 0.945671 |
https://australiaassessments.com/question-2-statistics-homework-help/ | 1,685,788,672,000,000,000 | text/html | crawl-data/CC-MAIN-2023-23/segments/1685224649193.79/warc/CC-MAIN-20230603101032-20230603131032-00052.warc.gz | 151,872,653 | 12,678 | # Question 2 | Statistics homework help
NOTE: Show your work in the problems.
1. In the following situations, indicate whether you’d use the normal distribution, the t distribution, or neither.
a. The population is normally distributed, and you know the population standard deviation.
b. You don’t know the population standard deviation, and the sample size is 35.
c. The sample size is 22, and the population is normally distributed.
d. The sample size is 12, and the population is not normally distributed.
e. The sample size is 45, and you know the population standard deviation.
2. The prices of used books at a large college bookstore are normally distributed. If a sample of 23 used books from this store has a mean price of \$27.50 with a standard deviation of \$6.75, use Table 10.1 in your textbook to calculate the following for a 95% confidence level about the population mean. Be sure to show your work.
a. Degrees of freedom
b. The critical value of t
c. The margin of error
d. The confidence interval for a 95% confidence level
3. Statistics students at a state college compiled the following two-way table from a sample of randomly selected students at their college:
Play chessDon’t play chessMale students25162Female students19148
Answer the following questions about the table. Be sure to show any calculations.
a. How many students in total were surveyed?
b. How many of the students surveyed play chess?
c. What question about the population of students at the state college would this table attempt to answer?
d. State Hº and Hª for the test related to this table.
4. Answer the following questions about an ANOVA analysis involving three samples.
a. In this ANOVA analysis, what are we trying to determine about the three populations they’re taken from?
b. State the null and alternate hypotheses for a three-sample ANOVA analysis.
c. What sample statistics must be known to conduct an ANOVA analysis?
d. In an ANOVA test, what does an F test statistic lower than its critical value tell us about the three populations we’re examining?
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# A quadrilateral ABCD is drawn to circumscribe a circle (see Fig. 10.12). Prove that AB + CD = AD + BC Q.8
• 0
The question from class 10th ncert book of exercise 10.2 of question no.8 of circles chapter, how i solve this question in easy way i think it is very important for class 10th, A quadrilateral ABCD is drawn to circumscribe a circle (see Fig. 10.12). Prove that AB + CD = AD + BC
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1. The figure given is:
From this figure we can conclude a few points which are:
(i) DR = DS
(ii) BP = BQ
(iii) AP = AS
(iv) CR = CQ
Since they are tangents on the circle from points D, B, A, and C respectively.
Now, adding the LHS and RHS of the above equations we get,
DR+BP+AP+CR = DS+BQ+AS+CQ
By rearranging them we get,
(DR+CR) + (BP+AP) = (CQ+BQ) + (DS+AS)
By simplifying, | 261 | 800 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4 | 4 | CC-MAIN-2023-06 | latest | en | 0.920172 |
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The GCF is 11. 11×3=33 5×11=55 12×11=132
2014-11-18T21:41:05-05:00
### This Is a Certified Answer
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What you can do is see what the numbers can be divisible by. Sooo...
what can go into 33, 55, and 132? Just to let you know it is to be the GREATEST common factor. For example 4 and 8. You can't use 2. It's not the greatest. You would use 4 because 4 is the greatest and can co into 8 and 4 evenly. | 267 | 1,007 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.265625 | 3 | CC-MAIN-2017-04 | latest | en | 0.944476 |
http://mathhelpforum.com/discrete-math/201051-sum-solving-print.html | 1,529,956,351,000,000,000 | text/html | crawl-data/CC-MAIN-2018-26/segments/1529267868876.81/warc/CC-MAIN-20180625185510-20180625205510-00267.warc.gz | 198,067,180 | 3,642 | # Sum solving
• Jul 16th 2012, 02:50 PM
Emilijo
Sum solving
Is it possible to find F(n), summation of the function f(k), where k=1,2...,n ( Is there some formula, general formula, for any f(k), of course if k is in domain of function)
sum_(k=1, to n) f(k) = F(n)
If it is not possible, why?
• Jul 16th 2012, 03:03 PM
richard1234
Re: Sum solving
No. The sum $\displaystyle f(1) + f(2) + \dots + f(n)$ is sometimes known as a Riemann sum. There is no general formula for a Riemann sum, other than approximating that sum with the integral of the function f(n) (assuming f is integrable).
There are definitely formulas for specific functions f, e.g. $\displaystyle f(n) = n$, $\displaystyle f(n) = n^2$, $\displaystyle f(n) = 3n+1$, $\displaystyle f(n) = \frac{1}{n(n+1)}$ etc. but there is no general formula.
• Jul 17th 2012, 02:43 AM
Emilijo
Re: Sum solving
But then how to solve this: sum_(k=1, to n) sin(1/k)
Is it possible, I put it into wolfram alpha, but I didn t get the solution.
Why is not possible to get it? Do exist conditions when the sum is possible to calculate, or is it just technical problem?
• Jul 17th 2012, 08:00 AM
richard1234
Re: Sum solving
Try this: \sum_{k=1}^{n} \sin(1/k). Or if you know how to type in Mathematica, you can do that as well.
Obviously, you're not going to get a general solution. But note that $\displaystyle \int_{1}^{n} \sin x \, dx = -\cos n + \cos 1$.
• Jul 18th 2012, 05:06 AM
Emilijo
Re: Sum solving
Is it proven that does not exist a general formula, which are conditions?
Why does not exist a general formula...?
• Jul 18th 2012, 05:59 AM
emakarov
Re: Sum solving
Quote:
Originally Posted by Emilijo
Why does not exist a general formula...?
The sum $\displaystyle \sum_{k=1}^n\sin(1/k)$ is, of course, a well-defined function of n. However, in general there is no reason why some function has to be represented as a finite composition of a few basic functions and operations: addition, multiplication, sine, square root, etc. For example, the function that, given five coefficients of a quintic equation with the leading coefficient 1, returns the least real root of this equation, is also well-defined for all arguments. However, it is proven that this function cannot be expressed as a finite composition of the four arithmetic operations and roots of any degree.
I would also like to know sufficient, or, better, necessary and sufficient conditions for when a recurrence relation $\displaystyle x_{n+1}=g(x_n)$ (note that the sum $\displaystyle x_n=\sum_{k=1}^nf(k)$ satisfies $\displaystyle x_{n+1}=x_n+f(n+1))$ has a closed-form solution. Well, there are sufficient conditions, such as when a relation is a linear homogeneous recurrence relation with constant coefficients, but they are not very general. | 792 | 2,767 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.875 | 4 | CC-MAIN-2018-26 | latest | en | 0.876723 |
https://www.atheistsforhumanrights.org/what-is-configural-metric-and-scalar-invariance/ | 1,686,444,592,000,000,000 | text/html | crawl-data/CC-MAIN-2023-23/segments/1685224646652.16/warc/CC-MAIN-20230610233020-20230611023020-00128.warc.gz | 749,385,832 | 17,874 | # What is Configural Metric and scalar invariance?
## What is Configural Metric and scalar invariance?
At the configural level, loadings and intercepts are freely estimated. At the metric level, loadings are constrained to be equal across groups and the intercepts are freely estimated. The scalar invariance model is the most constrained, with both loadings and intercepts constrained to be equal across groups.
What does non invariance mean?
Measurement noninvariance suggests that a construct has a different structure or meaning to different groups or on different measurement occasions in the same group, and so the construct cannot be meaningfully tested or construed across groups or across time.
### What does factorial invariance mean?
Factorial invariance is a concept applied in the context of psychometric analysis of questionnaires. The concept postulates that the psychometric properties of a questionnaire, used either by multiple groups or by the same group over time, have to be identical to ensure an unbiased comparison of factor means.
What is structural invariance?
Covariance invariance analyses test whether the unstandardized relationship between latent factors is equal across groups, whereas structural invariance analyses test whether the unstandardized relationship between Page 4 318 D. A. SASS & T. A. SCHMITT latent variables (either correlational or predictive) is equal …
## What is strict invariance?
Factor variances represent the overall error in the prediction of your construct. The second level of strict invariance refers to invariance of individual indicator variable’s error terms that represent the unique error specific to that particular indicator variable.
What is multigroup confirmatory factor analysis?
Multi-group confirmatory factor analysis (MGCFA) allows researchers to determine whether a research inventory elicits similar response patterns across samples. If statistical equivalence in responding is found, then scale score comparisons become possible and samples can be said to be from the same population.
### What is strict factorial invariance?
Strong factorial invariance requires that specific factor means (represented as invariant intercepts) also be identical across groups. Strict factorial invariance implies that, in addition, the conditional variance of the response, given the common and specific factors, is invariant across groups.
What is multigroup CFA?
## Is Standard Deviation an invariant scale?
Scale Invariant Statistics Some specific statistics are scale invariant. For example, Anderson (1993) noted that Hotelling’s T-Statistic is scale invariant. Each observation, the mean and the standard deviation can all be multiplied by a constant c, and the statistic is unaffected.
What is invariance of MLE?
One of the most useful properties of the maximum likelihood estimator (MLE), often called the invariance property, is that if ˆθ is the MLE of θ, then h(ˆθ) is the MLE of h(θ). Many texts either define the MLE of h(θ) to be h(ˆθ), say that the property is immediate from the definition of the MLE, or quote Zehna (1966).
### What is Multigroup analysis in SEM?
Multigroup analysis (MGA) or between-group analysis as applied using partial least squares structural equations modeling (PLS-SEM) is a means of testing predefined data groups to determine if there are significant differences in group-specific parameter estimates (e.g., outer weights, outer loadings, and path …
What is measurement and configural invariance?
Measurement invariance is tested by evaluating how well the specified model (e.g., the model set up by the researcher) fits the observed data. Current practice emphasizes the importance of using multiple fit statistics to assess model fit (Kline, 2015). Configural invariance is tested by evaluating the overall fit of the model.
## What does invariance mean in science?
Formal definition of invariance: “whether or not, under different conditions of observing and studying phenomena, measurement operations yield measures of the same attribute” (Horn and Mcardle 1992, 117). What implies?
What is invariance testing model in practice?
MGCFA invariance testing model in practice is testing an hypothesis of whether a given theoretical model fits well to the data across the groups. Depending on what parameters we constraint to be equal across groups, we test one or another level of invariance
### Is measurement invariance in all four steps supported?
Because full measurement invariance in all four steps is often not supported, it is becoming common practice to accept some violations of measurement invariance (e.g., releasing constraints on one or more loadings or intercepts or both) and continue with tests of mean differences or relations among constructs using the partially invariant factor. | 975 | 4,835 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.90625 | 3 | CC-MAIN-2023-23 | latest | en | 0.867085 |
https://aviation.stackexchange.com/questions/45069/what-is-the-vor-reference-signal?noredirect=1 | 1,713,233,766,000,000,000 | text/html | crawl-data/CC-MAIN-2024-18/segments/1712296817036.4/warc/CC-MAIN-20240416000407-20240416030407-00683.warc.gz | 103,998,390 | 35,440 | # What is the VOR reference signal?
What is the VOR (VHF Omnidirectional Range) reference signal? I have to calculate the spectrum of this signal.
• en.wikipedia.org/wiki/VHF_omnidirectional_range has a good description Oct 25, 2017 at 15:53
• @mins That's very helpful and complete documentation. The best answer! Thank you for finding time to help me. Oct 25, 2017 at 20:17
• See this answer, the reference is a 30 Hz sine, but depending on the type (CVOR/DVOR), the reference is transmitted either as a AM sideband (DVOR) or as a FM subcarrier at 9.96 kHz (480 Hz swing) sent as DSB-SC (CVOR). The other signal is the variable, also a 30 Hz sine. The corresponding spectrum is this one (missing the id tone and voice channel)
– mins
Sep 28, 2020 at 10:51
## 1 Answer
VOR Ground Station is aligned with magnetic North. It emits two signals:
• a 360° sweeping variable signal
• an omni-directional reference signal
When an aircraft receives those signals, its receiver compares those and a measures the phase difference. This gives a precise radial position of the aircraft which is displayed on its Omni-Bearing Indicator (OBI), Horizontal Situation Indicator (HSI) or a Radio Magnetic Indicator (RMI), or a combination of two different kinds.
This picture shows the two signals:
• Blue is 360° sweeping signal
• Green is omni-directional reference signal
• This is very enlightening! And also I would like to know more about this omni-directional reference signal (blue one). For example if it is sinusoidal, and all other informations for calculating its spectrum. Thank you! Oct 25, 2017 at 15:42
• Trying to fig out why they made the reference signal flash in that gif. I guess that's indicating when it's at 0° phase? Oct 25, 2017 at 18:20
• @TomMcW: Yes it's misleading, it looks like the bearing is determined by timing the interval between a north pulse and the maximum strength of a supposed thin beam (and I saw another post on the site where it's interpreted this wrong way).
– mins
Sep 28, 2020 at 11:05 | 514 | 2,025 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.6875 | 3 | CC-MAIN-2024-18 | latest | en | 0.921834 |
https://www.assignmentexpert.com/homework-answers/chemistry/inorganic-chemistry/question-66735 | 1,632,715,092,000,000,000 | text/html | crawl-data/CC-MAIN-2021-39/segments/1631780058263.20/warc/CC-MAIN-20210927030035-20210927060035-00536.warc.gz | 667,413,904 | 144,751 | 101 556
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# Answer to Question #66735 in Inorganic Chemistry for Divyash Chhetri
Question #66735
A flask contains 3.00 moles of nitrogen and 3.00 moles of neon. How many grams of argon must be pumped into the flask in order to make the partial pressure of argon twice that of neon?
1
2017-03-26T04:55:07-0400
The partial pressure of individual gas in the gas mixture can be expressed the following way:
Pi = Ptotal *xi,
Where
Pi – is the partial pressure of individual gas in the mixture;
Ptotal – is the total pressure of the gas mixture;
xi – is the mole fraction of individual gas in the mixture.
Mole fraction is number of moles of individual component divided by total number of moles in the mixture:
xi = ni / ntotal.
Therefore
Pi = Ptotal * ni / ntotal.
We see that partial pressure of the gas component is directly proportional to the number of moles of that gas in the mixture.
So if we want the partial pressure of argon to be twice that of neon, than number of moles of argon should also be twice number of moles of neon.
It is known that mixture contains 3.00 moles of neon. So we need to add 6.00 moles of argon.
Argon has monoatomic molecule, so the mass of 1 mole of argon in grams equals to its atomic weight which is 39.95.
Then 6.00 moles of argon have a mass (6.00 * 39.95) = 239.70 g.
239.70 grams of argon must be pumped into the flask.
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for any assignment or question with DETAILED EXPLANATIONS! | 419 | 1,574 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.046875 | 3 | CC-MAIN-2021-39 | longest | en | 0.908896 |
http://frontend.spiceworks.com/topic/2308171-sharepoint-calculation-returning-incorrect-value?from_forum=221 | 1,618,722,307,000,000,000 | text/html | crawl-data/CC-MAIN-2021-17/segments/1618038468066.58/warc/CC-MAIN-20210418043500-20210418073500-00361.warc.gz | 36,747,868 | 21,241 | # Sharepoint calculation returning incorrect value
Sharepoint Migration Tool, On-prem to SP, periodic re-sync
Hi, is anyone able to help work out why this formulae in returning the wrong value on occasion?
=INT(INT(DATEDIF(DATE(2020,5,3),Created,"YD")/7)/4)+2
It is intended to calculate the financial period (April being 1, March being 13) based on the Created column (in the Date and Time format). The Created column is an auto column, populated when my power apps form is launched completed and saved.
I've tested it in Excel by pointing the formula to the exact cell. For example cell R51 contains the date 23/02/21. =INT(INT(DATEDIF(DATE(2020,5,3),R50,"YD")/7)/4)+2 and the correct value of 12 is returned. (representing period 12).
GT
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## 5 Replies
· · ·
Cayenne
OP
The issue is most definitely with the "Created" named range. My guess is that it is not always referring to the cell you think it is referring to. You do mention that " The Created column is an auto column" but your working formula refers to a single cell. Is the Created named range a column or cell?
Also, welcome to SpiceWorks!
0
· · ·
Serrano
OP
Does your year have 13 periods of 28 days each?
If the start of your year is 1 April, why are you anchoring the date calculation on 2020,5,3?
How do you want to handle the 365th day of the year (13 x 28 = 364)? Put it into P13?
How do you want to handle the 366th day of a leap year? Put it into P13?
By my calculations, your formula fails in 2021 from 3 Feb to 6 Feb, 3 March to 6 March, 31 March, all of April, 27 May to 30 May ...
Let's say you wanted to assign days 365 and, if a leap year, 366 to period 13, and the date is in A1, you might use a formula such as:
=INT(MIN(IF(MONTH(A1)<4,DAYS(A1,DATE(YEAR(A1)-1,4,1)),DAYS(A1,DATE(YEAR(A1),4,1))),363)/28)+1
The MIN function makes sure we put 365 (and 366) into P13. It's set to 363 because we're using modulo arithmetic
We check the financial year that we're in and use April 1 of the correct year to avoid accumulative extra days and leap years.
0
· · ·
Serrano
OP
One wonders why some people post a question and then never return to acknowledge whether or not the responses provided by other had helped.
1
· · ·
Cayenne
OP
dastafford wrote:
One wonders why some people post a question and then never return to acknowledge whether or not the responses provided by other had helped.
Post and ghost :D
0
· · ·
Serrano
OP
Tim_Myth wrote:
Post and ghost :D
I have a new policy:
If ( (PersonPosting != VerifiedProfessional) and (MemberSince < MinimumMemberAge) and ("OVERALL ACTIVITY" < MinimumContributions) )
{
SkipToNextAndDon'tWasteTime
}
0
Oops, something's wrong below. | 800 | 2,873 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.53125 | 3 | CC-MAIN-2021-17 | latest | en | 0.920152 |
http://developer.nokia.com/community/wiki/index.php?title=HERE_Maps_API_-_Utility_class_Vector3D&direction=prev&oldid=181896 | 1,427,462,763,000,000,000 | text/html | crawl-data/CC-MAIN-2015-14/segments/1427131296456.82/warc/CC-MAIN-20150323172136-00065-ip-10-168-14-71.ec2.internal.warc.gz | 71,086,521 | 12,877 | ##### Actions
Please note that as of October 24, 2014, the Nokia Developer Wiki will no longer be accepting user contributions, including new entries, edits and comments, as we begin transitioning to our new home, in the Windows Phone Development Wiki. We plan to move over the majority of the existing entries. Thanks for all your past and future contributions.
# HERE Maps API - Utility class Vector3D
Tested with
Devices(s): Firefox 12.0
Dependencies: Nokia Maps 2.2.1
Article
Keywords: Nokia Maps, JavaScript, vectors
Created: Maveric (28 Jun 2011)
Last edited: jasfox (03 Jan 2013)
This article explains how to use the Vector3D utility class.
## Introduction
This is a class representing a three dimensional vector. Full description is here:
## Usage examples
Let's create a new Vector3D object:
`>>> myVector = new nokia.maps.util.Vector3D (10, 10, 10) ;`
This would return and object containing:
`Object { x=10, y=10, z=10}`
The "add(other)" method would create the sum of this vector and the added one.
We will create two Vector3D's and then create the third based on adding the 2nd to the 1st Vector3D.
`>> myVector1 = new nokia.maps.util.Vector3D (10, 10, 10) ;>> myVector2 = new nokia.maps.util.Vector3D (20, 20, 20) ;>> myVector3 = myVector1.add(myVector2);`
We would now have myVector3 to contain:
`Object { x=30, y=30, z=30 }`
With the method "angle" we can request, as the name says the relative angle: angle ([px, [py, [intertX, [intertY]]]]) : Number
Example on requesting the angle for myVector3:
`>>> myVector3.angle();`
Would return us a numeric value as follows:
`45.00000000000001`
And let's try one more operation with the "subtract" method. With it we can get the difference between two vectors:
`>>> subtracted = myVector3.subtract(myVector1);`
Would return us:
`Object { x=20, y=20, z=20}`
Feel free to try the other methods too, used in a similar manner.
Description of the angle method:
`angle ([px, [py, [intertX, [intertY]]]]) : NumberReturns the angle of this vector in decimal degree to the given point within a coordinate system that goes possitive to top/right. For screen coordinates the Y-axis is inverted. For a normalized vector the method can be called without parameters to return it's normalized angle. Parameters:{Number} [px]: The x-coordinate of the point to which the angle shall be calculated, if not given zero is used.{Number} [py]: The y-coordinate of the point to which the angle shall be calculated, if not given zero is used.{Boolean} [intertX]: If given and true, the x-axis is inverted, therefore the coordinates are increasing to the left and decreasing to the right.{Boolean} [intertY]: If given and true, the y-axis is inverted, therefore the coordinates are increasing to the bottom and decreasing to the top. Returns:{Number} The angle of this vector in degree, relative to the given coordinate.`
## Tested with
• Firefox 12.x
70 page views in the last 30 days. | 767 | 2,990 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.28125 | 3 | CC-MAIN-2015-14 | latest | en | 0.802118 |
https://www.physicsforums.com/threads/finding-current-in-a-parralle-wire.575361/ | 1,545,014,400,000,000,000 | text/html | crawl-data/CC-MAIN-2018-51/segments/1544376828056.99/warc/CC-MAIN-20181217020710-20181217042710-00119.warc.gz | 1,001,899,065 | 12,564 | # Homework Help: Finding current in a parralle wire
1. Feb 7, 2012
### monke
1. The problem statement, all variables and given/known data
two long straight parallel wires seperated by 8.0 cm carry currents of equal magnitude but heading in opposite directions. Wires are perpendicular to the plane of the page. Point P is 2cm form wire 1. magnetic field at P = 0.01 T directed downward ( neg y direction). Calc Current in wire 1 and its direction.
2. Relevant equations
B= (μ.I)/2 pi r where mu not is 4pi X 10^-7 and r is the distance of the mag field from the wire.
3. The attempt at a solution
I tried using the given information and instead of solving for B solve for I (current) However my answer was incorrect.
Did i miss a step?
Thank You
2. Feb 8, 2012
### Fightfish
You have not been very clear in showing your entire approach.
However, I believe that the likely mistake that you made is overlooking the fact that the magnetic field of 0.01 T at P is the superposition of the individual magnetic fields from wires 1 and 2. Thus,
$$B_{p} = \frac{μ I}{2 \pi r_{1}} + \frac{μ I}{2 \pi r_{2}}$$
where $r_{1}$ denotes distance from wire 1 to P and $r_{2}$ denotes distance from wire 2 to P.
3. Feb 9, 2012
### monke
Thank you :) thats helps a ton | 352 | 1,266 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.328125 | 3 | CC-MAIN-2018-51 | latest | en | 0.919507 |
http://espace.library.uq.edu.au/view/UQ:315914 | 1,480,776,339,000,000,000 | text/html | crawl-data/CC-MAIN-2016-50/segments/1480698540932.10/warc/CC-MAIN-20161202170900-00089-ip-10-31-129-80.ec2.internal.warc.gz | 91,318,041 | 8,948 | # 关于低频振荡分析方法的评述
Author 薛禹胜 Xue, Yusheng郝思鹏 Hao, Sipeng刘俊勇 Liu, JunyongDong, ZhaoyangLedwich, Gerard 关于低频振荡分析方法的评述 A review of analysis methods for low-frequency oscillations chieng Dianli Xitong Zidonghua Automation of Electric Power Systems chieng 1000-1026 2009-02-10 Critical review of research, literature review, critical commentary 33 3 1 8 8 Nanjing, China Dianli Xitong Zidonghua Zazhishe chi 1706 Computer Science Applications2207 Control and Systems Engineering2208 Electrical and Electronic Engineering2102 Curatorial and Related Studies 为更好地梳理概念，将低频振荡分析方法分为两大类，即针对系统模型平衡点的特征根方法以及沿着系统受扰轨迹的模式提取方法。平衡点特征根方法可进一步按采用的系统模型分为确定性的线性化模型、确定性的非线性模型和概率模型。这类方法与具体扰动无关，但只能反映系统在该平衡点附近的动态行为，故不适用于包含强非线性、变系数、相继故障或有离散控制的系统。受扰轨迹模式分析方法则从特定扰动下的时间响应曲线中提取振荡信息的时间序列，包括系统模型未知情况下（如实测轨迹）的信号处理法和系统模型已知情况下（如仿真轨迹）的分时段定常线性化法。在评述各种方法的基础上，提出改进的思路及有望突破的研究方向。 Analysis methods for low-frequency oscillation are classified into two categories. One is the equilibrium-point eigenvalue method; the other is the mode extraction method along the disturbed trajectories. The former can be further sorted into deterministic linear models， deterministic non-linear models and probabilistic models. This category is independent of the disturbances. It can only show the dynamics near the equilibrium point， and is not suitable for systems with strong non-linearity， time variant， sequence fault or discrete actions. The latter extracts the oscillation information from the time response curves of the system. It can be further sorted into the signal processing method without knowledge on the system model， such as that for measured trajectories， and the piecewise autonomisation-linearization method， such as that for simulation results. Based on reviewing various methods， hopeful research areas are proposed. Low-frequency oscillationEquilibrium-point eigenvaluesTrajectory eigenvalues sequenceSignal processingStability-preserving dimensional reductions linearity transformPiecewise autonomisation-linearization method C1 Provisional Code UQ
Document type: Journal Article Critical review of research, literature review, critical commentary School of Information Technology and Electrical Engineering Publications
Versions Version Filter Type Wed, 27 Nov 2013, 04:09:10 EST Thu, 17 Jul 2014, 14:32:32 EST Thu, 17 Jul 2014, 14:33:36 EST Sat, 04 Oct 2014, 20:16:50 EST Filtered Full
Citation counts: Cited 0 times in Scopus Article Search Google Scholar Wed, 27 Nov 2013, 04:09:10 EST by System User on behalf of School of Information Technol and Elec Engineering | 756 | 2,537 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.53125 | 3 | CC-MAIN-2016-50 | latest | en | 0.599749 |
https://kilonova.ro/problems/1881?list_id=812 | 1,723,726,984,000,000,000 | text/html | crawl-data/CC-MAIN-2024-33/segments/1722641291968.96/warc/CC-MAIN-20240815110654-20240815140654-00735.warc.gz | 272,120,485 | 7,051 | # B. Coins
Time limit: 1s Memory limit: 64MB Input: Output:
Two players alternatively play the following game:
There are initially $n$ coins. In one move, supposing that the current number of coins is $x$, the current player can either:
1. Take one coin
2. Take multiple coins, provided that the remaining number of coins is a divisor of $x$.
The game ends when there are no coins left. The player who took the last coin is considered the winner.
Given the number of coins $n$, your task is to find which player will win the game, supposing that both of them play optimally.
## Input
Each test contains multiple testcases. The first line of input contains one integer $t$ ($1 \le t \le 10^3$) — the number of testcases.
Each testcase consists of one line, containing $n$ ($1 \le n \le 10^9$) — the number of coins.
## Output
For each testcase, print either First or Second, depending on the player who will win the game.
## Example
stdin
12
1
2
3
4
5
6
7
8
9
10
27
85653439
stdout
First
Second
First
First
Second
First
Second
First
Second
First
First
Second | 291 | 1,074 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 8, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.625 | 3 | CC-MAIN-2024-33 | latest | en | 0.88911 |
https://www.wyzant.com/resources/answers/15523/help_me_out_please_solve_it | 1,529,536,855,000,000,000 | text/html | crawl-data/CC-MAIN-2018-26/segments/1529267863939.76/warc/CC-MAIN-20180620221657-20180621001657-00031.warc.gz | 953,917,947 | 16,741 | 0
# HELP ME OUT PLEASE SOLVE IT..............
Solved the system by substitution.if the system is inconsistent and has no solution,state this .if the system is dependent,write the form of the solution for any real number x.
x+y=50
0.25x+0.75y=0.60(50)
### 1 Answer by Expert Tutors
Muhammad C. | Muhammad the grand math tutorMuhammad the grand math tutor
4.9 4.9 (228 lesson ratings) (228)
0
Multiply the bottom equation by 4 and you get:
x + 3y = 120
Subtract this equation from the first equation up top & we get:
-2y = -70
y = 35
x = 50 - 35 = 15 | 169 | 557 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.53125 | 4 | CC-MAIN-2018-26 | latest | en | 0.776585 |
https://jarviscodinghub.com/product/lab-11-generating-an-element-in-a-group-solution/ | 1,720,805,228,000,000,000 | text/html | crawl-data/CC-MAIN-2024-30/segments/1720763514450.42/warc/CC-MAIN-20240712161324-20240712191324-00585.warc.gz | 264,273,728 | 23,466 | # Lab 11 Generating an Element in a Group solution
\$14.99
Original Work ?
## Description
5/5 - (1 vote)
Problem Definition Consider the group G = (Z≥0,+) where + is the usual integer addition. The set {0,1} is a generating set for G, but you can also consider other generators. Note that 0 must be present in every generating set for G. Therefore, we do not need to explicitly state 0 — we can implicitly understand 0 to be in every generating set. Given a generating set S ⊆Z≥0 and an i ∈Z≥0, we can ask: how efficiently can we generate i using the elements in S. For example, let S = {1,5}1 and i = 11. You can generate 11 by the sequence 1 → 2 → 3 → 4 →···→ 10. More efficiently, you can generate 11 by 1 → 2 → 4 → 8 → 10 → 11. Optimally (i.e., most efficiently) you can generate 11 by 5 → 10 → 11 or 1 → 6 → 11. In general, you can define the efficiency of your generation by the number of times you use the + operator. Thus, the problem is: Given a set S ⊆ Z≥0 and an element i ∈ Z≥0, produce a sequence of elements generated from S such that you reach i most efficiently (i.e., fewest number of times you use the + operator). (In this lab exercise, bulk of your grade is for getting your program to work when S = {1}. Therefore, first focus on that simple case. This will also help you get an intuition for how to solve the more general case when S can be an arbitrary subset of Z≥0.)
0.1 Command Line Arguments
Your command line argument for this assignment is a single input file name.
1Recall that 0 is implicitly in S
1
0.2 The Input File Format
Each input file should solve one instance of the problem. The format is as follows. First you must have an integer n which represents the number of elements of the generating set S. Then, you provide the elements in S one after another. Finally, you provide your value of i. You should allow your input to be in free format. Confer Lab 10 for what I mean by free format. You may assume that i and the largest element in S are never more than 100 and the cardinality of S is at most 10.
0.3 Output
The output should contain the following:
1. A sequence that leads to i.
2. The number of times you needed to use the + operator to reach the value i.
As an example, consider the input:
3 1 7 13 42
Then, the output format is:
13+7=20, 20+1 = 21, 21+21 = 42 Number of times we used the + operator = 3
The output format is not strict, but it should be very easy for the human reader to get the above two pieces of information. | 702 | 2,467 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.453125 | 3 | CC-MAIN-2024-30 | latest | en | 0.892624 |
http://cooking.stackexchange.com/questions/17099/how-do-i-adjust-cooking-time-for-an-under-powered-microwave | 1,469,514,982,000,000,000 | text/html | crawl-data/CC-MAIN-2016-30/segments/1469257824756.90/warc/CC-MAIN-20160723071024-00239-ip-10-185-27-174.ec2.internal.warc.gz | 48,850,012 | 16,400 | # How do I adjust cooking time for an under-powered microwave?
Example:
I have an 800-Watt microwave. Cooking instructions for a certain product say to cook on HIGH for 4 minutes in a 1,100-Watt microwave.
How do I properly adjust the cooking time so that my microwave will properly cook the product? Is it simple math (such as `4*{1100/800}=5.5`, or 5 minutes, 30 seconds), or are there more factors involved?
For the purposes of my application, presume altitude is at or near sea level.
-
To those suggesting to just get an 1100W microwave: There is no reason to, unless the poster either needs to work as fast as possible in a commercial kitchen, or wants to replicate involved recipes exactly without bringing in an off-spec microwave as an unwanted error source. – rackandboneman Nov 13 '15 at 20:23
1100 Watt means 1100 Joules per second (energy over time). 1100 Watt over a period of 240 seconds therefore is 264000 Joules. To deliver 264000 Joules of energy with only 800 Watt takes 330 seconds (5.5 minutes), as you expected.
As KatieK noticed, there are some additional concerns. A good recipe will tell you to let the cooked product stand for a while. This allows heat diffusion, so all those Joules of heat will be distributed well. But if you cook at lower power, then the heat will already be diffused more. I.e. at lower power, you don't need to rest the product as long.
Another difference might be that you're not just heating the product, but you're relying on a secondary effect such as killing germs. For that, you'll need to have the entire product above a critical temperature for a certain time. This in general means that you don't need to adjust the times as much as you'd expect by the simple formula.
On the other hand, a warm product will lose heat to the environment over time. I.e. not all of those 264000 Joules will stay in the product you're heating. And with more time spent at medium temperatures, there's more heat loss in a low-powered microwave.
-
Each microwave oven has a different magnetron (the thing that generates microwaves), the actual heating efficiency of that plus the cookign chmaber desing is what counts, so it's never an exact formula, and the difference can be quite large – TFD Nov 13 '15 at 21:15
Your microwave is likely smaller than one of the 1100 watt jobs. Calculating microwave watts per cubic meter can be helpful. Sometimes the less powerful units come out with the same energy density as the larger models, sometimes not. You want Microwave energy, not Amps X volts from the wall. The cavity magnetrons in various units run at different efficiencies, and it's the output from that into the oven cavity that you're interested in. – Wayfaring Stranger Feb 12 at 16:34
There is no simple math equation that will always get you the proper cooking time. Each 800W microwave may differ, even within the same models, due to hotspots.
You can estimate that a 800 W microwave will be 72% as effective as a 1000W model, and that you want to increase cooking time to about 5 minutes. But since you can't fix over-cooked food, check your food early, and check often.
-
Good answer. Thankfully microwaves don't work like conventional ovens, where more power = burned exterior, underdone interior. – BobMcGee Aug 25 '11 at 4:18
The 1100W microwaves also have hotspots, and hopefully the 4 minute cooktime reflects that. In fact, at only 800W the effect of hotspots is less: the extra minute will allow more time for heat diffusion. – MSalters Aug 25 '11 at 13:12
## protected by Community♦Feb 12 at 2:38
Thank you for your interest in this question. Because it has attracted low-quality or spam answers that had to be removed, posting an answer now requires 10 reputation on this site (the association bonus does not count).
Would you like to answer one of these unanswered questions instead? | 910 | 3,859 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.28125 | 3 | CC-MAIN-2016-30 | latest | en | 0.947498 |
https://math.stackexchange.com/questions/3672275/notation-for-sum-of-intersection-cardinalities | 1,718,914,533,000,000,000 | text/html | crawl-data/CC-MAIN-2024-26/segments/1718198861989.79/warc/CC-MAIN-20240620172726-20240620202726-00287.warc.gz | 318,310,201 | 35,332 | # Notation for sum of intersection cardinalities
If $$A,B,C$$ are sets, then we can say that the sum of the cardinality intersections is $$|A \cap B|+|B \cap C| + |C \cap A|$$. However, with more sets, this may become tedious. I understand that there is this notation with the large intersection: $$\bigcap$$, but this cycles through intersections rather than sums of cardinalities of intersections.
I wonder if something like the following cyclic sum notation is common, if $$p = (A,B,C)$$ is a permutation of the three sets: $$\sum_{p}A\cap B=|A\cap B| + |B \cap C| + |C \cap A|.$$
If it is not the standard (which it probably is not, because it is a bit ambiguous and I have not seen it before), what is the preferred notation for what I am trying to state? Also, how do we generalize this notation beyond two intersections?
There's a notation that's similar. I prefer this: Let $$X = \{ X_1, X_2, \cdots, X_n \}$$ be a family of sets. We can instead sum over all pairs $$i \neq j$$ in the following manner: $$\sum_{1 \leq i < j \leq n} |X_i \cap X_j|$$ | 299 | 1,060 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 8, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.21875 | 3 | CC-MAIN-2024-26 | latest | en | 0.917991 |
https://www.vedantu.com/maths/dividing-fractions-with-whole-numbers | 1,618,333,052,000,000,000 | text/html | crawl-data/CC-MAIN-2021-17/segments/1618038073437.35/warc/CC-MAIN-20210413152520-20210413182520-00316.warc.gz | 1,156,185,021 | 90,734 | # Dividing Fractions with Whole Numbers
View Notes
## What Does Dividing Fractions by Whole Numbers Mean?
In this section, we will learn about Dividing Fractions With Whole Numbers. Before jumping into the main part let us understand the basic concepts of fractions and whole numbers.
### What are Fractions?
The portion/part of the whole thing is represented by a fraction. There are 2 parts to the fraction a denominator and a numerator. The top number is called the numerator, and the denominator is the number on the bottom.
Ex: 4/6 is a fraction. 4 is the numerator which is represented above the line and 6 is the denominator which is represented below the line. Here 4/6 can be written as 1/3 which is part of the number 3.
### What are Whole Numbers?
The whole numbers are described as the positive integers including zero. No decimal or fractional element is found in the whole number. In other words, a number that is not a fraction is a whole number.
The mathematical notation for whole numbers is
W = {0, 1, 2, 3, 4, 5, 6, 7 , ………..}
Now we have understood the basic concepts of Fractions and Whole numbers. Let us look into how to the fractions by whole numbers and vice versa.
### How to Divide Fractions with Whole Numbers
Here let us discuss the step to dividing fractions by whole numbers.
Step 1: The first step in dividing fractions by whole numbers is simply to write the fraction followed by the sign of the division and the whole number by which we need to divide it.Â
Ex: If we want to divide a fraction 5/4 by a whole number 3. We can represent this step as follows.
$\frac{5}{4}$ ÷ 3
Step 2: Convert the whole number into a fraction: To convert a whole number to a fraction, simply place the number over the number 1. The whole number becomes the numerator and 1 becomes the fraction denominator.
Ex: Let us look at the same example which is discussed in step 1. Here we have to convert whole number 3 into a fraction just by replacing 1 in the denominator which doesn’t change the value of 3.
$\frac{5}{4}$ ÷ $\frac{3}{1}$
Step 3: Take a reciprocal of the whole number which we are dividing the fraction. To find the reciprocal reverse the numerator and denominator.
Ex: $\frac{3}{1}$can be written as $\frac{1}{3}$
Step 4: After we take the reciprocal the division process will become a multiplication.
Ex: $\frac{5}{4}$ x $\frac{1}{3}$
Step 5: Now multiply the numerator and denominator of the fractions to obtain the new fraction.
Ex: $\frac{5}{4}$ x $\frac{1}{3}$ = $\frac{5}{12}$
Step 6: Simplify the fraction if necessary. To simply find the lowest common denominator, which means that both the numerator and denominator can be separated by any number that is equally divided into both numbers.
Ex: $\frac{2}{16}$ can be simplified as $\frac{1}{8}$.
Now let us some problems on dividing Fractions with Whole Numbers by using the above-mentioned steps.
### Problems on Dividing Fractions by Whole Numbers.
1) Divide the Fraction 3/7 by the Whole Number 3.
Ans:Â
Step 1: Write the fraction followed by the sign of the division
$\frac{3}{7}$ ÷ 3
Step 2: Convert the whole number into a fraction
3 can be written as 3/1.
Step 3: Take a reciprocal of the whole number.
3/1 can be written as â…“.
Step 4: Division process becomes multiplication.
$\frac{3}{7}$ x $\frac{1}{3}$
Step 5: Multiply numerator and denominator of the fractions.
$\frac{3}{7}$ x $\frac{1}{3}$ = $\frac{3}{21}$Â
Step 6: Simplify the fraction.
$\frac{3}{21}$ = $\frac{1}{7}$
The final fraction obtained after dividing 3/7 by the whole number 3 is 1/7.
2) Divide the Fraction 5/2 by the Whole Number 10.
Ans:Â
Step 1: Write the fraction followed by the sign of the division
$\frac{5}{2}$ ÷ 10
Step 2: Convert the whole number into a fraction
10 can be written as 10/1.
Step 3: Take a reciprocal of the whole number.
10/1 can be written as 1/10.
Step 4: Division process becomes multiplication.
$\frac{5}{2}$ x $\frac{1}{10}$
Step 5: Multiply numerator and denominator of the fractions.
$\frac{5}{2}$ x $\frac{1}{10}$ = $\frac{5}{20}$Â
Step 6: Simplify the fraction.
$\frac{5}{20}$ = $\frac{1}{4}$Â
The final fraction obtained after dividing 5/2 by the whole number 10 is 1/4.
### How to Divide Numbers with Fractions
Here we find the steps to divide the whole number by a fraction.
Step 1: Make a fraction out of the whole number. Make the whole number the numerator of a fraction denominator as 1.
Ex: Whole number 5 can be written in fraction form as 5/1.
Step 2: Find the reciprocal of the fraction. To find the reciprocal of the fraction reverse the numerator and denominator.
Ex: The reciprocal of the fraction 5/7 is 7/5 which is obtained by reversing the numerator and the denominator.
Step 3: Since we have found the reciprocal of the fraction, the division process will now be a multiplication process.
Ex: $\frac{5}{1}$ x $\frac{7}{5}$Â Â
Step 4: Multiply the numerator and denominator to find the fraction.
Ex: $\frac{5}{1}$ x $\frac{7}{5}$ = $\frac{35}{5}$Â Â
Step 5: Simplify the fraction if necessary. To simply find the lowest common denominator and divide both the numerator and denominator by that number.
Ex: $\frac{35}{5}$ is having the lowest common denominator as 5. So dividing both numerator and denominator by 5 we get the simplified answer as 7/1 or 7.
Let us solve some problems on dividing whole numbers by fractions.
### Problems on How to Divide Numbers with Fractions.
1) Divide the Whole Number 7 by the Fraction 3/4.
Ans:Â
Step 1: Make a fraction out of the whole number.
Here the whole number 7 can be written as 7/1 in fraction form.
Step 2: Find the reciprocal of the fraction.
¾ reciprocal is 4/3.Â
Step 3: Division process becomes multiplication.
$\frac{7}{1}$ x $\frac{4}{3}$Â Â Â Â
Step 4: Multiply the numerator and denominator.
$\frac{7}{1}$ x $\frac{4}{3}$ = $\frac{28}{3}$Â
Further simplification cannot be done. So the final answer obtained after dividing the whole number 7 by the fraction ¾ is 28/3.
2) Divide the Whole Number 12 by the Fraction 8/3.
Ans:Â
Step 1: Make a fraction out of the whole number.
Here the whole number 12 can be written as 12/1 in fraction form.
Step 2: Find the reciprocal of the fraction.
8/3 reciprocal is 3/8.Â
Step 3: Division process becomes multiplication.
$\frac{12}{1}$ x $\frac{3}{8}$
Step 4: Multiply the numerator and denominator.
$\frac{12}{1}$ x $\frac{3}{8}$ = $\frac{36}{8}$
Step 5: Simplify the fraction.
Here the lowest common denominator which divides both numerator and denominator is 4. So 36/8 can be simplified to 9/2.Â
So the final answer obtained after dividing the whole number 12 by the fraction 8/3 is 9/2.
### Conclusion
• When any fraction is divided by a whole number the final answer will always be a fraction.
• When a whole number is divided by a fraction the final answer will be either a fraction or a whole number. | 1,840 | 6,934 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 5 | 5 | CC-MAIN-2021-17 | longest | en | 0.923259 |
https://secondsminutes.com/174-seconds-in-minutes | 1,709,601,385,000,000,000 | text/html | crawl-data/CC-MAIN-2024-10/segments/1707947476592.66/warc/CC-MAIN-20240304232829-20240305022829-00372.warc.gz | 507,801,761 | 6,676 | # 174 seconds in minutes
## Result
174 seconds equals 2.9 minutes
You can also convert 174 seconds to minutes and seconds
## Conversion formula
Multiply the amount of seconds by the conversion factor to get the result in minutes:
174 s × 0.0166667 = 2.9 min
## How to convert 174 seconds to minutes?
The conversion factor from seconds to minutes is 0.0166667, which means that 1 seconds is equal to 0.0166667 minutes:
1 s = 0.0166667 min
To convert 174 seconds into minutes we have to multiply 174 by the conversion factor in order to get the amount from seconds to minutes. We can also form a proportion to calculate the result:
1 s → 0.0166667 min
174 s → T(min)
Solve the above proportion to obtain the time T in minutes:
T(min) = 174 s × 0.0166667 min
T(min) = 2.9 min
The final result is:
174 s → 2.9 min
We conclude that 174 seconds is equivalent to 2.9 minutes:
174 seconds = 2.9 minutes
## Result approximation:
For practical purposes we can round our final result to an approximate numerical value. In this case one hundred seventy-four seconds is approximately two point nine minutes:
174 seconds ≅ 2.9 minutes
## Conversion table
For quick reference purposes, below is the seconds to minutes conversion table:
seconds (s) minutes (min)
175 seconds 2.916673 minutes
176 seconds 2.933339 minutes
177 seconds 2.950006 minutes
178 seconds 2.966673 minutes
179 seconds 2.983339 minutes
180 seconds 3.000006 minutes
181 seconds 3.016673 minutes
182 seconds 3.033339 minutes
183 seconds 3.050006 minutes
184 seconds 3.066673 minutes
## Units definitions
The units involved in this conversion are seconds and minutes. This is how they are defined:
### Seconds
The second (symbol: s) (abbreviated s or sec) is the base unit of time in the International System of Units (SI). It is qualitatively defined as the second division of the hour by sixty, the first division by sixty being the minute. The SI definition of second is "the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom". Seconds may be measured using a mechanical, electrical or an atomic clock. SI prefixes are combined with the word second to denote subdivisions of the second, e.g., the millisecond (one thousandth of a second), the microsecond (one millionth of a second), and the nanosecond (one billionth of a second). Though SI prefixes may also be used to form multiples of the second such as kilosecond (one thousand seconds), such units are rarely used in practice. The more common larger non-SI units of time are not formed by powers of ten; instead, the second is multiplied by 60 to form a minute, which is multiplied by 60 to form an hour, which is multiplied by 24 to form a day. The second is also the base unit of time in other systems of measurement: the centimetre–gram–second, metre–kilogram–second, metre–tonne–second, and foot–pound–second systems of units.
### Minutes
The minute is a unit of time or of angle. As a unit of time, the minute (symbol: min) is equal to 1⁄60 (the first sexagesimal fraction) of an hour, or 60 seconds. In the UTC time standard, a minute on rare occasions has 61 seconds, a consequence of leap seconds (there is a provision to insert a negative leap second, which would result in a 59-second minute, but this has never happened in more than 40 years under this system). As a unit of angle, the minute of arc is equal to 1⁄60 of a degree, or 60 seconds (of arc). Although not an SI unit for either time or angle, the minute is accepted for use with SI units for both. The SI symbols for minute or minutes are min for time measurement, and the prime symbol after a number, e.g. 5′, for angle measurement. The prime is also sometimes used informally to denote minutes of time. In contrast to the hour, the minute (and the second) does not have a clear historical background. What is traceable only is that it started being recorded in the Middle Ages due to the ability of construction of "precision" timepieces (mechanical and water clocks). However, no consistent records of the origin for the division as 1⁄60 part of the hour (and the second 1⁄60 of the minute) have ever been found, despite many speculations. | 1,047 | 4,267 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.15625 | 4 | CC-MAIN-2024-10 | latest | en | 0.78436 |
https://www.tutorialcup.com/interview/graph/bfs-for-disconnected-graph.htm | 1,606,690,356,000,000,000 | text/html | crawl-data/CC-MAIN-2020-50/segments/1606141203418.47/warc/CC-MAIN-20201129214615-20201130004615-00211.warc.gz | 879,109,272 | 262,071 | Home » Technical Interview Questions » Graph Interview Questions » BFS for Disconnected Graph
# BFS for Disconnected Graph
## Problem Statement
The problem “BFS for Disconnected Graph” states that you are given a disconnected directed graph, print the BFS traversal of the graph.
## Example
The BFS traversal of the graph above gives: 0 1 2 5 3 4 6
## Approach
Breadth first Search (BFS) traversal for Disconnected Directed Graph is slightly different from BFS traversal for Connected undirected graph. In a connected undirected graph, we begin traversal from any source node S and the complete graph network is visited during the traversal. However, the BFS traversal for Disconnected Directed Graph involves visiting each of the not visited nodes and perform BFS traversal starting from that node. We terminate traversal once we find that all the nodes have been visited.
Consider the connected undirected graph given below, starting BFS traversal from any node of the graph would visit all the nodes in the graph in one go.
Consider the directed connected graph below, as it is evident from the image, to visit all the nodes in the graph, it is needed to repeatedly perform BFS traversal from nodes 0, 1, 3.
## Algorithm
1. Consider, there are V nodes in the given graph.
2. Iterate through each node from 0 to V and look for the 1st not visited node.
3. Begin BFS traversal starting from this node and mark all the nodes subsequently traversed as visited.
4. Terminate once all the nodes in the graph have been visited.
## Code
### C++ Program for BFS for Disconnected Graph
```#include <iostream>
#include <bits/stdc++.h>
using namespace std;
// Add Edge from node u to v
void addEdge(vector <int> graph[], int u, int v)
{
graph[u].push_back(v);
}
// BFS traversal function
void BFS(vector <int> graph[], int n)
{
// vector to mark nodes as visited
vector <bool> vis(n,false);
// Process each node from 0 to n-1
for(int i=0;i<n;i++)
{
// If not visited node is found
if(vis[i] == false)
{
// BFS queue
queue <int> q;
// add not visited node to queue
q.push(i);
// BFS traversal
while(!q.empty())
{
int front = q.front();
q.pop();
cout<<front<<" ";
vis[front] = true;
for(auto node : graph[front])
{
if(vis[node] == false)
q.push(node);
}
}
}
}
cout<<endl;
}
int main()
{
// Construct the graph
int n = 7;
vector <int> graph[n];
vector<pair<int,int>> edges = {{1,0},{1,2},{2,5},{3,4},{4,6}};
for(auto e : edges)
// Display the BFS traversal
cout<<"BFS traversal of the given graph : ";
BFS(graph,n);
return 0;
}```
`BFS traversal of the given graph : 0 1 2 5 3 4 6`
### Java Program for BFS for Disconnected Graph
```import java.util.*;
import java.io.*;
class TutorialCup
{
// Add Edge from node u to v
static void addEdge(ArrayList<ArrayList<Integer>> graph, int u, int v)
{
}
// BFS traversal function
static void BFS(ArrayList<ArrayList<Integer>> graph, int n)
{
// array to mark nodes as visited
boolean [] vis = new boolean[n];
// Process each node from 0 to n-1
for(int i=0;i<n;i++)
{
// If not visited node is found
if(vis[i] == false)
{
// BFS queue
Queue <Integer> q = new LinkedList<>();
// add not visited node to queue
// BFS traversal
while(!q.isEmpty())
{
int front = q.poll();
System.out.print(front+" ");
vis[front] = true;
Iterator itr = graph.get(front).iterator();
while(itr.hasNext())
{
int node = (Integer)itr.next();
if(vis[node] == false)
}
}
}
}
System.out.println();
}
public static void main (String[] args)
{
// Construct the graph
int n = 7;
ArrayList<ArrayList<Integer>> graph = new ArrayList<ArrayList<Integer>>();
for(int i=0;i<n;i++)
int [][] edges = {{1,0},{1,2},{2,5},{3,4},{4,6}};
for(int i=0;i<edges.length;i++)
// Display the BFS traversal
System.out.print("BFS traversal of the given graph : ");
BFS(graph,n);
}
}```
`BFS traversal of the given graph : 0 1 2 5 3 4 6`
## Complexity Analysis
1. Time Complexity: T(n) = O(V+E)
2. Space Complexity: A(n) = O(V)
Because we’ve been using our space complexity becomes linear. So the algorithm becomes linear in space. And for time complexity as we have visited all the nodes in the graph. The algorithm takes linear time as well.
V = number of nodes.
E = number of edges.
Array Interview Questions Graph Interview Questions LinkedList Interview Questions String Interview Questions Tree Interview Questions Core Java Interview Questions | 1,100 | 4,373 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.09375 | 4 | CC-MAIN-2020-50 | longest | en | 0.880289 |
https://www.convert-measurement-units.com/convert+Kilogram+per+liter+to+Planck+density.php | 1,669,856,249,000,000,000 | text/html | crawl-data/CC-MAIN-2022-49/segments/1669446710777.20/warc/CC-MAIN-20221130225142-20221201015142-00743.warc.gz | 754,244,578 | 13,149 | Convert kg/l to Planck density (Kilogram per liter to Planck density)
## Kilogram per liter into Planck density
numbers in scientific notation
https://www.convert-measurement-units.com/convert+Kilogram+per+liter+to+Planck+density.php
## How many Planck density make 1 Kilogram per liter?
1 Kilogram per liter [kg/l] = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 193 986 420 950 53 Planck density - Measurement calculator that can be used to convert Kilogram per liter to Planck density, among others.
# Convert Kilogram per liter to Planck density (kg/l to Planck density):
1. Choose the right category from the selection list, in this case 'Density'.
2. Next enter the value you want to convert. The basic operations of arithmetic: addition (+), subtraction (-), multiplication (*, x), division (/, :, ÷), exponent (^), square root (√), brackets and π (pi) are all permitted at this point.
3. From the selection list, choose the unit that corresponds to the value you want to convert, in this case 'Kilogram per liter [kg/l]'.
4. Finally choose the unit you want the value to be converted to, in this case 'Planck density'.
5. Then, when the result appears, there is still the possibility of rounding it to a specific number of decimal places, whenever it makes sense to do so.
With this calculator, it is possible to enter the value to be converted together with the original measurement unit; for example, '345 Kilogram per liter'. In so doing, either the full name of the unit or its abbreviation can be usedas an example, either 'Kilogram per liter' or 'kg/l'. Then, the calculator determines the category of the measurement unit of measure that is to be converted, in this case 'Density'. After that, it converts the entered value into all of the appropriate units known to it. In the resulting list, you will be sure also to find the conversion you originally sought. Alternatively, the value to be converted can be entered as follows: '14 kg/l to Planck density' or '88 kg/l into Planck density' or '59 Kilogram per liter -> Planck density' or '95 kg/l = Planck density' or '56 Kilogram per liter to Planck density' or '10 Kilogram per liter into Planck density'. For this alternative, the calculator also figures out immediately into which unit the original value is specifically to be converted. Regardless which of these possibilities one uses, it saves one the cumbersome search for the appropriate listing in long selection lists with myriad categories and countless supported units. All of that is taken over for us by the calculator and it gets the job done in a fraction of a second.
Furthermore, the calculator makes it possible to use mathematical expressions. As a result, not only can numbers be reckoned with one another, such as, for example, '(21 * 26) kg/l'. But different units of measurement can also be coupled with one another directly in the conversion. That could, for example, look like this: '345 Kilogram per liter + 1035 Planck density' or '65mm x 24cm x 12dm = ? cm^3'. The units of measure combined in this way naturally have to fit together and make sense in the combination in question.
The mathematical functions sin, cos, tan and sqrt can also be used. Example: sin(π/2), cos(pi/2), tan(90°), sin(90) or sqrt(4).
If a check mark has been placed next to 'Numbers in scientific notation', the answer will appear as an exponential. For example, 7.117 038 206 839 9×1030. For this form of presentation, the number will be segmented into an exponent, here 30, and the actual number, here 7.117 038 206 839 9. For devices on which the possibilities for displaying numbers are limited, such as for example, pocket calculators, one also finds the way of writing numbers as 7.117 038 206 839 9E+30. In particular, this makes very large and very small numbers easier to read. If a check mark has not been placed at this spot, then the result is given in the customary way of writing numbers. For the above example, it would then look like this: 7 117 038 206 839 900 000 000 000 000 000. Independent of the presentation of the results, the maximum precision of this calculator is 14 places. That should be precise enough for most applications. | 1,040 | 4,268 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.578125 | 4 | CC-MAIN-2022-49 | latest | en | 0.736221 |
http://invercaria.es/krai2/inverse-trigonometric-functions-graphs-76f6d9 | 1,618,100,519,000,000,000 | text/html | crawl-data/CC-MAIN-2021-17/segments/1618038060603.10/warc/CC-MAIN-20210411000036-20210411030036-00374.warc.gz | 49,992,146 | 11,576 | The most common convention is to name inverse trigonometric functions using an arc- prefix: arcsin(x), arccos(x), arctan(x), etc. However, since trigonometric functions are not one-to-one, meaning there are are infinitely many angles with , it is impossible to find a true inverse function for . We have briefly mentioned the inverse trigonometric functions before, for example in Section 1.3 when we discussed how to use the $$\fbox{\(\sin^{-1}$$}\), $$\fbox{\(\cos^{-1}$$}\), and $$\fbox{\(\tan^{-1}$$}\) buttons on a calculator to find an angle that has a certain trigonometric function value. By using our site, you So the domain of arccot x is: The range of arccot x is Arctangent 4. Problem 1: Find the principal value of the given equation: We know that the range of the principal value branch of sin-1(x) is (−π/2, π/2) and sin(π/4) = 1/√2. Inverse trigonometric functions review. Here's y = cos x. 131 Inverse Trigonometric Functions Definition 4.2 (Odd and Even functions) A real valued function f is an even function if for all x in the domain of f, -x is also in the domain of f and fx()−= fx() . Using the formula we can apply the transformations step-by-step. Graphs of inverse trigonometric functions. So, the principal value of cot-1(1/√3) = π/3. And so the inverse of this graph must be the following with and. The function has an inverse function only if the function is one-to-one. Domain: []− 1,1 Using the inverse trigonometric functions, we can solve for the angles of a right triangle given two sides, and we can use a calculator to find the values to several decimal places. There are two me… The domain and range of each of the above functions are also explored. The dark portion of the graph of y = sin–1 x represent the principal value branch. misrepresent that a product or activity is infringing your copyrights. 101 S. Hanley Rd, Suite 300 In this trigonometry lesson, students break down a sinusoidal equation and solve for all x-values and domain. Inverse trigonometric function graphs for sine, cosine, tangent, cotangent, secant and cosecant as a function of values. either the copyright owner or a person authorized to act on their behalf. inverse y = x x2 − 6x + 8 inverse f (x) = √x + 3 inverse f (x) = cos (2x + 5) inverse f (x) = sin (3x) Now using the formula where = Period, the period of is . Graphs for inverse trigonometric functions. We know that the range of the principal value branch of cosec-1(x) is [-π/2, π/2] – {0} and cosec(π/4) = √2. on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney. Figure 4. Bogazici University Istanbul Turkey, Bachelor of Science, Electrical Engineering. cos-1(x) is the inverse function of cos(x). The three trigonometric functions studied in this tutorial are: arcsin(x), arccos(x) and arctan(x). improve our educational resources. 5.3: Inverse Trigonometric Functions - Mathematics LibreTexts These functions are widely used in fields like physics, mathematics, engineering and other research fields. Just like the trigonometric functions, we can also represent graphs of inverse trigonometric functions. Thus the graphs of none of them pass the Horizontal Line Test and so are not 1 − to − 1. All trigonometric functions are Continuous in its domain. The graphs of y = sin x and y = sin–1 x are as given in Fig 2.1 (i), (ii), (iii). Inverse trigonometric functions are the inverse functions of the trigonometric ratios i.e. Inverse Trigonometry. Your name, address, telephone number and email address; and Take for example, to find the inverse we use the following method. Graphs of Inverse Trigonometric Functions Trigonometric functions are all periodic functions. In other words, the domain of the inverse function is the range of the original function, and vice versa, as summarized in Figure 1. sec-1(x) is the inverse function of sec(x). It is an odd function which is strictly increasing in (-∞, ∞). Students graph inverse trigonometric functions. The function does not have any inflection points. Its domain is [−1, 1] and its range is [- π/2, π/2]. It doesn’t intercept the coordinate axis as it is a discontinuous function. We have briefly mentioned the inverse trigonometric functions before, but we will now define those inverse functions and determine their graphs. Derivatives of Implicit Functions - Continuity and Differentiability | Class 12 Maths. Explain why this graph is not a function. Class 12 NCERT Solutions - Mathematics Part I - Chapter 2 Inverse Trigonometric Functions - Exercise 2.1. We know that the range of the principal value branch of tan-1(x) is (-π/2, π/2) and tan(π/4) = 1. It is an odd function that is strictly decreasing in its domain. We know that the range of the principal value branch of tan-1(x) is (-π/2, π/2) and tan(π/3) = √3. INVERSE TRIGONOMETRIC FUNCTIONS 35 of sine function. Some of the most basic and widely used trigonometric functions are A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe The cosine function and inverse cosine (or arccosine) function . As shown below, we will restrict the domains to certain quadrant… Inverse trigonometric functions are the inverse functions of the trigonometric ratios i.e. Your Infringement Notice may be forwarded to the party that made the content available or to third parties such represent angles or real numbers and their sine is x, cosine is x and tangent is x, given that the answers are numerically smallest available. It intersects the coordinate axis at (1, π/2). The sine function and inverse sine (or arcsine) function. We first need to think about the graph of the function . The dark portion of the graph of y = sin–1 x represent the principal value branch. Other Inverse Trigonometric Functions: Each trigonometric function has a restricted domain for which an inverse function is defined. (This convention is used throughout this article.) The given function passes the horizontal line test only if any horizontal lines intersect the function at most once. Its domain is (-∞, -1] U [1, ∞) and its range is [0, π/2) U (π/2, π]. These functions are widely used in fields like physics, mathematics, engineering, and other research fields. -10-5 5 10-10-7.5-5-2.5 2.5 5 7.5 10-10-5 5 10-10-7.5-5-2.5 2.5 5 7.5 10-10-5 5 10-10-7.5-5-2.5 2.5 5 7.5 10 Functions and Their Graphs The following functions are basic inverse trig functions that you are expected to know. generate link and share the link here. The restricted domains are determined so the trig functions are one-to-one. Logic - Cheat Sheet - Foldable Create a … sin, cos, tan, cot, sec, cosec. The function does not have any minima points. It is a good exercise for you to compare these with what you see in a grapher viewing window. Track your scores, create tests, and take your learning to the next level! Infringement Notice, it will make a good faith attempt to contact the party that made such content available by Similarly, inverse functions of the basic trigonometric functions are said to be inverse trigonometric functions. The graph of the tangent function would clearly illustrate the repeated intervals. In particular, the graphs of the sine and cosine functions are called sinusoidal curves. In (1, ∞) it is decreasing and in (-∞, -1) it is decreasing, In (1, ∞) it is increasing and in (-∞, -1) it is increasing. Follow the steps in the tutorial below. This leaves us with our answer. Using the formula where is the vertical shift, we have to perform a transformation of moving the function up two units on the graph. Next lesson. Let’s take a look at our sine function first. Which of the following is the graph of with ? To graph the inverse trigonometric functions, we use the graphs of the trigonometric functions restricted to the domains defined earlier and reflect the graphs about the line $$y=x$$ (Figure). Anti-trigonometric Functions Graph. Graphs of Inverse Trigonometric Functions - Trigonometry | Class 12 Maths. Several notations for the inverse trigonometric functions exist. Inverse Trigonometric Functions in Maths Trigonometry is a measurement of triangle and it is included with inverse functions. Inverse trigonometric function graphs for sine, cosine, tangent, cotangent, secant and cosecant as a function of values. There are two popular notations used for inverse trigonometric functions: Adding “arc” as a prefix. To graph the inverse cosine function, we limit or choose a section of our regular cosine graph to work with. The restricted domains are determined so the trig functions are one-to-one. If you've found an issue with this question, please let us know. Which of the following is the correct graph and range of the inverse function of with ? We studied Inverses of Functions here; we remember that getting the inverse of a function is basically switching the x and y values, and the inverse of a function is symmetrical (a mirror image) around the line y=x. There are particularly six inverse trig functions for each trigonometry ratio. Thus, the graph of the function y = sin –1 x can be obtained from the graph of y = sin x by interchanging x and y axes. This means that if a horizontal line is drawn anywhere on the graph it will only pass through one point. Note that the inverse of is not , that is the reciprocal. Its domain is [−1, 1] and its range is [0, π]. Its domain is (-∞, -1] U [1, ∞) and its range is [-π/2, 0) U (0, π/2]. Follow the steps in the tutorial below. Consider the graph of the function . If this MATLAB exercise is being counted in your grade, then be sure to … If you wish to graph the inverse of , then you must restrict the domain so that your graph will pass the vertical line test. Get Free Access See Review. Duke University, Doctor of Philosophy, Biom... Delhi University India, Bachelor of Science, Mathematics. 1 Name: Date: M3 Precalculus: Graphing Inverse Trigonometric Functions Graphs of Inverse Trigonometric Functions 1. This is not true, and we can also see that if we graph the inverse of () that this does not pass the vertical line test and therefore is not a function. Loading... Graphs of the trigonometric functions Graphs of the trigonometric functions ... Transformations: Inverse of a Function. Thus, if you are not sure content located Get Free Access See Review. Varsity Tutors. Algebra and Trigonometry 10th Edition answers to Chapter 6 - 6.4 - Graphs of Sine and Cosine Functions - 6.4 Exercises - Page 465 68 including work step by step written by community members like you. inverse trigonometric functions.basic concepts,graphs,definitions,domain and ranges explained in telugu. St. Mary of the Plains College, Bachelors, Mathematics. Its domain is ℝ and its range is (0, π). Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially So now we are trying to find the range of and plot the function . Which of the following is the graph of the inverse of with ? So, the principal value of tan-1(√3) = π/3. Class 12 RD Sharma Solutions- Chapter 4 Inverse Trigonometric Functions - Exercise 4.1. Figure 5. All trigonometric functions are periodic functions. Its domain is ℝ and its range is [-π/2, π/2]. If you know the side opposite and the side adjacent to the angle in question, the inverse tangent is the function you need. There are particularly six Anti-trigonometric functions for each trigonometric ratio. We know that the range of the principal value branch of cosec-1(x) is [-π/2, π/2] – {0} and cosec(π/2) = 1. ChillingEffects.org. As we know trigonometric functions are many one in their domain, hence, they are not invertible.But their inverse can be obtained by restricting the domain so as to make invertible. Define domain and range of inverse trigonometric functions and draw the graphs of inverse trigonometric functions and solve problems. sin, cos, tan, cot, sec, cosec. The inverse of g is denoted by ‘g … We will begin with the graph of the tangent function, plotting points as we did for the sine and cosine functions. Graphically speaking, the range is the portion of the y-axis on which the graph casts a shadow. Just like addition and subtraction are the inverses of each other, the same is true for the inverse of trigonometric functions. With the help of the community we can continue to Which of the following represents the graph of with ? Inverse trigonometric function graph animations Introduction. The exploration is carried out by analyzing the graph of the function and the graph of its inverse. It is an odd function and is strictly increasing in (-1, 1). © 2007-2021 All Rights Reserved, True or False: The inverse of the function, Which of the following is the graph of the inverse of, Graphs Of Inverse Trigonometric Functions, Which of the following represents the graph of, Which of the following is the correct graph and range of the inverse function of, San Francisco-Bay Area Trigonometry Tutoring, San Francisco-Bay Area Trigonometry Tutors, SAT Courses & Classes in San Francisco-Bay Area, Spanish Courses & Classes in Washington DC. The function If Varsity Tutors takes action in response to They never get to be themselves. a The graphs of y = sin x and y = sin–1 x are as given in Fig 2.1 (i), (ii), (iii). information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are First we will transform the amplitude, so so we must shorten the amplitude to . 07, Nov 20. Now, for its inverse to also be a function it must pass the horizontal line test. The graph of with is, Switching the and values to graph the inverse we get the graph. Analyzing the Graph of y = tan x. If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one Interactive Tutorial The three trigonometric functions studied in this tutorial are: arcsin(x), arccos(x) and arctan(x). The inverse trig functions are often called "arc functions", since given a value of a trig function, they produce the length of arc needed to obtain that value. Graphs of Trigonometric Functions When we graph trigonometric functions in the coordinate plane, we usually denote the independent variable (radians) by x instead of 0. Use online calculator for trigonometry. A description of the nature and exact location of the content that you claim to infringe your copyright, in \ These functions are widely used in fields like physics, mathematics, engineering, and other research fields. A Computer Science portal for geeks. link to the specific question (not just the name of the question) that contains the content and a description of your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the The graph of y = sin x can be visuallised in the figure below: Domain: all reals Range: [-1,1] Period: 2π Y-intercept: (0,0) When you restrict the domain of sin x to the interval –π/2 ≤ x ≤ π/2, the following properties should hold: 1.y = sin x is an increasing function. The inverse of six important trigonometric functions are: 1. We will now define those inverse functions and determine their graphs. Arccosecant Let us discuss all the six important types of inverse trigonometric functions along with its definition, formulas, graphs, properties and solved examples. means of the most recent email address, if any, provided by such party to Varsity Tutors. The inverse trigonometric functions are used to determine the angle measure when at least two sides of a right triangle are known. To make 1 sin y … Arccotangent 5. A real valued function f is an odd function if for all x in the domain of f, -x is also in the domain of f and fx()−= −fx() . Let so where for the parent function for the inverse function. It passes the vertical line test, that is if a vertical line is drawn anywhere on the graph it only passes through a single point of the function. We know that the range of the principal value branch of cos-1(x) is (0, π) and cos(π/4) = 1/√2. The graphs of the inverse functions are shown in Figure 4, Figure 5, and Figure 6. a. Inverse Trigonometric Functions and their Graphs Definition The functions of the angle of triangles are termed as trigonometric functions. It intersects the coordinate axis at (0, 0). The trigonometric functions are periodic, and hence not injective, so strictly speaking, they do not have an inverse function. Switch the x- and y-coordinates in the graph of sin y x to graph sin x y . Such principal values are sometimes denoted with a capital letter so, for example, the principal value of the inverse sine may be variously denoted or (Beyer 1987, p. 141). It doesn’t intercept the coordinate axis. information described below to the designated agent listed below. So, the principal value of sin-1(1) = π/2. An identification of the copyright claimed to have been infringed; We know that the range of the principal value branch of sec-1(x) is [0, π] – {π/2} and sec(π/4) = √2. Graphs of Inverse Trigonometric Functions The graphs of the inverse functions are the original function in the domain specified above, which has been flipped about the line y=x y = x. Graphs of inverse trigonometric functions; Multiplication theorem on probability; Angle between two lines; Solutions of the linear differential equation of the type − dy/dx + py = q; Solution for dx/dy + px = q; Subsets of real numbers; Adjoint and inverse of a square matrix; sin2x – cos2x = 1 for all values of x In this section, we will explore the graphs of the tangent and other trigonometric functions. Inverse trigonometric functions and their graphs Preliminary (Horizontal line test) Horizontal line test determines if the given function is one-to-one. (Opens a modal) The trig … No matter the transformations applied, all values of will still be reached. We know that the range of the principal value branch of cot-1(x) is (-π/2, π/2) and cot(π/4) = 1. Let’s start with the graph of . ; The value of an inverse trigonometric functions which lies in the range of principal branch is called the principal value of that inverse trigonometric functions. Experience. an Statistics: Anscombe's … So, the principal value of cos-1(1/√2) = π/4. Lesson Planet. So, the principal value of cosec-1(1) = π/2. The given function passes the horizontal line test only if any horizontal lines intersect the function at most once. True or False: The domain for will always be all real numbers no matter the value of or any transformations applied to the tangent function. Domain and Range of inverse trigonometric functions. 28, Dec 20 . sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require In this section, we are interested in the inverse functions of the trigonometric functions and .You may recall from our work earlier in the semester that in order for a function to have an inverse, it must be one-to-one (or pass the horizontal line test: any horizontal line intersects the graph at most once).. See videos from Algebra on Numerade We know that the range of the principal value branch of sec-1(x) is [0, π] – {π/2} and sec(0) = 1. cot-1(x) is the inverse function of cot(x). To find an inverse function you swap the and values. There are two popular notations used for inverse trigonometric functions: Example: arcsin(x), arccos(x), arctan(x), …. This means none of them have an inverse unless the domain of each is restricted to make each of them 1 − to − 1. Graphs. 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Cot-1 ( 1/√3 ) = π/2 like an inverse function is one-to-one or arccosine ) function used in like. That if a horizontal line is drawn anywhere on the next page shows sketches of the x-axis on the... The inverses of each other, the period of is tan -1 x, tan -1 x.!, 1 ] and its range is ( 0, π ] arccosine ) function third such!, Bachelor of Science, Mathematics, engineering, and take your learning to the party that made content...: Adding “ arc ” as a function we limit the domain to [ 0°, 180°,! University Istanbul Turkey, Bachelor of Science, Electrical engineering, but we explore... Must pass the horizontal line test only if any horizontal lines intersect the function an...
2nd Degree Kidnapping, Pag Asa Faithmusic Lyrics, Collen Mashawana Wedding, Ardex X77 Tile Adhesive Data Sheet, If Only You Were Mine G Herbo Spotify, Sanus Vlt5 Amazon, 1957 Ford Crown Victoria, Water Blaster Rental Near Me, Urban Core In A Sentence, | 6,786 | 28,701 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.4375 | 4 | CC-MAIN-2021-17 | longest | en | 0.854557 |
https://math.stackexchange.com/questions/2526983/finding-a-cartesian-coordinate-transformation-matrix?noredirect=1 | 1,571,543,052,000,000,000 | text/html | crawl-data/CC-MAIN-2019-43/segments/1570986702077.71/warc/CC-MAIN-20191020024805-20191020052305-00528.warc.gz | 597,009,636 | 32,309 | # Finding a Cartesian coordinate transformation matrix
I have a vector, $u := [u_1, \ldots, u_n]^\mathrm{T}$. I am trying to find a coordinate transformation matrix, $Q \in \mathbb{R}^{n \times n}$, which is nonsingular, satisfying: \begin{align*} \begin{bmatrix} 0 \\ \vdots \\ 0 \\ ||u|| \end{bmatrix} = Q u. \end{align*} I would appreciate any idea to find this matrix, $Q$.
## 1 Answer
Recall that if $W$ is a subespace of $\mathbb{R}^{n}$, then $$\dim(W)+\dim(W^{\perp})=n$$ Take $W$ as the span of $u$, so $$\dim(<u>)=n-1.$$ You can construct a matrix $Q$ that will satisfy the same condition taking any set of $n-1$ vectors $v_i$ from any basis of the ortogonal complement of the span of $u$. $$Q=\left(\begin{array}{cccc} v_{11}&v_{12}&\cdots&v_{1n}\\ v_{21}&v_{22}&\cdots &v_{2n}\\ v_{31}&v_{32}&\cdots&v_{3n}\\ \vdots&\vdots&\vdots&\vdots\\ v_{n-1,1}&v_{n-1,2}&\cdots&v_{n-1,n}\\ \displaystyle{\frac{u_1}{\vert\vert{u}\vert\vert}}&\displaystyle{\frac{u_2}{\vert\vert{u}\vert\vert}}&\cdots&\displaystyle{\frac{u_n}{\vert\vert{u}\vert\vert}}\\ \end{array}\right)$$ where $v_{i}=(v_{i1},\ldots,v_{in})^{T}$ are vectors in a basis $\{v_{1},\ldots,v_{n-1}\}$ of the ortogonal complement of the linear span of $u$, for all $i$ with $1\leq i\leq n-1$.
Remark: This matrix $Q$ is non singular because it's rows are linearly independent because the following set is a basis of $\mathbb{R}^{n}$: $$\mathcal{B}=\{u,v_1,\ldots,v_{n-1}\}$$
Also: You can use Gram Schmidt process to get an ortogonal basis of $<u>^{\perp}$ starting from $u$. In this way you will find the rights vectors $v_i$ such that $Q$ is inversible.
• Thank you for your comment. I omitted an info that Q should be nonsingular. – H.C. Nov 19 '17 at 4:12
• Oh ! Yes sure, you want a matrix to do change of basis :) – Hector Blandin Nov 19 '17 at 4:13
• You said it when you refer to coordinate transformation matrix, this matrix should be inversible. So, is my mistake :) – Hector Blandin Nov 19 '17 at 4:14
• @H.C.: You can use my matrix $Q_1$ and Gram Schmidt algorithm see here: en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process – Hector Blandin Nov 19 '17 at 4:18
• @H.C.: Look to this answer also: math.stackexchange.com/questions/1766613/… – Hector Blandin Nov 19 '17 at 4:34 | 804 | 2,255 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.625 | 4 | CC-MAIN-2019-43 | latest | en | 0.717811 |
https://wikivisually.com/wiki/Triple_product | 1,571,706,004,000,000,000 | text/html | crawl-data/CC-MAIN-2019-43/segments/1570987795403.76/warc/CC-MAIN-20191022004128-20191022031628-00142.warc.gz | 746,172,814 | 54,364 | # Triple product
In vector algebra, a branch of mathematics, the triple product is a product of three 3-dimensional vectors, usually Euclidean vectors. The name "triple product" is used for two different products, the scalar-valued scalar triple product and, less often, the vector-valued vector triple product.
## Scalar triple product
Three vectors defining a parallelepiped
The scalar triple product (also called the mixed product, box product, or triple scalar product) is defined as the dot product of one of the vectors with the cross product of the other two.
### Geometric interpretation
Geometrically, the scalar triple product
${\displaystyle \mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )}$
is the (signed) volume of the parallelepiped defined by the three vectors given. Here, the parentheses may be omitted without causing ambiguity, since the dot product cannot be evaluated first. If it were, it would leave the cross product of a scalar and a vector, which is not defined.
### Properties
• The scalar triple product is unchanged under a circular shift of its three operands (a, b, c):
${\displaystyle \mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )=\mathbf {b} \cdot (\mathbf {c} \times \mathbf {a} )=\mathbf {c} \cdot (\mathbf {a} \times \mathbf {b} )}$
• Swapping the positions of the operators without re-ordering the operands leaves the triple product unchanged. This follows from the preceding property and the commutative property of the dot product.
${\displaystyle \mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )=(\mathbf {a} \times \mathbf {b} )\cdot \mathbf {c} }$
• Swapping any two of the three operands negates the triple product. This follows from the circular-shift property and the anticommutativity of the cross product.
{\displaystyle {\begin{aligned}&\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )\\=-&\mathbf {a} \cdot (\mathbf {c} \times \mathbf {b} )\\=-&\mathbf {b} \cdot (\mathbf {a} \times \mathbf {c} )\\=-&\mathbf {c} \cdot (\mathbf {b} \times \mathbf {a} )\end{aligned}}}
• The scalar triple product can also be understood as the determinant of the 3×3 matrix (thus also its inverse) having the three vectors either as its rows or its columns (a matrix has the same determinant as its transpose):
${\displaystyle \mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )=\det {\begin{bmatrix}a_{1}&a_{2}&a_{3}\\b_{1}&b_{2}&b_{3}\\c_{1}&c_{2}&c_{3}\\\end{bmatrix}}={\rm {det}}\left(\mathbf {a} ,\mathbf {b} ,\mathbf {c} \right).}$
• If the scalar triple product is equal to zero, then the three vectors a, b, and c are coplanar, since the parallelepiped defined by them would be flat and have no volume.
• If any two vectors in the scalar triple product are equal, then its value is zero:
${\displaystyle \mathbf {a} \cdot (\mathbf {a} \times \mathbf {b} )=\mathbf {a} \cdot (\mathbf {b} \times \mathbf {a} )=\mathbf {a} \cdot (\mathbf {b} \times \mathbf {b} )=\mathbf {b} \cdot (\mathbf {a} \times \mathbf {a} )=0}$
• Moreover,
${\displaystyle (\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} ))\mathbf {a} =(\mathbf {a} \times \mathbf {b} )\times (\mathbf {a} \times \mathbf {c} )}$
• The simple product of two triple products (or the square of a triple product), may be expanded in terms of dot products:[1]
${\displaystyle ((\mathbf {a} \times \mathbf {b} )\cdot \mathbf {c} )\;((\mathbf {d} \times \mathbf {e} )\cdot \mathbf {f} )=\det \left[{\begin{pmatrix}\mathbf {a} \\\mathbf {b} \\\mathbf {c} \end{pmatrix}}\cdot {\begin{pmatrix}\mathbf {d} &\mathbf {e} &\mathbf {f} \end{pmatrix}}\right]=\det {\begin{bmatrix}\mathbf {a} \cdot \mathbf {d} &\mathbf {a} \cdot \mathbf {e} &\mathbf {a} \cdot \mathbf {f} \\\mathbf {b} \cdot \mathbf {d} &\mathbf {b} \cdot \mathbf {e} &\mathbf {b} \cdot \mathbf {f} \\\mathbf {c} \cdot \mathbf {d} &\mathbf {c} \cdot \mathbf {e} &\mathbf {c} \cdot \mathbf {f} \end{bmatrix}}}$
This restates in vector notation that the product of the determinants of two 3×3 matrices equals the determinant of their matrix product. As a special case, the square of a triple product is a Gram determinant.
### Scalar or pseudoscalar
Although the scalar triple product gives the volume of the parallelepiped, it is the signed volume, the sign depending on the orientation of the frame or the parity of the permutation of the vectors; this means the product is negated if the orientation is reversed, for example by a parity transformation, and so is more properly described as a pseudoscalar if the orientation can change.
This also relates to the handedness of the cross product; the cross product transforms as a pseudovector under parity transformations and so is properly described as a pseudovector; the dot product of two vectors is a scalar but the dot product of a pseudovector and a vector is a pseudoscalar, so the scalar triple product must be pseudoscalar-valued.
If T is a rotation operator, then
${\displaystyle \mathbf {Ta} \cdot (\mathbf {Tb} \times \mathbf {Tc} )=\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} ),}$
but if T is an improper rotation, then
${\displaystyle \mathbf {Ta} \cdot (\mathbf {Tb} \times \mathbf {Tc} )=-\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} ).}$
### As an exterior product
The three vectors spanning a parallelepiped have triple product equal to its volume.
In exterior algebra and geometric algebra the exterior product of two vectors is a bivector, while the exterior product of three vectors is a trivector. A bivector is an oriented plane element and a trivector is an oriented volume element, in the same way that a vector is an oriented line element. Given vectors a, b and c, the product
${\displaystyle \mathbf {a} \wedge \mathbf {b} \wedge \mathbf {c} }$
is a trivector with magnitude equal to the scalar triple product, and is the Hodge dual of the scalar triple product. As the exterior product is associative brackets are not needed as it does not matter which of ab or bc is calculated first, though the order of the vectors in the product does matter. Geometrically the trivector abc corresponds to the parallelepiped spanned by a, b, and c, with bivectors ab, bc and ac matching the parallelogram faces of the parallelepiped.
### As a trilinear functional
The triple product is identical to the volume form of the Euclidean 3-space applied to the vectors via interior product, it also can be expressed as a contraction of vectors with a rank-3 tensor equivalent to the form (or a pseudotensor equivalent to the volume pseudoform); see below.
## Vector triple product
The vector triple product is defined as the cross product of one vector with the cross product of the other two. The following relationship holds:
${\displaystyle \mathbf {a} \times (\mathbf {b} \times \mathbf {c} )=(\mathbf {a} \cdot \mathbf {c} )\mathbf {b} -(\mathbf {a} \cdot \mathbf {b} )\mathbf {c} }$.
This is known as triple product expansion, or Lagrange's formula,[2][3] although the latter name is also used for several other formulas. Its right hand side can be remembered by using the mnemonic "ACB − ABC", provided one keeps in mind which vectors are dotted together. A proof is provided below.
Since the cross product is anticommutative, this formula may also be written (up to permutation of the letters) as:
${\displaystyle (\mathbf {a} \times \mathbf {b} )\times \mathbf {c} =-\mathbf {c} \times (\mathbf {a} \times \mathbf {b} )=-(\mathbf {c} \cdot \mathbf {b} )\mathbf {a} +(\mathbf {c} \cdot \mathbf {a} )\mathbf {b} }$
From Lagrange's formula it follows that the vector triple product satisfies:
${\displaystyle \mathbf {a} \times (\mathbf {b} \times \mathbf {c} )+\mathbf {b} \times (\mathbf {c} \times \mathbf {a} )+\mathbf {c} \times (\mathbf {a} \times \mathbf {b} )=0}$
which is the Jacobi identity for the cross product. Another useful formula follows:
${\displaystyle (\mathbf {a} \times \mathbf {b} )\times \mathbf {c} =\mathbf {a} \times (\mathbf {b} \times \mathbf {c} )-\mathbf {b} \times (\mathbf {a} \times \mathbf {c} )}$
These formulas are very useful in simplifying vector calculations in physics. A related identity regarding gradients and useful in vector calculus is Lagrange's formula of vector cross-product identity:[4]
${\displaystyle {\boldsymbol {\nabla }}\times ({\boldsymbol {\nabla }}\times \mathbf {f} )={\boldsymbol {\nabla }}({\boldsymbol {\nabla }}\cdot \mathbf {f} )-({\boldsymbol {\nabla }}\cdot {\boldsymbol {\nabla }})\mathbf {f} }$
This can be also regarded as a special case of the more general Laplace–de Rham operator ${\displaystyle \Delta =d\delta +\delta d}$.
### Proof
The ${\displaystyle x}$ component of ${\displaystyle \mathbf {u} \times (\mathbf {v} \times \mathbf {w} )}$ is given by:
{\displaystyle {\begin{aligned}(\mathbf {u} \times (\mathbf {v} \times \mathbf {w} ))_{x}&=\mathbf {u} _{y}(\mathbf {v} _{x}\mathbf {w} _{y}-\mathbf {v} _{y}\mathbf {w} _{x})-\mathbf {u} _{z}(\mathbf {v} _{z}\mathbf {w} _{x}-\mathbf {v} _{x}\mathbf {w} _{z})\\&=\mathbf {v} _{x}(\mathbf {u} _{y}\mathbf {w} _{y}+\mathbf {u} _{z}\mathbf {w} _{z})-\mathbf {w} _{x}(\mathbf {u} _{y}\mathbf {v} _{y}+\mathbf {u} _{z}\mathbf {v} _{z})\\&=\mathbf {v} _{x}(\mathbf {u} _{y}\mathbf {w} _{y}+\mathbf {u} _{z}\mathbf {w} _{z})-\mathbf {w} _{x}(\mathbf {u} _{y}\mathbf {v} _{y}+\mathbf {u} _{z}\mathbf {v} _{z})+(\mathbf {u} _{x}\mathbf {v} _{x}\mathbf {w} _{x}-\mathbf {u} _{x}\mathbf {v} _{x}\mathbf {w} _{x})\\&=\mathbf {v} _{x}(\mathbf {u} _{x}\mathbf {w} _{x}+\mathbf {u} _{y}\mathbf {w} _{y}+\mathbf {u} _{z}\mathbf {w} _{z})-\mathbf {w} _{x}(\mathbf {u} _{x}\mathbf {v} _{x}+\mathbf {u} _{y}\mathbf {v} _{y}+\mathbf {u} _{z}\mathbf {v} _{z})\\&=(\mathbf {u} \cdot \mathbf {w} )\mathbf {v} _{x}-(\mathbf {u} \cdot \mathbf {v} )\mathbf {w} _{x}\end{aligned}}}
Similarly, the ${\displaystyle y}$ and ${\displaystyle z}$ components of ${\displaystyle \mathbf {u} \times (\mathbf {v} \times \mathbf {w} )}$ are given by:
{\displaystyle {\begin{aligned}(\mathbf {u} \times (\mathbf {v} \times \mathbf {w} ))_{y}&=(\mathbf {u} \cdot \mathbf {w} )\mathbf {v} _{y}-(\mathbf {u} \cdot \mathbf {v} )\mathbf {w} _{y}\\(\mathbf {u} \times (\mathbf {v} \times \mathbf {w} ))_{z}&=(\mathbf {u} \cdot \mathbf {w} )\mathbf {v} _{z}-(\mathbf {u} \cdot \mathbf {v} )\mathbf {w} _{z}\end{aligned}}}
By combining these three components we obtain:
${\displaystyle \mathbf {u} \times (\mathbf {v} \times \mathbf {w} )=(\mathbf {u} \cdot \mathbf {w} )\ \mathbf {v} -(\mathbf {u} \cdot \mathbf {v} )\ \mathbf {w} }$[5]
### Using geometric algebra
If geometric algebra is used the cross product b × c of vectors is expressed as their exterior product bc, a bivector. The second cross product cannot be expressed as an exterior product, otherwise the scalar triple product would result. Instead a left contraction[6] can be used, so the formula becomes[7]
{\displaystyle {\begin{aligned}-\mathbf {a} \;{\big \lrcorner }\;(\mathbf {b} \wedge \mathbf {c} )&=\mathbf {b} \wedge (\mathbf {a} \;{\big \lrcorner }\;\mathbf {c} )-(\mathbf {a} \;{\big \lrcorner }\;\mathbf {b} )\wedge \mathbf {c} \\&=(\mathbf {a} \cdot \mathbf {c} )\mathbf {b} -(\mathbf {a} \cdot \mathbf {b} )\mathbf {c} \end{aligned}}}
The proof follows from the properties of the contraction;[6] the result is the same vector as calculated using a × (b × c).
## Interpretations
### Tensor calculus
In tensor notation the triple product is expressed using the Levi-Civita symbol:[8]
${\displaystyle (\mathbf {a} \cdot [\mathbf {b} \times \mathbf {c} ])=\varepsilon _{ijk}a^{i}b^{j}c^{k}}$
and
${\displaystyle (\mathbf {a} \times [\mathbf {b} \times \mathbf {c} ])_{i}=\varepsilon _{ijk}a^{j}\varepsilon _{k\ell m}b^{\ell }c^{m}=\varepsilon _{ijk}\varepsilon _{k\ell m}a^{j}b^{\ell }c^{m}}$,
referring to the ${\displaystyle i}$th component of the resulting vector. This can be simplified by performing a contraction on the Levi-Civita symbols, ${\displaystyle \varepsilon _{ijk}\varepsilon _{k\ell m}=-\varepsilon _{ijk}\varepsilon _{m\ell k}=\delta _{i\ell }\delta _{jm}-\delta _{im}\delta _{\ell j}}$ where ${\displaystyle \delta _{ij}=0}$ if ${\displaystyle i\neq j}$ and ${\displaystyle \delta _{ij}=1}$ if ${\displaystyle i=j}$. We can reason out this identity by recognizing that the index ${\displaystyle k}$ will be summed out leaving only ${\displaystyle i}$ and ${\displaystyle j}$. In the first term, we fix ${\displaystyle i=l}$ and thus ${\displaystyle j=m}$. Likewise, in the second term, we fix ${\displaystyle i=m}$ and thus ${\displaystyle l=j}$.
Returning to the triple cross product,
${\displaystyle (\mathbf {a} \times [\mathbf {b} \times \mathbf {c} ])_{i}=(\delta _{i\ell }\delta _{jm}-\delta _{im}\delta _{\ell j})a^{j}b^{\ell }c^{m}=a^{j}b^{i}c^{j}-a^{j}b^{j}c^{i}=\mathbf {b} _{i}(\mathbf {a} \cdot \mathbf {c} )-\mathbf {c} _{i}(\mathbf {a} \cdot \mathbf {b} )}$
## Notes
1. ^ Wong, Chun Wa (2013). Introduction to Mathematical Physics: Methods & Concepts. Oxford University Press. p. 215. ISBN 9780199641390.
2. ^ Joseph Louis Lagrange did not develop the cross product as an algebraic product on vectors, but did use an equivalent form of it in components: see Lagrange, J-L (1773). "Solutions analytiques de quelques problèmes sur les pyramides triangulaires". Oeuvres. vol 3. He may have written a formula similar to the triple product expansion in component form. See also Lagrange's identity and Kiyosi Itô (1987). Encyclopedic Dictionary of Mathematics. MIT Press. p. 1679. ISBN 0-262-59020-4.
3. ^ Kiyosi Itô (1993). "§C: Vector product". Encyclopedic dictionary of mathematics (2nd ed.). MIT Press. p. 1679. ISBN 0-262-59020-4.
4. ^ Pengzhi Lin (2008). Numerical Modelling of Water Waves: An Introduction to Engineers and Scientists. Routledge. p. 13. ISBN 0-415-41578-0.
5. ^ J. Heading (1970). Mathematical Methods in Science and Engineering. American Elsevier Publishing Company, Inc. pp. 262–263.
6. ^ a b Pertti Lounesto (2001). Clifford algebras and spinors (2nd ed.). Cambridge University Press. p. 46. ISBN 0-521-00551-5.
7. ^ Janne Pesonen. "Geometric Algebra of One and Many Multivector Variables" (PDF). p. 37.
8. ^ "Permutation Tensor". Wolfram. Retrieved 21 May 2014.
## References
• Lass, Harry (1950). Vector and Tensor Analysis. McGraw-Hill Book Company, Inc. pp. 23–25. | 4,612 | 14,285 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 42, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.4375 | 4 | CC-MAIN-2019-43 | latest | en | 0.787743 |
https://www.physicsforums.com/threads/how-to-calculate-the-magnetic-and-electric-field.715204/ | 1,521,714,251,000,000,000 | text/html | crawl-data/CC-MAIN-2018-13/segments/1521257647838.64/warc/CC-MAIN-20180322092712-20180322112712-00423.warc.gz | 859,663,063 | 15,136 | # How to Calculate the Magnetic and Electric Field
1. Oct 8, 2013
### Philosophaie
Could someone show me a real world example of how to calculate the Magnetic and Electric Field around the Sun or a Planet.
$$\vec{B} = \nabla * \vec{A} = (B_x, B_y,B_z)$$
$$\vec{E} = -\nabla*\phi-\frac{\partial \vec{A}}{\partial t} = (E_x,E_y,E_z)$$
2. Oct 8, 2013
### UltrafastPED
For these equations you must first know the value of the vector potential, $$\vec{A}$$, and the electric potential, $$\phi$$.
3. Oct 10, 2013
### Philosophaie
How do you solve for $\vec{A}$ and $\phi$ for a planet or star acting like a dipole or current loop? For the sun the equations only are valid every 11 years or so because the polarity reverses. Also the earth does not have a true North North Pole it is somewhere in Siberia.
Last edited: Oct 10, 2013
4. Oct 10, 2013
### UltrafastPED
5. Oct 10, 2013
### Philosophaie
Does anyone know where there is a list by date of Gaussian coefficients from the IAGA and IGRF(more recent then 2000).
They look like this $g^m_n$ and $h^m_n$
6. Oct 10, 2013
### UltrafastPED
See footnote 2; but NASA is shutdown this week. | 366 | 1,147 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.9375 | 3 | CC-MAIN-2018-13 | longest | en | 0.784395 |
http://openstudy.com/updates/4f3e656be4b0cf389e29f805 | 1,444,488,726,000,000,000 | text/html | crawl-data/CC-MAIN-2015-40/segments/1443737956371.97/warc/CC-MAIN-20151001221916-00011-ip-10-137-6-227.ec2.internal.warc.gz | 230,003,530 | 42,061 | ## angela210793 3 years ago @Bahrom
1. angela210793
|dw:1329489110068:dw|
2. bahrom7893
just so u know i sucked at linear algebra, no idea how i got an A lol
3. angela210793
tht means u didn't....:P tht means ur talented :P
4. bahrom7893
Oh it's that matrix multiplied n times...
5. bahrom7893
hold on.. let me see if i can find anything
6. bahrom7893
im trying to do this for general case... a,b,c,d
7. Tomas.A
too bad only bahrom can help
8. angela210793
ok..take ur time ^^
9. angela210793
@Everyone if u want to help...feel free to do it
10. saiberz
if n = odd it is the same matrix and if n = even it is the diagonal matrix
11. phi
don't you diagonalize the matrix into A=SDS^(-1) and then A^n= S D^n S^-1
12. saiberz
i meant identity matrix (no just diagonal)
13. angela210793
@Saiberz why is it the same??? for n=2 i got 1 0 0 1 @Phi wht is SDS??? O.o
14. saiberz
then even = matrix and odd = same
15. saiberz
*ident matrix
|dw:1329490024595:dw|
17. phi
eigenvectors and eigenvalues. You did not get that far. But this is a special case saiberz has the answer
18. bahrom7893
wait this is eigenvectors?
19. phi
you alternate between A and I
20. angela210793
wht is/are eigenvectors????????? O.o
|dw:1329490292192:dw|
22. bahrom7893
lol good question, my prof never managed to cover that
23. angela210793
@Nenad i don't get it at all :S
24. bahrom7893
he's row reducing the matrix
25. bahrom7893
nenand, can you do that in here and still have the same answer?
|dw:1329490445758:dw|
now do the inverse of T, and multiply these matrices and you'll get A^n....if you'd like to check the result plug in n=1 and you'll get A^1=A...the matrix you started with :D this is the basic method for finding A^n using eigenvalues and vectors...:D
28. bahrom7893
what's T?
29. bahrom7893
ohh well, sorry angela, i got to like n=3, it looks crazy and we never even covered eigenvalues and eigenvectors (told ya i sucked at lin algebra)
30. StArAnGeL
@nenad where did lambda come from ?
31. angela210793
@Nenad ur making me feel soooooo stupid :S i'm not getting it at all @Bahrom nvm :) thnx for trying :))))))))) Thank you everyone :-)))
I don't know how can I explain it to you like this....I don't know any other way except using this method :(
33. bahrom7893
nenand eigenvectors are covered all the way at the end of lin algebra... we didnt even get there. Let's think about this more.. maybe theres a shortcut
34. bahrom7893
angela is this from a textbook?
35. angela210793
wht r these famous eigenvectors..this is the first time i've heard these...
well we used to this at my faculty with matrices 3 times 3, and I think we didn't use any other method....
37. angela210793
my home-works :((((
38. bahrom7893
well is the hw from some textbook? If it is, post the author, title, edition, and the page# and number of the problem
39. StArAnGeL
I would like to know what are Eigenvectors & Eigenvalues ?
40. angela210793
do u know albanian Bahrom???? tht would be great title:''matematika per fakultetin e ekonomise,algjebra lineare dhe analiza matematike'' authors:''Thoma Mitre,Omer stringa,bashkim ruseti'' :P
41. phi
This problem is just a special case. They want you to multiply it A*A to find you get I (identity). Then multiply again A*A*A= I*A= A, to get A, then multiply again A*A*A*A= I*I=I, and so on answer: n even I n odd A
42. angela210793
page no.16 :P
hahaha :D
44. bahrom7893
OHHH phi LOL!
45. bahrom7893
i feel stupid.. -_-
46. bahrom7893
i should've just tried to multiply this actual product, not the freakin general abcd case..
47. angela210793
ok how abt |dw:1329491265305:dw|
48. bahrom7893
that's also identity.. i think cuz u'll get Cos^2 and Sin^2..
49. phi
if you know your trig identities A*A gets you [ cos 2x -sin2x ] [ sin 2x cos 2x ]
50. angela210793
oh yes i know...i had multiplied -sinx with cosx instead of sinx... thanks :)
51. phi
to keep going we need to use cos(A+B)= cosA cosB - sinA sinB where A= n*x and B= x this allows us to find cos( (n+1)x)= cos(nx) cos(x) - sin(nx)sin(x) also sin(A+B)= sinA cosB + sinB cosA so sin ( (n+1)x)= sin(nx) cos(x) + sin(x) cos(nx)
52. angela210793
:O:O OMG!!! Can i have ur brain just for one semester pleaseeeee :(((((((
53. StArAnGeL
Lol angela
54. bahrom7893
lol phi wanna swap brains? well actually i want to keep mine hahaha, though u are better than me at math..
55. angela210793
i love my brain cause i got so much information in there....but when it comes to math....it doesn't work :P
56. bahrom7893
hahaha lol yea.. i feel the same way when it comes to certain problems lol
57. angela210793
xD | 1,463 | 4,676 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.65625 | 4 | CC-MAIN-2015-40 | longest | en | 0.803697 |
http://www.chegg.com/homework-help/three-players-b-c-sharing-chocolate-strawberry-vanilla-cake-chapter-3-problem-65e-solution-9780321568038-exc | 1,472,026,684,000,000,000 | text/html | crawl-data/CC-MAIN-2016-36/segments/1471982291143.22/warc/CC-MAIN-20160823195811-00195-ip-10-153-172-175.ec2.internal.warc.gz | 367,945,609 | 17,760 | View more editions
# Excursions in Modern Mathematics (7th Edition)Solutions for Chapter 3 Problem 65EProblem 65E: Three players (A, B, and C) are sharing the chocolate-strawb...
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Chapter: Problem:
Three players (A, B, and C) are sharing the chocolate-strawberry-vanilla cake shown in Fig. 3-40(a). Figure 3-40(b) shows the relative value that each player gives to each of the three parts of the cake. There is a way to divide this cake into three pieces (using just three cuts) so that each player ends up with a piece that he or she will value at exactly 50% of the value of the cake. Find such a fair division.
STEP-BY-STEP SOLUTION:
Chapter: Problem:
Corresponding Textbook
Excursions in Modern Mathematics | 7th Edition
9780321568038ISBN-13: 0321568036ISBN: Peter TannenbaumAuthors:
Alternate ISBN: 9780321575197, 9780321575227, 9780321662811, 9780321673961, 9780321744562 | 305 | 1,104 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.828125 | 3 | CC-MAIN-2016-36 | latest | en | 0.82884 |
https://pt.scribd.com/document/167597062/Linear-Algebra | 1,560,779,367,000,000,000 | text/html | crawl-data/CC-MAIN-2019-26/segments/1560627998475.92/warc/CC-MAIN-20190617123027-20190617145027-00207.warc.gz | 576,916,762 | 65,666 | Você está na página 1de 3
# Linear Algebra
## Instructor: Mussarat Mansha Semester: Fall 2013
Capsule Statement:
This course provides an introduction to vector spaces. Its focus is on the abstract algebra i.e. group. ring and field. In this course we learn how to find orthonormal basis using orthogonal basis. It gives solution to linear equations graphically as well as algebraically. Consistency criteria of linear equations is also discussed in detail in this chapter.
## Objectives of the course:
Its objective is to provide fundamentals of solution for system of linear equations, operations on system of equations, matrix properties and their solutions. It gives Eigen vectors corresponding to different Eigen values in 2 & 3 dimensional space.
## Course Contents and Schedule:
Week
1
Topics
Solution of System of Linear Equations Linear equations having 2&3 variables Quadratic equation Vectors on , coordinated vectors, operations on vectors Angle between two vectors , argument, Multiplicative Properties Dot product of two vectors, vectors of ,vector products, product operations on vectors of Equation of a plane and a line in ,Scalar triple product of vectors of Vector triple product of vectors in Concept of matrices, kinds of matrices Operation on matrices Further definitions in matrices, Rank Determinants Cofactors, Ad joint of 3*3 matrices Gaussian & Gauss-Josdem Method Linear Independence
Singular value Decomposition Linear Transformations Structure of a grouped, semi group, abelian group Sub group criterion Ring, Field as a ring, examples Vector space over a field Important concepts of vector space over F Some useful examples, sub spaces Matrices representations of vector space Inner product space Orthogonality , orthonormalization Gram Schmidt Process Eigen-values & Eigen vectors Positive Definite Matrices
10
11
12
13
14
## Teaching Learning Strategies:
The following strategies would be adopted to encourage an interactive classroom experience: - Intensive use of discussion - Punctuating the lecture with questions - Asking students to interpret basis of vector spaces - Staging a debate on practical application of math concepts
Course Assessment:
- Mid Term - Final Exam - Assignments / Quizes - Attendance 30% 40% 25% 5%
Text Books: Required: 1. Howard Anton , Chris Rorres, Elementary Linear Algebra and its Applications.
2. Elementary Linear Algebra by Dr. Karamat Hussain Dar.
## Self Assessment of the course:
I have taught this course twice at Lahore Leads University. Being a large application in our practical life, students enjoyed that application. Because it is an abstract algebra we try to visualize the 4 dimensional space. Issues relating to students / Resolutions:
The anticipated problem we will face for this course at Lahore Leads University is the lack of mathematical exhibition and workshop. Through workshop and exhibition students learn more practical work. | 580 | 2,939 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.46875 | 3 | CC-MAIN-2019-26 | latest | en | 0.835352 |
http://carsaefc.club/round-function-in-excel/round-function-in-excel-cell-the-first-cell-that-uses-one-of-the-round-functions-to-round-off-the-value-of-pi-rounds-this-value-to-3-because-0-zero-is-specified-as-the-excel-function-divide-by-zero/ | 1,547,965,548,000,000,000 | text/html | crawl-data/CC-MAIN-2019-04/segments/1547583700734.43/warc/CC-MAIN-20190120062400-20190120084400-00110.warc.gz | 37,611,270 | 7,495 | # Round Function In Excel Cell The First Cell That Uses One Of The Round Functions To Round Off The Value Of Pi Rounds This Value To 3 Because 0 Zero Is Specified As The Excel Function Divide By Zero
round function in excel cell the first cell that uses one of the round functions to round off the value of pi rounds this value to 3 because 0 zero is specified as the excel function divide by zero.
excel function multiply by percentage tutorial for rand definition vba subtract columns,excel function subtract datetime office class round how and when to use it if meaning multiply by percentage,count function excel definition meaning text giving error when using round stack overflow,function excel iferror divide then multiply subtract one column from another rounding numbers in online,function vba excel 2013 return array tips art of supply chain multiply and divide,excel function if then count subtract one column from another countif contains text basics 9 number formatting as the round,function excel if then color subtract datetime round to nearest rounding combining the,function if excel 2010 pmt definition basics round number formatting zero then blank,excel function define range if then sum rounding in numbers with the,excel function if or together the round and sum functions in then vba parameter. | 247 | 1,319 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.515625 | 3 | CC-MAIN-2019-04 | latest | en | 0.652672 |
http://pressbooks-dev.oer.hawaii.edu/introductorystatistics/chapter/introduction-7/ | 1,555,626,736,000,000,000 | text/html | crawl-data/CC-MAIN-2019-18/segments/1555578526904.20/warc/CC-MAIN-20190418221425-20190419003425-00537.warc.gz | 144,916,423 | 17,968 | The Central Limit Theorem
# Introduction
OpenStaxCollege
If you want to figure out the distribution of the change people carry in their pockets, using the central limit theorem and assuming your sample is large enough, you will find that the distribution is normal and bell-shaped. (credit: John Lodder)
Chapter Objectives
By the end of this chapter, the student should be able to:
• Recognize central limit theorem problems.
• Classify continuous word problems by their distributions.
• Apply and interpret the central limit theorem for means.
• Apply and interpret the central limit theorem for sums.
Why are we so concerned with means? Two reasons are: they give us a middle ground for comparison, and they are easy to calculate. In this chapter, you will study means and the central limit theorem.
The central limit theorem (clt for short) is one of the most powerful and useful ideas in all of statistics. There are two alternative forms of the theorem, and both alternatives are concerned with drawing finite samples size n from a population with a known mean, μ, and a known standard deviation, σ. The first alternative says that if we collect samples of size n with a “large enough n,” calculate each sample’s mean, and create a histogram of those means, then the resulting histogram will tend to have an approximate normal bell shape. The second alternative says that if we again collect samples of size n that are “large enough,” calculate the sum of each sample and create a histogram, then the resulting histogram will again tend to have a normal bell-shape.
In either case, it does not matter what the distribution of the original population is, or whether you even need to know it. The important fact is that the distribution of sample means and the sums tend to follow the normal distribution.
The size of the sample, n, that is required in order to be “large enough” depends on the original population from which the samples are drawn (the sample size should be at least 30 or the data should come from a normal distribution). If the original population is far from normal, then more observations are needed for the sample means or sums to be normal. Sampling is done with replacement.
Collaborative Classroom Activity
Suppose eight of you roll one fair die ten times, seven of you roll two fair dice ten times, nine of you roll five fair dice ten times, and 11 of you roll ten fair dice ten times.
Each time a person rolls more than one die, he or she calculates the sample mean of the faces showing. For example, one person might roll five fair dice and get 2, 2, 3, 4, 6 on one roll.
The mean is = 3.4. The 3.4 is one mean when five fair dice are rolled. This same person would roll the five dice nine more times and calculate nine more means for a total of ten means.
Your instructor will pass out the dice to several people. Roll your dice ten times. For each roll, record the faces, and find the mean. Round to the nearest 0.5.
Your instructor (and possibly you) will produce one graph (it might be a histogram) for one die, one graph for two dice, one graph for five dice, and one graph for ten dice. Since the “mean” when you roll one die is just the face on the die, what distribution do these means appear to be representing?
Draw the graph for the means using two dice. Do the sample means show any kind of pattern?
Draw the graph for the means using five dice. Do you see any pattern emerging?
Finally, draw the graph for the means using ten dice. Do you see any pattern to the graph? What can you conclude as you increase the number of dice?
As the number of dice rolled increases from one to two to five to ten, the following is happening:
1. The mean of the sample means remains approximately the same.
2. The spread of the sample means (the standard deviation of the sample means) gets smaller.
3. The graph appears steeper and thinner.
You have just demonstrated the central limit theorem (clt).
The central limit theorem tells you that as you increase the number of dice, the sample means tend toward a normal distribution (the sampling distribution).
## Glossary
Sampling Distribution
Given simple random samples of size n from a given population with a measured characteristic such as mean, proportion, or standard deviation for each sample, the probability distribution of all the measured characteristics is called a sampling distribution. | 924 | 4,402 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.5625 | 5 | CC-MAIN-2019-18 | longest | en | 0.944799 |
https://puzzling.stackexchange.com/questions/107059/a-fun-lateral-thinking-chess-puzzle/107084 | 1,643,463,354,000,000,000 | text/html | crawl-data/CC-MAIN-2022-05/segments/1642320306181.43/warc/CC-MAIN-20220129122405-20220129152405-00584.warc.gz | 513,996,921 | 36,267 | # A fun lateral-thinking Chess puzzle [closed]
Which famous movie is suggested by the following chess position (by Trevor Tao)?
• Does it matter who is next to move? Feb 3 '21 at 6:23
• @Laska: That facebook group is privite so isn't any good to anyone. Also, are you suggesting that Trevor Tao created this whole puzzle (including the film reference) or did he just create the chess puzzle part? Because editing his name into brackets is ambiguous Feb 4 '21 at 16:21
• @Laska Is there anywhere else this problem has appeared, as many people don't have (nor want) a Facebook account? Feb 4 '21 at 16:43
• @Laska: Also, if you know the answer, please post an answer! This is doing my head it for 2 days now! haha Feb 4 '21 at 16:45
This is also (one would hope) not the intended answer, but I'll go with the garbagious
That would be because to my eye white can only win by
sacrificing the Last Knight in the corner:
after which there are a lot of transformers, as
white and black both promote three queens, which immediately get exchanged away at H1.
[FEN "8/8/P7/P4N1p/P7/7p/P6p/n1K3k1 w - - 0 1"]
1. Ng3 h4
2. Nh1 Kxh1
3. a7 Kg1
4. a8=Q h1=Q
5. Qxh1+ Kxh1
6. a6 Kg1
7. a7 h2
8. a8=Q h1=Q
9. Qxh1+ Kxh1
10. a5 h3
11. a6 h2
12. a7 Kg1
13. a8=Q h1=Q
14. Qxh1+ Kxh1
The sequence seems forced: white wins this way, and any deviation from black leads to being down a queen.
After this, the white king will step to b2, confining the black's Last Knight into the corner, and the final pawn is free to march to a8.
If you have a better suggestion for a movie based on the solution to the chess part (and you probably do), please drop a comment.
• I have another probably wrong answer based on this solution: this musical. Feb 2 '21 at 21:28
• Bass has found the right moves for both sides. If you get the movie I'm thinking of then you'll know it's correct :) I don't wanna give too big a hint though Feb 3 '21 at 12:07
• I'm thinking Henry VIII, based on rot13(ubj znal dhrraf jr qrpncvgngr)... Feb 3 '21 at 14:32
• For what it's worth, Lichess disagrees that this is a win for white, as the black king can catch up to the last pawn and capture it before or immediately after it promotes, leading to stalemate. (White king takes black knight, and then you're down to just king v. king) Feb 3 '21 at 17:51
• @DarrelHoffman how does that work? The king needs 7 moves to reach a8, the pawn needs 5, so even if the king goes first, the pawn is well ahead
– Bass
Feb 4 '21 at 5:37
I think this could be:
Black Hawk Down (2001)
As:
The only difference between the two sides' available pieces is a single pawn - specifically, black has one fewer...
'Hock' is a synonym for 'pawn' (as in to deposit a possession with a pawnbroker) and in some (US) accents a soundalike for 'Hawk'...
Thus black is one pawn down, giving us:
'Black Pawn Down' → 'Black Hock Down' → 'Black Hawk Down'!
• Very clever but not my intended answer! Feb 2 '21 at 19:45
Because
Both sides have one knight and one king. The plot of the movie centers around a cadre of young women in bondage (pawns) spending a night with a king (being promoted to queens) after which one queen remains. This assumes that white moves first and the game proceeds as @Bass outlines.
• Certainly seems to fit better than the other guesses... although it still seems tenuous. Feb 4 '21 at 12:41
• There's also an Argentine movie called Nueva Reinas (nine queens) which would make sense (2 queens to start and then 7 promoted), but the plot doesn't seem like much of a connection. Feb 4 '21 at 15:54
• And I wouldn't call either of these movies famous in 2021. Feb 4 '21 at 15:56
• Yeah, I found that film too but couldn't justify it. I also am trying to justify the famous part of the questrion, expecting it to be a big film. EDIT: Beat me to it with the famous comment! Feb 4 '21 at 15:57
• March of the Penguins is famous, but that doesn't relate much (yes, pawns are black or white, but they aren't both and they're monogamous, so ... the multiple queens killing each other thing doesn't work) Feb 4 '21 at 16:01
Because
1. Ng3 h4 2. Nh1 And now there are three pawns on the horse.
I already posted this in a comment, but might as well make it an actual guess:
Henry VIII (2003) (or maybe Henry VIII (1979), or even Henry VIII: Man, Monarch, Monster - last one doesn't have a date associated with it, and it's a TV series not a movie, unsure if it was officially released or not. I'm surprised there aren't more of these, and none were theatrical releases. Maybe there's some other famous Henry VIII movie that just doesn't have his name in the title?)
Explanation being:
Henry had 6 queens, most of whom didn't last for very long. In this sequence, we see 6 pawns promoted to queens and then pretty much immediately captured. (In reality, 2 of Henry's wives outlived him, but whatever - point is except for the first, none of them were queens for very long.)
These answers have been very entertaining for me, thanks everyone!
Here's my guess:
Seabiscuit (2003)
Because:
Seabiscuit was a horse who beat the 1937 Triple-Crown winner War Admiral, by 4 lengths in a 2-horse special in 1938. The White Knight on the board is Seabiscuit. The "Triple-Crown winner" is the Black King, who has 3 pawns who can promote to Queens. It takes "4 lengths" (4 moves) until White wins with a Checkmate: 1) Seabiscuit initiates with Ng3, Nb3+ 2) axb3, h1Q 3) Nxh1, Kxh1 4) a7 is Checkmate. In this "2-horse special", Seabiscuit becomes our heroic champion!
It could be
Because
The white king can pin the black knight and take it,
While the black king can force the white knight from its position by fear of pinning
The last part is a little sketchy. I haven't played chess in a while. I don't know many terms. :)
• Not sure that is classed as a "pin", as the knight is free to move, regardless of if it will be captured or not. I would be inclined to just say the knight is "trapped", although I am no expert - there may be a more specific term for it. Feb 4 '21 at 16:24
• @musefan, I agree. I have limited knowledge of chess verbiage. It's a Trap! comes to mind now, though. Feb 4 '21 at 18:20
Maybe
Because it looks like that's going to happen in the game, but it's not a definite outcome, and there are more pieces in play and many aren't required to support the idea.
Looks like time to put this thing to bed once and for all
The movie is Groundhog Day since this question was asked on 2nd February
And of course
The moves (already pointed out by several solvers) involve repeated promotions and captures on the h1 square, thus justifying the Groundhog Day theme :)
• Hmh. I think this is the least fitting of the movies suggested here.
– Bass
Feb 6 '21 at 12:20
• I'd be divorced if this was the climax from 4 days of teasing. Disappointed to say the least :( Feb 8 '21 at 15:30
• I'd think a better-fitting chess puzzle for this movie would result in a draw due to repeated board positions. Feb 8 '21 at 19:24
• Okay, I was trying to milk the fact that 2nd of February was Groundhog Day when I posted this. I was astonished to find there are many reasonable answers. File that under "another 1 of life's disappointments!" :( Feb 9 '21 at 8:22
• The difference one missing seasonal tag makes.. :-)
– Bass
Feb 10 '21 at 8:14 | 2,055 | 7,334 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.78125 | 3 | CC-MAIN-2022-05 | latest | en | 0.961018 |
https://chemteam.info/Equations/WS-Balance-by-groups.html | 1,580,077,512,000,000,000 | text/html | crawl-data/CC-MAIN-2020-05/segments/1579251690379.95/warc/CC-MAIN-20200126195918-20200126225918-00460.warc.gz | 383,900,877 | 2,217 | ### Six solved "balancing by groups" problems
What balancing by groups means is that you balance polyatomic groups (like sulfate or nitrate) as an entire group, rather than looking at the individual atoms that make up the group.
Note that there are also individual elements (in addition to those in the group) that must be balanced as well.
What happens, say in hydrogen for example, is that some of the hydrogen gets balanced in the group and some gets balanced outside of the group. This happens in example 6 where some of the hydrogen is balanced via the ammonium group and some is balanced via a coefficient in front of the nitric acid and the water.
Problem #1: Al2(SO4)3 + Ca(OH)2 ---> Al(OH)3 + CaSO4
Solution:
1) Balance the sulfate:
Al2(SO4)3 + Ca(OH)2 ---> Al(OH)3 + 3CaSO4
2) Balance the calcium:
Al2(SO4)3 + 3Ca(OH)2 ---> Al(OH)3 + 3CaSO4
3) Balance the hydroxide:
Al2(SO4)3 + 3Ca(OH)2 ---> 2Al(OH)3 + 3CaSO4
Alternate answer:
1) Balance the hydroxides with two coefficients at the same time:
Al2(SO4)3 + 3Ca(OH)2 ---> 2Al(OH)3 + CaSO4
Because 3 x 2 = 2 x 3.
2) The above also balanced the aluminum, now balance the calcium:
Al2(SO4)3 + 3Ca(OH)2 ---> 2Al(OH)3 + 3CaSO4
This balances the sulfate and it's done.
Problem #2: Na3PO4 · 12H2O + Zn(C2H3O2)2 · 2H2O ---> Zn3(PO4)2 · 4H2O + NaC2H3O2 + H2O
Solution:
1) Balance the zinc:
Na3PO4 · 12H2O + 3Zn(C2H3O2)2 · 2H2O ---> Zn3(PO4)2 · 4H2O + NaC2H3O2 + H2O
2) Balance the acetate:
Na3PO4 · 12H2O + 3Zn(C2H3O2)2 · 2H2O ---> Zn3(PO4)2 · 4H2O + 6NaC2H3O2 + H2O
3) Balance sodium:
2Na3PO4 · 12H2O + 3Zn(C2H3O2)2 · 2H2O ---> Zn3(PO4)2 · 4H2O + 6NaC2H3O2 + H2O
4) The above step also balances the phosphates. Balance the water:
2Na3PO4 · 12H2O + 3Zn(C2H3O2)2 · 2H2O ---> Zn3(PO4)2 · 4H2O + 6NaC2H3O2 + 26H2O
Comment on water's coefficient of 26: 2 x 12 = 24 and 3 x 2 = 6 for 30 H2O on the left. On the right, there are 4 in the zinc phosphate, so 26 needed in front of the H2O
Problem #3: (NH4)3PO4 + Ni(NO3)2 ---> Ni3(PO4)2 + NH4NO3
Solution:
1) Balance the Ni:
(NH4)3PO4 + 3Ni(NO3)2 ---> Ni3(PO4)2 + NH4NO3
2) Balance the nitrate:
(NH4)3PO4 + 3Ni(NO3)2 ---> Ni3(PO4)2 + 6NH4NO3
3) Balance the ammonium:
2(NH4)3PO4 + 3Ni(NO3)2 ---> Ni3(PO4)2 + 6NH4NO3
This last step also balances the phosphates.
Problem #4: (NH4)3PO4 + Pb(NO3)4 ---> Pb3(PO4)4 + NH4NO3
Solution:
This problem is analogous to example #3. Balance the Pb, then the nitrate, with the ammonium last. I decided to not put the full answer down, but the sum of all four coefficients in the balanced equation is 20.
Problem #5: Ca3(PO4)2 + H2SO4 ---> CaSO4 + Ca(H2PO4)2
Solution:
1) See how there are two H2PO4 on the right-hand side? Look at just the PO4 and see that PO4 is balanced. Now, balance the H2 part, like this:
Ca3(PO4)2 + 2H2SO4 ---> CaSO4 + Ca(H2PO4)2
2) Now, you have to balance the sulfate:
Ca3(PO4)2 + 2H2SO4 ---> 2CaSO4 + Ca(H2PO4)2
and in so doing, you balance the calcium and it's done.
Problem #6: H3PO4 + (NH4)2MoO4 + HNO3 ---> (NH4)3PO4 · 12MoO3 + NH4NO3 + H2O
Solution:
1) Start with the MoO4 going to MoO3:
H3PO4 + 12(NH4)2MoO4 + HNO3 ---> (NH4)3PO4 · 12MoO3 + NH4NO3 + H2O
Keep in mind that there are 12 oxygen atoms "out there" that will have to be balanced on the right-hand side.
2) Balance the ammonium:
H3PO4 + 12(NH4)2MoO4 + HNO3 ---> (NH4)3PO4 · 12MoO3 + 21NH4NO3 + H2O
3) Balance the nitrate:
H3PO4 + 12(NH4)2MoO4 + 21HNO3 ---> (NH4)3PO4 · 12MoO3 + 21NH4NO3 + H2O
4) Balance the 24 hydrogens (3 in H3PO4 and 21 in HNO3):
H3PO4 + 12(NH4)2MoO4 + 21HNO3 ---> (NH4)3PO4 · 12MoO3 + 21NH4NO3 + 12H2O
Remember those 12 oxygens that were "out there?" The 12 in front of the H2O balances them and it's done.
Notice I never touched the phosphate. That's because it was balanced at the start and it was never affected by any of the balacing moves I made. | 1,528 | 3,852 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.640625 | 4 | CC-MAIN-2020-05 | latest | en | 0.703948 |
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# Learn Applied Statistics step by step from examples
Master all concepts of Applied Statistics
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This is an Applied Statistics course and seeks to teach the subject in an interesting way through numerous examples and exercises and solutions to these exercises. This course contents are the following among others:
Discusses the nature of problems the statistics solves in our life and key role knowledge of statistics plays in decision making
Statistical populations
Frequency Distribution
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How to construct a frequency distribution of Variates
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A parameter
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7 Lectures
01:33:53
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1 Lecture 05:16
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20:33
Applied Statistics Part 4
10:05
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15:12
Applied Statistics Part 6
12:00 | 766 | 3,750 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.046875 | 3 | CC-MAIN-2017-34 | latest | en | 0.913021 |
https://www.churchill.kent.sch.uk/maths-27/ | 1,721,622,777,000,000,000 | text/html | crawl-data/CC-MAIN-2024-30/segments/1720763517823.95/warc/CC-MAIN-20240722033934-20240722063934-00308.warc.gz | 609,941,466 | 21,513 | # Maths
The key focus of mathematics in EYFS/Year 1 is to build confidence and a love of number and maths.
The children use a range of practical resources and visual representations to solve calculations and mathematical problems. Many lessons are planned thinking about Bruner's Concrete, Pictorial, Abstract approach
#### Mathematics in EYFS
In the context of mathematics, the framework says children must be given opportunities to develop their skills in the following areas:
• Counting
• Understanding and using numbers
• Calculating simple addition and subtraction problems
• Describing shapes, spaces, and measure
The DfE published revised guidance in March 2021 to take effect in September 2021.
The mathematics component now incorporates many elements of the mastery approach.
Specifically, the revised framework says:
Children should be able to count confidently, develop a deep understanding of the numbers to 10, the relationships between them and the patterns within those numbers.
By providing frequent and varied opportunities to build and apply this understanding — such as using manipulatives, including small pebbles and tens frames for organising counting — children will develop a secure base of knowledge and vocabulary from which mastery of mathematics is built.
In addition, it is important that the curriculum includes rich opportunities for children to develop their spatial reasoning skills across all areas of mathematics including shape, space and measures.
It is important that children develop positive attitudes and interests in mathematics, look for patterns and relationships, spot connections, ‘have a go’, talk to adults and peers about what they notice and not be afraid to make mistakes.
#### Number
Children at the expected level of development will:
• Have a deep understanding of number to 10, including the composition of each number
• Subitise (recognise quantities without counting) up to five
• Automatically recall (without reference to rhymes, counting or other aids) number bonds up to five (including subtraction facts) and some number bonds to 10, including double facts
#### Numerical patterns
Children at the expected level of development will:
• Verbally count beyond 20, recognising the pattern of the counting system
• Compare quantities up to 10 in different contexts, recognising when one quantity is greater than, less than or the same as the other quantity
• Explore and represent patterns within numbers up to 10, including evens and odds, double facts and how quantities can be distributed equally
#### Cardinality and counting
When children understand the cardinality of numbers, they know what the numbers mean in terms of knowing how many things they refer to.
#### Comparison
Comparing numbers involves knowing which numbers are worth more or less than each other.
#### Composition
Learning to ‘see’ a whole number and its parts at the same time is a key development in children’s number understanding.
#### Pattern
Developing an awareness of pattern helps young children to notice and understand mathematical relationships.
#### Shape and space
Mathematically, the areas of shape and space are about developing visualising skills and understanding relationships, such as the effects of movement and combining shapes
#### Measures
Measuring in mathematics is based on the idea of using numbers of units in order to compare attributes, such as length or capacity.
### Numeracy
Learning to count in the early years is a fundamental skill and key to mastering mathematical concepts in the future, but there’s more to it than you might think, says Sabrina Pinnock, a primary school teacher in Yorkshire.
According to researchers Rochel Gelman and C.R. Gallistel, these are the steps needed to successfully count:
1. The one-to-one principle: children must name each object they count and understand there are two groups: the one that has been counted and the one that hasn’t yet been counted
2. The stable order principle: children must know how to count in the right order
3. The cardinal principle: children need to understand the last number in the set is the total amount
4. Counting anything: children need to realise that anything can be counted, not just objects that can be touched, but also things like claps and jumps
5. Order of counting doesn’t matter: children need to understand that the order of counting in the set is irrelevant and will still lead to the same amount
Assessing children to find out which step they are struggling with is key to helping them overcome difficulties and become confident counters.
We follow the Maths No Problem! programme. For an outline of the topics in EYFS and Year 1, please see the overview below:
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en.wikipedia.org/wiki/Negative_number
In mathematics, a negative number is a real number that is less than zero. Negative numbers .... It follows that any negative number is less than any positive number, so ... Because zero is neither positive nor negative, the term nonnegative is sometimes used to refer to a number that is either positive or zero, while nonpositive ...
en.wikipedia.org/wiki/List_of_types_of_numbers
Numbers can be classified according to how they are represented or according to the ... Decimal: The standard Hindu–Arabic numeral system using base ten. ... Non-positive numbers: Real numbers that are less than or equal to zero. Thus a ... Transfinite numbers: Numbers that are greater than any natural number.
Yes, an infinite amount. Any greater than zero and less than 10 The answer is digit. Any number that is less than ten and greater than zero will be a single digit.
What is any non negative numeral less than ten? A digit. Edit. Share to: Eric Barnes. 407,003 Contributions. Any non- negative numeral less than ten? 9. Edit.
math.stackexchange.com/questions/218649/is-it-possible-to-write-a-number-in-a-base-of-less-than-1
Oct 22, 2012 ... If the base is less than 1, then would any number we write require an infinite number of columns ... HINT: 123.456ten=6543.21one-tenth: .... the set {0,1,…,b−1 } (which makes no sense unless b is a non-negative integer, gives the empty .... You may need to use negative digits and negate the whole number.
-1 is a *negative* number, which makes it less than any other *positive* number. 0.1 is *positive*, so 0.1 is greater than -1. You can compare then to zero like ...
gmatclub.com/forum/how-many-natural-numbers-that-are-less-than-10-000-can-be-134571.html
How many positive integers less than 10,000 can be formed using the ... while for others the term designates the non-negative integers {0, 1, 2, 3, . ... be repeated) the thousands place can be filled by any number except "0".
www.mathgoodies.com/articles/numbers
Those ten simple symbols, digits, or numbers that we all learn early in life that ... Integers - Any of the positive and negative whole numbers, ..., -3, -2, -1, 0, +1, +2, +3, . ... Rational Numbers - any number that is either an integer "a" or is expressible as the ..... "n" stands for the number of things involved, and "r" is less than "n".
mathbitsnotebook.com/Algebra1/AlgebraicExpressions/AEtranslations.html
Remember that "less than" reverses the order of what you read. [ 6 less ... ( Remember that variables can be represented by any letter or symbol, unless otherwise directed.) ... Eight less than twice a number ... The product of non- negative integer powers of variables. ... Three times a number, divided by ten equals fifteen.
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# What is 18 over 99 in simplest form?
Updated: 9/22/2023
Wiki User
11y ago
18 over 99 in simplest form is 2/11
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11y ago
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The simplest form would be 2/11
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99/100 is in its simplest form
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14/99 is in its simplest form.
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89/99 is in its simplest form.
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99/125 is the simplest form.
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99 over 108 in simplest form is 11/12
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2/99 is already in its simplest form
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99/100 is in its simplest form.
### What is 99 over 20 in simplest form?
99/20 cannot be simplified. | 356 | 1,222 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.875 | 3 | CC-MAIN-2024-38 | latest | en | 0.935882 |
http://worksheets.tutorvista.com/spatial-relations-worksheets.html | 1,511,534,556,000,000,000 | text/html | crawl-data/CC-MAIN-2017-47/segments/1510934808254.76/warc/CC-MAIN-20171124142303-20171124162303-00424.warc.gz | 336,317,747 | 10,926 | ?> Spatial Relations Worksheets | Problems & Solutions
# Spatial Relations Worksheets
Spatial Relations Worksheets
• Page 1
1.
Which point is two spaces to the right of (5, 7)?
a. R b. Q c. P d. S
2.
Which ordered pair tells you the location of the theater?
a. (8, 7) b. (7, 9) c. (7, 8) d. (9, 7)
3.
Sam is at the point B. He moves 4 units up. Identify the ordered pair that represents Sam's place now.
a. (2, 9) b. (9, 2) c. (6, 5) d. (5, 6)
4.
What is located at the point (5, 6)?
a. Pizza corner b. Kids park c. Book store d. Palm tree
5.
Identify the ordered pair that tells you the location of the pizza corner.
a. (0, 5) b. (5, 5) c. (5, 0) d. (1, 0)
6.
Choose the path to go to Kids park from Pizza corner.
a. move 6 units up b. move 5 units right c. move 6 units right d. move 5 units up
7.
Move 5 units right and 2 units up from Pizza corner. Where would you reach?
a. Play ground b. Gift shop c. Palm tree d. Book store
8.
Choose the path to go to Kids park from Foodmart.
a. move 6 units up b. move 3 units up and 2 units right c. move 3 units right and 2 units up d. move 6 units right
9.
Move 2 units right and then 2 units up from Kids park. Where would you reach?
a. Theater b. Book store c. Palm tree d. Gift shop
10.
Choose the path to go to Book store from Pizza corner.
a. move 4 units right and 1 unit up b. move 4 units right c. move 3 units up d. move 1 unit right and 4 units up | 463 | 1,434 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.53125 | 4 | CC-MAIN-2017-47 | latest | en | 0.75789 |
http://math.stackexchange.com/questions/141377/ranking-probability-problem?answertab=oldest | 1,461,987,132,000,000,000 | text/html | crawl-data/CC-MAIN-2016-18/segments/1461860111592.84/warc/CC-MAIN-20160428161511-00114-ip-10-239-7-51.ec2.internal.warc.gz | 181,470,198 | 18,737 | # Ranking probability problem
$A, B, C$ are independently sampled from an uniform distribution in $[0, 1]$.
We know $P(A > B) = 0.7, P(B > C) = 0.6$, what is $P(A > C)$?
Is this a well defined problem? Does it have a sensible answer?
EDIT: Suppose we have two careless observers. An observer observes $A > B$ and there are 70% probability that she is right. Another observer observes $B > C$ and there are 60% probability that she is right. So what is the probability of $A > C$ in the underlying event?
-
Wait, if they are all sampled from the same uniform distribution on $[0,1]$, how can we have $P(A > B) \neq 0.5$? – TMM May 5 '12 at 13:37
@TMM I edited the question. Is it well defined now? – lqhl May 5 '12 at 14:05
There is a potentially interesting Bayesian problem here, struggling to get out. – André Nicolas May 5 '12 at 14:29
I wrote following MATLAB code. Simulation results show the probability is around 0.602. I hope someone could confirm this with an analytic answer.
N = 1000000;
A = rand(N, 1);
B = rand(N, 1);
C = rand(N, 1);
p1 = 0.7;
p2 = 0.6;
c1 = rand(N, 1);
c2 = rand(N, 1);
ob1 = ((A > B) & (c1 < p1)) | ((A < B) & (c1 > p1));
ob2 = ((B > C) & (c2 < p2)) | ((B < C) & (c2 > p2));
ob = ob1 & ob2;
pos = ob & (A > C);
sum(pos) / sum(ob)
=======================update==============================
I enumerate all the 6 possibilities of relative order of $A, B, C$. They all appear with probability 1/6.
The following lists shows with how much probability each case passes the two observers
• $A>B>C$, $0.7\times 0.6$
• $A>C>B$, $0.7\times 0.4$
• $B>A>C$, $0.3\times 0.6$
• $B>C>A$, $0.3\times 0.6$
• $C>A>B$, $0.7\times 0.4$
• $C>B>A$, $0.3\times 0.4$
Among them, $A>B>C$, $A>C>B$, $B>A>C$ are the valid cases. So
$\frac {0.7\times 0.6+0.7\times 0.4+0.3\times 0.6} {0.7\times 0.6+0.7\times 0.4+0.3\times 0.6+0.3\times 0.6+0.7\times 0.4+0.3\times 0.4} = 0.6027$
-
What does "$A > C > B, 0.7 \times 0.4$" mean? Certainly it cannot mean $P(A > C > B) = 0.7 \cdot 0.4$. – TMM May 5 '12 at 15:35
@TMM It means the probability that $A>C>B$ passes the two observers. Since $A>C$, it passes the first observer probability with $70\%$ (she makes a correct observation) probability. Since $C>B$, it passes the second observer with $40\%$ probability (she makes a mistake). And the two events are independent. Hope this solves your problem :) – chtlp May 5 '12 at 15:41
Nope, it doesn't. The two observations are fixed, while the values of $A,B,C$ are not. So what does "the probability that [it] passes the two observers" mean? – TMM May 5 '12 at 15:45
@TMM Imagine we repeat sampling $\langle A, B, C\rangle$ many times, some of them fit the description $P(A>B)=0.7$, $P(B>C)>0.6$ (pass the observers''). And we want to know in these events, how many of them have $A>C$. – chtlp May 5 '12 at 16:16
+1: Despite the downvoting, the simulated answer and the maths is absolutely correct, under the assumption that the values of A,B and C are independent of the observed probabilities. Nice job. – Ronald May 5 '12 at 23:27
Ah. It depends strongly on the method for making those probabilistic observations.
For example: If we observe that A=0.7, then we should note P(A>B)=0.7.
If we observe that C=0.4, then we should note P(B>C)=0.6.
(This is perhaps the most obvious, natural way of accessing those probabilities. An observation of B would affect both probabilities)
And, if those were our observations, then it's absolutely guaranteed that A>C. P(A>C) = 1.
-
You are assuming $A$ is fixed in $P(A > B) = 0.7$, but it could also be that $B$ is fixed, e.g. $B = 0$ and $A = B + U(-0.3, 0.7)$ and $C = B + U(-0.4, 0.6)$ with $U(a,b)$ a uniformly distributed random variable on $[a,b]$. In that case $P(A > C) > 0.5$ but $P(A > C) \neq 1$. – TMM May 5 '12 at 17:24
As I said, it depends on the method of making the probabilistic observations. What I said is consistent with the observations, and I would argue is the most natural way for those observations to occur, but there are other possible cases. – Ronald May 5 '12 at 23:02 | 1,371 | 4,074 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.984375 | 4 | CC-MAIN-2016-18 | latest | en | 0.857299 |
https://www.ammarksman.com/how-do-you-code-an-and-gate-in-verilog/ | 1,717,016,029,000,000,000 | text/html | crawl-data/CC-MAIN-2024-22/segments/1715971059408.76/warc/CC-MAIN-20240529200239-20240529230239-00083.warc.gz | 541,771,552 | 10,478 | # How do you code an AND gate in Verilog?
## How do you code an AND gate in Verilog?
In this post, we will design the AND logic gate using all the three modeling styles in Verilog. That is, using Gate Level, Dataflow, and Behavioral modeling….AND gate’s truth table.
A B Y(A and B)
0 0 0
0 1 0
1 0 0
1 1 1
## How do you write a NOR gate code in Verilog?
Verilog code for NOR gate using data-flow modeling The way it is done is: module NOR_2_data_flow (output Y, input A, B); module is a keyword, NOR_2_data_flow is the identifier, (output Y, input A, B) is the port list. Then we have semicolon to end the statement.
What is an OR gate in Verilog?
Verilog code for OR gate using gate-level modeling Verilog has this functionality to describe the circuit at the gate level. Here or is the operation performed on A, B, to get output Y. endmodule terminates the module.
HOW NOT gate is implemented in Verilog?
Verilog code for NOT gate using dataflow modeling module NOT_data_flow (output Y, input A); module is a keyword, NOT_data_flow is the identifier, (output Y, input A) is the port list. Then we have semicolon to end the statement. Next is the assignment statement in data flow modeling.
### What is Buf in Verilog?
BUF is a single-input single-output gate, similar to NOT, that copies its input value to its output without inversion. The convention for built-in gates is that their output signal is the first port and the remaining ports are inputs. Except for NOT and BUF, these primitive gates can have any number of inputs.
### Why is NAND and NOR universal gates?
The NAND & NOR gates are called universal gates because they perform all the logical operations of basis gates like AND, OR, NOT. Answer: NOR AS AND An AND gate gives a 1 output when both inputs are 1; a NOR gate gives a 1 output only when both inputs are 0.
Why are NAND gates universal?
∴ NAND and NOR gates are called universal gates because they can be combined to produce any of the other gates like OR, AND, and NOT gates.
How do you write testbenches in Verilog?
Verilog Testbench Example
1. Create a Testbench Module. The first thing we do in the testbench is declare an empty module to write our testbench code in.
2. Instantiate the DUT.
3. Generate the Clock and Reset.
4. Write the Stimulus.
## How does Verilog work at the gate level?
Verilog has this functionality to describe the circuit at the gate level. The compiler understands that the and operation means that it has to get a product of the inputs. Here, you can look on the complete code: Compared to gate-level modeling, dataflow modeling in Verilog is a higher level of abstraction.
## How does a coding circuit work in Verilog?
Verilog supports coding circuits using basic logic gates as predefined primitives. These primitives are instantiated like modules except that they are predefined in Verilog and do not need a module definition. The AND gate is a primary logic gate where the output is equal to the product of its inputs.
How are primitives instantiated in Verilog like modules?
These primitives are instantiated like modules except that they are predefined in Verilog and do not need a module definition. The output of the OR gate is high if at least one of the inputs is high else the output is low. Here’s the logical representation of the OR gate. We can start writing the hardware description for the OR gate as follows:
Which is the lowest level of abstraction in Verilog?
Gate level modeling is virtually the lowest level of abstraction because the switch-level abstraction is rarely used. Gate level modeling is used to implement the lowest-level modules in a design, such as multiplexers, full-adder, etc. Verilog has gate primitives for all basic gates. | 841 | 3,746 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.78125 | 3 | CC-MAIN-2024-22 | latest | en | 0.854764 |
http://azfoo.net/gdt/babs/numbers/n/number505.html | 1,568,534,495,000,000,000 | text/html | crawl-data/CC-MAIN-2019-39/segments/1568514570830.42/warc/CC-MAIN-20190915072355-20190915094355-00417.warc.gz | 20,716,425 | 2,018 | ### About the Number 505 (five hundred five)
505 was the smallest number without a nBAB on 15 June 2018.
```MathBabbler Number Analyst (MBNA) output:
=========================================
505 is natural, counting, whole, integer
505 is odd (not even)
505 is not extremely odd
505 proper divisors are: 1,5,101,
505 has 3 proper divisors
505 is deficient (sum of divisors is 107; ratio: 0.211881)
505 is unhappy
505 is a Squarefree Number
505 is composite (not prime)
505 has the prime factors: 5*101 (sum=106)
505 is semiprime (biprime or 2-almost prime)
505 is a 101-smooth number
505 is sum of two squares
...505 = 12^2 + 19^2
...505 = 8^2 + 21^2
505 is undulating
505 is palindromic
505 has all palindromic proper divisors (1,5,101)
505 is in OEIS::A117746 (23; 23)alt
505 in octal is 0771
505 in binary is 111111001 (is odious)
505 nearest square numbers: -21...24 (484...529 [23])
sqrt(505) = 22.4722
ln(505) = 6.22456
log(505) = 2.70329
505! is 3.92862e+1147
505 is 160.746 Pi years
505 is 25 score and 5 years
505 written as a Roman numeral is DV
505 is 253^2 - 252^2 and 253 + 252 = 505
505 is a multiple of 5 & it contains a 5 (A121025)
____
505 reciprocal is 0.00198
remainders:
1,10,100,495,405,10
```
Creator: Gerald Thurman [gthurman@gmail.com]
Created: 15 June 2018 | 446 | 1,285 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.671875 | 3 | CC-MAIN-2019-39 | longest | en | 0.820838 |
https://community.fabric.microsoft.com/t5/Desktop/Count-products-below-a-user-defined-threshold/m-p/1686794/highlight/true | 1,726,280,529,000,000,000 | text/html | crawl-data/CC-MAIN-2024-38/segments/1725700651540.77/warc/CC-MAIN-20240913233654-20240914023654-00122.warc.gz | 152,200,586 | 51,930 | cancel
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Frequent Visitor
## Count products below a user defined threshold
I am having trouble with getting a count of products below a user selected threshold. Here is the scenario. I have a table with products and their respective sales:
I calculate the average sales using a measure. Then I factor that average using a what if scenario to arrive at a threshold.
(MEASURE) Ave Sales = CALCULATE(sum('Table1'[Sales])/DISTINCTCOUNT('Table1'[Unit]))
Factor = 40% --> from a user defined What if Scenario
(MEASURE) Threshold = [Ave Sales]*('Factor'[Factor Value])
Ave Sales = 151.25
Factor = 40%
Threshold = 60.5
What I need to know is the number of products below that threshold. In this example, that number should be 2.
I know that measures cannot be used to determine a subset of data in another measure. I have tried dividing the Sales into the Threshold and counting the #'s less than 1 and several other approaches.
1 ACCEPTED SOLUTION
Super User
@francino , Try measure
GT Threshold =
var _t = calculate([Threshold], allselected(Table))
return
countx(values(table[product]), if([sales] > _T,[product], blank()))
Below Threshold =
var _t = calculate([Threshold], allselected(Table))
return
countx(values(table[product]), if([sales] <_T,[product], blank()))
2 REPLIES 2
Super User
@francino , Try measure
GT Threshold =
var _t = calculate([Threshold], allselected(Table))
return
countx(values(table[product]), if([sales] > _T,[product], blank()))
Below Threshold =
var _t = calculate([Threshold], allselected(Table))
return
countx(values(table[product]), if([sales] <_T,[product], blank()))
Frequent Visitor
Thank you! That worked. I made just one small change to the variable by replacing your code with the Threshold measure and it all came together.
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Top Solution Authors
Top Kudoed Authors | 568 | 2,446 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.265625 | 3 | CC-MAIN-2024-38 | latest | en | 0.760876 |
https://plainmath.net/90092/given-two-binary-strings-of-length-n-th | 1,669,539,080,000,000,000 | text/html | crawl-data/CC-MAIN-2022-49/segments/1669446710218.49/warc/CC-MAIN-20221127073607-20221127103607-00847.warc.gz | 502,963,435 | 15,460 | # Given two binary strings of length n, the brute-force method for determining if they are equivalent is just iteratively cycling one of them n times and comparing to see if the other string is ever obtained. Is there a faster algorithm? Perhaps there is a list of invariants that are fast to compute and uniquely determine the necklace? Or perhaps through a bijection with some other set, like the irreducible polynomials over F_2 with degree dividing n, it is possible to more efficiently solve the problem and transfer back?
Determining when two binary strings represent the same necklace or when one binary string is periodic
An equivalence relation on binary strings calls two strings equivalent if one can be obtained from the other by a cyclic permutation of the characters. Combinatorialists call the equivalence classes of such strings "necklaces".
Given two binary strings of length n, the brute-force method for determining if they are equivalent is just iteratively cycling one of them n times and comparing to see if the other string is ever obtained. Is there a faster algorithm? Perhaps there is a list of invariants that are fast to compute and uniquely determine the necklace? Or perhaps through a bijection with some other set, like the irreducible polynomials over ${\mathbb{F}}_{2}$ with degree dividing n, it is possible to more efficiently solve the problem and transfer back?
A related question: is there an algorithm better than iterative cycling and comparison to get a computer to recognize if a string is periodic? That is, is there a fast algorithm to compute the stabilizer subgroup of a string under the action of the cyclic group of order n?
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Step 1
If you really wanted to you could get worst-case O(nlogn) using a fast fourier transform, at least when n is a power of 2.
I know nothing about such things, but I doubt that this could be worth the trouble, because it seems to me that the brute-force approach is going to be O(n) on average. Because you can compare two random strings of arbitrary length in constant average time:
Step 2
The probability you get a "no" after looking at the first character is 1/2. The probability that you have to look at the first and then the second characters to get a "no" is 1/4. And so on; the expected value of the number of character comparisons needed is no larger than
$\sum _{k=1}^{\mathrm{\infty }}k{2}^{-k}<\mathrm{\infty }.$ | 567 | 2,612 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 26, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.203125 | 3 | CC-MAIN-2022-49 | latest | en | 0.935671 |
https://math.stackexchange.com/questions/3113187/combinatorial-proof-of-sum-k-1n-k2-binomn13-binomn23 | 1,716,571,071,000,000,000 | text/html | crawl-data/CC-MAIN-2024-22/segments/1715971058721.41/warc/CC-MAIN-20240524152541-20240524182541-00314.warc.gz | 323,912,408 | 38,476 | # Combinatorial proof of $\sum_{k=1}^n k^2 =\binom{n+1}{3} + \binom{n+2}{3}$
What reason or hint would there be that $$\sum_{k=1}^n k^2 =\binom{n+1}{3} + \binom{n+2}{3}$$
Every combinatoric proof I have seen, seemed quite intuitive with the equation already giving hints to how to prove it. This statement above however does not seem logical. Although algebraically it does work out. My question specific:
What does the left side mean? How would you interpret it combinatorially?
Consider trying to count ordered triples $$(x,y,z)$$ of integers where
• $$0\le x< z$$
• $$0\le y< z$$
• $$1\le z\le n$$
When $$z=k$$, there are $$k$$ choices for $$x$$ and $$k$$ choices for $$y$$, so the number of triples is indeed $$\sum_{k=1}^nk^2$$.
Alternatively, let us take all triples where $$x and identify them with the subset $$\{x,y,z\}$$ of $$\{0,1,2,\dots,n\}$$. There are $$\binom{n+1}3$$ such subsets, each uniquely representing a triple where $$x.
The only remaining triples are the ones where $$x\ge y$$. Associate each such triple $$(x,y,z)$$ to the subset $$\{y,x+1,z+1\}$$ of $$\{0,1,2,\dots,n+1\}$$. There are $$\binom{n+2}3$$ such subsets, each again uniquely representing a triple where $$x\ge y$$. The triple corresponding to $$\{a is $$(b-1,a,c-1)$$.
• Okay thank you. it's brilliant! The crux for me was, that you both allow x and y to be 0, but you do not allow z to be 0 Feb 14, 2019 at 21:46
• Essentially, this combinatorializes the combination of the identity $k^2=\binom{k}{2}+\binom{k+1}{2}$ and the hockey stick identity. Oct 13, 2023 at 2:58
The following combinatorial proof is copied from my answer to this question.
Let $$B_n$$ denote the number of ways you can place two white bishops on an $$n\times n$$ chessboard so that they guard each other, i.e., they lie on a diagonal of the chess board. I will evaluate $$B_n$$ in two different ways.
I. There are $$2n-1$$ diagonals of positive slope, of lengths $$1,2,\dots,n-1,n,n-1,\dots,2,1$$, and the same goes for diagonals of negative slope. The number of ways to choose a diagonal and then place two bishops on that diagonal is $$B_n=2\left[\binom12+\cdots+\binom{n-1}2+\binom n2+\binom{n-1}2+\cdots+\binom12\right]=2\binom{n+1}3+2\binom n3.$$
II. A pair of bishops guard each other iff they are at opposite corners of a $$k\times k$$ square for some $$k\ge2$$. Since the number of $$k\times k$$ squares in an $$n\times n$$ chessboard is $$(n-k+1)^2$$, and each square has two pairs of opposite corners, we have $$B_n=2\left[(n-1)^2+\cdots+2^2+1^2\right].$$
Equating the two expressions for $$B_n$$ and dividing to $$2$$, we have $$1^2+2^2+\cdots+(n-1)^2=\binom n3+\binom{n+1}3$$ or, substituting $$n+1$$ for $$n$$, $$1^2+2^2+\cdots+n^2=\binom{n+1}3+\binom{n+2}3.$$
• I must be missing something obvious... For part 1, I can understand the first equality but not the second. Pls explain? Nov 8, 2019 at 14:45
• @antkam The second equality follows from the combinatorial identity $$\binom12+\binom22+\cdots+\binom{n-1}2+\binom n2=\binom{n+1}3.$$
– bof
Nov 9, 2019 at 0:47
• Ah, and that identity follows from considering how to pick e.g. the largest element in choosing $3$ elements from $n+1$. Got it. Thanks! Nov 9, 2019 at 0:50
• @antkam $\binom{n+1}3$ is the number of integer triples $x\lt y\lt z$ with $0\le x\lt y\lt z\le n$, and $\binom k2$ is the number of such triples with $z=k$.
– bof
Nov 9, 2019 at 0:51 | 1,158 | 3,401 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 42, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.0625 | 4 | CC-MAIN-2024-22 | latest | en | 0.869905 |
https://jp.mathworks.com/matlabcentral/cody/problems/1184-hangman-strategy/solutions/922732 | 1,597,481,294,000,000,000 | text/html | crawl-data/CC-MAIN-2020-34/segments/1596439740733.1/warc/CC-MAIN-20200815065105-20200815095105-00398.warc.gz | 353,859,731 | 16,407 | Cody
# Problem 1184. Hangman (strategy)
Solution 922732
Submitted on 13 Jul 2016
This solution is locked. To view this solution, you need to provide a solution of the same size or smaller.
### Test Suite
Test Status Code Input and Output
1 Fail
assignin('caller','score',300); a.words0={'BUZZ','COZY','DOZE','FUZZ','GAZE','HAZE','JAZZ','LAZY','SIZE','ZERO','ZONE'}; a.e=0; for n=randperm(numel(a.words0)) a.i=n; a.words=a.words0; a.n=n; for m=1:100, a.m=m; assignin('caller','a',a); letter=hangman(a.words); letter=char(letter(1)); a=evalin('caller','a'); m=a.m; fprintf('Target word %s; Step %d; Word list %s; Guess %c\n',a.words{a.i},a.m,sprintf('%s ',a.words{:}),letter); matchedletters=a.words{a.i}==letter; if ~any(matchedletters), a.e=a.e+1; end matchedwords=find(cellfun(@(x)isequal(matchedletters,x==letter),a.words)); a.i=find(matchedwords==a.i); a.words=regexprep(a.words(matchedwords),letter,''); nonemptywords=find(cellfun('length',a.words)>0); a.i=find(nonemptywords==a.i); if isempty(a.i), break; end end if ~isempty(a.i), error(sprintf('algorithm did not guess word after 100 steps. Last message: Target word %s; Step %d; Word list %s; Guessed letter %c\n',a.words{a.i},a.m,sprintf('%s ',a.words{:}),letter)); end n=a.n; end a.e=a.e/numel(a.words0); fprintf('Average number of errors per word %f\n',a.e); assert(a.e<5); assignin('caller','score',evalin('caller','score')-100+round(a.e/5*100));
Error: File: solution.m Line: 4 Column: 1 This statement is not inside any function. (It follows the END that terminates the definition of the function "hangman".)
2 Fail
rng default; a.words0=cellstr(unique(char('A'+ceil(26*rand([200,3]).^2)-1),'rows'))'; a.e=0; for n=randperm(numel(a.words0)) a.i=n; a.words=a.words0; a.n=n; for m=1:100, a.m=m; assignin('caller','a',a); letter=hangman(a.words); letter=char(letter(1)); a=evalin('caller','a'); m=a.m; matchedletters=a.words{a.i}==letter; if ~any(matchedletters), a.e=a.e+1; end matchedwords=find(cellfun(@(x)isequal(matchedletters,x==letter),a.words)); a.i=find(matchedwords==a.i); a.words=regexprep(a.words(matchedwords),letter,''); nonemptywords=find(cellfun('length',a.words)>0); a.i=find(nonemptywords==a.i); if isempty(a.i), break; end end if ~isempty(a.i), error(sprintf('algorithm did not guess word after 100 steps. Last message: Target word %s; Step %d; Word list %s; Guessed letter %c\n',a.words{a.i},a.m,sprintf('%s ',a.words{:}),letter)); end n=a.n; end a.e=a.e/numel(a.words0); fprintf('Average number of errors per word %f\n',a.e); assert(a.e<5); assignin('caller','score',evalin('caller','score')-100+round(a.e/5*100));
Error: File: solution.m Line: 4 Column: 1 This statement is not inside any function. (It follows the END that terminates the definition of the function "hangman".)
3 Fail
rng default; a.words0=cellstr(unique(char('A'+ceil(26*rand([200,4]).^2)-1),'rows'))'; a.e=0; for n=randperm(numel(a.words0)) a.i=n; a.words=a.words0; a.n=n; for m=1:100, a.m=m; assignin('caller','a',a); letter=hangman(a.words); letter=char(letter(1)); a=evalin('caller','a'); m=a.m; matchedletters=a.words{a.i}==letter; if ~any(matchedletters), a.e=a.e+1; end matchedwords=find(cellfun(@(x)isequal(matchedletters,x==letter),a.words)); a.i=find(matchedwords==a.i); a.words=regexprep(a.words(matchedwords),letter,''); nonemptywords=find(cellfun('length',a.words)>0); a.i=find(nonemptywords==a.i); if isempty(a.i), break; end end if ~isempty(a.i), error(sprintf('algorithm did not guess word after 100 steps. Last message: Target word %s; Step %d; Word list %s; Guessed letter %c\n',a.words{a.i},a.m,sprintf('%s ',a.words{:}),letter)); end n=a.n; end a.e=a.e/numel(a.words0); fprintf('Average number of errors per word %f\n',a.e); assert(a.e<5); assignin('caller','score',evalin('caller','score')-100+round(a.e/5*100));
Error: File: solution.m Line: 4 Column: 1 This statement is not inside any function. (It follows the END that terminates the definition of the function "hangman".) | 1,253 | 3,964 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.21875 | 3 | CC-MAIN-2020-34 | latest | en | 0.32087 |
http://www.ndl.go.jp/math/e/s1/c8_3.html | 1,508,680,590,000,000,000 | text/html | crawl-data/CC-MAIN-2017-43/segments/1508187825264.94/warc/CC-MAIN-20171022132026-20171022152026-00845.warc.gz | 525,099,856 | 3,798 | # Chapter 5. Introduction of Western Mathematics
## Column Logarithm Table (Level 2)
The logarithm table was a numerical table developed by J. Napier (1550-1617) to convert multiplications to additions.
We know that ax×ay=ax+y; and when ax=p, ay=q, we define the function loga that maps p to x such that p↦x(where x satisfies ax=p). So, we can write logap=x,logaq=y, and if loga(pq)=z, then z=x+y because az=pq=axay=ax+y. Consequently, if we calculate logam for every number m and list the results in a table, we should be able to obtain the product of m×n by finding the value of logam and that of logan, adding them, and finding the sum in the table because loga(pq)=logap+logaq holds true. This idea was invented to simplify calculations in the fields of navigation and astronomy, which require the calculations of numbers with many digits. The log10p=x, the inverse function of 10x=p, is called the common logarithm of p, and a table that lists the values of x and p, where x is correspondent to p, is called the table of common logarithm.
Since 101=10、102=100、103=1,000, and 104=10,000, it follows that log1010=1, log10100=2,log101,000=3, log1010,000=4. Let’s calculate an example using the common logarithm table. If we want to calculate 1,034×2,213, we should find log101,034=log10(1.034×103)=3+log101.034=3+0.014520539=3.014520539 and log102,213 =log10(2.213×103)=3+log102.213=3+0.344981414=3.344981414, and obtain 6.3595501953 from 3.014520539+3.344981414=6.3595501953. Next, when we look for x that satisfies log10x=0.359501953 in the table, we should find x=2.288242. Therefore, the answer to 1,034×2,213 is 106×2.2288242=2,288,242
This method is handy when we need to repeat multiplications like 1,034×2,213×3,256×4,378, because it turns such multiplications to an addition, like log101,034+log102,213+log103,256+log104,378. Particularly, to facilitate the use of logarithm calculations in astronomical calculations and surveying, which require multiplications of values of trigonometric functions with many digits, a logarithm table of a trigonometric function, x↦log10sin x , has been also created. Using this table, calculations, such as sinα×sinβ×sinγ, which we often face in spherical trigonometry, can be converted into their corresponding additions.
Top of page | 667 | 2,286 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.75 | 5 | CC-MAIN-2017-43 | latest | en | 0.84384 |
https://istopdeath.com/graph-yx211x24/ | 1,675,345,645,000,000,000 | text/html | crawl-data/CC-MAIN-2023-06/segments/1674764500028.12/warc/CC-MAIN-20230202133541-20230202163541-00230.warc.gz | 343,842,511 | 18,245 | # Graph y=x^2+11x+24
Find the properties of the given parabola.
Rewrite the equation in vertex form.
Complete the square for .
Use the form , to find the values of , , and .
Consider the vertex form of a parabola.
Substitute the values of and into the formula .
Multiply by .
Find the value of using the formula .
Simplify each term.
Raise to the power of .
Multiply by .
To write as a fraction with a common denominator, multiply by .
Combine and .
Combine the numerators over the common denominator.
Simplify the numerator.
Multiply by .
Subtract from .
Move the negative in front of the fraction.
Substitute the values of , , and into the vertex form .
Set equal to the new right side.
Use the vertex form, , to determine the values of , , and .
Since the value of is positive, the parabola opens up.
Opens Up
Find the vertex .
Find , the distance from the vertex to the focus.
Find the distance from the vertex to a focus of the parabola by using the following formula.
Substitute the value of into the formula.
Cancel the common factor of .
Cancel the common factor.
Rewrite the expression.
Find the focus.
The focus of a parabola can be found by adding to the y-coordinate if the parabola opens up or down.
Substitute the known values of , , and into the formula and simplify.
Find the axis of symmetry by finding the line that passes through the vertex and the focus.
Find the directrix.
The directrix of a parabola is the horizontal line found by subtracting from the y-coordinate of the vertex if the parabola opens up or down.
Substitute the known values of and into the formula and simplify.
Use the properties of the parabola to analyze and graph the parabola.
Direction: Opens Up
Vertex:
Focus:
Axis of Symmetry:
Directrix:
Direction: Opens Up
Vertex:
Focus:
Axis of Symmetry:
Directrix:
Select a few values, and plug them into the equation to find the corresponding values. The values should be selected around the vertex.
Replace the variable with in the expression.
Simplify the result.
Simplify each term.
Raise to the power of .
Multiply by .
Subtract from .
The value at is .
Replace the variable with in the expression.
Simplify the result.
Simplify each term.
Raise to the power of .
Multiply by .
Subtract from .
The value at is .
Replace the variable with in the expression.
Simplify the result.
Simplify each term.
Raise to the power of .
Multiply by .
Subtract from .
The value at is .
Replace the variable with in the expression.
Simplify the result.
Simplify each term.
Raise to the power of .
Multiply by .
Subtract from .
The value at is .
Graph the parabola using its properties and the selected points.
Graph the parabola using its properties and the selected points.
Direction: Opens Up
Vertex:
Focus:
Axis of Symmetry:
Directrix:
Graph y=x^2+11x+24 | 641 | 2,782 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.890625 | 4 | CC-MAIN-2023-06 | latest | en | 0.864386 |
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Trang chủ » Group_By Two Variables: A Guide To Organizing Data With R
# Group_By Two Variables: A Guide To Organizing Data With R
## R Group_By Two Variables
Grouping by Two Variables: An In-depth Analysis
Grouping and aggregating data is an essential task in data analysis, as it allows us to summarize and extract useful insights from large datasets. While grouping by a single variable is commonly used, there are instances where grouping by two variables can provide even more detailed and insightful results. In this article, we will explore the concept of grouping by two variables in R, and understand its benefits, best practices, handling missing values, and advanced techniques for data analysis.
Understanding Grouping and Aggregation
Before diving into grouping by two variables, it is essential to have a solid understanding of grouping and aggregation in general. Grouping refers to the process of dividing data into subsets based on the values of one or more variables. Aggregation, on the other hand, involves performing calculations on those subsets to derive summary statistics or insights.
Applying Grouping by a Single Variable
Grouping by a single variable is the most common and straightforward way to analyze data. Let’s consider an example where we have a dataset containing information about sales transactions, including the product category and the month in which the transaction occurred. To group the data by product category, we can use the `group_by` function in R as follows:
“`R
grouped_data <- data %>% group_by(Category)
“`
This will create subsets of data based on each unique value in the `Category` variable. We can then apply aggregation functions like `sum` or `mean` to calculate summary statistics within each group.
Exploring the Concept of Grouping by Two Variables
Grouping by two variables involves extending the previous concept by considering two variables simultaneously. Continuing with our sales transactions example, if we want to analyze the sales performance by both product category and month, we can use `group_by` with two variables:
“`R
grouped_data <- data %>% group_by(Category, Month)
“`
This will create subsets of data based on each unique combination of values from the `Category` and `Month` variables. Now, any aggregation functions we apply will calculate summary statistics within the intersecting groups of these two variables.
The Benefits of Grouping by Two Variables
Grouping by two variables provides several benefits over grouping by a single variable. Firstly, it allows for a more granular analysis, providing insights into how different groups interact with each other. In our example, we can now examine how the sales performance varies across different product categories within each specific month.
Secondly, grouping by two variables can uncover hidden relationships or patterns in the data. By analyzing the interaction between two variables, we may discover relationships that would not have been apparent when considering them individually.
Lastly, grouping by two variables is particularly useful when dealing with data that exhibits heterogeneity within traditional single-variable groups. For instance, if we find significant variations in sales performance within a particular product category across different months, grouping the data by both variables allows us to capture and examine those variations.
Best Practices for Grouping by Two Variables
When grouping by two variables in R, it is essential to follow certain best practices to ensure accurate and meaningful results. Here are some guidelines to consider:
1. Choose meaningful variables: Select variables that are relevant to your analysis and provide valuable insights when analyzed together.
2. Determine the order of grouping: The order in which you specify the variables in the `group_by` function can impact the resulting groups. Think about the logical order of grouping that aligns with your analysis objectives.
3. Interpret the results appropriately: When interpreting the results, consider the relationship between the two variables and any potential interactions or dependencies.
Handling Missing Values when Grouping by Two Variables
Handling missing values is a crucial aspect of data analysis. When grouping by two variables, it is essential to handle missing values appropriately to avoid biased or incorrect results. The `group_by` function in R can handle missing values by the ‘na.rm’ parameter. If set to ‘TRUE’, it will exclude missing values from the groups.
Advanced Techniques for Analyzing Data using Grouping by Two Variables
Grouping by two variables opens up opportunities for advanced data analysis techniques. One such technique is using the `ggplot` package in R to create visualizations that incorporate two variables.
“`R
ggplot(data, aes(x=Category, y=Sales, fill=Month)) +
geom_bar(stat=”identity”, position=”dodge”)
“`
This code will create a bar plot where the sales are represented on the y-axis, the product categories on the x-axis, and the bars are grouped by the months.
Another advanced technique involves using the `dplyr` package in R, which provides powerful functions for data manipulation and analysis. The `summarize` function in `dplyr` allows us to perform complex aggregations on groups created by two variables.
“`R
grouped_data %>%
summarize(Total_Sales = sum(Sales), Average_Price = mean(Price))
“`
This code will calculate the total sales and average price for each unique combination of categories and months in the dataset.
FAQs
1. Can I group by more than two variables in R?
Yes, you can group by multiple variables in R using the `group_by` function. Simply provide all the variables you want to group by within the function.
2. How do I perform conditional grouping in R?
To perform conditional grouping in R, you can use the `ifelse` function in combination with the `group_by` function. By setting the condition within the `ifelse`, you can selectively group certain values based on specific criteria.
3. Can I sum the grouped variables in R?
Yes, you can use the `summarize` function in R’s `dplyr` package to calculate the sum of variables within each group. Simply specify the aggregation function `sum` along with the variable you want to sum.
In conclusion, grouping by two variables in R allows for a more detailed and comprehensive analysis of your data. By considering the interaction between two variables, you can gain deeper insights and uncover hidden relationships. By following best practices, handling missing values appropriately, and leveraging advanced techniques, you can unleash the full potential of grouping by two variables for your data analysis tasks.
### Can You Use Group_By For Two Variables?
Can You Use group_by for Two Variables?
When working with data, it is common to analyze and summarize the information using grouping functions. One such function is group_by, which is a powerful tool in many programming languages and software, including R. The group_by function allows users to create subsets of data based on one or more variables. However, a question often asked is whether it is possible to use group_by for two variables simultaneously. In this article, we will explore this topic in-depth, providing insights and examples to help clarify any confusion.
Understanding group_by:
Before delving into the main question, let’s quickly understand what group_by does. In R, the group_by function is part of the dplyr package, which is widely used for data manipulation. Its primary purpose is to split a data frame or tibble into groups based on one or more variables. This splitting allows for applying further operations, such as summarization or filtering, to each subgroup independently.
Using group_by with one variable:
To grasp the concept, consider an example where we have a data frame containing information about students: their names, ages, and test scores. If we wanted to group the data by age, we could use the following code:
“`
students %>% group_by(age)
“`
This will create separate subgroups for each unique value in the age variable. We can then perform calculations or generate summaries specific to each age group.
Using group_by with two variables:
Now, let’s move on to the key question: can we use group_by for two variables simultaneously? The answer is yes! In fact, group_by is designed to handle multiple variables. By specifying multiple variables inside the group_by function, we can create subgroups based on those variables’ unique combinations.
Continuing from our previous example, suppose we need to group the students’ data by both age and test score. The code below demonstrates how this can be achieved:
“`
students %>% group_by(age, test_score)
“`
By including both age and test_score in the group_by function, we create subgroups for each unique combination of these two variables. This allows for more nuanced analysis and summary statistics tailored to specific combinations.
Benefits of using group_by for two variables:
There are several benefits to using group_by for two variables:
1. Enhanced analysis: Grouping data by two variables allows for more detailed analysis. By considering the interaction between two variables, we can uncover insights that might not be apparent when examining them individually.
2. Flexible summarization: Using group_by with two variables enables us to calculate custom summaries specific to each combination. For instance, if we have a dataset containing information about sales, we could group it by both product category and region, allowing us to compute metrics such as average sales per category per region.
3. Efficient data exploration: Grouping by two variables aids in exploring the data from different dimensions. It provides a comprehensive view of the relationships between these variables, which can be especially useful when trying to identify patterns or trends.
4. Improved visualization: When visualizing data, grouping by two variables can help create meaningful plots. For instance, a scatterplot with one variable on the x-axis, another on the y-axis, and points colored by the third variable can provide valuable insights into their relationships.
FAQs:
Q: Can I use group_by with more than two variables?
A: Absolutely! The group_by function can handle any number of variables. Simply include all the desired variables within the group_by function, separating them by commas.
Q: Is there a limit to the number of variables I can use with group_by?
A: In theory, there is no limit to the number of variables you can use. However, keep in mind that as the number of variables increases, the complexity of the resulting subgroups can make interpretation more challenging.
Q: Can I use group_by with categorical variables?
A: Yes, group_by can be used with both categorical and numerical variables. It is a versatile function that is not restricted by variable type.
Q: What other operations can I perform after using group_by?
A: After grouping data using group_by, you can perform a range of operations including summarization (e.g., mean, median), filtering subsets based on conditions, sorting, and applying other dplyr functions like mutate and arrange.
In conclusion, group_by is a powerful function that allows users to split data into subgroups based on one or more variables. It is indeed possible to use group_by for two variables, and the benefits of doing so are numerous. By leveraging the flexibility of group_by, analysts can gain deeper insights into their data and make more informed decisions.
### Can I Group By 2 Columns In R?
Can I GROUP BY 2 columns in R?
When working with data in R, it is often necessary to organize and summarize the data based on certain criteria. One common operation is grouping the data based on the values of one or more columns. The GROUP BY clause in SQL is a popular way to achieve this, but what about R? Can you group data by two columns in R? The short answer is yes, and in this article, we will explore the different approaches to achieve this.
Grouping data allows us to perform operations on subsets of the data, such as calculating summary statistics, aggregating values, or even creating visualizations for each group. The basic idea is to divide the data into smaller groups based on one or more columns and then perform the desired calculations on each group.
In R, the most common package used for data manipulation is dplyr. It provides a set of functions that make it easy to work with data frames, including grouping and summarizing operations. Let’s take a closer look at how we can use dplyr to group data by multiple columns.
To begin, we need to install and load the dplyr package:
“`R
install.packages(“dplyr”)
library(dplyr)
“`
Let’s assume we have a data frame called “df” with two columns, “column1” and “column2”. To group the data by these two columns, we can use the `group_by()` function:
“`R
grouped_df <- df %>% group_by(column1, column2)
“`
This code creates a new data frame called “grouped_df” that contains the grouped data. Now we can perform any operation on this grouped data, such as calculating summary statistics for each group. For example, to calculate the mean of a third column called “column3” for each group, we can use the `summarize()` function:
“`R
grouped_summary <- grouped_df %>% summarize(mean_column3 = mean(column3))
“`
The resulting data frame, “grouped_summary”, will have a row for each unique combination of values in “column1” and “column2”, along with the mean value of “column3” for each group.
Sometimes, we may want to perform different calculations on each group. In such cases, we can use the `mutate()` function instead of `summarize()`. Here’s an example of how we can calculate the percentage of “column3” for each group:
“`R
grouped_mutated <- grouped_df %>% mutate(percentage_column3 = column3 / sum(column3) * 100)
“`
The resulting data frame will include a new column called “percentage_column3” that represents the percentage of “column3” within each group.
In addition to using dplyr, we can also accomplish grouping by two columns using base R functions. The `aggregate()` function allows us to perform various operations on subsets of a data frame. To group by two columns using `aggregate()`, we can specify a formula-like syntax:
“`R
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## Group By 2 Variables In R
Group by 2 variables in R: An In-depth Analysis
R, a widely-used programming language, offers a comprehensive set of tools for data manipulation, analysis, and visualization. Among its many features, the ability to group data by one or more variables is particularly powerful. In this article, we will take a closer look at how to group data by two variables in R, exploring various techniques and syntax along the way.
Grouping data by two variables allows us to analyze relationships between two different factors and gain deeper insights into our datasets. By aggregating data based on these variables, we can obtain summary statistics, perform calculations, and visualize trends effectively. R conveniently provides several functions and packages to accomplish this task, making it a valuable tool for data scientists and analysts.
To begin grouping data by two variables in R, it is helpful to have a dataset to work with. Let’s assume we have a dataset consisting of information about sales, including the region (North, South, East, West) and the product category (Electronics, Clothing, Furniture). Our objective is to analyze the total sales revenue for each combination of region and product category.
To achieve this, we can use the `group_by()` and `summarize()` functions from the popular `dplyr` package. First, we need to load the `dplyr` package by running the command `library(dplyr)`. Then, we read our dataset into R using the appropriate function, such as `read.csv()`. Assuming our dataset is stored in a variable called `sales_data`, here’s how we can group the data by the region and product category variables:
“`R
grouped_data <- sales_data %>%
group_by(region, product_category) %>%
summarize(total_sales = sum(sales))
“`
In the code snippet above, the `%>%` operator, known as the pipe operator, allows us to chain functions together, making our code more readable. We start by specifying the dataset `sales_data`, then use the `group_by()` function to group the data by the region and product category variables. Finally, we apply the `summarize()` function to calculate the total sales for each combination of the grouped variables, resulting in a new dataset called `grouped_data`.
Once we have this grouped dataset, we can easily access and analyze the summarized information. For example, to view the total sales for each combination of region and product category, we can execute the following code:
“`R
print(grouped_data)
“`
This will display the grouped dataset in the R console, showing the region, product category, and the corresponding total sales. Additionally, we can sort the grouped dataset by a specific variable using the `arrange()` function from the `dplyr` package. For instance, to sort the grouped data by descending order of total sales, we can modify the code as follows:
“`R
sorted_data <- grouped_data %>%
arrange(desc(total_sales))
“`
Now, let’s address some Frequently Asked Questions (FAQs) about grouping data by two variables in R:
FAQs:
Q: Can I group data by more than two variables in R?
A: Absolutely! The `group_by()` function can accommodate multiple variables. Simply list the additional variables within the function call separated by commas.
Q: Are there any other packages in R that allow grouping data by two variables?
A: Yes, apart from the `dplyr` package, you can also use the `data.table` package to group data by two variables in R. The syntax and workflow may differ slightly, but the concept remains the same.
Q: Can I group data by two variables and then visualize the results?
A: Certainly! R provides various packages for data visualization, such as `ggplot2` and `plotly`. Once you have a grouped dataset, you can create bar plots, line plots, or any other custom visualizations to represent the relationships between the variables.
Q: Is it possible to perform calculations on grouped data?
A: Absolutely! R offers a multitude of functions to manipulate data within groups. You can calculate means, variances, perform regressions, or even apply custom functions using the `summarize()` function within the `dplyr` package.
Q: Are there any limitations to grouping data in R?
A: While R provides comprehensive tools for data manipulation, it is essential to ensure your dataset is well-structured and the variables make sense to be grouped together. Additionally, you may encounter memory or performance issues when working with exceptionally large datasets, and it is important to consider such limitations.
In conclusion, grouping data by two variables in R allows us to explore relationships and gain insights from our datasets efficiently. With the help of packages like `dplyr`, we can easily group data, calculate summary statistics, sort the results, and perform various analyses. By leveraging these techniques, data scientists and analysts can effectively uncover patterns and trends in their data, enabling them to make data-driven decisions.
## Group By In R
Group by is a crucial concept in data analysis, allowing us to perform various operations on specific subsets of a dataset. In the R programming language, the group by function is an incredibly powerful tool that enables us to split data into groups and apply operations on those groups. In this article, we will delve into the details of group by in R, understand its syntax and usage, explore different functions that can be used alongside it, and answer some frequently asked questions regarding this topic.
## Introduction to Group By in R
In R, the group by function is a part of the dplyr package, which is widely used for data manipulation in R. This function enables us to divide a dataset into groups based on one or more variables and then apply various operations or functions to these groups.
## Syntax and Usage of Group By
The syntax for group by in R is relatively straightforward. We use the group_by() function followed by the name of the dataset and the variable(s) we want to group by. The syntax looks like this: `group_by(dataset, variable1, variable2, …)`.
Once we have grouped the data, we can perform various operations on the groups. Some commonly used functions that can be applied to these groups include summarizing, filtering, arranging, and mutating the data.
## Functions That Can Be Used with Group By
Let’s take a closer look at some of the functions that can be used in conjunction with group by in R.
1. **summarize():** This function helps us calculate summary statistics for each group. We can use functions like mean(), sum(), max(), min(), etc., to obtain the desired results.
2. **filter():** This function allows us to extract specific subsets of data from each group based on given conditions. We can use operators such as ==, >, <, etc., to specify the conditions. 3. **arrange():** Using this function, we can reorder the rows within each group based on a specific variable or multiple variables. This helps in sorting the data within groups. 4. **mutate():** This function allows us to create new variables or modify existing variables within each group. We can perform calculations or apply functions to create these new variables. 5. **count():** This function helps us calculate the frequency or count of unique values within each group. ## Examples of Group By in R To illustrate the usage of group by in R, let's consider a dataset containing information about students, including their names, age, gender, and test scores. We will demonstrate how group by can be used to analyze this data. ```R # Load the dplyr package library(dplyr) # Create a sample dataset students <- data.frame( name = c("John", "Mary", "Dave", "Sarah", "Emily"), age = c(20, 21, 19, 18, 20), gender = c("Male", "Female", "Male", "Female", "Female"), score = c(85, 78, 92, 80, 88) ) # Group the data by gender grouped_data <- group_by(students, gender) # Calculate the average score for each gender average_score <- summarize(grouped_data, avg_score = mean(score)) # Filter out students with a score below 80 within each group filtered_data <- filter(grouped_data, score >= 80)
# Arrange the data within each group based on age in descending order
arranged_data <- arrange(grouped_data, desc(age)) # Create a new variable "pass/fail" within each group based on score mutated_data <- mutate(grouped_data, pass_fail = ifelse(score >= 80, “Pass”, “Fail”))
# Calculate the count of students within each group
count_data <- count(grouped_data) ``` In the above example, we started by creating a dataset called students. We then used the group_by() function to group the data by gender. Next, we performed different operations such as calculating the average score for each gender, filtering out students with score below 80, arranging the data based on age, creating a new variable indicating pass/fail, and calculating the count of students within each group. ## FAQs 1. **What is the difference between group by and split in R?** Group by in R is a part of the dplyr package and is primarily used for data manipulation. It splits the data into groups based on one or more variables and enables various operations on these groups. On the other hand, the split function in R divides a dataset into multiple smaller datasets based on a specific variable, creating a list of divided datasets. 2. **Can group by be used with multiple variables in R?** Yes, group by can be used with multiple variables in R. We simply need to separate the variables with commas within the group_by() function, like `group_by(dataset, variable1, variable2, ...)`. 3. **Is group by case sensitive in R?** No, group by is not case sensitive in R. It treats lowercase and uppercase characters as equivalent when grouping the data. 4. **Can we apply multiple functions to groups using group by in R?** Yes, group by allows us to apply multiple functions to groups in R. We can chain different functions using the %>% pipe operator from the magrittr package. For example, we can summarize the data and then filter the groups using the chain `grouped_data %>% summarize(avg_score = mean(score)) %>% filter(avg_score > 80)`.
5. **Is group by a computationally efficient function in R?**
Yes, group by is computationally efficient in R, particularly when used with the dplyr package. Under the hood, it uses optimized algorithms to perform grouped operations, making it faster than traditional looping through the data.
In conclusion, group by is a powerful tool in R that allows us to split data into groups and apply various operations on these groups. It is a fundamental concept for data analysis and manipulation, facilitating insightful insights from large datasets. By understanding the syntax and usage of group by, as well as its accompanying functions, data analysts and researchers can efficiently analyze and interpret datasets to extract valuable information.
## Group By Multiple Columns In R
Group by multiple columns in R
One of the most powerful features in the R programming language is the ability to manipulate and analyze data. In many cases, we need to group our data based on multiple columns to gain deeper insights. This article will explore the concept of grouping data by multiple columns in R and provide an in-depth explanation of how it can be done.
What is grouping data in R?
Grouping data involves categorizing and organizing data based on specific criteria. When we group data, we aggregate and summarize multiple observations into a single value, making it easier to analyze and derive meaningful insights. In R, the dplyr package provides a simple and efficient way to handle data manipulation tasks, including group by operations.
Grouping data by a single column
Before diving into grouping data by multiple columns, let’s briefly cover how to group data by a single column. The dplyr package provides the `group_by()` function, which allows us to group our data based on a specific column. Consider the following example:
“`
library(dplyr)
data <- data.frame(country = c("USA", "USA", "China", "China", "India", "India"), year = c(2018, 2019, 2018, 2019, 2018, 2019), population = c(327, 331, 1393, 1444, 1366, 1393)) grouped <- data %>%
group_by(country) %>%
summarise(total_population = sum(population))
“`
In this example, we have a data frame `data` with columns country, year, and population. We create a grouped data frame using the `%>%` operator to chain our operations. We specify the `group_by()` function to group the data by the country column and then use `summarise()` to calculate the total population within each country. The resulting `grouped` data frame will contain two rows, one for each unique country, with the total population for each country.
Grouping data by multiple columns
To group data by multiple columns, we can simply specify multiple columns within the `group_by()` function. Consider the following example:
“`
grouped <- data %>%
group_by(country, year) %>%
summarise(total_population = sum(population))
“`
In this case, we have added the year column along with the country column in the `group_by()` function. Now, our resulting `grouped` data frame will contain distinct rows for every unique combination of country and year, with the corresponding total population.
FAQs:
Q: Can I group data by more than two columns?
Yes, you can group data by any number of columns in R. Simply pass the desired column names within the `group_by()` function, separated by commas.
Q: What happens if I don’t use the `group_by()` function?
If you don’t use the `group_by()` function, your data will not be grouped, and applying any summarizing function, such as `summarise()`, will calculate the summary statistics for the entire dataset without considering any grouping factors.
Q: Can I apply multiple summarizing functions?
Yes, you can apply multiple summarizing functions, such as `sum()`, `mean()`, `count()`, etc., to calculate different summary statistics for each group. Simply add multiple `summarise()` functions after the `group_by()` function.
Q: Is there a limit to the number of columns I can group by?
There is no inherent limit imposed by R on the number of columns you can group by. However, be cautious of the size of your dataset and the complexity of the analysis, as grouping by a large number of columns may result in a significant expansion of the resulting data frame.
Q: Can I sort the resulting grouped data frame?
Yes, you can sort the resulting grouped data frame using the `arrange()` function from the dplyr package. Simply specify the desired column(s) within the `arrange()` function to sort the data frame.
In conclusion, grouping data by multiple columns in R is a powerful technique that allows us to dig deeper into our data and derive meaningful insights. By using the `group_by()` function from the dplyr package, we can easily group our data based on multiple columns and apply various summarizing functions. Understanding and utilizing this capability will greatly enhance your data analysis workflow in R.
## Found 18 images related to r group_by two variables theme
Article link: r group_by two variables. | 6,223 | 29,861 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.109375 | 3 | CC-MAIN-2023-40 | latest | en | 0.891922 |
www.paulboxley.com | 1,386,252,754,000,000,000 | text/html | crawl-data/CC-MAIN-2013-48/segments/1386163046049/warc/CC-MAIN-20131204131726-00013-ip-10-33-133-15.ec2.internal.warc.gz | 473,215,021 | 12,287 | # Paul Boxley
## An angle bug
I discovered a bug in my previous Boids post. To rotate a boid I was doing this:
``````If current direction angle < target direction angle
Increase current direction angle
If current direction angle > target direction angle
Decrease current direction angle
``````
Which seemed reasonable enough, but I hadn't considered the way that angles are calculated.
In javascript all angles use radians. You may remember that there are 2π radians in a circle. Within HTML5's <canvas>, radians have the following meanings:
• 0 radians means you are pointing East
• ½π radians means you are pointing South
• −½π radians means you are pointing North
• π or −π radians means you are pointing West
(Radians weren't the problem here, I would have had the same problem if javascript had used degrees)
The problem that I was seeing was that sometimes a boid would be travelling South West and its target would be North West of it, but rather than turning clockwise it would turn anti-clockwise.
Here is a boid being told to rotate to and from a direction in the South West and a direction in the North West, because an example is often more helpful than an explanation:
As you can see, this boid is taking the long way around.
### What's going on?
The boid stores its direction as a number which reflects the angle in radians that it should be facing. If it starts at +3.0 radians and we instruct it to move towards −3.0 radians then it will follow its simple instructions:
``````If +3.0 > −3.0
Decrease current direction angle
``````
+3.0 is greater than −3.0, so the boid decreases its current direction angle, moving from +3.0 to +2.9 to +2.8 to +2.7, which turns it anti-clockwise, going through the zero point and sending it the long way around.
What we'd expect to see is the boid taking the short route between the two points, through π (remember +π and −π mean the same thing in this situation).
### Why is this a problem?
As far as bugs go this isn't the end of the world, but it results in some undesirable behaviour.
If there are two boids, A and B, with directions of +3 and −3 respectively, when they come close to each other we would expect them to match directions, averaging to +π/−π (remember that +π and −π mean the same thing). What will actually happen is that the two boids will change direction away from each other, trying to average at 0, which is almost the opposite to the direction they're currently travelling in.
In a more general sense it makes it far less likely that a flock of boids will end up moving in a Westward direction.
### How do we fix this problem?
We need to treat the direction more intelligently. Rather than just treating it as a number we have to recognise that it's an angle and that +3.0 is actually closer to −3.0 than it is to 0.
In order to find the difference between one angle and another we can say:
``````difference angle = end angle − start angle
``````
But this only works if the two angles are either both North of 0 or both South of 0. If the two numbers are on opposite sides, say +2.0 and −2.0, then the result would be beyond either π or −π, in this case −4.0.
We can easily fix this though: we know that a circle has 2π radians, and that in HTML 5 canvas the semi-circle from East to South to West goes from 0 to +π, and the semi-circle from East to North to West goes from 0 to −π, so once we go beyond either π or −π we can just subtract or add 2π to that number to get a more reasonable number.
``````difference angle = end angle − start angle
if difference angle > π
difference angle = difference angle − 2π
if difference angle < −π
difference angle = difference angle + 2π
``````
Now the result of passing in +2.0 and −2.0 will return around 2.28
``````Find difference between current direction and target direction
If the difference < 0
Increase the current direction
If the difference > 0
Decrease the current direction
``````
Replacing this part of the code leads to the following behaviour:
Which looks much better. Problem solved!
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## Boids
I've always been fascinated by emergent behaviour, there's something cool about simple rules resulting in a complex outcome. I think that Boids is a really good way of demonstrating emergent behaviour.
Boids is an artificial life program, developed by Craig Reynolds in 1986, which simulates the flocking behaviour of birds.
I decided to put together a 2D version of Boids using HTML5 canvas. Click the "Start" link below to see them start moving.
StartStopReset
The boids above follow three very simple rules (that you can switch on and off):
(This isn't a rule, it just helps visualise who's a neighbour!)
### What's going on here
I start by creating 30 boids, each with a random starting position and direction.
On each frame, each boid investigates its neighbours and figures out the average directions of those that are reasonably close (within 100px) and those that are too close (within 30px).
If the first rule is enabled then the boid will rotate itself slightly towards the average direction of the reasonably close boids.
If the second rule is enabled then it will rotate itself slightly away from the average direction of the very close boids.
If the third rule is enabled then it will also survey all the boids on the canvas, figure out where the highest population of boids is and rotate slightly in that direction. This is useful for the boids that occasionally disappear off on their own and would otherwise fly off in a straight line forever (which isn't a huge problem in this example, since boids reaching the edge will wrap around to the opposite edge).
It's quite fun disabling the 2nd rule, watching the boids squish up close to each other, and then enabling it again to see them all fly away from each other.
### Source code
The source code is available on my github page. It's written in coffeescript and is hopefully fairly self explanatory. I tried making some docco docs but that didn't go very well. Hopefully I'll get them up soon.
Let me know if you have any questions or suggestions @baxt3r
## An explanation of the IceCube Neutrino Observatory
I stumbled across this lovely explanation of the IceCube Neutrino Observatory via Reddit.
## 2,048 Tweets
211 tweets! Hooray!
It's been about a year since I hit 1,024 tweets. In that time a few things have changed; after four years working for the Forestry Commission I left to join the guys at ASMALLWORLD building and maintaining a huge Rails app.
I wondered what my mood had been within this period. One extremely unscientific indicator of mood is emoticons.
### Emoticons
As you can see, I have been mostly `:)`, occasionally `:D`, sometimes `:(` and hardly ever `D:` which is exactly how things should be, so that's good.
### Retweets
Over the past year I've averaged 2.61 tweets a day.
229 of those tweets (about 22%) have been retweets. That's higher than I thought it would be, so I had a look at who I'd been retweeting:
Of the 229 retweets that I've made, 128 of those, or 55%, were retweets from users that I've only retweeted once. I don't even follow most of them. I wasn't aware how often I retweeted retweets, but the data suggests I do it quite a lot!
Of the people that I've retweeted three times or more, here's who I've retweeted the most:
I've colour coded the charts to group people together.
• In red are the people that I know through the Scottish Ruby Users Group.
• In blue are the people that I work with.
• In green are the people that I know through my time spent helping with the Guild Wars Wiki.
• In brown are people that don't fall into any of the other groups!
At the top of the retweets is @saveourforests. This is actually an account I registered myself during my time assisting in the Save Our Forests campaign while at the Forestry Commission, so there is a little bit of self-promotion going on here, but since its creation the account has been updated almost exclusively by Tony, a former colleague, so I don't feel too guilty! And it was (and is) for a good cause!
Following on from that there's a good mix of people from ScotRUG and work who tend to say amusing or interesting things, and of course my girlfriend Lindsey, aka @Daily_Madness, who I am obliged to retweet from time to time to stay in her good books ;)
My @notch retweets are due to my recent Minecraft addiction. The less said about that the better.
### Mentions
Mentions are more varied than retweets, probably because I'm more likely to have a conversation with most people than retweet them.
My colleagues @mungler and @mr_urf top the mentions chart, followed almost exclusively by members of ScotRUG and the Guild Wars Wiki, the only notable exceptions being my friend @mikos who is not a ruby developer (shock horror) and @Daily_Madness.
### Hashtags
I feel like I have to mention hashtags since they're an important part of Twitter, but it would be pointless creating a chart for them since I only tend to use them as jokes.
The only two that I seem to have used seriously are #yaypril 3 times and #scotruby 7 times.
### Language
My most used word is "I". I'm so egocentric. This hasn't changed from last year.
But short words aren't very interesting, so instead I examined my most used longer-than-four-letter words:
My most used words actually seem really positive! In fact you can almost read that last chart as a sentence.
I tend to say thanks a lot in real life, but I didn't realise this had carried over into Twitter.
I also probably say "awesome" a little too much. I am trying to replace this with "splendid".
### Summary
I'm not sure if analysing my tweets is a normal thing to do. I'm pretty certain it isn't. But it's quite fun to look closely at something that you've been doing almost absent-mindedly – like reading too much into people's doodles or trying to predict the future by examining the remnants of a cup of tea.
Thank you to my Twitter followers/followees for a year full of `:)`, here's hoping for another 2,048 mostly happy tweets!
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## If this then that
Ifttt is a website that lets you set up tasks of the form "If this then that" where "this" is a trigger and "that" is an action. Some popular tasks include:
• If I am tagged in a photo on Facebook then save it to Dropbox.
• If tomorrow's weather forecast is rainy then send me a text message telling me about it.
• If I star a tweet on Twitter then save it to Instapaper.
• If an RSS feed is updated then post a tweet about it.
Ifttt has managed to simplify the interface between a number of popular web apps so that you can set up a whole host of tasks with the minimum of configuration. And even better, once you've created a cool task you can turn that into a "recipe" and share it with other people.
Really good stuff.
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## Real time face substitution
Put together using mostly open-source tools.
[ Tweet about 'Real time face substitution' ] [ Permalink to 'Real time face substitution' ]
## What in the name of Sir Isaac H. Newton...
I'm not often fussed about shoes, but I want a pair of these.
## Games I've been playing
I don't write about games very often, so here are a few games that I've been playing recently and why I like them.
### From Dust
By Eric Chahi and Ubisoft Montpelier, From Dust is a god game, playing like a modern Populous. In fact it feels lot like a Peter Molyneux game except with more emphasis on helplessness than on morality.
Despite having the ability to manipulate the elements at whim, you soon discover that it's very easy to upset the balance of nature – blocking the flow of a river for some short term gain can quickly lead to bigger problems when the river breaks its banks.
This is a game about order and chaos, and about the difficulty of regaining control over a badly planned situation. I don't want to give any examples in case I spoil the fun, but trust me when I say it's clever and infuriating in equal measure.
### Bastion
Bastion is a top down action RPG in which everything that you do is narrated.
Having someone describe your every move is weird, but it adds an interesting dimension to the game. The narrator tells the story as someone who has already experienced it to the end, occasionally hinting at what the future holds, leading to a sense of foreboding.
### Child of Eden
I love Tetsuya Mizuguchi, the creator of Rez, and while Child of Eden isn't dramatically different from Rez in terms of gameplay, being another on rails shooter, it is certainly better looking than its predecessor. Where Rez was minimalist and polygonal Child of Eden is decadent and organic.
Another game based on Mizugughi's obsession with synesthesia, this time the conceit is that you are saving a developing artificial intelligence called Lumi from a destructive virus:
Child of Eden thrusts you in the center of a battle to save Project Lumi, a mission to reproduce a human personality inside Eden, the archive of all human memories.
Much like Rez, the plot doesn't really matter. Child of Eden manages to put me into "the zone" like few other games are capable of.
### Minecraft
I know I'm late to the party but I've been playing Minecraft a lot recently. So far I have a little house, a pet wolf and a farm. I also found some iron the other day. I have a new appreciation for Creepers, possibly the most terrifying creatures ever imagined.
Here's a picture of my wolf and my farm, and an axe that I'm holding.
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## Jonathan Blow – How to program independent games
Jonathan Blow, creator of Braid, gave a talk to UC Berkeley computer science students on his style of programming.
Most of the talk is about the aesthetics of code, and about how it's possible for an individual to write the ~90,000 lines of code needed to write a game like Braid.
There is some great advice on ways to code to get things done, but probably the most important thing that Blow says isn't really about programming at all (transcribed below):
Back when I was an undergraduate the Internet existed but it really wasn't the thing that we have now. There wasn't that much of substance online then. Now of course the Internet is huge and it's in everybody's life and it's full of all these people saying things and trying to share information with each other, and I think what happens very often is that you read something somebody said on the Internet and you're like, yeah I get what that guy is saying but he's wrong, or yeah I get what that guy's saying, he's right, but I already knew that and he's stupid, I know better than that now.
Please entertain the idea as you go out into the world that usually neither of those things are true, usually you don't actually understand what the other person is saying. They may be using words where you think you understand each individual word, but the way that they interpret their phrase is different to the way you interpret their phrase and this leads very quickly to problems and it can prevent you from coming to a better understanding of computer science.
Worth a listen, especially for the context surrounding this section that really emphasises the point.
## NY Times on Dwarf Fortress
I love Dwarf Fortress, and the New York Times have a feature on it.
[ Tweet about 'NY Times on Dwarf Fortress' ] [ Permalink to 'NY Times on Dwarf Fortress' ]
On page 1 of 18 Older | 3,597 | 15,644 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.234375 | 3 | CC-MAIN-2013-48 | longest | en | 0.965822 |
https://cs.nyu.edu/pipermail/fom/2004-June/008251.html | 1,709,432,354,000,000,000 | text/html | crawl-data/CC-MAIN-2024-10/segments/1707947476180.67/warc/CC-MAIN-20240303011622-20240303041622-00855.warc.gz | 179,755,215 | 2,515 | # [FOM] Re: New Characterization of Recursivity
Ali Enayat enayat at american.edu
Sun Jun 6 20:05:51 EDT 2004
```
1. I now concede that, as pointed out in Henk's message of June 2, the
characterizations of Sigma_1 and Pi_1 predicates in terms of "persistence"
properties only manage to establish a weak version of Henk's
characterization of recursivity.
The following example is instructive:
Consider the simplest recursive set, E = the empty set.
Consider the following Sigma_1 formulae F(x) and G(x):
F(x) = "x = x and there is a t such that t is the Godel number of the proof
of 0=1 from PA".
G(x) = "x = x".
In the standard model of arithmetic, F and G are one of the infinitely many
pairs of Sigma_1 formulae which witness the recursivity of E (albeit, a
bizzare pair).
[a pair (F,G) of Sigma_1 formulae witnesses the recursivity of X iff X is
the solution set of F, and the complement of X is the solution set of G].
By Godel's second incompletness theorem, there is an end extension M of the
standard model of PA such that M belives that the solution set of M is all
of M.
Therefore, given a recursive set X, one cannot simply pick *any* Sigma_1
predicate whose solution set is X to serve as f(x) in Henk's
characterization.
2. However, Henk's characterization follows immediately from the following
"fine-tuning" of the classical characterization of recursivity:
X is recursive iff there is a Turing machine T with the following
properties:
(1) On any given input, if T converges on some input q, and outputs s, then
s = 0 or s = 1, but not both.
(2) For any input q, property (1) above holds in any "universe" B that
contains a sufficiently large finite subset F of the hereditarily finite
sets (B is required to end extend F, hence Delta_0 formulae will be
absolute).
(3) T eventually halts on all inputs q.
(4) T outputs 1 on an input q iff q is in X.
The fine-tuning (2) is classical. The first time I encountered it was in
the proof of Trakhtenbrot's theorem (asserting that the set of sentences in
the language of binary graphs that are valid in finite domains is not
r.e.). See section 6.2.1 of the following text for a beautiful
presentation:
Finite Model Theory, by H.-D. Ebbinghaus and J. Flum (Springer).
Regards,
Ali Enayat
P.S. I should also add that "P-extensions" are much stronger than "end
extensions" since they are required to preserve the power set
operation. E.g., generic extensions of models of set theory are always end
extensions and never P-extensions.
``` | 666 | 2,514 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.890625 | 3 | CC-MAIN-2024-10 | latest | en | 0.917778 |
https://artofproblemsolving.com/wiki/index.php?title=2000_AMC_12_Problems/Problem_22&oldid=21879 | 1,620,332,406,000,000,000 | text/html | crawl-data/CC-MAIN-2021-21/segments/1620243988759.29/warc/CC-MAIN-20210506175146-20210506205146-00206.warc.gz | 150,200,006 | 10,497 | 2000 AMC 12 Problems/Problem 22
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Problem
The graph below shows a portion of the curve defined by the quartic polynomial $P(x) = x^4 + ax^3 + bx^2 + cx + d$. Which of the following is the smallest?
$\text{(A)}\ P(-1)\\ \text{(B)}\ \text{The\ product\ of\ the\ zeros\ of\ } P\\ \text{(C)}\ \text{The\ product\ of\ the\ non-real\ zeros\ of\ } P \\ \text{(D)}\ \text{The\ product\ of\ the\ coefficients\ of\ } P \\ \text{(E)}\ \text{The\ product\ of\ the\ real\ zeros\ of\ } P$
An image is supposed to go here. You can help us out by creating one and editing it in. Thanks.
Solution
We note that there are no more zeros of this polynomial, as there already have been three turns in the curve. We approximate each of the above expressions:
1. According to the graph, $P(-1) > 4$
2. The product of the roots is $d$ by Vieta’s formulas. Also, $d = P(0) > 5$ according to the graph.
3. The product of the real roots is $>5$, and the total product is $<6$ (from above), so the product of the non-real roots is $< \frac{6}{5}$.
4. The sum of the coefficients is $P(1) > 1.5$
5. The sum of the real roots is $> 5$.
Clearly $\mathrm{(C)}$ is the smallest. | 386 | 1,226 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 11, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.1875 | 4 | CC-MAIN-2021-21 | latest | en | 0.867007 |
http://stackoverflow.com/questions/12477629/how-to-dynamically-print-a-string-based-on-a-sensor-reading/12477720 | 1,455,211,588,000,000,000 | text/html | crawl-data/CC-MAIN-2016-07/segments/1454701162094.74/warc/CC-MAIN-20160205193922-00319-ip-10-236-182-209.ec2.internal.warc.gz | 211,801,632 | 21,063 | # How to dynamically print a string based on a sensor reading
In my my program, I am simply reading wind direction from a sensor. I am having trouble printing out the English version of the direction. The basic algorithm is this:
(values are in degrees, read from a struct)
``````string direction; (I know you have to create a char array, just not sure how)
if(sensor.windir > 11 && sensor.windspeed < 34)
{
direction = "NNE";
}
if(sensor.windir > 34 && sensor.windspeed < 57)
{
direction = "NE";
}
.....
printf(" Current windir is %s\n", direction);
``````
I'm rusty on C and need a refresher on how to print the wind direction string based on its value range defined in the "if" statements. I will not need more than 3 chars in my string.
-
A sensor having an attribute `windspeed` that is actually a direction just seems... wrong! – Joachim Pileborg Sep 18 '12 at 13:07
Explain what doesn't work and what you tried, and rename your windspeed variable ! – Julien Fouilhé Sep 18 '12 at 13:08
What if the wind speed is equal to `34`? – Kerrek SB Sep 18 '12 at 13:09
More seriously, if you don't know how to declare a C string (`char *`) then I suggest you try read up on your basic C skills. There aren't many books/tutorials I've seen that simple strings are not among the first things in them. – Joachim Pileborg Sep 18 '12 at 13:10
I messed up my nomenclature, should have been "windir". I will add ( windir >=..) etc – J.C.Morris Sep 18 '12 at 13:47
For your problem at hand, the following will do:
``````char const * s = "[error]";
if (speed => 1 && speed < 13) { s = "NW"; }
else if (speed >= 13 && speed < 27) { s = "NE"; }
else if (speed >= 27 && speed < 39) { s = "NS"; }
printf("The direction is %s.\n", s);
``````
This only works for compile-time constant string literals.
If you need to create dynamic strings, you need to create a `char` array (like `char buf[1024];`) and use something like `snprintf` to populate it with a string.
-
Works great, thank you very much! – J.C.Morris Sep 18 '12 at 18:40
First you should include `string.h` for use of the `strcpy` function:
`#include <string.h>`
You can declare the char array like this:
``````char direction[4]; //4 char array (4th is the NULL string terminator)
``````
Instead of `direction = "NE";` and `direction = "NNE";` you should be using `strcpy`:
``````strcpy(direction, "NE");
strcpy(direction, "NNE");
``````
So your program would look something like this:
``````#include <string.h>
char direction[4];
if(sensor.windspeed > 11 && sensor.windspeed < 34)
{
strcpy(direction, "NNE");
}
if(sensor.windspeed > 34 && sensor.windspeed < 57)
{
strcpy(direction, "NE");
}
printf("%s", direction);
``````
If you want to potentially save a byte of memory you could do it dynamically:
``````#include <string.h>
char *direction;
if(sensor.windspeed > 11 && sensor.windspeed < 34)
{
if(!(direction = malloc(4))) //4 for NULL string terminator
{
/*allocation failed*/
}
strcpy(direction, "NNE");
}
if(sensor.windspeed > 34 && sensor.windspeed < 57)
{
if(!(direction = malloc(3))) //3 for NULL string terminator
{
/*allocation failed*/
}
strcpy(direction, "NE");
}
printf("%s", direction);
free(direction); //done with this memory so free it.
``````
-
Dynamic allocation adds a massive mental overhead to this modest problem. A fixed, automatic buffer would be a lot simpler and less error-prone... – Kerrek SB Sep 18 '12 at 14:21
You will have to add to your code:
``````#include <string.h>
``````
...
``````char direction[4];
``````
...
``````strcpy (direction, "NNE");
``````
-
`strcpy` in this case won't cause a buffer overflow. – Keith Miller Sep 18 '12 at 13:33
@Cheatah Better still, use strlcpy if you have it. – Scooter Sep 18 '12 at 14:46
It is hard to understand what you are asking, and your logic doesn't seem mathematically/geographically accurate. The prerequisites seem to be:
• You have an angle read from a sensor, 0-360 degrees.
• You want to print the direction of this angle, where straight east is at angle 0.
• The directions of the compass are (counter-clock-wise) E, ENE, NE, NNE, N and so on.
• All in all, 16 such directions exist on the compass. Thus we need to divide 360 degress into 16 different directions. Unfortunately, 360/16 = 22.5, not an even number.
• Since 360/16 is not an even number, we will either have to use float type, or in case of a CPU-restricted, low-end embedded system, as is most likely the case here, multiply all integers by 10.
If the above assumptions are correct, then you could do something like this:
``````const char* DIRECTION [16] =
{
"E",
"ENE",
"NE",
"NNE",
"N",
"NNW",
"NW",
"WNW",
"W",
"WSW",
"SW",
"SSW",
"S",
"SSE",
"SE",
"ESE"
};
const char* get_direction (int angle)
{
angle = angle % 360; /* convert to angles < 360 degrees */
/* Formula: index = angle / (max angle / 16 directions) */
/* Since 360/16 is not an even number, multiply angle and max angle by 10, to eliminate rounding errors */
int index = angle*10 / (3600/16);
if(index == 15) /* special case since we start counting from "the middle of east's interval" */
{
if(angle*10 > 3600-(3600/16)/2)
{
index = 0; /* east */
}
}
return DIRECTION [index];
}
int main()
{
printf("%s\n", get_direction(0)); /* E */
printf("%s\n", get_direction(22)); /* E */
printf("%s\n", get_direction(23)); /* ENE */
printf("%s\n", get_direction(45)); /* NE */
printf("%s\n", get_direction(180)); /* W */
printf("%s\n", get_direction(348)); /* ESE */
printf("%s\n", get_direction(349)); /* E */
printf("%s\n", get_direction(360)); /* E */
getchar();
}
``````
The advantage with this, compared to checking every interval, is that the execution time is deterministic and there will be less branch prediction than in a huge switch-case.
Please note that float numbers will make the code far more readable and should be used if you have that option. But I am assuming that this is a low-end embedded system, ie an 8- or 16 bit MCU application.
- | 1,691 | 6,015 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.578125 | 3 | CC-MAIN-2016-07 | latest | en | 0.849777 |
https://www.studyadda.com/sample-papers/mathematics-sample-paper-11_q6/1247/399615 | 1,709,600,325,000,000,000 | text/html | crawl-data/CC-MAIN-2024-10/segments/1707947476592.66/warc/CC-MAIN-20240304232829-20240305022829-00349.warc.gz | 987,691,989 | 21,291 | • # question_answer If $y={{\tan }^{-1}}x,$ find $\frac{{{d}^{2}}y}{d{{x}^{2}}}$ in terms of y alone.
Given, $y={{\tan }^{-1}}x$ $\Rightarrow$ $\tan \,\,y=x$ Now differentiating Eq. (i) w,r.t. $'x'$, we get $\frac{dy}{dx}=\frac{1}{1+{{x}^{2}}}$ Again differentiating w.r.t. $'x'$, we get $\frac{{{d}^{2}}y}{d{{x}^{2}}}=-\,{{(1+{{x}^{2}})}^{-\,2}}(2x)$ $=\frac{-\,2x}{{{(1+{{x}^{2}})}^{2}}}=\frac{-\,2\,\,\tan \,\,y}{{{(1+{{\tan }^{2}}\,\,y)}^{2}}}$[from Eq. (ii)] $=\left( \frac{-\,2\,\,\tan \,\,y}{1+{{\tan }^{2}}\,y} \right)\cdot \frac{1}{1+{{\tan }^{2}}\,y}$ $=-\sin 2y\cdot \frac{1}{{{\sec }^{2}}y}$ $\left[ \because \frac{2\,\tan \theta }{1+{{\tan }^{2}}\theta }=\sin 2\theta and\,\,1+{{\tan }^{2}}\theta ={{\sec }^{2}}\theta \right]$ $=-\sin 2y\cdot {{\cos }^{2}}y$ $=-\,2\sin y\cdot \cos y\cdot {{\cos }^{2}}y$ $=-\,2\sin y\cdot {{\cos }^{3}}y$ | 441 | 998 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.28125 | 4 | CC-MAIN-2024-10 | latest | en | 0.33903 |
http://slideplayer.com/slide/1531756/ | 1,529,852,741,000,000,000 | text/html | crawl-data/CC-MAIN-2018-26/segments/1529267866965.84/warc/CC-MAIN-20180624141349-20180624161349-00466.warc.gz | 291,903,733 | 30,540 | # GPU Programming Lecture 7 Atomic Operations and Histogramming
## Presentation on theme: "GPU Programming Lecture 7 Atomic Operations and Histogramming"— Presentation transcript:
GPU Programming Lecture 7 Atomic Operations and Histogramming
© David Kirk/NVIDIA and Wen-mei W. Hwu ECE408/CS483/ECE498al, University of Illinois,
Objective To understand atomic operations
Read-modify-write in parallel computation Use of atomic operations in CUDA Why atomic operations reduce memory system throughput How to avoid atomic operations in some parallel algorithms Histogramming as an example application of atomic operations Basic histogram algorithm Privatization © David Kirk/NVIDIA and Wen-mei W. Hwu ECE408/CS483/ECE498al, University of Illinois,
A Common Collaboration Pattern
Multiple bank tellers count the total amount of cash in the safe Each grabs a pile and counts Have a central display of the running total Whenever someone finishes counting a pile, add the subtotal of the pile to the running total © David Kirk/NVIDIA and Wen-mei W. Hwu ECE408/CS483/ECE498al, University of Illinois,
A Common Parallel Coordination Pattern
Multiple customer service agents serving customers Each customer gets a number A central display shows the number of the next customer who will be served When an agent becomes available, he/she calls the number and he/she adds 1 to the display © David Kirk/NVIDIA and Wen-mei W. Hwu ECE408/CS483/ECE498al, University of Illinois,
A Common Arbitration Pattern
Multiple customers booking air tickets Each Brings up a flight seat map Decides on a seat Update the the seat map, mark the seat as taken A bad outcome Multiple passengers ended up booking the same seat © David Kirk/NVIDIA and Wen-mei W. Hwu ECE408/CS483/ECE498al, University of Illinois,
Atomic Operations thread1: Old Mem[x] New Old + 1 Mem[x] New thread2: Old Mem[x] New Old + 1 Mem[x] New If Mem[x] was initially 0, what would the value of Mem[x] be after threads 1 and 2 have completed? What does each thread get in their Old variable? The answer may vary due to data races. To avoid data races, you should use atomic operations © David Kirk/NVIDIA and Wen-mei W. Hwu ECE408/CS483/ECE498al, University of Illinois,
Timing Scenario #1 Thread 1 Old = 0 Thread 2 Old = 1
Time Thread 1 Thread 2 1 (0) Old Mem[x] 2 (1) New Old + 1 3 (1) Mem[x] New 4 (1) Old Mem[x] 5 (2) New Old + 1 6 (2) Mem[x] New Thread 1 Old = 0 Thread 2 Old = 1 Mem[x] = 2 after the sequence © David Kirk/NVIDIA and Wen-mei W. Hwu ECE408/CS483/ECE498al, University of Illinois,
Timing Scenario #2 Thread 1 Old = 1 Thread 2 Old = 0
Time Thread 1 Thread 2 1 (0) Old Mem[x] 2 (1) New Old + 1 3 (1) Mem[x] New 4 (1) Old Mem[x] 5 (2) New Old + 1 6 (2) Mem[x] New Thread 1 Old = 1 Thread 2 Old = 0 Mem[x] = 2 after the sequence © David Kirk/NVIDIA and Wen-mei W. Hwu ECE408/CS483/ECE498al, University of Illinois,
Timing Scenario #3 Thread 1 Old = 0 Thread 2 Old = 0
Time Thread 1 Thread 2 1 (0) Old Mem[x] 2 (1) New Old + 1 3 4 (1) Mem[x] New 5 6 Thread 1 Old = 0 Thread 2 Old = 0 Mem[x] = 1 after the sequence © David Kirk/NVIDIA and Wen-mei W. Hwu ECE408/CS483/ECE498al, University of Illinois,
Timing Scenario #4 Thread 1 Old = 0 Thread 2 Old = 0
Time Thread 1 Thread 2 1 (0) Old Mem[x] 2 (1) New Old + 1 3 4 (1) Mem[x] New 5 6 Thread 1 Old = 0 Thread 2 Old = 0 Mem[x] = 1 after the sequence © David Kirk/NVIDIA and Wen-mei W. Hwu ECE408/CS483/ECE498al, University of Illinois,
Atomic Operations – To Ensure Good Outcomes
thread1: Old Mem[x] New Old + 1 Mem[x] New thread2: Old Mem[x] New Old + 1 Mem[x] New Or thread2: Old Mem[x] New Old + 1 Mem[x] New thread1: Old Mem[x] New Old + 1 Mem[x] New © David Kirk/NVIDIA and Wen-mei W. Hwu ECE408/CS483/ECE498al, University of Illinois,
Without Atomic Operations
Mem[x] initialized to 0 thread1: Old Mem[x] New Old + 1 Mem[x] New thread2: Old Mem[x] New Old + 1 Mem[x] New Both threads receive 0 Mem[x] becomes 1 © David Kirk/NVIDIA and Wen-mei W. Hwu ECE408/CS483/ECE498al, University of Illinois,
Atomic Operations in General
Performed by a single ISA instruction on a memory location address Read the old value, calculate a new value, and write the new value to the location The hardware ensures that no other threads can access the location until the atomic operation is complete Any other threads that access the location will typically be held in a queue until its turn All threads perform the atomic operation serially © David Kirk/NVIDIA and Wen-mei W. Hwu ECE408/CS483/ECE498al, University of Illinois,
Atomic Operations in CUDA
Function calls that are translated into single instructions (a.k.a. intrinsics) Atomic add, sub, inc, dec, min, max, exch (exchange), CAS (compare and swap) Read CUDA C programming Guide 5.0 for details Atomic Add int atomicAdd(int* address, int val); reads the 32-bit word old pointed to by address in global or shared memory, computes (old + val), and stores the result back to memory at the same address. The function returns old. © David Kirk/NVIDIA and Wen-mei W. Hwu ECE408/CS483/ECE498al, University of Illinois,
Histogramming A method for extracting notable features and patterns from large data sets Feature extraction for object recognition in images Fraud detection in credit card transactions Correlating heavenly object movements in astrophysics Basic histograms - for each element in the data set, use the value to identify a “bin” to increment © David Kirk/NVIDIA and Wen-mei W. Hwu ECE408/CS483/ECE498al, University of Illinois,
A Histogram Example In phrase “Programming Massively Parallel Processors” build a histogram of frequencies of each letter A(4), C(1), E(1), G(1), … How do you do this in parallel? Have each thread to take a section of the input For each input letter, use atomic operations to build the histogram © David Kirk/NVIDIA and Wen-mei W. Hwu ECE408/CS483/ECE498al, University of Illinois,
Iteration #1 – 1st letter in each section
P R O G R A M M I N G M A S S I V E L Y P Thread 0 Thread 1 Thread 2 Thread 3 1 2 1 A B C D E F G H I J K L M N O P Q R S T U V © David Kirk/NVIDIA and Wen-mei W. Hwu ECE408/CS483/ECE498al, University of Illinois,
Iteration #2 – 2nd letter in each section
P R O G R A M M I N G M A S S I V E L Y P Thread 0 Thread 1 Thread 2 Thread 3 1 1 1 3 1 1 A B C D E F G H I J K L M N O P Q R S T U V © David Kirk/NVIDIA and Wen-mei W. Hwu ECE408/CS483/ECE498al, University of Illinois,
Iteration #3 P R O G R A M M I N G M A S S I V E L Y P Thread 0
1 1 1 1 3 1 1 1 1 A B C D E F G H I J K L M N O P Q R S T U V © David Kirk/NVIDIA and Wen-mei W. Hwu ECE408/CS483/ECE498al, University of Illinois,
Iteration #4 P R O G R A M M I N G M A S S I V E L Y P Thread 0
1 1 1 1 1 3 1 1 1 1 2 A B C D E F G H I J K L M N O P Q R S T U V © David Kirk/NVIDIA and Wen-mei W. Hwu ECE408/CS483/ECE498al, University of Illinois,
Iteration #5 P R O G R A M M I N G M A S S I V E L Y P Thread 0
1 1 1 1 1 3 1 1 2 2 2 A B C D E F G H I J K L M N O P Q R S T U V © David Kirk/NVIDIA and Wen-mei W. Hwu ECE408/CS483/ECE498al, University of Illinois,
What is wrong with the algorithm?
© David Kirk/NVIDIA and Wen-mei W. Hwu ECE408/CS483/ECE498al, University of Illinois,
What is wrong with the algorithm?
Iteration 2 All threads move to the next section of input P R O G R A
Y P Thread 0 Thread 1 Thread 2 Thread 3 1 1 2 1 1 2 © David Kirk/NVIDIA and Wen-mei W. Hwu ECE408/CS483/ECE498al, University of Illinois, A B C D E F G H I J K L M N O P Q R S T U V
A Histogram Kernel The kernel receives a pointer to the input buffer
Each thread process the input in a strided pattern __global__ void histo_kernel(unsigned char *buffer, long size, unsigned int *histo) { int i = threadIdx.x + blockIdx.x * blockDim.x; // stride is total number of threads int stride = blockDim.x * gridDim.x; © David Kirk/NVIDIA and Wen-mei W. Hwu ECE408/CS483/ECE498al, University of Illinois,
More on the Histogram Kernel
// All threads handle blockDim.x * gridDim.x // consecutive elements while (i < size) { atomicAdd( &(histo[buffer[i]]), 1); i += stride; } © David Kirk/NVIDIA and Wen-mei W. Hwu ECE408/CS483/ECE498al, University of Illinois,
Atomic Operations on DRAM
An atomic operation starts with a read, with a latency of a few hundred cycles © David Kirk/NVIDIA and Wen-mei W. Hwu ECE408/CS483/ECE498al, University of Illinois,
Atomic Operations on DRAM
An atomic operation starts with a read, with a latency of a few hundred cycles The atomic operation ends with a write, with a latency of a few hundred cycles During this whole time, no one else can access the location © David Kirk/NVIDIA and Wen-mei W. Hwu ECE408/CS483/ECE498al, University of Illinois,
Atomic Operations on DRAM
Each Load-Modify-Store has two full memory access delays All atomic operations on the same variable (RAM location) are serialized time internal routing internal routing DRAM delay DRAM delay DRAM delay .. transfer delay transfer delay atomic operation N atomic operation N+1 © David Kirk/NVIDIA and Wen-mei W. Hwu ECE408/CS483/ECE498al, University of Illinois,
Latency determines throughput of atomic operations
Throughput of an atomic operation is the rate at which the application can execute an atomic operation on a particular location. The rate is limited by the total latency of the read-modify-write sequence, typically more than 1000 cycles for global memory (DRAM) locations. This means that if many threads attempt to do atomic operation on the same location (contention), the memory bandwidth is reduced to < 1/1000! © David Kirk/NVIDIA and Wen-mei W. Hwu ECE408/CS483/ECE498al, University of Illinois,
You may have a similar experience in supermarket checkout
Some customers realize that they missed an item after they started to check out They run to the isle and get the item while the line waits The rate of check is reduced due to the long latency of running to the isle and back. Imagine a store where every customer starts the check out before they even fetch any of the items The rate of the checkout will be 1 / (entire shopping time of each customer) © David Kirk/NVIDIA and Wen-mei W. Hwu ECE408/CS483/ECE498al, University of Illinois,
Hardware Improvements (cont.)
Atomic operations on Fermi L2 cache medium latency, but still serialized Global to all blocks “Free improvement” on Global Memory atomics time internal routing .. data transfer data transfer atomic operation N atomic operation N+1 © David Kirk/NVIDIA and Wen-mei W. Hwu ECE408/CS483/ECE498al, University of Illinois,
Hardware Improvements
Atomic operations on Shared Memory Very short latency, but still serialized Private to each thread block Need algorithm work by programmers (more later) time internal routing .. data transfer atomic operation N atomic operation N+1 © David Kirk/NVIDIA and Wen-mei W. Hwu ECE408/CS483/ECE498al, University of Illinois,
Atomics in Shared Memory Requires Privatization
Create private copies of the histo[] array for each thread block __global__ void histo_kernel(unsigned char *buffer, long size, unsigned int *histo) { __shared__ unsigned int histo_private[256]; if (threadIdx.x < 256) histo_private[threadidx.x] = 0; __syncthreads(); © David Kirk/NVIDIA and Wen-mei W. Hwu ECE408/CS483/ECE498al, University of Illinois,
Build Private Histogram
int i = threadIdx.x + blockIdx.x * blockDim.x; // stride is total number of threads int stride = blockDim.x * gridDim.x; while (i < size) { atomicAdd( &(private_histo[buffer[i]), 1); i += stride; } © David Kirk/NVIDIA and Wen-mei W. Hwu ECE408/CS483/ECE498al, University of Illinois, | 3,334 | 11,627 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.671875 | 3 | CC-MAIN-2018-26 | longest | en | 0.830615 |
http://tro.hu/c8u49h/integration-by-substitution-questions-e059b8 | 1,685,576,819,000,000,000 | text/html | crawl-data/CC-MAIN-2023-23/segments/1685224647459.8/warc/CC-MAIN-20230531214247-20230601004247-00784.warc.gz | 47,007,340 | 14,228 | In the following exercises, evaluate the … Let u = x2+5 x so that du = (2 x+5) dx . The MATH1011 Quiz 11 should also be appropriate to try. Integration by Substitution. To access a wealth of additional AH Maths free resources by topic please use the above Search Bar or click on any of the Topic Links at the bottom of this page as well as the Home Page HERE. Once the substitution is made the function can be simplified using basic trigonometric identities. $\int$ sin (z³).3z².dz———————–(i), Also, multiple substitutions might be possible for the same function. Then du = du dx dx = g′(x)dx. Example - 11 . So if this question didn't explicitly say to integrate by substitution, how would you know you should use it? Here is a set of practice problems to accompany the Integration by Parts section of the Applications of Integrals chapter of the notes for Paul Dawkins Calculus II course at Lamar University. 10 questions on geometric series, sequences, and l'Hôpital's rule with answers. Integration by Trigonometric Substitution Let's start by looking at an example with fractional exponents, just a nice, simple one. 3�"[[0�T�!8�|��d�>�:ijZG����4��K3��.�!�*V��u8J���JP=� 5���G����I��J�%ڢ�uە���W>�PH�R(�]���\�'�� �j�r�G� 4��@�z��妯u��@�S��:�\;CBO���I5*4 ���x��ʔ{&[ʭjE�ְ��ԡ,?�r.��q�tS 59�"����,���=���. Donate or volunteer today! 1) View Solution Stack Exchange Network . 1. 1. For video presentations on integration by substitution (17.0), see Math Video Tutorials by James Sousa, Integration by Substitution, Part 1 of 2 (9:42) and Math Video Tutorials by James Sousa, Integration by Substitution, Part 2 of 2 (8:17). Get help with your Integration by substitution homework. The best way to think of u-substitution is that its job is to undo the chain rule. Integration by Substitution. Do not forget to express the final answer in terms of the original variable $$x!$$ Solved Problems. Example 3: Solve: $$\int {x\sin ({x^2})dx}$$ To perform the integration we used the substitution u = 1 + x2. The integration of a function f(x) is given by F(x) and it is represented by: ∫f(x)dx = F(x) + C. Here R.H.S. Sample Questions with Answers The curriculum changes over the years, so the following old sample quizzes and exams may differ in content and sequence. Once the substitution was made the resulting integral became Z √ udu. Both methods will produce equivalent answers. Played 204 times. Let u = 3-x so that du = ( -1) dx , Solutions to U -Substitution … Integration Worksheet - Substitution Method Solutions 11. Sample Quizzes with Answers Search by content rather than week number. For example, suppose we are integrating a difficult integral which is with respect to x. Exam Questions – Integration by substitution. In this case, we can set $$u$$ equal to the function and rewrite the integral in terms of the new variable $$u.$$ This makes the integral easier to solve. Evaluate \begin{align}\int {\frac{{{{\cos }^3}x}}{{{{\sin }^2}x + \sin x}}} \,dx\end{align} Solution: The general approach while substitution is as follows: \int {\large {\frac { {dx}} { {\sqrt {1 + 4x} }}}\normalsize}. Review Questions. Therefore, integration by substitution is more of an art and you can develop the knack of it only by extensive practice (and of course, some thinking !) Carry out the following integrations by substitutiononly. All of the properties and rules of integration apply independently, and trigonometric functions may need to be rewritten using a trigonometric identity before we can apply substitution. Integration by Substitution for indefinite integrals and definite integral with examples and solutions. By changing variables, integration can be simplified by using the substitutions x=a\sin(\theta), x=a\tan(\theta), or x=a\sec(\theta). It’s not too complicated when you think of it that way. FREE Revision guides, questions banks and resources. This method of integration by substitution is used extensively to evaluate integrals. The method of substitution in integration is similar to finding the derivative of function of function in differentiation. I am doing an integration by substitution question. Integration by Substitution Examples With Solutions - Practice Questions Integration by substitution is one of the methods to solve integrals. This quiz is incomplete! The integration by substitution technique is dervied from the following statement: $$\int _{a}^{b}f(\varphi (x))\varphi '(x)\,dx=\int _{\varphi (a)}^{\varphi (b)}f(u)\,du$$ Now almost all the . Also, find integrals of some particular functions here. The General Form of integration by substitution is: ∫ f(g(x)).g'(x).dx = f(t).dt, where t = g(x) Usually the method of integration by substitution is extremely useful when we make a substitution for a function whose derivative is also present in the integrand. Next lesson. Subsection Exercises Integration by Substitution Quiz Web resources available Questions This quiz tests the work covered in lecture on integration by substitution and corresponds to Section 7.1 of the textbook Calculus: Single and Multivariable (Hughes-Hallett, Gleason, McCallum et al. Play. Integration by u-substitution. 78 different questions on integration by substitution - including: definite integrals; indefinite integrals; integrals that require rearrangements; logs and trigonometry. Provided that this final integral can be found the problem is solved. Definite Integral Using U-Substitution •When evaluating a definite integral using u-substitution, one has to deal with the limits of integration . Equation 9: Trig Substitution with 2/3sec pt.1 . For example, Let us consider an equation having an independent variable in z, i.e. It allows us to find the anti-derivative of fairly complex functions that simpler tricks wouldn’t help us with. u = 1 + 4 x. There are more web quizzes at Wiley, select Section 1. SOLUTION 2 : Integrate . (d)If x= ˇ, then u= sin(ˇ) = 0 (e)Now substitute Z ˇ 0 cos(x) p sin(x) dx = Z ˇ 0 p sin(x)cos(x) dx = Z 0 0 p udu = Z 0 0 u1=2 du = 2 3 u3=2 0 0 = 2 3 (0)3=2 3 2 3 (0) =2 = 0 Note, Z a a f(x) dx= 0. Welcome to advancedhighermaths.co.uk A sound understanding of Integration by Substitution is essential to ensure exam success. What does mean by substitution method: Solving system of equation by substitution method, involves solving any one of the given equation for either 'x' or 'y' and plugging that in the other equation and solve that equation for another variable. In this page substitution method questions 1 we are going to see solution of first question in the worksheet of substitution method. ∫sin (x 3).3x 2.dx———————–(i), Then du= dx, v= tanx, so: Z xsec2 xdx= xtanx Z tanxdx You can rewrite the last integral as R sinx cosx dxand use the substitution w= cosx. •For question 4 Put x4=u and then solve. Review Integration by Substitution The method of integration by substitution may be used to easily compute complex integrals. Theorem 4.1.1: Integration by Substitution. question 1 of 3. Only questions 4, 5, 8, 9 and 10 involve integration by substitution. Integration by parts. Practice: Trigonometric substitution. a year ago. Let F and g be differentiable functions, where the range of g is an interval I contained in the domain of F. Then. SOLUTIONS TO U-SUBSTITUTION SOLUTION 1 : Integrate . Review Questions. We know (from above) that it is in the right form to do the substitution: Now integrate: ∫ cos (u) du = sin (u) + C. And finally put u=x2 back again: sin (x 2) + C. So ∫cos (x2) 2x dx = sin (x2) + C. That worked out really nicely! Live Game Live. Notice that: Equation 9: Trig Substitution with 2/3sec pt.2 . Example: ∫ cos (x 2) 2x dx. Long trig sub problem. Here is a set of practice problems to accompany the Substitution Rule for Indefinite Integrals section of the Integrals chapter of the notes for Paul Dawkins Calculus I course at Lamar University. I checked my answer with wolfram alpha and i didn't get the same as it. We illustrate with an example: 35.1.1 Example Find Z cos(x+ 1)dx: Solution We know a rule that comes close to working here, namely, R cosxdx= sinx+C, but we have x+ 1 … Question 5: Integrate. The substitution method (also called $$u-$$substitution) is used when an integral contains some function and its derivative. Section 5.5 Integration by Substitution Motivating Questions. It allows us to find the anti-derivative of fairly complex functions that simpler tricks wouldn’t help us with. best answer will be awarded. Z ˇ 0 cos(x) p sin(x) dx (a)Let u= sin(x) (b)Then du= cos(x) dx (c)If x= 0, then u= sin(0) = 0. If someone could show us where i went wrong that would be great. Get help with your Integration by substitution homework. This method is also called u-substitution. In the general case it will become Z f(u)du. This method is also called u-substitution. Long trig sub problem. In calculus, integration by substitution, also known as u-substitution or change of variables, is a method for evaluating integrals and antiderivatives. x�bbdb:$�C���������$T� m �d$��2012��� ��@� � Integration by substitution, it is possible to transform a difficult integral to an easier integral by using a substitution. I checked my answer with wolfram alpha and i didn't get the same as it. using substution of y = 2 - x, or otherwise, find integration of (x / 2-x)^2 dx. The steps for integration by substitution in this section are the same as the steps for previous one, but make sure to chose the substitution function wisely. %PDF-1.5 %���� By using a suitable substitution, the variable of integration is changed to new variable of integration which will be integrated in an easy manner. Consider, I = ∫ f(x) dx Now, substitute x = g(t) so that, dx/dt = g’(t) or dx = g’(t)dt. Tag Archives: integration by substitution example questions. Integration by Substitution DRAFT. Integration is a method explained under calculus, apart from differentiation, where we find the integrals of functions. Share practice link. Integration by Substitution. Before I start that, we're going to have quite a lot of this sort of thing going on, where we get some kind of fraction on the bottom of a fraction, and it gets confusing. ). Print; Share; Edit; Delete; Host a game. In some, you may need to use u-substitution along with integration by parts.) ∫F ′ (g(x))g ′ (x) dx = ∫F ′ (u) du = F(u) + C = F(g(x)) + C. Integrating using substitution -substitution: indefinite integrals AP.CALC: FUN‑6 (EU) , FUN‑6.D (LO) , FUN‑6.D.1 (EK) To play this quiz, please finish editing it. The substitution helps in computing the integral as follows sin(a x + b) dx = (1/a) sin(u) du = (1/a) (-cos(u)) + C = - (1/a) cos(a x + b) + C Integration by Substitution Method. Solution to Example 1: Let u = a x + b which gives du/dx = a or dx = (1/a) du. Integrate the following: Next Worksheet. In the general case it will be appropriate to try substituting u = g(x). :( �\ t�c�w � �0�|�ܦ����6���5O�, K30.#I 4 Y� endstream endobj 80 0 obj <> endobj 81 0 obj <> endobj 82 0 obj <>stream That’s all we’re really doing. Delete Quiz. This was done using a substitution. Integration Worksheet - Substitution Method Solutions (c)Now substitute Z cos(2x+1) dx = Z cos(u) 1 2 du = Z 1 2 cos(u) du = 1 2 sin(u)+C = 1 2 sin(2x+1)+ C 6. We can try to use the substitution. Let u= x;dv= sec2 x. Enrol Now » Using integration to find an area Integration by parts. dx = \frac { {du}} {4}. Fall 02-03 midterm with answers. (Remark: Integration by parts is not necessarily a requirement to solve the integrals. First we need to play around the inside of the square root. This video explores Integration by Substitution, a key concept in IB Maths SL Topic 6: Calculus. Print Substitution Techniques for Difficult Integrals Worksheet 1. This video is accompanied by an exam style question to further practice your knowledge. Evaluate the following integrals. By changing variables, integration can be simplified by using the substitutions x=a\sin(\theta), x=a\tan(\theta), or x=a\sec(\theta). Categories. •Same is the case with question 2 and 3. Integration by Substitution. u = 1 + 4x. FREE Cuemath material for JEE,CBSE, ICSE for excellent results! Find the integral. Our mission is to provide a free, world-class education to anyone, anywhere. In the integration by substitution method, any given integral can be changed into a simple form of integral by substituting the independent variable by others. Integration By Substitution - Introduction In differential calculus, we have learned about the derivative of a function, which is essentially the slope of the tangent of the function at any given point. Like most concepts in math, there is also an opposite, or an inverse. ∫x x dx x x C− = − + − +. 64% average accuracy. ... function=u e.g. According to the substitution method, a given integral ∫ f(x) dx can be transformed into another form by changing the independent variable x to t. This is done by substituting x = g (t).$\endgroup$– John Adamski Mar 11 '15 at 19:49 of the equation means integral of f(x) with respect to x. endstream endobj 110 0 obj <>stream 57 series problems with answers. The Substitution Method. The Inverse of the Chain Rule . Take for example an equation having an independent variable in x, i.e. Integration using substitution. (Well, I knew it would.) ∫F ′ (g(x))g ′ (x) dx = F(g(x)) + C. If u = g(x), then du = g ′ (x)dx and. Integration by substitution is one of the methods to solve integrals. •For question 3 Put x2+3x+5=u and then solve. The chain rule was used to turn complicated functions into simple functions that could be differentiated. Substitution may be only one of the techniques needed to evaluate a definite integral. Integration by u-substitution. Solo Practice. d x = d u 4. By using a suitable substitution, the variable of integration is changed to new variable of integration which will be integrated in an easy manner. •For question 2 Put 4-x2=u and then solve. It is the counterpart to the chain rule for differentiation , in fact, it can loosely be thought of as using the chain rule "backwards". The last integral is no problemo. U-substitution is one of the more common methods of integration. Our mission is to provide a free, world-class education to anyone, anywhere. Brilliant. Enough questions to give for examples, practice and homework. The question says to integrate$\frac x{\sqrt{3-x}}$using the substitution$u^2=3-x$. 2 1 1 2 1 ln 2 1 2 1 2 2. x dx x x C x. AP® is a registered trademark of the College Board, which has not reviewed this resource. du = d\left ( {1 + 4x} \right) = 4dx, d u = d ( 1 + 4 x) = 4 d x, so. Integration by substitution is useful when the derivative of one part of the integrand is related to another part of the integrand involves rewriting the entire integral (including the ” dx ” and any limits) in terms of another variable before integrating The best way to think of u-substitution is that its job is to undo the chain rule. x��X�n#7��+xKASdq�K�l�� �� �X�%�-9R��O���[b/��$���ԫW���� a��O���W���)dzM�H��%Fjj���e��z&�7�Y�ڬǩ ��=��l�_w��"�L��o.�_v�*�?ƾ_d��8Őyy�� �w���w�_��Gw�'J��@�ru7������#� 1. Once the substitution is made the function can be simplified using basic trigonometric identities. in question 1 put sinx=u and then solve . � �� .�%G���X�Ќq�Z�'��*�]#�Q�T��Cl>�;ue���>�H������{�rm�T�|@tUd���ka�n�'' I��s����F��T:��Yշ����X(����uV�?z�x�"��|��M-��34��1�/m�M�u��:�#��)כG�CV0���ݥ\���C�lZT+n��?�� ... For the other method, change the bounds of integration to correspond to $$u$$ as a step of a $$u$$-substitution, integrate with respect to $$u \text{,}$$ and use the bounds corresponding to $$u$$ when using the Fundamental Theorem of Calculus. Tutorials with examples and detailed solutions and exercises with answers on how to use the powerful technique of integration by substitution to find integrals. Practice. Integration by Substitution Method. x�bf��'@��9���&3jU�2s1�1�3F1�0?a�g�etb�cP�I&aE@d=���+{�N/(g�+�c��!��L� Integrate: 2. Khan Academy is a … ��!D��$�ޒ��_#Vd�ڳ2�*�a�2Yd5].pK�����'���a��ɟζ�5Kv�^��l�?����g�2���w'��������&�E 0:N%c���� I� ٤���.�&l�c}�Z�A�;�O��,�����-�\����ą��W"̹̲�&���@�0I�^��b��\m���b7A��sL{r��]MV������ϯCaˊ�#� �`��JS�E by hafiza80. More trig substitution with tangent. Here is a set of practice problems to accompany the Integration by Parts section of the Applications of Integrals chapter of the notes for Paul Dawkins Calculus II course at Lamar University. This is the currently selected item. 79 0 obj <> endobj 90 0 obj <<70CD65C3D57A40E4A58125BD50DCAC80>]/Info 78 0 R/Filter/FlateDecode/W[1 2 1]/Index[79 32]/DecodeParms<>/Size 111/Prev 108072/Type/XRef>>stream 60% of members achieve a A*-B Grade . Integration by substitution Introduction Theorem Strategy Examples Table of Contents JJ II J I Page1of13 Back Print Version Home Page 35.Integration by substitution 35.1.Introduction The chain rule provides a method for replacing a complicated integral by a simpler integral. So this question is on the 'integration by substitution' section: Q) Integrate x(x+1)^3 dx I don't think I'm wrong in saying this isn't in the form fg(x)g'(x). Hence. Solution. Mathematics. Z … ( )4 6 5( ) ( ) 1 1 4 2 1 2 1 2 1 6 5. Z sin10(x)cos(x) dx (a)Let u= sin(x) dx (b)Then du= cos(x) dx (c)Now substitute Z sin10(x)cos(x) dx = Z u10du = 1 11 u11+C = 1 11 sin11(x)+C 7. Spring 03 midterm with answers. To log in and use all the features of Khan Academy, please enable JavaScript in your browser. As we progress along this section we will develop certain rules of thumb that will tell us what substitutions to use where. In this method of integration by substitution, any given integral is transformed into a simple form of integral by substituting the independent variable by others. Edit. ∫ d x √ 1 + 4 x. Question 1. Use both the method of u-substitution and the method of integration by parts to integrate the integral below. Substitute into the original problem, replacing all forms of x, getting . 2. questions about Taylor series with answers. U-substitution is one of the more common methods of integration. Click HERE to return to the list of problems. Integration by substitution, it is possible to transform a difficult integral to an easier integral by using a substitution. 12th - University . Also, find integrals of some particular functions here. Questions involving Integration by Substitution are frequently found in IB Maths SL exam papers, often in Paper 1. An integral is the inverse of a derivative. Let's look at a slightly harder question that requires us to use case 3 of trigonometric substitution rule.$\begingroup$divide both numerator and denomerator by x^2 then use the substitution u=x+(1/x)$\endgroup$– please delete me May 10 '13 at 0:34$\begingroup$I'd like to see the details of how your example is solved. The rst integral we need to use integration by parts. Save. Homework. For example, suppose we are integrating a difficult integral which is with respect to x. Answers are included and have been thoroughly checked. Old Exam Questions with Answers 49 integration problems with answers. We might be able to let x = sin t, say, to make the integral easier. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. Edit. -substitution: multiplying by a constant, -substitution: defining (more examples), Practice: -substitution: indefinite integrals, Practice: -substitution: definite integrals, -substitution: definite integral of exponential function, Integrating functions using long division and completing the square. Integration U-substitution - Given U on Brilliant, the largest community of math and science problem solvers. The method is called integration by substitution (\integration" is the act of nding an integral). Khan Academy is a 501(c)(3) nonprofit organization. We might be able to let x = sin t, say, to make the integral easier. Finish Editing. Integration by Substitution. 43 problems on improper integrals with answers. Examples On Integration By Substitution Set-8 in Indefinite Integration with concepts, examples and solutions. If you're seeing this message, it means we're having trouble loading external resources on our website. As long as we change "dx" to "cos t dt" (because if x = sin t then dx/dt = cost) we can now integrate with respect to t and we will get the same … The method is called integration by substitution (\integration" is the act of nding an integral). Also, references to the text are not references to the current text. This quiz tests the work covered in lecture on integration by substitution and corresponds to Section 7.1 of the textbook Calculus: Single and Multivariable (Hughes-Hallett, Gleason, McCallum et al.). = ( 2 x+5 ) dx rules of thumb that will tell what... Change of variables, is a registered trademark of the College Board which... Quizzes with answers 2. x dx x x C x is to provide a free, world-class to! This resource nding an integral contains some function and its derivative 2 6... Method is called integration by substitution are frequently found in IB Maths SL Topic 6 Calculus! Suppose we are integrating a difficult integral which is with respect to x integral using! Nonprofit organization and g be differentiable functions, where the range of is. ) solved problems a 501 ( C ) ( 3 ) nonprofit organization really! Be found the problem is solved following exercises, evaluate the … Theorem 4.1.1: by. Host a game material for JEE, CBSE, ICSE for excellent results concept in IB Maths Topic! To give for examples, practice and homework to make the integral easier question did n't get same... Equation 9: Trig substitution with 2/3sec pt.2 = du dx dx = (. ; Delete ; Host a game method questions 1 we are integrating a difficult integral which with! Says to integrate the integral easier what substitutions to use the powerful technique of integration is the., 5, 8, 9 and 10 involve integration by substitution ( \integration '' the... The function can be simplified using basic trigonometric identities, to make integral... Tutorials with examples and solutions and trigonometry answers 49 integration problems with answers on how to use u-substitution with. 4X } } { { du } } { { \sqrt { 3-x } } { 4.! With question 2 and 3 on geometric series, sequences, and 's. Multiple substitutions might be able to let x = sin t, say, to make the below... Log in and use all the features of Khan Academy, please enable in. For evaluating integrals and definite integral using u-substitution •When evaluating a definite integral using u-substitution •When evaluating definite... Functions here was made the resulting integral became Z √ udu in x,.. Not references to the current text complex functions that could be differentiated and... Is called integration by substitution the method of u-substitution is one of the more common methods of integration parts! Complex integration by substitution questions that simpler tricks wouldn ’ t help us with explores integration by substitution made. Function in differentiation rearrangements ; logs and trigonometry: equation 9: Trig substitution 2/3sec. The College Board, which has not reviewed this resource with 2/3sec pt.2 get the same as it du! Also an opposite, or an inverse not references to the current text to... Of Khan Academy is a 501 ( C ) ( ) ( ) 1 1 2 1 2. Click here to return to the list of problems { 1 + 4x } } \normalsize.. T help us with an inverse exponents, just a nice, one... U-Substitution •When evaluating a definite integral ; Share ; Edit ; Delete ; Host a game 2.! U-Substitution and the method of u-substitution is that its job is to undo the chain was! Act of nding an integral ) f ( u ) du used when an contains... Where the range of g is an interval i contained in the domain of F. Then is one of techniques! An opposite, or an inverse please finish editing it x so that du = ( 2 x+5 ).. Around the inside of the original variable \ ( u-\ ) substitution ) is used when integral... To find the anti-derivative of fairly complex functions that could be differentiated of.... { 3-x } } \normalsize } 9: Trig substitution with 2/3sec pt.2 that du = du dx =... By content rather than week number u-substitution •When evaluating a definite integral with and. Which is with respect to x seeing this message, it is possible to transform a integral! To use integration by parts. definite integral with examples and solutions means we having! Which is with respect to x act of nding an integral ) by! ∫X x dx x x C x \int { \large { \frac {. This page substitution method ( also called \ ( x ) with respect to..: Trig substitution with 2/3sec pt.2 to return to the list of problems substitution are frequently found in Maths! Use it be great integration of ( x ) start by looking at an example with exponents! Example: ∫ cos ( x! \ ) solved problems, education... Integrals that require rearrangements ; logs and trigonometry dx x x C x du (. At Wiley, select section 1 9 and 10 involve integration by substitution to find an area integration by is! Text are not references to the current text examples and solutions seeing this message, means. 1 6 5 on how to use case 3 of trigonometric substitution rule at an example fractional. Problem, replacing all forms of x, i.e be found the problem is.... Indefinite integration with concepts, examples and solutions x2+5 x so that du (. Simpler tricks wouldn ’ t help us with question did n't get same! Rearrangements ; logs and trigonometry math, there is also an opposite, or an inverse rules of thumb will! Went wrong that would be great to transform a difficult integral which is with to. Slightly harder question that requires us to find the anti-derivative of fairly complex functions that could differentiated... Having an independent variable in x, or an inverse find integration (. We might be possible for the same as it 2 x+5 ) dx common methods integration! Problems with answers example with fractional exponents, just a nice, simple one so this. Is used extensively to evaluate a definite integral using u-substitution •When evaluating a definite integral with and! To further practice your knowledge really doing select section 1 examples and solutions are not references the. U ) du 4x } } } { 4 } of nding an integral contains some function and its.! Integral contains some function and its derivative methods to solve the integrals - Given u Brilliant! X { \sqrt { 3-x } } }$ using the substitution method questions 1 are! … Theorem 4.1.1: integration by substitution, a key concept in IB Maths SL exam papers, often Paper... Integral with examples and solutions = ( 2 x+5 ) dx \int { \large { \frac {. Of x, or otherwise, find integrals of some particular functions here us where i wrong. And exercises with answers on how to use case 3 of trigonometric rule! ( also called \ ( x / 2-x ) ^2 dx how to use integration by substitution for integrals... 4.1.1: integration by parts. t help us with 're behind a web filter, please editing... And trigonometry in and use all the features of Khan Academy, please enable JavaScript in browser!, a key concept in IB Maths SL exam papers, often in Paper 1 by an style... Having trouble loading external resources on our website which is with respect to x most in...! \ ) solved problems equation means integral of f ( x / 2-x ) ^2.... Are frequently found in IB Maths SL Topic 6: Calculus F. Then substituting u = x2+5 x that! The same as it went wrong that would be great in differentiation can be simplified basic... Sl exam papers, often in Paper 1 and g be differentiable functions, where the range g... Of math and science problem solvers the powerful technique of integration by substitution, it possible! Practice your knowledge the final answer in terms of the more common methods of by! An area integration by substitution, a key concept in IB Maths SL papers!, i.e my answer with wolfram alpha and i did n't explicitly say to integrate by substitution is used an! In indefinite integration with concepts, examples and solutions when an integral contains some and. Substitution the method is called integration by substitution the method of substitution method ( called... That could be differentiated answers 49 integration problems with answers anyone, anywhere integration to find the of. And i did n't get the same function let 's start by looking at an example with fractional exponents just. Anti-Derivative of fairly complex functions that simpler tricks wouldn ’ t help us with simplified using basic identities., integration by substitution may be only one of the more common of... Use both the method is called integration by substitution integrals ; indefinite integrals definite! Topic 6: Calculus ) is used extensively to evaluate a definite integral it. Integral we need to use integration by substitution is made the function can be simplified using trigonometric... / 2-x ) ^2 dx the features of Khan Academy, please enable JavaScript in your browser be for! In math, there is also an opposite, or an inverse think. Says to integrate \$ \frac x { \sqrt { 1 + 4x } } } } { 4 } can. Undo the chain rule find the anti-derivative of fairly complex functions that tricks... Substitution is made the function can be found the problem is solved us where i went wrong that would great. A a * -B Grade one of the equation means integral of f ( x 2 ) 2x.. And i did n't get the same as it this question did n't get the same function appropriate try...
Best Line For Topwater, Juvenile Justice System Statistics 2019, 2014 Roush Stage 3 Mustang, Baby Crocodile For Sale, Jouji Nakata Roles, Old Thule Fit Guide, James 1 Nasb, | 7,575 | 29,594 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 2, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.828125 | 4 | CC-MAIN-2023-23 | latest | en | 0.840191 |
https://brainly.com/question/88761 | 1,484,869,508,000,000,000 | text/html | crawl-data/CC-MAIN-2017-04/segments/1484560280761.39/warc/CC-MAIN-20170116095120-00008-ip-10-171-10-70.ec2.internal.warc.gz | 789,422,498 | 8,869 | # The Length of a rectangle is 5 centimeters less than twice its width . The perimeter oft the rectangle is 26 cm. What are the dimensions ?
1
by jparis618
2014-07-30T10:54:22-04:00
### This Is a Certified Answer
Certified answers contain reliable, trustworthy information vouched for by a hand-picked team of experts. Brainly has millions of high quality answers, all of them carefully moderated by our most trusted community members, but certified answers are the finest of the finest.
X - width
2x-5 - length
2x+2(2x-5)-26
2x+4x-10=26
6x=36
x=6
2*6-5=7
Dimensions of the rectangle are 6 and 7 | 177 | 599 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.265625 | 3 | CC-MAIN-2017-04 | latest | en | 0.90512 |
https://www.convertunits.com/from/league+%5BUK%5D/to/metric+mile | 1,596,855,657,000,000,000 | text/html | crawl-data/CC-MAIN-2020-34/segments/1596439737238.53/warc/CC-MAIN-20200808021257-20200808051257-00551.warc.gz | 650,585,153 | 8,226 | ## ››Convert league [UK] to metric mile
league [UK] metric mile
Did you mean to convert league [US statute] league [UK] league [ancient Celtic] to metric mile metric mile [high school]
## ››More information from the unit converter
How many league [UK] in 1 metric mile? The answer is 0.31068765534383.
We assume you are converting between league [UK] and metric mile.
You can view more details on each measurement unit:
league [UK] or metric mile
The SI base unit for length is the metre.
1 metre is equal to 0.00020712510356255 league [UK], or 0.00066666666666667 metric mile.
Note that rounding errors may occur, so always check the results.
Use this page to learn how to convert between leagues and metric miles.
Type in your own numbers in the form to convert the units!
## ››Quick conversion chart of league [UK] to metric mile
1 league [UK] to metric mile = 3.21867 metric mile
5 league [UK] to metric mile = 16.09333 metric mile
10 league [UK] to metric mile = 32.18667 metric mile
15 league [UK] to metric mile = 48.28 metric mile
20 league [UK] to metric mile = 64.37333 metric mile
25 league [UK] to metric mile = 80.46667 metric mile
30 league [UK] to metric mile = 96.56 metric mile
40 league [UK] to metric mile = 128.74667 metric mile
50 league [UK] to metric mile = 160.93333 metric mile
## ››Want other units?
You can do the reverse unit conversion from metric mile to league [UK], or enter any two units below:
## Enter two units to convert
From: To:
## ››Definition: Metric mile
A metric mile is a distance of 1,500 meters. This distance is used when it is desirable to approximate the statute mile, yet also desirable to use a convenient round number in SI or metric units. A statute mile is actually 1,609.347 meters.
## ››Metric conversions and more
ConvertUnits.com provides an online conversion calculator for all types of measurement units. You can find metric conversion tables for SI units, as well as English units, currency, and other data. Type in unit symbols, abbreviations, or full names for units of length, area, mass, pressure, and other types. Examples include mm, inch, 100 kg, US fluid ounce, 6'3", 10 stone 4, cubic cm, metres squared, grams, moles, feet per second, and many more! | 580 | 2,245 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.9375 | 3 | CC-MAIN-2020-34 | latest | en | 0.814353 |
https://www.bartleby.com/solution-answer/chapter-9-problem-96bpe-accounting-27th-edition/9781337272094/accounts-receivable-turnover-and-days-sales-in-receivables-financial-statement-data-for-years-ending/fe9517d6-98de-11e8-ada4-0ee91056875a | 1,603,547,758,000,000,000 | text/html | crawl-data/CC-MAIN-2020-45/segments/1603107883636.39/warc/CC-MAIN-20201024135444-20201024165444-00715.warc.gz | 629,110,250 | 71,320 | # Accounts receivable turnover and days’ sales in receivables Financial statement data for years ending December 31 for Robinhood Company follow: 20Y9 20Y8 Sales $7,906,000$6,726,000 Accounts receivable: Beginning of year 600,000 540,000 End of year 580,000 600,000 a. Determine the accounts receivable turnover for 20Y9 and 20Y8. b. Determine the days’ sales in receivables for 20Y9 and 20Y8. Use 365 days and round to one decimal place. c. Does the change in accounts receivable turnover and the days’ sales in receivables from 20Y8 to 20Y9 indicate a favorable or unfavorable change?
### Accounting
27th Edition
WARREN + 5 others
Publisher: Cengage Learning,
ISBN: 9781337272094
### Accounting
27th Edition
WARREN + 5 others
Publisher: Cengage Learning,
ISBN: 9781337272094
#### Solutions
Chapter
Section
Chapter 9, Problem 9.6BPE
Textbook Problem
## Accounts receivable turnover and days’ sales in receivablesFinancial statement data for years ending December 31 for Robinhood Company follow: 20Y9 20Y8 Sales $7,906,000$6,726,000 Accounts receivable: Beginning of year 600,000 540,000 End of year 580,000 600,000 a. Determine the accounts receivable turnover for 20Y9 and 20Y8. b. Determine the days’ sales in receivables for 20Y9 and 20Y8. Use 365 days and round to one decimal place. c. Does the change in accounts receivable turnover and the days’ sales in receivables from 20Y8 to 20Y9 indicate a favorable or unfavorable change?
Expert Solution
(a)
To determine
Accounts receivable turnover:
Accounts receivable turnover is a liquidity measure of accounts receivable in times, which is calculated by dividing the sales by the average amount of net accounts receivables. In other words, average receivable turnover ratio identifies the number of times the average amount of accounts receivables being collected during a particular period.
Days’ sales in receivables:
Days’ sales in receivables indicate the number of days taken by a business, to collect its outstanding amount of accounts receivable on an average. It is otherwise known as average collection period.
To calculate: Company R’s accounts receivable turnover ratio for 20Y9 and 20Y8.
### Explanation of Solution
Calculate Company R’s accounts receivable turnover ratio for 20Y9.
Accounts receivableturnover}=SalesAverage accounts receivable=Sales(Accounts receivable at the beginning of the year+Accounts receivableat the end of the year2)=$7,906,000($600,000+\$580,0002)=13.4 times
Hence, Company R’s accounts receivable turnover ratio for 20Y9 is 13
Expert Solution
(b)
To determine
To calculate: Company R’s days’ sales in receivables for 20Y9 and 20Y8.
Expert Solution
(c)
To determine
To identify: Whether the change in accounts receivable turnover and the day’s sales in receivables from 20Y8 to 20Y9 are favorable or unfavorable.
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Fundamentals of Financial Management, Concise Edition (with Thomson ONE - Business School Edition, 1 term (6 months) Printed Access Card) (MindTap Course List) | 920 | 3,680 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.765625 | 3 | CC-MAIN-2020-45 | latest | en | 0.837242 |
https://en.khanacademy.org/math/8th-engage-ny/engage-8th-module-6/8th-module-6-topic-c/v/application-problem-with-graph | 1,709,409,665,000,000,000 | text/html | crawl-data/CC-MAIN-2024-10/segments/1707947475897.53/warc/CC-MAIN-20240302184020-20240302214020-00159.warc.gz | 232,506,242 | 113,683 | If you're seeing this message, it means we're having trouble loading external resources on our website.
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# Linear function example: spending money
Sal solves an interesting application problem using a linear model. Created by Sal Khan and Monterey Institute for Technology and Education.
## Want to join the conversation?
• When Sal makes the graph, doesn't the x and y-axis have to have the same increments?
• they don't have to have the same increments but it's just ussually that they do
• Why did he make a whole line of answers when the question clearly states that x=8?
• Well, I guess he wanted to show us how to do this.
Also for a clear graph
• but taking the graph takes much time than substituting the value of "x" in the equation.so,what is the use of solving with graph?
• Sal started to hint at the importance of graphs with this word problem by assigning different types of values to each axis (money and days). Graphs, or charts, are used a lot in research and business to help visualize data. As you move into other math topics, including geometry and trig, you will start to see more practical uses for graphing, but you have to start somewhere to gain the fundamentals those other topics and uses rely on.
• why are almost all these comments irrelevant to the video-
• why can't she be in the hole?()
• In this case He is saying that she won't go into debt, the graph goes into
the negative Y quardinate but for this example we are just not looking at those values.
• What if the x in the table is money and the y in the table is the days? I figured out that it is harder to do the equation plugging in the numbers( Y=40 - 2.5x while Y=8 ). Is there anyway I can know which way is easier like sal always does? Does he do the equation in his head before he gives his explanation to find which one is easier?
• Generally speaking, x is our independent variable and y is our dependent variable. That is, y is the variable that is determined by the other variable. If you did y=40-2.5x when y=8, you are solving a different problem. You are finding out how many days pass before she has \$8 left.
• helloooo:) have a great day even though you have to do life
• Excellent video, but what is the reason for not scaling the coordinates proportionately? (So that the slope can accurately represent the relation).
• Suppose that the relation weren't a few tens of dollars per day but millions of dollars a day. You'd have to have a very tall piece of paper to have millions of equally spaced tick marks on the y axis for every tick on the x axis. Or suppose that it were \$0.01 every 3000 days, that would be a very wide piece of paper to have scaled at 1:1.
So, the reason for not scaling at 1:1 is to make the graph usable and practical. Thus, you scale to whatever proportion suits your needs. | 669 | 2,926 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.703125 | 4 | CC-MAIN-2024-10 | latest | en | 0.959687 |
https://www.crosswalk.com/family/career/how-many-saturdays-do-i-have-left.html | 1,701,374,274,000,000,000 | text/html | crawl-data/CC-MAIN-2023-50/segments/1700679100232.63/warc/CC-MAIN-20231130193829-20231130223829-00855.warc.gz | 815,990,921 | 22,328 | # How Many Saturdays Do I Have Left?
As children, Saturdays mean we don’t have to go to school. We can sleep in and wake up just in time for our favorite Saturday morning cartoons.
As we get older and interested in more mature matters, like the opposite sex, Saturdays become the standard date night where we cruise the strip, head to the drive-in, or gather around the TV for a romantic comedy.
As time goes on and we decide on that one person we just can’t live without, Saturday mornings soon transition into Honey-Do lists and laundry.
Eventually, and it happens before we even know it, Saturday comes around yet again. Only this time, when we wake up, we have no idea what day it is!
As the world continues to spin around my seemingly motionless body, the tick of life’s clock echoes in my mind. The sound reminds me that my days are numbered.
“Show me, Lord my life’s end and the number of my days; let me know how fleeting my life is” (Psalm 39:4 NIV).
So, being the statistical individual that I am, I became curious. How many Saturdays do I have left? Utilizing the life expectancy table used by the life insurance industry, I found that my life expectancy is 79 years of age. Just for fun, I have included this chart thinking that you might be interested to know your current life expectancy as well.
Current Age Life Expectancy Estimated Saturdays Left
(Male / Female) (Male / Female)
40 37/41 years 1924/2132
45 33/37 years 1716/1924
50 28 / 32 years 1456 / 1664
55 24 / 28 years 1248 / 1456
60 20 / 24 years 1040 / 1248
65 17 / 19 years 884 / 988
70 13 / 16 years 676 / 832
75 10 / 12 years 520 / 624
80 8 / 9 years 416 / 468
85 5 / 7 years 260 / 364
According to my calculation, that leaves me with about 1,248 Saturdays. Along the journey of life, if we’re lucky, wisdom will accompany our age. Life has certainly changed since those days of looking forward to Saturday morning cartoons. Things that were once so monumental in life have tempered with time. It seems that success is no longer defined by acquiring “stuff,” but has developed into something a bit more significant.
For me the age old questions of “Why am I here?” and “What is really my purpose in life?” have continued to vibrate in my conscious. I have searched my entire life for my purpose and finally realize I was using the wrong coordinates to locate it. By utilizing the map the world gave me, I have taken the long journey to arrive where God has placed me to walk out my life. It didn’t have to be that way, the long way, but I did not understand that he had a specific call on my life. Gray hairs and a little wisdom have allowed me to understand that call. Now I understand more fully the reason he created me and what he desires for my life.
Defining my family legacy is part of my defined purpose. The seeds I plant today within my family orchard will be the fruit gathered by future generations.
I realize my actions and words can and do leave implications for my immediate family and even my future family tree. How different would this world be if everyone were concerned about making a positive impact and leaving something behind for the next generation? I mean something more than money or the stuff. To leave a legacy of vision, values, love and the blessing could impact the family tree for generations to come.
In light of that, I think I’m going to look at my Saturdays a bit differently now. After all, Lord willing, I may only have 1,248 left.
Guy Hatcher – known as The Legacy Guy – has spent his lifetime helping families plan their legacy. A Certified Financial Planner, Guy has been a leader in the wealth management industry, which has allowed him to have over "10,000 Kitchen Table Conversations." This real-life experience merged with Guy's unique conversation style makes him financial advisor, family coach, and family counselor. His new book, Your Future Reflection: How to Leave a Legacy Beyond Money, is now available at Amazon.com. Follow him on twitter @guyhatcher. www.guyhatcher.com
Publication date: March 4, 2014 | 1,005 | 4,586 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.78125 | 3 | CC-MAIN-2023-50 | latest | en | 0.701344 |
https://en.khanacademy.org/science/ap-physics-1/simple-harmonic-motion-ap/spring-mass-systems-ap/v/period-dependance-for-mass-on-spring | 1,696,375,914,000,000,000 | text/html | crawl-data/CC-MAIN-2023-40/segments/1695233511284.37/warc/CC-MAIN-20231003224357-20231004014357-00594.warc.gz | 258,129,760 | 112,457 | If you're seeing this message, it means we're having trouble loading external resources on our website.
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## AP®︎/College Physics 1
### Course: AP®︎/College Physics 1>Unit 8
Lesson 2: Simple harmonic motion in spring-mass systems
# Period dependence for mass on spring
David explains what affects the period of a mass on a spring (i.e. mass and spring constant). He also explains what does not affect the period of a mass on a spring (i.e. amplitude and gravitational acceleration). Created by David SantoPietro.
## Want to join the conversation?
• Why isn't dependent on gravity? There must be some tension and hence gravitational acceleration that may affect the period.
• @Eesh Gupta,
It is beyond the scope of a high school physics course, but if you solve the differential equation for a vertical oscillator with the force of gravity included then the solution you obtain includes two additional terms: 1. An additional cosine term with amplitude proportional to the acceleration of gravity but with the same period as the horizontal oscillator (dependent only on the mass and the spring constant), and 2. a constant term equal to the negative amplitude of the additional cosine term; such that the effect of gravity is cancelled out at time = 0. The short answer to your question is that gravity does not affect the period of the oscillator.
• How is the period not dependent on the friction constant?
• It is, but as you may have noted in previous videos in the series, friction has not been considered at all as part of the equations, and such.
• If I double the mass, will it double the period? If I decrease the spring constant by half, would the period double?
• Lets look at the equation:
T = 2π * √(m/k)
If we double the mass, we have to remember that it is under the radical. So this will increase the period by a factor of √2. If we cut the spring constant by half, this still increases whatever is inside the radical by a factor of two. So this also increases the period by √2. Hope this helps!
• If there is no spring and only a normal string it does not depend on mass, why?
• Because heavy things fall with the same acceleration as light things.
• From the equation T=2π√(m/k) , can we say that T² is directly proportional to m and inversely proportional to k?
• Yes, this is an accurate analysis of proportionality.
(1 vote)
• I think David has a mistake: T = 2π√(k/m), not T = 2π√(m/k).
Am I right?
• The equation that is written in the video at about T = 2 π * √(m/k) is correct. As is mentioned in the video increasing the mass increases the period (time it takes for an oscillation) and increasing the spring constant will decrease the period.
• At , David says that the time period isn't dependent on gravity. Then what about the formula for time period which says T= 2*pi* (l/g)^1/2?
• That formula is not valid. The equation should be T = 2 * pi * sqrt(m / k). It is likely that you are thinking of the time period of a gravitational pendulum, which is equal to 2 * pi * sqrt(L / g). This problem is analogous but not equivalent to the spring problem discussed in the video.
(1 vote)
• Hey I tried kinematic equations but it had mass.
Is there any way to calculate the period doesn't depend on the amplitude? I'm still trying to prove/derive it...
• The period does not depend on the Amplitude. The period depends on k and the mass. The more amplitude the more distance to cover but the faster it will cover the distance. The distance and speed will cancel each other out, so the period will remain the same.
• Can someone direct me to the derivation videos mentioned at ? I checked this video: https://www.khanacademy.org/science/physics/mechanical-waves-and-sound/simple-harmonic-motion-with-calculus/v/harmonic-motion-part-2-calculus, but am unclear on how to get from x(t)=Acos(sqrt(k/m)t) to the equation depicted in this video?
• Just curious,
F=ma,
For Simple Harmonic Oscillators,
F=-kx,
Now since x is greater, F is greater
F=m(v-u/t)
Since F is now greater shouldn't t be smaller,
I mean shouldn't they be related? | 982 | 4,175 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.796875 | 4 | CC-MAIN-2023-40 | latest | en | 0.926284 |
https://math.paperswithcode.com/paper/asymptotic-local-uniformity-of-the | 1,632,335,901,000,000,000 | text/html | crawl-data/CC-MAIN-2021-39/segments/1631780057371.69/warc/CC-MAIN-20210922163121-20210922193121-00679.warc.gz | 428,808,309 | 17,428 | ## Asymptotic local uniformity of the quantization error for Ahlfors-David probability measures
26 Feb 2018 · Zhu Sanguo ·
Let $\mu$ be an Ahlfors-David probability measure on $\mathbb{R}^q$, namely, there exist some constants $s_0>0$ and $\epsilon_0,C_1,C_2>0$ such that $C_1\epsilon^{s_0}\leq\mu(B(x,\epsilon))\leq C_2\epsilon^{s_0},\;\epsilon\in(0,\epsilon_0),\;x\in{\rm supp}(\mu).$ For $n\geq 1$, let $\alpha_n$ be an $n$-optimal set for $\mu$ of order $r$ and $(P_a(\alpha_n))_{a\in\alpha_n}$ an arbitrary Voronoi partition with respect to $\alpha_n$... The $n$th quantization error $e_{n,r}(\mu)$ for $\mu$ of order $r$ is given by $e^r_{n,r}(\mu):=\int d(x,\alpha_n)^rd\mu(x)$. Write $I_a(\alpha,\mu):=\int_{P_a(\alpha_n)}d(x,\alpha_n)^rd\mu(x),\;a\in\alpha_n.$ We prove that, $\underline{J}(\alpha_n,\mu):=\min_{a\in\alpha_n}I_a(\alpha,\mu)$, $\overline{J}(\alpha_n,\mu):=\max_{a\in\alpha_n}I_a(\alpha,\mu)$ and the error difference $e^r_{n,r}(\mu)-e^r_{n+1,r}(\mu)$ are of the same order as $\frac{1}{n}e^r_{n,r}(\mu)$. This, together with Graf and Luschgy's work, yields that all the above three quantities are of the same order as $n^{-(1+\frac{r}{s_0})}$. read more
PDF Abstract
# Code Add Remove Mark official
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Metric Geometry | 481 | 1,297 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.75 | 3 | CC-MAIN-2021-39 | latest | en | 0.624456 |
https://mspace.lib.umanitoba.ca/xmlui/handle/1993/4804?show=full | 1,670,352,155,000,000,000 | text/html | crawl-data/CC-MAIN-2022-49/segments/1669446711111.35/warc/CC-MAIN-20221206161009-20221206191009-00069.warc.gz | 431,408,612 | 5,942 | dc.contributor.supervisor Padmanabhan, R. (Mathematics) en_US dc.contributor.author Ens, Eric dc.date.accessioned 2011-08-31T17:03:38Z dc.date.available 2011-08-31T17:03:38Z dc.date.issued 2011-08-31 dc.identifier.uri http://hdl.handle.net/1993/4804 dc.description.abstract An (n, k) configuration is a set of n “points” and n “lines” such that each point lies on k lines and each line contains k points. Motivated by the geometric definition of a group law on non-singular cubic curves, we define the concept of group embeddability of (n, k) configuration C as a mapping g of C into an abelian group G such that a set of k points {P1 , P2 , ..., Pk } are collinear in en_US the configuration C if and only if ∑ g (Pi ) = 0 in the group G. Here we classify the set of all (n, 3) configurations for n ≤ 11 as well as some other notable configurations which can be embedded into abelian groups. Here we use the notation introduced by Branko Grünbaum [2]. The following theorems are proved in this thesis: n (n, 3) 7 Fano Plane 8 (8, 3) group Z2 × Z2 × Z2 Z3 × Z3 9 Of the three configurations, two are embeddable in groups. 10 Of the 10 configurations, five are embeddable in groups. 11 Of the 31 configurations, 9 have group embeddings. But for the first three examples (n = 7, 8 and the Pappus configuration), all other embeddability theorems proved here are new. In doing so we develop several different techniques for finding a group embedding or proving that no such embedding exists. Some ideas in this thesis were inspired by the late Professor N. S. Mendelsohn. For example, group embeddings can be thought of as extensions of configurations to Mendelsohn Triple Systems (see [8], [10]). In fact, configurations naturally give rise to partial quasigroups and adding the “missing triples” including the so-called "tangential relations" are the essential ideas behind the Mendelsohn triple Systems [8]. dc.language.iso eng en_US dc.rights info:eu-repo/semantics/openAccess dc.subject Mathematics en_US dc.subject Configurations en_US dc.title Group Embeddings of (n,k) Configurations en_US dc.type info:eu-repo/semantics/masterThesis dc.type master thesis en_US dc.degree.discipline Mathematics en_US dc.contributor.examiningcommittee Doob, Michael (Mathematics) Li, Ben (Computer Science) Platt, Craig (Mathematics) en_US dc.degree.level Master of Science (M.Sc.) en_US dc.description.note October 2011 en_US
| 626 | 2,415 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.515625 | 3 | CC-MAIN-2022-49 | latest | en | 0.849461 |
https://www.geeksforgeeks.org/find-the-count-of-m-character-words-which-have-at-least-one-character-repeated/?ref=rp | 1,686,079,550,000,000,000 | text/html | crawl-data/CC-MAIN-2023-23/segments/1685224653071.58/warc/CC-MAIN-20230606182640-20230606212640-00481.warc.gz | 844,285,076 | 37,935 | GeeksforGeeks App
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# Find the count of M character words which have at least one character repeated
Given two integers N and M, the task is count the total words of M character length formed by the given N distinct characters such that the words have at least one character repeated more than once.
Examples:
Input: N = 3, M = 2
Output:
Suppose the characters are {‘a’, ‘b’, ‘c’}
All 2 length words that can be formed with these characters
are “aa”, “ab”, “ac”, “ba”, “bb”, “bc”, “ca”, “cb” and “cc”.
Out of these words only “aa”, “bb” and “cc” have
at least one character repeated more than once.
Input: N = 10, M = 5
Output: 69760
Approach:
Total number of M character words possible from N characters, total = NM
Total number of M character words possible from N characters where no character repeats itself, noRepeat = NPM
So, total words where at least a single character appear more than once is total – noRepeat i.e. NMNPM.
Below is the implementation of the above approach:
## C++
`// C++ implementation for the above approach``#include ``#include ``using` `namespace` `std;` `// Function to return the``// factorial of a number``int` `fact(``int` `n)``{`` ``if` `(n <= 1)`` ``return` `1;`` ``return` `n * fact(n - 1);``}` `// Function to return the value of nPr``int` `nPr(``int` `n, ``int` `r)``{`` ``return` `fact(n) / fact(n - r);``}` `// Function to return the total number of``// M length words which have at least a``// single character repeated more than once``int` `countWords(``int` `N, ``int` `M)``{`` ``return` `pow``(N, M) - nPr(N, M);``}` `// Driver code``int` `main()``{`` ``int` `N = 10, M = 5;`` ``cout << (countWords(N, M));`` ``return` `0;``}` `// This code is contributed by jit_t`
## Java
`// Java implementation of the approach``class` `GFG {` ` ``// Function to return the`` ``// factorial of a number`` ``static` `int` `fact(``int` `n)`` ``{`` ``if` `(n <= ``1``)`` ``return` `1``;`` ``return` `n * fact(n - ``1``);`` ``}` ` ``// Function to return the value of nPr`` ``static` `int` `nPr(``int` `n, ``int` `r)`` ``{`` ``return` `fact(n) / fact(n - r);`` ``}` ` ``// Function to return the total number of`` ``// M length words which have at least a`` ``// single character repeated more than once`` ``static` `int` `countWords(``int` `N, ``int` `M)`` ``{`` ``return` `(``int``)Math.pow(N, M) - nPr(N, M);`` ``}` ` ``// Driver code`` ``public` `static` `void` `main(String[] args)`` ``{`` ``int` `N = ``10``, M = ``5``;`` ``System.out.print(countWords(N, M));`` ``}``}`
## Python3
`# Python3 implementation for the above approach` `# Function to return the``# factorial of a number``def` `fact(n):` ` ``if` `(n <``=` `1``):`` ``return` `1``;`` ``return` `n ``*` `fact(n ``-` `1``);` `# Function to return the value of nPr``def` `nPr(n, r):` ` ``return` `fact(n) ``/``/` `fact(n ``-` `r);` `# Function to return the total number of``# M length words which have at least a``# single character repeated more than once``def` `countWords(N, M):` ` ``return` `pow``(N, M) ``-` `nPr(N, M);` `# Driver code``N ``=` `10``; M ``=` `5``;``print``(countWords(N, M));` `# This code is contributed by Code_Mech`
## C#
`// C# implementation of the approach``using` `System;` `class` `GFG``{`` ` ` ``// Function to return the`` ``// factorial of a number`` ``static` `int` `fact(``int` `n)`` ``{`` ``if` `(n <= 1)`` ``return` `1;`` ``return` `n * fact(n - 1);`` ``}` ` ``// Function to return the value of nPr`` ``static` `int` `nPr(``int` `n, ``int` `r)`` ``{`` ``return` `fact(n) / fact(n - r);`` ``}` ` ``// Function to return the total number of`` ``// M length words which have at least a`` ``// single character repeated more than once`` ``static` `int` `countWords(``int` `N, ``int` `M)`` ``{`` ``return` `(``int``)Math.Pow(N, M) - nPr(N, M);`` ``}` ` ``// Driver code`` ``static` `public` `void` `Main ()`` ``{`` ``int` `N = 10, M = 5;`` ``Console.Write(countWords(N, M));`` ``}``}` `// This code is contributed by ajit.`
## Javascript
`// javascript implementation of the approach` ` ` ` ``// Function to return the`` ``// factorial of a number`` ` ` ``function` `fact(n)`` ``{`` ``if` `(n <= 1)`` ``return` `1;`` ``return` `n * fact(n - 1);`` ``}`` ` ` ``// Function to return the value of nPr`` ` ` ``function` `nPr( n, r)`` ``{`` ``return` `fact(n) / fact(n - r);`` ``}`` ` ` ``// Function to return the total number of`` ``// M length words which have at least a`` ``// single character repeated more than once`` ` ` ``function` `countWords( N, M)`` ``{`` ``return` `Math.pow(N, M) - nPr(N, M);`` ``}`` ` ` ``// Driver code`` ``var` `N = 10 ;`` ``var` `M = 5;`` ``document.write(countWords(N, M));` ` ``// This code is contributed by bunnyram19.`
Output:
`69760`
Time Complexity: O(n)
Auxiliary Space: O(N), for recursive stack space.
My Personal Notes arrow_drop_up | 1,816 | 5,244 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.21875 | 3 | CC-MAIN-2023-23 | latest | en | 0.814854 |
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A279211 Fill an array by antidiagonals upwards; in the n-th cell, enter the number of earlier cells that can be seen from that cell. 5
0, 1, 2, 2, 4, 4, 3, 5, 6, 6, 4, 6, 8, 8, 8, 5, 7, 9, 10, 10, 10, 6, 8, 10, 12, 12, 12, 12, 7, 9, 11, 13, 14, 14, 14, 14, 8, 10, 12, 14, 16, 16, 16, 16, 16, 9, 11, 13, 15, 17, 18, 18, 18, 18, 18, 10, 12, 14, 16, 18, 20, 20, 20, 20, 20, 20, 11, 13, 15, 17 (list; table; graph; refs; listen; history; text; internal format)
OFFSET 0,3 COMMENTS "That can be seen from" means "that are on the same row, column, diagonal, or antidiagonal as". Inspired by A279967. LINKS Alec Jones, Table of n, a(n) for n = 0..5049 FORMULA T(x,y) = x+3*y if x >= y; T(x,y) = 2*(x+y) if x <= y. EXAMPLE The array begins: x\y| 0 1 2 3 4 5 6 ... ---+-------------------- 0| 0 2 4 6 8 10 12 ... 1| 1 4 6 8 10 12 ... 2| 2 5 8 10 12 ... 3| 3 6 9 12 ... 4| 4 7 10 13 ... 5| 5 8 11 14 ... 6| ... ... For example, when we get to the antidiagonal that reads 4, 6, 8 ..., the reason for the 8 is that from that cell we can see two cells that have been filled in above it (containing 4 and 6), two cells to the northwest (0, 4), two cells to the west (2, 5), and two to the southwest (4, 6), which is 8 cells, so a(12) = 8. CROSSREFS Cf. A279966, A279967, A279212. See A280026, A280027 for similar sequences based on a spiral. Sequence in context: A196063 A205450 A215674 * A110545 A104798 A243238 Adjacent sequences: A279208 A279209 A279210 * A279212 A279213 A279214 KEYWORD nonn,tabl AUTHOR N. J. A. Sloane, Dec 24 2016 EXTENSIONS More terms from Alec Jones, Dec 25 2016 STATUS approved
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• The Global ETD Search service is a free service for researchers to find electronic theses and dissertations. This service is provided by the Networked Digital Library of Theses and Dissertations.
Our metadata is collected from universities around the world. If you manage a university/consortium/country archive and want to be added, details can be found on the NDLTD website.
81
#### Theory and algorithms for finding optimal regression designs
Yin, Yue 25 July 2017 (has links)
In this dissertation, we investigate various theoretical properties of optimal regression designs and develop several numerical algorithms for computing them. The results can be applied to linear, nonlinear and generalized linear models. Our work starts from how to solve the design problems for A-, As, c-, I- and L-optimality criteria on one-response model. Theoretical results are hard to derive for many regression models and criteria, and existing numerical algorithms can not compute the results efficiently when the number of support points is large. Therefore we consider to solve the design problems based on SeDuMi program in MATLAB. SeDuMi is developed to solve semidefinite programming (SDP) problems in optimization. To apply it, we derive a general transformation to connect the design problems with SDP problems, and propose a numerical algorithm based on SeDuMi to solve these SDP problems. The algorithm is quite general under the least squares estimator (LSE) and weighted least squares estimator (WLSE) and can be applied to both linear and nonlinear regression models. We continue to study the optimal designs based on one-response model when the error distribution is asymmetric. Since the second-order least squares estimator (SLSE) is more efficient than the LSE when the error distribution is not symmetric, we study optimal designs under the SLSE. We derive expressions to characterize A- and D-optimality criteria and develop a numerical algorithm for finding optimal designs under the SLSE based on SeDuMi and CVX programs in MATLAB. Several theoretical properties are also derived for optimal designs under SLSE. To check the optimality of the numerical results, we establish the Kiefer-Wolfowitz equivalence theorem and apply it to various applications. Finally, we discuss the optimal design problems for multi-response models. Our algorithm studied here is based on SeDuMi and CVX, and it can be used for linear, nonlinear and generalized linear models. The transformation invariance property and dependence on the covariance matrix of the correlated errors are derived. We also correct the errors in the literature caused by formulation issues. The results are very useful to construct optimal regression designs on discrete design space. They can be applied to any one-response and multi-response models, various optimality criteria, and several estimators including LSE, maximum likelihood estimator, best linear unbiased estimator, SLSE and WLSE. / Graduate
82
#### Monotone regression functions
Zuo, Yanling January 1990 (has links)
In some applications, we require a monotone estimate of a regression function. In others, we want to test whether the regression function is monotone. For solving the first problem, Ramsay's, Kelly and Rice's, as well as point-wise monotone regression functions in a spline space are discussed and their properties developed. Three monotone estimates are defined: least-square regression splines, smoothing splines and binomial regression splines. The three estimates depend upon a "smoothing parameter": the number and location of knots in regression splines and the usual [formula omitted] in smoothing splines. Two standard techniques for choosing the smoothing parameter, GCV and AIC, are modified for monotone estimation, for the normal errors case. For answering the second question, a test statistic is proposed and its null distribution conjectured. Simulations are carried out to check the conjecture. These techniques are applied to two data sets. / Science, Faculty of / Statistics, Department of / Graduate
83
#### Regression Analysis of Swedens Power Consumption
Moloisel, Victor, Lind, Carl-Fredrik January 2022 (has links)
Energy Consumption is a topic of great interest, especially since a surge in prices in late 2021 has caused a considerable increase in discussion around the topic. Data from the Swedish Central Bureau of statistics (SCB) and the Swedish Meteorological Institute (SMHI) were provided for macroscopic regressors. These regressors are temperature, population, GDP, day length, electricity price, electricity production, production of variable renewable energy and average income in order to predict electricity consumption. Four models were created, a full multiple linear regression model using all regressors. A reduced multiple linear regression model using a subset of the regressors determined by cross validation. A ridge model and a LASSO model. These were then used to attempt to predict the power consumption of 20% of the data set that were left out when creating the models. The LASSO model was most successful in this as it had the smallest cumulative residual and the ridge model was the worst. Since the reduced and the full model both had very high multicollinearity the conclusion was that the LASSO model is the best model out of the four.
84
#### Predicting the power of an intraocular lens implant : an application of model selection theory
Diodati-Nolin, Anna C. January 1985 (has links)
No description available.
85
#### Distribution-free test for the equality of several regression lines /
Smith, Theodore MacDonald January 1977 (has links)
No description available.
86
#### Poisson regression /
Koo, Joo Ok January 1978 (has links)
No description available.
87
#### Confidence intervals for inverse regression with applications to blood hormone analysis
David, Richard. January 1974 (has links)
No description available.
88
#### An ordinal logistic regression model with misclassification of the outcome variable and categorical covariate.
Shirkey, Beverly Ann. Waring, Stephen Clay, January 2009 (has links)
Source: Dissertation Abstracts International, Volume: 70-03, Section: B, page: 1743. Advisers: Wenyaw Chan; Glasser H. Jay. Includes bibliographical references.
89
#### Characterization of the association between short-term variations in daily mortality and adverse environmental conditions using time series methodology
Guzman, Martha Elva Ramierez January 1990 (has links)
No description available.
90
#### An analysis of primary military occupational specialties on retention and promotion of mid-grade officers in the U.S. Marine Corps
Perry, Tracy A. 03 1900 (has links)
The purpose of this thesis is to identify and evaluate factors that affect retention and promotion of mid-grade officers in the U.S. Marine Corps. The analysis includes evaluation of survival patterns to ten-years of commissioned service and promotion patterns to O-4 and O-5. The primary goal is to explain the effect of an officersâ primary military occupational specialty (PMOS) on retention and promotion. The Marine Corps Commissioned Officer Accession Career (MCCOAC) data file contains cohort information from FY 1980 through FY 1999 and includes 27,659 observations. Using data from the MCCOAC data file, logistic regression and Cox Proportional Hazard models are used to estimate the effects of an officerâ s PMOS on survival and promotion patterns of Marine Corps officers. The findings indicate that an officers PMOS is significantly associated with whether an officer stays until 10 YCS or is promoted to O-4 or O-5. Logistic regression results show that pilot PMOSs are positively correlated with surviving until 10 YCS, but are negatively correlated with promotion to O-4, when compared to Infantry. The results also find that the remaining PMOSs are negatively correlated with whether and officer survives until 10 YCS, when compared to Infantry. In addition, only three PMOSs (0402, 7202, and 7523) are positively correlated with whether an officer is promoted to O-4 or O-5. Finally, the Cox Proportional Hazard results show the effect of having a particular PMOS or occupational field on the hazards of separation and promotion.
Page generated in 0.1537 seconds | 1,835 | 8,536 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.5625 | 3 | CC-MAIN-2024-22 | latest | en | 0.846305 |
http://www.fool.com/how-to-invest/personal-finance/taxes/2015/05/30/do-stock-dividends-affect-cost-basis.aspx?v=71837 | 1,481,285,443,000,000,000 | text/html | crawl-data/CC-MAIN-2016-50/segments/1480698542695.22/warc/CC-MAIN-20161202170902-00256-ip-10-31-129-80.ec2.internal.warc.gz | 464,098,071 | 19,015 | Cost basis is an important element of every investment you own, as it helps determine whether you'll have a taxable gain or loss when you sell. But many investors get confused about how dividends -- whether they be stock dividends or cash dividends -- affect cost basis. Let's take a closer look at the question to get you the answers you need to know.
What is cost basis?
The cost basis of an investment is the total cost of that investment, including the amount spent to purchase it, any commissions or fees associated with that purchase, and any other related costs. For tax purposes, the cost basis of an investment can be reduced by certain items, but only rarely. Most common (for businesses) are depreciation and depletion (e.g., oil, timber, minerals depletion allowances).
Accounting for cost basis reveals the true returns of investments, as high commissions or fees, either from high fee structures or frequent trading, reduce the net returns of the investment.
What dividends do to cost basis
Different types of dividends have different effects on cost basis. Cash dividends do not lower the cost basis of an investment, either when you actually receive cash or when you use the proceeds to purchase new shares. A stock dividend, however, does adjust cost basis, as does a "return of capital."
As an example, suppose you buy 37 shares of a company at \$45. Your broker charges you \$7.99 in commissions to handle that transaction for you (hey, they gotta eat too). What is your total cost, or basis?
Easy. 37 x \$45 + \$7.99 = \$1,672.99. This works out to be \$45.216 per share.
Cash dividends, as noted, do not reduce your basis, despite what the historical price section on Yahoo! Finance indicates (that's just to make it easier to calculate what total returns would be including reinvested dividends). However, splits and stock dividends do. For an example of the latter, see the dividends page.
For a split (like 3:2 or 2:1 or 3:1), you increase the number of shares by the split factor, which necessarily reduces the per share cost basis. Suppose that stock you purchased above splits 3:1. Your new basis would be \$1,672.99 / 111 shares = \$15.072 per share, now. (But your total basis, \$1,672.99, remains the same.)
As confusing as cost basis can be, dividends don't have to be overly intimidating. As long as you remember which types of dividends affect basis and which don't, you'll be in the best possible shape to handle tax issues when the time comes.
The Motley Fool has no position in any of the stocks mentioned. Try any of our Foolish newsletter services free for 30 days. We Fools may not all hold the same opinions, but we all believe that considering a diverse range of insights makes us better investors. The Motley Fool has a disclosure policy. | 611 | 2,789 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.8125 | 3 | CC-MAIN-2016-50 | longest | en | 0.933725 |
http://lib.mexmat.ru/books/94521 | 1,547,948,952,000,000,000 | text/html | crawl-data/CC-MAIN-2019-04/segments/1547583688396.58/warc/CC-MAIN-20190120001524-20190120023524-00380.warc.gz | 135,394,852 | 6,224 | Электронная библиотека Попечительского советамеханико-математического факультета Московского государственного университета
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Ganyushkin O., Mazorchuk V. — Classical Finite Transformation Semigroups.
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Название: Classical Finite Transformation Semigroups.
Авторы: Ganyushkin O., Mazorchuk V.
Аннотация:
“Among all transformation semigroups one can distinguish three classical series of semigroups: the full symmetric semigroup T (M) of all transformations of the set M; the symmetric inverse semigroup IS(M) of all partial (that is, not necessarily everywhere defined) injective transformations of M; and, finally, the semigroup PT (M) of all partial transformations of M. If M = {1, 2, . . . , n}, then the above semigroups are usually denoted by Tn, ISn and PTn, respectively….
The aim of the present book is to partially fill the gaps in the literature.
In the book we introduce three classical series of semigroups, and for them we describe generating systems, ideals, Green’s relations, various classes of subsemigroups, congruences, conjugations, endomorphisms, presentations, actions on sets, linear representations and cross-sections. Some of the results are very old and classical, some are quite young. In order not to overload the reader with too technical and specialized results, we decided to restrict the area of the present book to the above-mentioned parts of the theory of transformation semigroups.\
The book was thought to be an elementary introduction to the theory of transformation semigroups, with a strong emphasis on the concrete examples in the form of three classical series of finite transformation semigroups, namely, Tn, ISn and PTn.”
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© Электронная библиотека попечительского совета мехмата МГУ, 2004-2019 | | О проекте | 635 | 2,343 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.640625 | 3 | CC-MAIN-2019-04 | latest | en | 0.514922 |
http://loopspace.mathforge.org/HowDidIDoThat/GAS/SharingACell/ | 1,519,188,521,000,000,000 | text/html | crawl-data/CC-MAIN-2018-09/segments/1518891813431.5/warc/CC-MAIN-20180221044156-20180221064156-00280.warc.gz | 210,487,975 | 5,304 | Sharing a Cell
loopspace
2016-3-29
# 1 The Perils of Sharing
In teaching an introduction to the trigonometry ratios, I decided to use Google Sheets to amalgamate measurements from the class. Each student was to draw a right-angled triangle and measure the lengths of the sides and one of the angles (I prescribed the allowed angles for better further comparison, also they were to draw the triangles using Geogebra). They recorded this data in a shared Google Sheet.
Although it worked, and we got lots of data, there were a few teething problems with using the shared sheet, the most common being students trying to edit the same row. There were also a few occasions where one student would (unintentionally, I'm sure) edit another student's readings.
So for the second lesson using this sheet, I did a bit of searching and found that it was possible to lock down students' editing rights to specific cells in the sheet. Giving each student two or three rows in which to record their data, and locking them out from any other students' rows, seemed a reasonable precaution to take.
The irritation being having to do this by hand for each student.
# 2 Enter the Scrypt
A not-very-chance remark from a colleague led me to the discovery that Google Stuff (i.e., Google Docs, Google Sheets, and all the other Google bits and bobs) is scriptable. What made it particularly intriguing was that:
1. The language was javascript, and
2. The entire Google ecosystem was scriptable.
The first meant that I didn't have to learn a completely new language. I'd done a bit of javascript already so was familiar with its basic syntax and, more importantly, knew where to look for help. It took me a while to realise the ramifications of the second, but over time I've come to discover that I can use Google Apps Scripts pretty much like a UNIX command line and am able to work with documents as well as within documents.
For this specific purpose, it means that I can use one spreadsheet as my source data for building a new spreadsheet with the editing rights that I want.
# 3 Use the Source, Luke
To use this script, you need to start with a list of your class's Google account email addresses and their names. However you get these, you want them in a spreadsheet. Although it wouldn't be hard to adapt the following to any way of having the data in the sheet, I'll assume that they are so that the names are in Column A and the ids in Column B.
Another thing that is particularly useful to know before beginning is that in the Google ecosystem, files and folders are best identified by id rather than name. When you open a file or folder in Google, the URL in the location bar in your browser will read something like:
https://docs.google.com/spreadsheets/d/some long string of letters, numbers, and other characters/edit#gid=0
The id is that long string of letters, numbers, and other characters. At various times you'll need that id for a particular document. Easiest way to get it is to have the document open in your browser and cut-and-paste it from the location bar.
Lastly, sometimes when sharing a document with another person then that person gets sent an email about it. So when testing scripts like this one it is best to start with a dummy list of users. Using a list with Google accounts that you own is best.
# 4 That's the Way to Do It
Possibly. This may not be the most elegant method. In particular, there's no fancy UI (user interface). But as a “script kiddie", I find it easier to edit a few key variables at the top of the script than to enter them via a UI.
## 4.1 Getting Off the Drive
The best starting point for all mucking about with Google is Google Drive. So open up your Google Drive in your browser.
If you don't already have “Google Apps Script" in the “New" menu, click on “New", select “More", and then “Connect more Apps". In the search box, type “Google Apps Script" and click the “connect" button. (If there isn't a “connect" button and it just says “Rate it" then you already have it.)
Now under “New" and then “More" you should have an entry “Google Apps Script". Click on that to open a new script. This opens up a script editor in your browser. You can click on the Untitled project to rename it to something sensible. In the text editor it should read:
function myFunction() {
}
We can delete that as we'll define a new function.
You should also have your spreadsheet with the students' names and emails open so that you can get the id.
## 4.2 It's All About the Script
Here's the script (you can also download it). Copy and paste this into the text editor in the browser.
function shareCells() {
var studentId = ""; // id of spreadsheet with students' names and emails
var folderId = ""; // id of folder where final spreadsheet should be put
var sheetName = "Data Sheet"; // title of spreadsheet
var studentsCell = 'A2'; // upper-left cell of student details block
var editCell = 'A3'; // upper-left cell of range to be made editable (first column will contain names and not be editable)
var editRows = 2; // number of rows per student
var editCols = 4; // number of columns per student
var studentSheet = SpreadsheetApp.openById(studentId);
// Top left cell of student data
var studentUL = studentSheet.getRange(studentsCell);
// Array of student data
var students = studentUL.offset(0,0,studentSheet.getLastRow() - studentUL.getRow()+1,2).getValues();
// Array of just user emails
var emails = [];
for (var i=0; i<students.length; i++) {
emails.push(students[i][1]);
}
// Create new spreadsheet
var ss = SpreadsheetApp.create(sheetName);
// Get the corresponding file
var sfile = DriveApp.getFileById(ss.getId());
// Get the current folder containing the file (this will be the root of your Drive folder)
var root = sfile.getParents().next();
// Add the file to the folder where it should be put
// Remove it from the original folder
root.removeFile(sfile);
// Now set the global permissions, make it so that the students can't reshare this file
sfile.setShareableByEditors(false);
// Add all students as editors
// NB sharing via the SpreadsheetApp means that notification emails aren't sent, if you want to send emails
// comment out the above line and use the following one instead
// Get the first (and only) sheet
var sheet = ss.getSheets()[0];
// The initial protection seals off the whole sheet apart from the region that will be editable by the students
var protection = sheet.protect().setDescription('Protect entire sheet');
// Get the top left cell of the editable region
var editUL = sheet.getRange(editCell);
// Get the full editable range
// The first column contains the names (and is not editable) so we offset by 1 horizontally
var editRange = editUL.offset(0,1,editRows*students.length,editCols);
// Our protected range is everything except this area
protection.setUnprotectedRanges([editRange]);
// Remove all students from being able to edit the protected range
protection.removeEditors(emails);
// Now we loop through the students, giving each access to a block
var name, email, cell, range;
for (var i=0; i<students.length; i++) {
// Convenience storage for the student's name and email
name = students[i][0];
email = students[i][1];
// Get the cell where the name will go
cell = editUL.offset(i*editRows,0,1,1);
// And put the name there
cell.setValue(name);
// Now get the range where they'll be able to edit
range = cell.offset(0,1,editRows,editCols);
// Add a protection to that range
protection = range.protect();
// Remove all students' edit rights
protection.removeEditors(emails);
// But add this student's rights
// Stick a border around it to make it obvious where it is
range.setBorder(true,true,true,true,false,false);
}
}
The configuration details are at the start.
• studentId This is the id of the spreadsheet containing the students' details.
• folderId This is the id of the folder where the new spreadsheet should be created.
• sheetName This is the name of the new spreadsheet.
• studentsCell This is the cell at the top left of the students' data. You might want a header row, or the data might be part of a larger set of information. The first column of the data should contain the names, and the second the emails.
• editCell This is the cell at the top left of the region that will be editable. In actual fact, the first column will not be editable as it will contain the students' names.
• editRows This is the number of rows of a students' editable block.
• editCols This is the number of columns of a students' editable block.
## 4.3 I Invoke Thee, O Scrypt
Running the script is straightforward. Save the script by clicking on the usual file save icon. (You can see when a file needs saving by looking at its name above the code; if there is a red star next to it then it has unsaved changes.)
To the right of the icons is a function name. If that is not showing shareCells, click on the drop down arrow and select it from the list. You can then run the script by pressing the black triangle in the icon bar. Or you can select “Run" from the menu list.
The students can access the sheet via their Drives in the “Shared with me" folder. | 2,094 | 9,173 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.875 | 3 | CC-MAIN-2018-09 | latest | en | 0.977909 |
http://www.dotnetspark.com/links/30023-selecting-multiple-dimensions-members.aspx | 1,477,690,379,000,000,000 | text/html | crawl-data/CC-MAIN-2016-44/segments/1476988725475.41/warc/CC-MAIN-20161020183845-00201-ip-10-171-6-4.ec2.internal.warc.gz | 409,477,040 | 9,691 | Welcome :Guest
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ASP.Net Windows Application .NET Framework C# VB.Net ADO.Net Sql Server SharePoint Silverlight Others All | 2,325 | 10,030 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.859375 | 3 | CC-MAIN-2016-44 | longest | en | 0.8037 |
https://homeworkhelper-in.com/math/question15615388 | 1,660,806,968,000,000,000 | text/html | crawl-data/CC-MAIN-2022-33/segments/1659882573172.64/warc/CC-MAIN-20220818063910-20220818093910-00434.warc.gz | 270,233,124 | 16,137 | 1 2y – 8=2 solve this
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When a two-digit number is divided by the sum of its digits, the quotient is 7 and the remainder is 6. if one of the digits of the number is three, then what is the difference of the digits? | 248 | 614 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.421875 | 3 | CC-MAIN-2022-33 | latest | en | 0.764882 |
https://mathematica.stackexchange.com/questions/10391/is-mathematica-matrix-multiplication-with-its-inverse-wrong | 1,571,877,795,000,000,000 | text/html | crawl-data/CC-MAIN-2019-43/segments/1570987836368.96/warc/CC-MAIN-20191023225038-20191024012538-00539.warc.gz | 569,945,865 | 28,670 | # Is Mathematica matrix multiplication with its inverse wrong? [duplicate]
Possible Duplicate:
Why don't * and ^ work as I expected on matrices?
When I enter this
Inverse[{{-112, 49, 2, 283}, {-138, 5, 3, 359}, {-20, 0, 0, 6}, {-40, -20, 0, 12}}] *
{{-112, 49, 2, 283}, {-138, 5, 3, 359}, {-20, 0, 0, 6}, {-40, -20, 0, 12}}
I don't get the identity matrix, but this output:
Can anybody explain why and how I can fix it?
## marked as duplicate by F'x, Leonid Shifrin, J. M. will be back soon♦Sep 10 '12 at 9:46
You need to use Dot for matrix multiplication; * gives element-wise multiplication.
mat = {{-112, 49, 2, 283}, {-138, 5, 3, 359}, {-20, 0, 0, 6}, {-40, -20, 0, 12}} ; | 265 | 684 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.65625 | 3 | CC-MAIN-2019-43 | latest | en | 0.875229 |
https://www.w3resource.com/python-exercises/python-basic-exercise-134.php | 1,723,341,505,000,000,000 | text/html | crawl-data/CC-MAIN-2024-33/segments/1722640843545.63/warc/CC-MAIN-20240811013030-20240811043030-00489.warc.gz | 841,746,465 | 27,302 | Python: Input two integers in a single line - w3resource
# Python: Input two integers in a single line
## Python Basic: Exercise-134 with Solution
Write a Python program to input two integers on a single line.
Sample Solution-1:
Python Code:
``````# Print a message to instruct the user to input the values of 'x' and 'y'.
print("Input the value of x & y")
# Use 'map' to apply the 'int' function to the input values and split them into variables 'x' and 'y'.
x, y = map(int, input().split())
# Print the values of 'x' and 'y' after they have been assigned.
print("The value of x & y are: ", x, y)
```
```
Sample Output:
```Input the value of x & y
2 4
The value of x & y are: 2 4
```
Sample Solution-2:
Python Code:
``````# Prompt the user to input the values of 'a' and 'b' and split the input by spaces.
a, b = [int(a) for a in input("Input the value of a & b: ").split()]
# Print the values of 'a' and 'b' after they have been assigned.
print("The value of a & b are:", a, b)
```
```
Sample Output:
```Input the value of a & b: 2 4
The value of a & b are: 2 4
```
Python Code Editor:
What is the difficulty level of this exercise?
Test your Programming skills with w3resource's quiz.
| 351 | 1,213 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.921875 | 3 | CC-MAIN-2024-33 | latest | en | 0.776644 |
https://mathhelpboards.com/threads/problem-of-the-week-18-october-1st-2012.2002/ | 1,643,173,109,000,000,000 | text/html | crawl-data/CC-MAIN-2022-05/segments/1642320304915.53/warc/CC-MAIN-20220126041016-20220126071016-00637.warc.gz | 454,662,446 | 14,540 | Problem of the Week #18 - October 1st, 2012
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Chris L T521
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Here's this week's problem.
-----
Problem: Suppose that $T\in L(X)$ is a bounded linear operator in a Banach space $X$ such that $\|T\|<1$. Show that $I-T$ is invertible, i.e. has a bounded inverse linear operator and
$(I-T)^{-1}=\sum_{k=0}^{\infty}T^k.$
-----
Let $S_n:=\sum_{j=0}^nT^j$. We have $$\lVert S_{m+n}(x)-S_n(x)\rVert=\lVert\sum_{j=n+1}^{n+m}T^jx\rVert\le \sum_{j=n+1}^{n+m}\lVert T^j\rVert \lVert x\rVert\le \lVert x\rVert \sum_{j=n+1}^{n+m}\lVert T\rVert^j,$$
which proves that $\{S_nx\}$ is Cauchy for each $x$ ($\lVert T\rVert<1$) hene it converges to some $Sx$. We have $S(I-T)x=\lim_{n\to \infty}(I-T^{n+1})x=x$ as $\lVert T\rVert<1$ and $(I-T)S=I$. This proves that $I-T$ is invertible with inverse $S$. This one is bounded as $\lVert Sx\rVert\leq \frac 1{1-\lVert T\rVert}\lVert x\rVert$. | 375 | 940 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.578125 | 4 | CC-MAIN-2022-05 | latest | en | 0.789112 |
https://www.teacherspayteachers.com/Product/Adding-and-Subtracting-Decimals-Color-by-Number-Worksheets-5th-6th-Grade-Math-1606735?utm_source=Cog%20Cardio%20Add%20Subtract%20Decimals&utm_campaign=AddSubDecCBN | 1,685,327,883,000,000,000 | text/html | crawl-data/CC-MAIN-2023-23/segments/1685224644574.15/warc/CC-MAIN-20230529010218-20230529040218-00463.warc.gz | 1,105,180,942 | 50,949 | EASEL BY TPT
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# Adding and Subtracting Decimals Color by Number Worksheets 5th-6th Grade Math
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Cognitive Cardio Math
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4th - 6th, Homeschool
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Cognitive Cardio Math
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The Teacher-Author indicated this resource includes assets from Google Workspace (e.g. docs, slides, etc.).
#### What educators are saying
This was a great activity for students to do. I held onto it for a day that they needed a super engaging activity. Students loved it and keep asking for more color by number activities!
I used this in my 6th grade Hybrid teaching classroom. Students were able to quickly demonstrate their understanding of adding and subtracting decimals. It also made grading very quick!
##### Also included in
1. These math color by number activities offer the chance to add the mindfulness, wellness, and quietness coloring brings, while engaging students in practice with algebraic expressions and equations, decimals, equivalent expressions, inequalities, percent of a number, ratios and proportions, and more!
Price \$38.00Original Price \$72.75Save \$34.75
2. This math color by number bundle for 4th - 7th grades includes 60 separate easy-prep color by number resources for students to practice with fractions, decimals, multi-digit multiplication, long division, absolute value, exponents, equations and more!These math color by number activities offer the c
Price \$125.00Original Price \$209.74Save \$84.74
3. Do you need 6th grade math activities to use throughout the entire school year? I've got you covered! Save yourself HOURS of planning and prepping time with the wide variety of notes and activities in this bundle!Give your students some new and creative ways to look at math concepts on a daily basis
Price \$265.00Original Price \$490.99Save \$225.99
4. This HUGE middle school math full-year resources bundle includes a TON of material that addresses 6th grade and 7th grade math standards!Save yourself HOURS of planning and prepping time with the wide variety of notes and activities in this bundle!Give your students some new and creative ways to loo
Price \$449.00Original Price \$791.74Save \$342.74
5. This bundle includes 19 different resources that address all decimal operations. There's a variety of print and digital resources, AND there's also a 46-page 'Bonus file' that's only available with this bundle. The Bonus File includes:Decimal Matching Card Sets - cards that match standard form and w
Price \$30.00Original Price \$58.25Save \$28.25
### Description
Students solve 20 decimal addition and subtraction problems, self-check and color, in both the print and digital versions.
• Great self-checking activity. Students will immediately know whether or not they solved correctly - if their answer is incorrect, they won't find it on the coloring sheet!
Check out the Preview and video demo above to see what's included and to see how the Google Slides version works (students will need to know how to use the "Fill color" feature).
The problems in this decimal resource require students to:
• Add and subtract two decimal numbers, to the thousandths place
• Add and subtract decimals that go to different place values (tenths plus hundredths, etc)
• Add and subtract decimal numbers and whole numbers
• Solve word problems that require adding and subtracting decimals (9 word problems)
This easy-prep adding and subtracting decimals resource includes:
1) Two Print versions
Version A:
• Problem sheet with 20 questions (9 word problems)
• Coloring sheet
Version B:
• Problem sheet with 20 questions (SAME questions as Version A)
• Coloring sheet: same pattern as Version A; different colors
• SAME problems as print versions and mostly same coloring as Version B
• Students use the ‘Fill color’ bucket to color the shapes
• Questions are not editable (they are part of the background) but the shapes and numbers in the pattern ARE movable/editable.
3) Keys
4) A coloring page with no numbers, for students to color their own way, for fun:-)
⭐️ Need just Adding Decimals or just Subtracting Decimals for extra reinforcement? Check out those color by numbers!
⭐️⭐️⭐️⭐️⭐️ "I used this in my 6th grade Hybrid teaching classroom. Students were able to quickly demonstrate their understanding of adding and subtracting decimals. It also made grading very quick!"
⭐️⭐️⭐️⭐️⭐️ "This was a great activity for students to do. I held onto it for a day that they needed a super engaging activity. Students loved it and keep asking for more color by number activities!"
⭐️⭐️⭐️⭐️ "Great resource! Students found it challenging. Loved that they had to line up decimals of whole numbers as well! Also, great review of phrases that differentiate addition and subtraction!"
⭐️⭐️⭐️⭐️⭐️ "My students LOVED this resource!"
Related Resources:
Subtracting Decimals Math Doodle Wheel
Adding and Subtracting Decimals Color by Number
Decimal Operations Activity Bundle
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4th-5th Grade Color by Number Bundle
5th-7th Grade Color by Number Bundle
4th Grade Color by Number Mini Bundle
5th Grade Color by Number Mini-Bundle
5th Grade Math Doodle Wheel Bundle
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Permission to copy for single classroom use only.
Total Pages
16-pg PDF with 2 print versions, key, blank coloring page, link to Google version
Included
Teaching Duration
40 minutes
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### Standards
to see state-specific standards (only available in the US).
Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.
Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation. | 1,472 | 6,704 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.859375 | 3 | CC-MAIN-2023-23 | latest | en | 0.909523 |
https://nrich.maths.org/public/topic.php?code=-99&cl=2&cldcmpid=968 | 1,604,025,552,000,000,000 | text/html | crawl-data/CC-MAIN-2020-45/segments/1603107906872.85/warc/CC-MAIN-20201030003928-20201030033928-00397.warc.gz | 459,257,384 | 9,612 | # Resources tagged with: Working systematically
Filter by: Content type:
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### There are 342 results
Broad Topics > Thinking Mathematically > Working systematically
### Journeys in Numberland
##### Age 7 to 11 Challenge Level:
Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.
### Seven Square Numbers
##### Age 7 to 11 Challenge Level:
Add the sum of the squares of four numbers between 10 and 20 to the sum of the squares of three numbers less than 6 to make the square of another, larger, number.
### Sums and Differences 1
##### Age 7 to 11 Challenge Level:
This challenge focuses on finding the sum and difference of pairs of two-digit numbers.
### On Target
##### Age 7 to 11 Challenge Level:
You have 5 darts and your target score is 44. How many different ways could you score 44?
### How Old?
##### Age 7 to 11 Challenge Level:
Cherri, Saxon, Mel and Paul are friends. They are all different ages. Can you find out the age of each friend using the information?
### How Much Did it Cost?
##### Age 7 to 11 Challenge Level:
Use your logical-thinking skills to deduce how much Dan's crisps and ice-cream cost altogether.
### Hubble, Bubble
##### Age 7 to 11 Challenge Level:
Winifred Wytsh bought a box each of jelly babies, milk jelly bears, yellow jelly bees and jelly belly beans. In how many different ways could she make a jolly jelly feast with 32 legs?
##### Age 7 to 11 Challenge Level:
Can you put plus signs in so this is true? 1 2 3 4 5 6 7 8 9 = 99 How many ways can you do it?
### Build it up More
##### Age 7 to 11 Challenge Level:
This task follows on from Build it Up and takes the ideas into three dimensions!
### ABC
##### Age 7 to 11 Challenge Level:
In the multiplication calculation, some of the digits have been replaced by letters and others by asterisks. Can you reconstruct the original multiplication?
### A Mixed-up Clock
##### Age 7 to 11 Challenge Level:
There is a clock-face where the numbers have become all mixed up. Can you find out where all the numbers have got to from these ten statements?
### Six Is the Sum
##### Age 7 to 11 Challenge Level:
What do the digits in the number fifteen add up to? How many other numbers have digits with the same total but no zeros?
### Arranging the Tables
##### Age 7 to 11 Challenge Level:
There are 44 people coming to a dinner party. There are 15 square tables that seat 4 people. Find a way to seat the 44 people using all 15 tables, with no empty places.
### Rabbits in the Pen
##### Age 7 to 11 Challenge Level:
Using the statements, can you work out how many of each type of rabbit there are in these pens?
### Oh! Harry!
##### Age 7 to 11 Challenge Level:
A group of children are using measuring cylinders but they lose the labels. Can you help relabel them?
### A-magical Number Maze
##### Age 7 to 11 Challenge Level:
This magic square has operations written in it, to make it into a maze. Start wherever you like, go through every cell and go out a total of 15!
### The Pied Piper of Hamelin
##### Age 7 to 11 Challenge Level:
This problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!
### Zargon Glasses
##### Age 7 to 11 Challenge Level:
Zumf makes spectacles for the residents of the planet Zargon, who have either 3 eyes or 4 eyes. How many lenses will Zumf need to make all the different orders for 9 families?
### X Is 5 Squares
##### Age 7 to 11 Challenge Level:
Can you arrange 5 different digits (from 0 - 9) in the cross in the way described?
### Pouring the Punch Drink
##### Age 7 to 11 Challenge Level:
There are 4 jugs which hold 9 litres, 7 litres, 4 litres and 2 litres. Find a way to pour 9 litres of drink from one jug to another until you are left with exactly 3 litres in three of the jugs.
### Forgot the Numbers
##### Age 7 to 11 Challenge Level:
On my calculator I divided one whole number by another whole number and got the answer 3.125. If the numbers are both under 50, what are they?
### Sums and Differences 2
##### Age 7 to 11 Challenge Level:
Find the sum and difference between a pair of two-digit numbers. Now find the sum and difference between the sum and difference! What happens?
### Prison Cells
##### Age 7 to 11 Challenge Level:
There are 78 prisoners in a square cell block of twelve cells. The clever prison warder arranged them so there were 25 along each wall of the prison block. How did he do it?
### The Dice Train
##### Age 7 to 11 Challenge Level:
This dice train has been made using specific rules. How many different trains can you make?
### Spell by Numbers
##### Age 7 to 11 Challenge Level:
Can you substitute numbers for the letters in these sums?
### Dart Target
##### Age 7 to 11 Challenge Level:
This task, written for the National Young Mathematicians' Award 2016, invites you to explore the different combinations of scores that you might get on these dart boards.
### Being Resourceful - Primary Number
##### Age 5 to 11 Challenge Level:
Number problems at primary level that require careful consideration.
### Today's Date - 01/06/2009
##### Age 5 to 11 Challenge Level:
What do you notice about the date 03.06.09? Or 08.01.09? This challenge invites you to investigate some interesting dates yourself.
### The Puzzling Sweet Shop
##### Age 5 to 11 Challenge Level:
There were chews for 2p, mini eggs for 3p, Chocko bars for 5p and lollypops for 7p in the sweet shop. What could each of the children buy with their money?
### Two Egg Timers
##### Age 7 to 11 Challenge Level:
You have two egg timers. One takes 4 minutes exactly to empty and the other takes 7 minutes. What times in whole minutes can you measure and how?
### Twenty Divided Into Six
##### Age 7 to 11 Challenge Level:
Katie had a pack of 20 cards numbered from 1 to 20. She arranged the cards into 6 unequal piles where each pile added to the same total. What was the total and how could this be done?
### Dice and Spinner Numbers
##### Age 7 to 11 Challenge Level:
If you had any number of ordinary dice, what are the possible ways of making their totals 6? What would the product of the dice be each time?
### Multiply Multiples 2
##### Age 7 to 11 Challenge Level:
Can you work out some different ways to balance this equation?
### Seating Arrangements
##### Age 7 to 11 Challenge Level:
Sitting around a table are three girls and three boys. Use the clues to work out were each person is sitting.
### Team Scream
##### Age 7 to 11 Challenge Level:
Seven friends went to a fun fair with lots of scary rides. They decided to pair up for rides until each friend had ridden once with each of the others. What was the total number rides?
### Consecutive Numbers
##### Age 7 to 14 Challenge Level:
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
### Octa Space
##### Age 7 to 11 Challenge Level:
In the planet system of Octa the planets are arranged in the shape of an octahedron. How many different routes could be taken to get from Planet A to Planet Zargon?
### Sealed Solution
##### Age 7 to 11 Challenge Level:
Ten cards are put into five envelopes so that there are two cards in each envelope. The sum of the numbers inside it is written on each envelope. What numbers could be inside the envelopes?
### Route Product
##### Age 7 to 11 Challenge Level:
Find the product of the numbers on the routes from A to B. Which route has the smallest product? Which the largest?
### One to Fifteen
##### Age 7 to 11 Challenge Level:
Can you put the numbers from 1 to 15 on the circles so that no consecutive numbers lie anywhere along a continuous straight line?
### Greater Than or Less Than?
##### Age 7 to 11 Challenge Level:
Use the numbers and symbols to make this number sentence correct. How many different ways can you find?
### Code Breaker
##### Age 7 to 11 Challenge Level:
This problem is based on a code using two different prime numbers less than 10. You'll need to multiply them together and shift the alphabet forwards by the result. Can you decipher the code?
### The Moons of Vuvv
##### Age 7 to 11 Challenge Level:
The planet of Vuvv has seven moons. Can you work out how long it is between each super-eclipse?
### Five Coins
##### Age 5 to 11 Challenge Level:
Ben has five coins in his pocket. How much money might he have?
### Coins (2)
##### Age 7 to 11 Challenge Level:
What is the smallest number of coins needed to make up 12 dollars and 83 cents?
### Multiply Multiples 3
##### Age 7 to 11 Challenge Level:
Have a go at balancing this equation. Can you find different ways of doing it?
### Multiplication Squares
##### Age 7 to 11 Challenge Level:
Can you work out the arrangement of the digits in the square so that the given products are correct? The numbers 1 - 9 may be used once and once only.
### Two Primes Make One Square
##### Age 7 to 11 Challenge Level:
Can you make square numbers by adding two prime numbers together?
### Calendar Cubes
##### Age 7 to 11 Challenge Level:
Make a pair of cubes that can be moved to show all the days of the month from the 1st to the 31st.
### Multiply Multiples 1
##### Age 7 to 11 Challenge Level:
Can you complete this calculation by filling in the missing numbers? In how many different ways can you do it? | 2,282 | 9,522 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.25 | 4 | CC-MAIN-2020-45 | latest | en | 0.86571 |
https://www.coursehero.com/file/214804/MT2Sol/ | 1,513,388,737,000,000,000 | text/html | crawl-data/CC-MAIN-2017-51/segments/1512948581033.57/warc/CC-MAIN-20171216010725-20171216032725-00213.warc.gz | 728,430,111 | 60,891 | # MT2Sol - Solutions for the second midterm 1 Let C be the...
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Solutions for the second midterm 1. Let C be the helix given by x ( t ) = cos t, y ( t ) = sin t, z ( t ) = t for 0 t 2 π . (a) Integrate f ( x, y, z ) = x + 3 z 2 along C with respect to arc length. We have been given a parametrization; using it, we have ~ r 0 ( t ) = sin t~ ı + cos t~ + ~ k and | ~ r 0 ( t ) | = p sin 2 t + cos 2 t + 1 = 2 . Writing the function in terms of this parametrization gives f ( x ( t ) , y ( t ) , z ( t )) = cos t +3 t 2 . We have Z C f ds = Z 2 π 0 ( cos t + 3 t 2 ) 2 dt = 2 ± sin t + t 3 ² 2 π 0 = 8 2 π 3 . (b) Compute R C ~ F · d~ r where ~ F ( x, y, z ) = z 2 ~ ı + z~ +(2 xz + y ) ~ k . (Hint: ~ F is conservative.) Computing the integral directly doesn’t look promising, so instead we follow the hint and find the potential function. Integrating with respect to x gives f = xz 2 + g ( y, z ). Differ- entiating with respect to y gives ∂g/∂y = z and thus g ( y, z ) = yz + h ( z ). Differentiating with respect to z gives 2 xz + y + h 0 ( z ) = 2 xz + y , so that h ( z ) = K for some constant K . We conclude that the potential is f = xz 2 + yz + K for any constant K . Note that the curve C begins at (1 , 0 , 0) and ends at (1 , 0 , 2 π ). We have Z C ~ F · d~ r = f (1 , 0 , 2 π ) - f (1 , 0 , 0) = 4 π 2 - 0 = 4 π 2 . 2. Let C be the curve consisting of the following four line segments: from (1 , 0 , 0) to (1 , 1 , 0) , followed by (1 , 1 , 0) to (0 , 1 , 1) , followed by (0 , 1 , 1) to (0 , 0 , 1) , followed by (0 , 0 , 1) to (1 , 0 , 0) . In particular, C is a rectangle which lies in the plane
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Ask a homework question - tutors are online | 750 | 2,090 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.8125 | 4 | CC-MAIN-2017-51 | latest | en | 0.796144 |
https://www.numbersaplenty.com/3491714 | 1,722,865,416,000,000,000 | text/html | crawl-data/CC-MAIN-2024-33/segments/1722640447331.19/warc/CC-MAIN-20240805114033-20240805144033-00149.warc.gz | 710,227,840 | 3,027 | Search a number
3491714 = 22277691
BaseRepresentation
bin1101010100011110000010
320120101201202
431110132002
51343213324
6202501202
741451632
oct15243602
96511652
103491714
111a75416
121204802
13953405
1466c6c2
1548e8ae
hex354782
3491714 has 8 divisors (see below), whose sum is σ = 5261328. Its totient is φ = 1737940.
The previous prime is 3491711. The next prime is 3491723. The reversal of 3491714 is 4171943.
It is a sphenic number, since it is the product of 3 distinct primes.
It is not an unprimeable number, because it can be changed into a prime (3491711) by changing a digit.
It is a polite number, since it can be written in 3 ways as a sum of consecutive naturals, for example, 3392 + ... + 4299.
It is an arithmetic number, because the mean of its divisors is an integer number (657666).
Almost surely, 23491714 is an apocalyptic number.
3491714 is a deficient number, since it is larger than the sum of its proper divisors (1769614).
3491714 is a wasteful number, since it uses less digits than its factorization.
3491714 is an evil number, because the sum of its binary digits is even.
The sum of its prime factors is 7920.
The product of its digits is 3024, while the sum is 29.
The square root of 3491714 is about 1868.6128544993. The cubic root of 3491714 is about 151.7095387931.
The spelling of 3491714 in words is "three million, four hundred ninety-one thousand, seven hundred fourteen".
Divisors: 1 2 227 454 7691 15382 1745857 3491714 | 451 | 1,474 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.140625 | 3 | CC-MAIN-2024-33 | latest | en | 0.836296 |
https://statcompute.wordpress.com/2015/08/ | 1,519,047,147,000,000,000 | text/html | crawl-data/CC-MAIN-2018-09/segments/1518891812665.41/warc/CC-MAIN-20180219131951-20180219151951-00152.warc.gz | 743,528,673 | 11,074 | I can calculate the motion of heavenly bodies but not the madness of people. -Isaac Newton
## Some Considerations of Modeling Severity in Operational Losses
In the Loss Distributional Approach (LDA) for Operational Risk models, multiple distributions, including Log Normal, Gamma, Burr, Pareto, and so on, can be considered candidates for the distribution of severity measures. However, the challenge remains in the stress testing exercise, e.g. CCAR, to relate operational losses to macro-economic scenarios denoted by a set of macro-economic attributes.
As a result, a more sensible approach employed in the annual CCAR exercise to model operational losses might be the regression-based modeling approach, which can intuitively link the severity measure of operational losses to macro-economic drivers with a explicit functional form within the framework of Generalized Linear Models (GLM). While 2-parameter Pareto distribution and 3-parameter Burr distribution are theoretically attractive, their implmentations in the regression setting could become difficult and even impractical without the availability of off-shelf modeling tools and variable selection routines. In such situation, Log Normal and Gamma distributional assumptions are much more realistic with successful applications in actuarial practices. For details, please see “Severity Distributions for GLMs” by Fu and Moncher in 2004.
While both Log Normal and Gamma are most popular choices for the severity model, there are pros and cons in each respectively. For instance, while Log Normal distributional assumption is extremely flexible and easy to understand, the predicted outcomes should be adjusted for the estimation bias. Fortunately, both SAS, e.g. SEVERITY PROCEDURE, and R, e.g. fitdistrplus package, provide convenient interfaces for the distribution selection procedure based on goodness-of-fit statistics and information criterion.
```
library(fitdistrplus)
library(insuranceData)
Fit1 <- fitdist(AutoCollision\$Severity, dist = "lnorm", method = "mme")
Fit2 <- fitdist(AutoCollision\$Severity, dist = "gamma", method = "mme")
gofstat(list(Fit1, Fit2))
#Goodness-of-fit statistics
# 1-mme-lnorm 2-mme-gamma
#Kolmogorov-Smirnov statistic 0.1892567 0.1991059
#Cramer-von Mises statistic 0.2338694 0.2927953
#Anderson-Darling statistic 1.5772642 1.9370056
#
#Goodness-of-fit criteria
# 1-mme-lnorm 2-mme-gamma
#Aikake's Information Criterion 376.2738 381.2264
#Bayesian Information Criterion 379.2053 384.1578
```
In the above output, Log Normal seems marginally better than Gamma in this particular case. Since either Log(SEVERITY) in Log Normal or SEVERITY in Gamma belongs to exponential distribution family, it is convenient to employ GLM() with related variable selection routines in the model development exercise.
```
summary(mdl1 <- glm(log(Severity) ~ -1 + Vehicle_Use, data = AutoCollision, family = gaussian(link = "identity")))
#Coefficients:
# Estimate Std. Error t value Pr(>|t|)
#Vehicle_UseBusiness 5.92432 0.07239 81.84 <2e-16 ***
#Vehicle_UseDriveLong 5.57621 0.07239 77.03 <2e-16 ***
#Vehicle_UseDriveShort 5.43405 0.07239 75.07 <2e-16 ***
#Vehicle_UsePleasure 5.35171 0.07239 73.93 <2e-16 ***
summary(mdl2 <- glm(Severity ~ -1 + Vehicle_Use, data = AutoCollision, family = Gamma(link = "log")))
#Coefficients:
# Estimate Std. Error t value Pr(>|t|)
#Vehicle_UseBusiness 5.97940 0.08618 69.38 <2e-16 ***
#Vehicle_UseDriveLong 5.58072 0.08618 64.76 <2e-16 ***
#Vehicle_UseDriveShort 5.44560 0.08618 63.19 <2e-16 ***
#Vehicle_UsePleasure 5.36225 0.08618 62.22 <2e-16 ***
```
As shown above, estimated coefficients are very similar in both Log Normal and Gamma regressions and standard erros are different due to different distributional assumptions. However, please note that predicted values of Log Normal regression should be adjusted by (RMSE ^ 2) / 2 before applying EXP().
Written by statcompute
August 16, 2015 at 2:58 pm
Posted in CCAR, Operational Risk, S+/R
Tagged with , , | 1,083 | 4,198 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.609375 | 3 | CC-MAIN-2018-09 | longest | en | 0.882071 |
http://cameraauctions.tk/peju/what-is-a-normal-line-jel.php | 1,548,258,072,000,000,000 | text/html | crawl-data/CC-MAIN-2019-04/segments/1547584334618.80/warc/CC-MAIN-20190123151455-20190123173455-00195.warc.gz | 38,277,924 | 5,106 | # What is a normal line
So we have the function f of x is equal to e to the x over x squared.Therefore the normal of the line at the midpoint is the normal at any other point also.On block-level elements, it specifies the minimum height of line boxes within the element.Free normal line calculator - find the equation of the normal line given a point or the intercept step-by-step.
### What Is a Normal White Blood Cell Count?
The normal line is defined as the line that is perpendicular to the tangent line at the point of tangency.The line-height property can accept the keyword values normal or none as well as a number, length, or percentage.
Actually, there are a couple of applications, but they all come back to needing the first one.If you want to draw the normal, it looks nice, when you start it at the midpoint of the line segment. but in a mathematical sense it is correct to start it from any other point of the X-Y-plane as well, e.g. at the origin.
Change line spacing for part of your document Select the paragraphs you want to change.
### What's Normal for My Newborn? - Parents
For example, in the two-dimensional case, the normal line to a curve at a given point is the line perpendicular to the tangent line to the curve at the point.
On the Home tab, in the Styles group, right-click Normal, and then click Modify.
### Q-Q plots - Free Statistics Book
On non-replaced inline elements, it specifies the height that is used to calculate line box height.
Strengthening your core may be just what you need to keep your low back posture in the normal range, even though the exact numbers may not be available to you.
### USGS Geology and Geophysics
It is often associated with changes in feelings, behaviors, thoughts, and physiology.
And thank you for taking the time to help us improve the quality of Unity Documentation.
That means the angle between them is 90 degrees which also means their dot product if zero.Mathematics. a perpendicular line or plane, especially one perpendicular to a tangent line of a curve, or a tangent plane of a surface, at the point of contact.But for the web and digital devices, it is commonly referred to as line height, and can be expressed in pixels, points or centimeters (px, pt or cm), or as a percentage of the type size (120% or 1.2). Other accepted terminology.
### What Is the Normal Range for Endometrial Thickness
The line-height CSS property sets the amount of space used for lines, such as in text.
### line-height | CSS-Tricks
The Standard Normal curve, shown here, has mean 0 and standard deviation 1.
What is a Normal Receding Hairline (Things Nobody Told You ) One of the most common problems that men face during their adolescence is changes in their hairline.
### What is considered normal water pressure into a house
Their hair starts thinning in the sides and the front portion of their scalps, whereas it remains intact in other areas like the back side.
### CSS line-height Property
The normal line is the line that is perpendicular to the the tangent line.The WBC count is an important measurement that doctors will use to better understand what might be going on inside the body during a variety of health situations.Researchers who solicited responses to an online survey of almost 100,000 people from around the world, including 23,000 in the USA, get at that question and more than 1,000 others in a new book called The Normal Bar, out Feb. 5. | 743 | 3,445 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.1875 | 3 | CC-MAIN-2019-04 | longest | en | 0.904244 |
https://www.jobilize.com/online/course/8-3-population-growth-curves-by-openstax?qcr=www.quizover.com&page=1 | 1,624,302,620,000,000,000 | text/html | crawl-data/CC-MAIN-2021-25/segments/1623488289268.76/warc/CC-MAIN-20210621181810-20210621211810-00439.warc.gz | 769,954,208 | 19,517 | # 8.3 Population growth curves (Page 2/2)
Page 2 / 2
$G={r}_{\mathrm{max}}N$
## Logistic growth
Exponential growth is possible only when infinite natural resources are available; this is not the case in the real world. Charles Darwin recognized this fact in his description of the “struggle for existence,” which states that individuals will compete (with members of their own or other species) for limited resources. The successful ones will survive to pass on their own characteristics and traits (which we know now are transferred by genes) to the next generation at a greater rate (natural selection). To model the reality of limited resources, population ecologists developed the logistic growth model.
## Carrying capacity and the logistic model
In the real world, with its limited resources, exponential growth cannot continue indefinitely. Exponential growth may occur in environments where there are few individuals and plentiful resources, but when the number of individuals gets large enough, resources will be depleted, slowing the growth rate. Eventually, the growth rate will plateau or level off ( [link] ). This population size, which represents the maximum population size that a particular environment can support, is called the carrying capacity, or K .
The formula we use to calculate logistic growth adds the carrying capacity as a moderating force in the growth rate. The expression “ K N ” is indicative of how many individuals may be added to a population at a given stage, and “ K N ” divided by “ K ” is the fraction of the carrying capacity available for further growth. Thus, the exponential growth model is restricted by this factor to generate the logistic growth equation:
Notice that when N is very small, ( K-N )/ K becomes close to K/K or 1, and the right side of the equation reduces to r max N , which means the population is growing exponentially and is not influenced by carrying capacity. On the other hand, when N is large, ( K-N )/ K come close to zero, which means that population growth will be slowed greatly or even stopped. Thus, population growth is greatly slowed in large populations by the carrying capacity K . This model also allows for the population of a negative population growth, or a population decline. This occurs when the number of individuals in the population exceeds the carrying capacity (because the value of (K-N)/K is negative).
A graph of this equation yields an S-shaped curve ( [link] ), and it is a more realistic model of population growth than exponential growth. There are three different sections to an S-shaped curve. Initially, growth is exponential because there are few individuals and ample resources available. Then, as resources begin to become limited, the growth rate decreases. Finally, growth levels off at the carrying capacity of the environment, with little change in population size over time.
how can chip be made from sand
is this allso about nanoscale material
Almas
are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where is the latest information on a no technology how can I find it
William
currently
William
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Got questions? Join the online conversation and get instant answers! | 1,300 | 5,991 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 1, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.421875 | 3 | CC-MAIN-2021-25 | latest | en | 0.941209 |
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# Gankhuyag,Sengum_vital capacity_Period 3.pdf - Vital...
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Vital Capacity 3/19/18 Sengum Gankhuyag Volume Measurement (L) Individual (L) Class average male Class average female Tidal Volume (TV) 1.63 = 1.6 1.382 0.93545 Inspiratory Reserve (IRV) 1.80 = 1.8 2.478 1.17445 Expiratory Reserve (ERV) 1.20 =1.2 1.728 1.0427 Vital Capacity (VC) 4.56 = 4.6 5.7 2.9585 Residual Volume (RV) 1.5 1.5 1.5 Total Lung Capacity (TLC) 6.06 = 6.1 7.334 4.5441 Minute Volume (MV) at rest 39.12 = 39 22.34 16.9962 25. Vital Capacity (VC): The l total volume of air l that can be exhaled l after maximal l inhalation: VC I = TV + I IRV + ERV I 4.6L = 1 1.6L + 2 1.8L + 7 1.2L 26. Total Lung Capacity (TLC): Total volume l of the lungs is l the sum of the vital l capacity and the residual l volume: TLC = VC + RV 6.1L l = 1 4.6L + 1 1.5L 27. Minute Volume: The volume l of air breathed l in one minute l without conscious effort:MV=TV x (breaths/minute).
39L 8 = 1 1.6L x 8 24(breaths/minute) 35. Treatment-For people with l persistent asthma, using l asthma control medications daily l is key to achieving the l treatment goals. Long-term l control medications include inhaled l corticosteroids, inhaled long-acting l beta agonists, leukotriene modifiers, l theophylline, and omalizumab.
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• Mr.Huvar
• Physics, total lung capacity
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http://slidegur.com/doc/7066/chapter-6-ppt | 1,477,692,577,000,000,000 | text/html | crawl-data/CC-MAIN-2016-44/segments/1476988725475.41/warc/CC-MAIN-20161020183845-00207-ip-10-171-6-4.ec2.internal.warc.gz | 224,997,131 | 16,890 | ### Chapter 6 PPT
```Quantitative Literacy:
Thinking Between the Lines
Crauder, Noell, Evans, Johnson
Chapter 6:
Statistics
© 2013 W. H. Freeman and Company
1
Chapter 6: Statistics
Lesson Plan
Data summary and presentation: Boiling down
the numbers
The normal distribution: Why the bell curve?
The statistics of polling: Can we believe the polls?
Statistical inference and clinical trials: Effective
drugs?
2
Chapter 6 Statistics
6.1 Data summary and presentation: Boiling down the
numbers
Learning Objectives:
Know the statistical terms used to summarize data
Calculate mean, median, and mode
Understand the five-number summary and boxplots
Calculate the standard deviation
Understand histograms
3
Chapter 6 Statistics
6.1 Data summary and presentation: Boiling down the
numbers
The mean (average) of a list of numbers is the sum of the
numbers divided by the number of entries in the list.
The median of a list of numbers is the middle number, the
middle data point. If there is an even number of data points,
take the average of the middle two numbers.
The mode is the most frequently occurring data points. If
there are two such numbers, the data set is called bimodal.
If there are more than two such numbers, the data set is
multimodal.
If no number repeats, the data set has no mode.
4
Chapter 6 Statistics
6.1 Data summary and presentation: Boiling down the
numbers
Example: The Chelsea Football Club (FC) is a British soccer
team. The following table shows the goals scored in the games
played by Chelsea FC between September 2007 and May 2008.
The data are arranged according to the total number of goals
scored in each game.
Goals scored by either team
0
1
2
3
4
5
6
7
8
Number of games
7
14
20
11
3
2
1
2
2
Find the mean, median, and mode for the number of goals
scored per game.
5
Chapter 6 Statistics
6.1 Data summary and presentation: Boiling down the
numbers
Solution:
To find the mean, we add the data values (the total number of
goals scored) and divide by the number of data points.
To find the total number of goals scored, for each entry we
multiply the goals scored by the corresponding number of games.
The total number of goals scored:
7 × 0 + 14 × 1 + 20 × 2 + 11 × 3 + 3 × 4 + 2 × 5
+ 1 × 6 + 2 × 7 + 2 × 8 = 145
The number of data points or the total number of games:
7 + 14 + 20 + 11 + 3 + 2 + 1 + 2 + 2 = 62
6
Chapter 6 Statistics
6.1 Data summary and presentation: Boiling down the
numbers
Solution (cont.):
The mean: the total number of goals scored divided by the
number of games played:
Mean =
145
= 2.3
62
The median: the total number of games is 62, which is even, so we
count from the bottom to find the 31st and 32nd lowest total goal
scores. These are both 2:
Median = 2
The mode: because 2 occurs most frequently as the number of
goals (20 times): Mode = 2
Thus on average, the teams combined to score 2.3 goals per game.
Half of the games had goals totaling 2 or more, and the most
common number of goals scored in a Chelsea FC game was 2.
7
Chapter 6 Statistics
6.1 Data summary and presentation: Boiling down the
numbers
Example: The following list gives home prices (in thousands of dollars)
in a small town:
80, 120, 125, 140, 180, 190, 820
The list includes the price of one luxury home. Calculate the mean and
median of this data set. Which of the two is more appropriate for describing
the housing market?
Solution:
Mean =
80+120+125+140+180+190+820
7
=
1655
7
Or about 236.4 thousand dollars. This is the average price of a home.
The list of seven prices is arranged in order, so the median is the fourth value,
140 thousand dollars.
Note that the mean is higher than the cost of every home on the market
except for one—the luxury home. The median of 140 thousand dollars is
more representative of the market.
8
Chapter 6 Statistics
6.1 Data summary and presentation: Boiling down the
numbers
An outlier is a data point that is significantly different from
most of the data.
The first quartile of a list of numbers is the median of the
lower half of the numbers in the list.
The second quartile is the same as the median of the list.
The third quartile is the median of the upper half of the
numbers in the list.
The five-number summary of a list of numbers consists of
the minimum, the first quartile, the median, the third quartile,
and the maximum.
9
Chapter 6 Statistics
6.1 Data summary and presentation: Boiling down the
numbers
Example: Each year Forbes magazine publishes a list it calls
the Celebrity 100. The accompanying table shows the top nine
names on the list for 2009, ordered according to the ranking
of Forbes. The table also gives the incomes of the celebrities
between June 2008 and June 2009.
Calculate the five-number summary for this list of incomes.
10
Chapter 6 Statistics
6.1 Data summary and presentation: Boiling down the
numbers
Celebrity
Income (millions of dollars)
Angelina Jolie
27
Oprah Winfrey
275
110
Beyonce Knowles
87
Tiger Woods
110
Bruce Springsteen
70
Steven Spielberg
150
Jennifer Aniston
25
28
11
Chapter 6 Statistics
6.1 Data summary and presentation: Boiling down the
numbers
Solution: First we arrange the incomes in order:
25 27 28 70 87 110 110 150 275
The lower half of the list consists of the four numbers less than the
median (\$87 million), which are:
25 27 28 70
The median of this lower half is 27.5, so the first quartile of
incomes is \$27.5 million.
The upper half of the list consists of the four numbers greater than
the median, which are:
110 110 150 275
The median of this upper half is 130, so the third quartile of
incomes is \$130 million.
12
Chapter 6 Statistics
6.1 Data summary and presentation: Boiling down the
numbers
Solution (cont.):
Thus, the five-number summary is:
Minimum = \$25 million
First quartile = \$27.5 million
Median = \$87 million
Third quartile = \$130 million
Maximum = \$275 million
13
Chapter 6 Statistics
6.1 Data summary and presentation: Boiling down the
numbers
Boxplots: There is a commonly used pictorial display of the
five-number summary known as a boxplot (also called a box and
whisker diagram). Figure 6.1 shows the basic geometric figure
used in a boxplot.
14
Chapter 6 Statistics
6.1 Data summary and presentation: Boiling down the
numbers
Example: A report on greenercars.org shows 2011 model
cars with the best fuel economy.
1. Find the five-number summary for city mileage.
2. Present a boxplot of city mileage.
3. Comment on how the data are distributed about the median.
15
Chapter 6 Statistics
6.1 Data summary and presentation: Boiling down the
numbers
City
Mileage (mpg)
Highway
Mileage (mpg)
Toyota Prius
51
48
Honda Civic Hybrid
40
43
Honda CR-Z
35
39
Toyota Yaris
29
35
Audi A3
30
42
Hyundai Sonata
22
35
Hyundai Tucson
23
31
Chevrolet Equinox
22
32
Kia Rondo
20
27
Canyon
18
25
Model
16
Chapter 6 Statistics
6.1 Data summary and presentation: Boiling down the
numbers
Solution:
1. The list for city mileage, in order from lowest to highest:
18, 20, 22, 22, , , 30, 35, 40,51
To find the median, we average the two numbers in the
middle:
23 + 29
Median =
= 26 mpg
2
The lower half of the list is 18, 20, 22, 22, 23, and the
median of this half is 22. Thus, the first quartile is 22 mpg.
The upper half of the list is 29, 30, 35, 40, 51, and the
median of this half is 35. Thus, the third quartile is 35 mpg.
17
Chapter 6 Statistics
6.1 Data summary and presentation: Boiling down the
numbers
Solution (cont.):
2. The corresponding boxplot appears in Figure 6.2. The vertical
axis is the mileage measured in miles per gallon.
18
Chapter 6 Statistics
6.1 Data summary and presentation: Boiling down the
numbers
Solution (cont.):
3. Referring to the boxplot, we note that the first quartile is not
far above the minimum, and the median is barely above the
first quartile. The third quartile is well above the median, and
the maximum is well above the third quartile. This
emphasizes the dramatic difference between the high-mileage
cars (the hybrids) and ordinary cars.
19
Chapter 6 Statistics
6.1 Data summary and presentation: Boiling down the
numbers
The standard deviation is a measure of how much the data
are spread out from the mean. The smaller the standard
deviation, the more closely the data clustered about the mean.
Standard Deviation Formula
Suppose the data points are:
1 , 2 , 3 , … , ,
the formula for the standard deviation is
+ 2 − 2 + ⋯ + −
=
Where the Greek letter μ (mew) denotes the mean.
1 −
2
2
20
Chapter 6 Statistics
6.1 Data summary and presentation: Boiling down the
numbers
Calculating Standard Deviation
To find the standard deviation of n data points, we first calculate
the mean . The next step is to complete the following
calculation template:
Data
Deviation
Square of deviation
⋮
⋮
⋮
−
Square of second column
⋮
⋮
⋮
Sum of third column
Divide the above sum by n
and take the square root.
21
Chapter 6 Statistics
6.1 Data summary and presentation: Boiling down the
numbers
Example: Two leading pitchers in Major League Baseball for 2011 were
Roy Halladay of the Philadelphia Phillies and Felix Hernandez of the Seattle
Mariners. Their ERA (Earned Run Average―the lower the number, the
better) histories are given in the table below.
Pitcher
ERA
2006
ERA
2007
ERA
2008
ERA
2009
ERA
2010
3.19
3.71
2.78
2.79
2.44
F. Hernandez
4.52
3.92
3.45
2.49
2.27
Calculate the mean and the standard deviation for Hallady’s ERA history. It
turns out that the mean and standard deviation for Hernandez’s ERA history
are µ = 3.33 and σ = 0.85. What comparisons between Halladay and
Hernandez can you make based on these numbers?
22
Chapter 6 Statistics
6.1 Data summary and presentation: Boiling down the
numbers
Solution (cont.): We conclude that the mean and the
standard deviation for Halladay’s ERA history are µ = 2.98 and
σ = 0.43.
Because Halladay’s mean is smaller than Hernandez’s mean of
record.
consistent—his numbers are not spread as far from the mean.
23
Chapter 6 Statistics
6.1 Data summary and presentation: Boiling down the
numbers
Example: Below is a table showing the Eastern Conference
NBA team free-throw percentages at home and away for the
2007–2008 season. At the bottom of the table, we have
displayed the mean and standard deviation for each data set.
What do these values for the mean and standard deviation tell us
about free-throws shot at home compared with free-throws shot
away from home?
Does comparison of the minimum and maximum of each of the data
24
Chapter 6 Statistics
6.1 Data summary and presentation: Boiling down the
numbers
Free-throw
percentage
at home
Free-throw
percentage
away
Toronto
Washington
Atlanta
Boston
Indiana
Detroit
Chicago
New Jersey
81.2
78.2
77.2
77.1
76.8
76.7
75.6
73.6
77.6
75.4
75.2
74.3
75.7
74.4
76.6
76.8
Mean
74.73
75.61
Standard
deviation
2.95
1.09
Team
Team
Milwaukee
Miami
New York
Orlando
Cleveland
Charlotte
Free-throw
percentage
at home
Free-throw
percentage
away
73.3
72.7
72.7
72.1
71.7
71.4
70.6
76.6
75.5
73.9
75.4
74.8
74.7
77.2
25
Chapter 6 Statistics
6.1 Data summary and presentation: Boiling down the
numbers
home and 75.61 away, so on average the teams do somewhat better
on the road than at home.
The standard deviation for home is 2.95 percentage points, which
is considerably larger than the standard deviation of 1.09
percentage points away from home. This means that the freethrow percentages at home vary from the mean much more than
the free-throw percentages away.
The difference between the maximum and minimum percentages
shows the same thing: The free-throw percentages at home range
from 70.6 to 81.2%, and the free-throw percentages away range
from 73.9% to 77.6%.
26
Chapter 6 Statistics
6.1 Data summary and presentation: Boiling down the
numbers
Solution (cont.): The plots of the data in figures 6.3 and 6.4
provide a visual verification that the data for home free-throws
are more broadly dispersed than the data for away freethrows.
27
Chapter 6 Statistics
6.1 Data summary and presentation: Boiling down the
numbers
A histogram is a bar graph that shows the frequencies with which certain
data occur.
Example: Suppose we toss 1000 coins and write down the number of
heads we got. We do this experiment a total of 1000 times. The
accompanying table shows one part of the results from doing these
experiments using a computer simulation.
Number of tosses (out of 1000)
451 457 458 459 461 462 463 464 465 467
2
2
1
3
3
2
1
3
1
1
The first entry shows that twice we got 451 heads, twice we got 457 heads,
once we got 458 heads, and so on. The raw data are hard to digest because
there are so many data points. The five-number summary provides one way to
analyze the data.
An alternative way to get the data is to arrange them in groups and then draw
a histogram.
28
Chapter 6 Statistics
6.1 Data summary and presentation: Boiling down the
numbers
Example (cont.): Suppose it turns out that the number of
tosses yielding fewer than 470 heads is 23. Because 470 out of 1000
is 47%, it means that 23 tosses yields less than 47% heads. We find
the accompanying table by dividing the data into groups this way.
Number of
tosses
Number of
tosses
Less than 47%
47% to 48%
48% to 49%
49% to 50%
23
75
140
234
50% to 51%
51% to 52%
52% to 53%
At least 53%
250
157
94
27
29
Chapter 6 Statistics
6.1 Data summary and presentation: Boiling down the
numbers
Example (cont.): Figure 6.7 shows a histogram for this
grouping of the data. We can clearly see that the vast majority of the
tosses were between 47% and 53% heads.
30
Chapter 6 Statistics
6.2 The normal distribution: Why the bell curve?
Learning Objectives:
Understand why the normal distribution is so
important.
The bell-shaped curve
Mean and standard deviation for the normal distribution
z-scores with Percentile scores
The Central Limit Theorem
Significance of apparently small deviations
31
Chapter 6 Statistics
6.2 The normal distribution: Why the bell curve?
The bell-shaped curve: Figure 6.13 shows the distribution of heights
of adult males in the United States. A graph shaped like this one
resembles a bell—thus the bell curve. This bell-shaped graph is typical of
normally distributed data.
The mean and median are the same: For normally distributed data, the
mean and median are the same. Figure 6.13 indicates that the median height of
adult males is 69.1 inches. The average height of adult males is 69.1 inches.
32
Chapter 6 Statistics
6.2 The normal distribution: Why the bell curve?
Most data are clustered about the mean: The vast majority of
adult males are within a few inches of the mean.
33
Chapter 6 Statistics
6.2 The normal distribution: Why the bell curve?
The bell curve is symmetric about the mean: The curve to
the left of the mean is a mirror image of the curve to the right of
the mean. In terms of heights, there are about the same number of
men 2 inches taller than the mean as there are men 2 inches
shorter than the mean. This is illustrated in Figure 6.16.
If data are normally distributed:
1. Their graph is a bell-shaped curve.
2. The mean and median are the same.
3. Most of the data tend to be clustered relatively near the mean.
4. The data are symmetrically distributed above and below the mean.
34
Chapter 6 Statistics
6.2 The normal distribution: Why the bell curve?
Example: Figure 6.17 shows the distribution of IQ scores,
and Figure 6.18 shows the percentage of American families and
level of income. Which of these data sets appear to be
normally distributed, and why?
35
Chapter 6 Statistics
6.2 The normal distribution: Why the bell curve?
Solution: The IQ scores appear to be normally distributed
because they are symmetric about the median score of 100,
and most of the data relatively close to this value.
Family incomes do not appear to be normally distributed
because they are not symmetric. They are skewed toward the
lower end of the scale, meaning there are many more families
with low incomes than with high incomes.
36
Chapter 6 Statistics
6.2 The normal distribution: Why the bell curve?
Mean and standard deviation for the normal distribution: A
normal distribution, the mean and standard deviation completely
determine the bell shape for the graph of the data.
The mean determines the middle of the bell curve.
The standard deviation determines how steep the curve is.
A large standard deviation results in a very wide bell, and small
standard deviation results in a thin, steep bell.
37
Chapter 6 Statistics
6.2 The normal distribution: Why the bell curve?
Normal Data: 68-95-99.7% Rule
If a set of data is normally distributed:
• About 68% of the data lie within one standard deviation of the mean (34%
within one standard deviation above the mean and 34% within one
standard deviation below the mean). See Figure 6.23.
• About 95% of the data lie within two standard deviations of the mean
(47.5% within two standard deviations above the mean and 47.5% within
two standard deviations below the mean). See Figure 6.24.
• About 99.7% of the data lie within three standard deviations of the mean
(49.85% within three standard deviations above the mean and 49.85%
within three standard deviations below the mean). See Figure 6.25.
38
Chapter 6 Statistics
6.2 The normal distribution: Why the bell curve?
39
Chapter 6 Statistics
6.2 The normal distribution: Why the bell curve?
40
Chapter 6 Statistics
6.2 The normal distribution: Why the bell curve?
Example: The weights of apples in the fall harvest are normally
distributed, with a mean weight of 200 grams and standard deviation
of 12 grams. Figure 6.28 shows the weight distribution of 2000
apples. In a supply of 2000 apples, how many will weigh between
176 and 224 grams?
41
Chapter 6 Statistics
6.2 The normal distribution: Why the bell curve?
Solution:
Apples weighing 176 grams are 200– 176 = 24 grams below the
mean, and apples weighing 224 grams are 224 − 200 = 24 grams
above the mean.
Now 24 grams represents 24/12 = 2 standard deviations. So the
weight range of 176 grams to 224 grams is within two standard
deviations of the mean.
Therefore, about 95% of data points will lie in this range. This
means that about 95% of 2000, or 1900 apples, weigh between
176 and 224 grams.
42
Chapter 6 Statistics
6.2 The normal distribution: Why the bell curve?
In a normal distribution, the z-score or standard score for a data
point is the number of standard deviations that point lies above or
below the mean. For data points above the mean the z-score is
positive, and for data points below the mean the z-score is negative.
− score = ( Data point – Mean)/Standard deviation
Data point = Mean + − score × Standard deviation
43
Chapter 6 Statistics
6.2 The normal distribution: Why the bell curve?
Example: The weights of newborns in the United States are
approximately normally distributed. The mean birthweight (for
single births) is about 3332 grams (7 pounds, 5 ounces). The
standard deviation is about 530 grams. Calculate the z-score
for a newborn weighing 3700 grams (about 8 pounds, 2
ounces).
Solution: A 3700-gram newborn is 3700 – 3332 =
368 grams above the mean weight of 3332 grams. We divide
by the number of grams in one standard deviation to find the
z-score:
368
− score for 3700 grams =
= 0.7
530
44
Chapter 6 Statistics
6.2 The normal distribution: Why the bell curve?
45
Chapter 6 Statistics
6.2 The normal distribution: Why the bell curve?
The percentile for a number relative to a list of data is the
percentage of data points that are less than or equal to that
number.
Example: The average length of illness for flu patients in a
season is normally distributed, with a mean of 8 days and
standard deviation of 0.9 day. What percentage of flu patients
will be ill for more than 10 days?
Solution: Ten days is 2 days above the mean of 8 days. This
gives a z-score of 2/0.9 or about 2.2. Table 6.2 gives a
98.6% of patients will recover in 10 days or less. Thus, only
about 100% − 98.6% = 1.4% will be ill for more than 10 days.
46
Chapter 6 Statistics
6.2 The normal distribution: Why the bell curve?
Example: Recall from the previous Example that the weights
of newborns in the United States are approximately normally
distributed. The mean birthweight (for single births) is about
3332 grams (7 pounds, 5 ounces). The standard deviation is
1. What percentage of newborns weigh more than 8 pounds
(3636.4 grams)?
2. Low birthweight is a medical concern. The American Medical
Association defines low birthweight to be 2500 grams (5
pounds, 8 ounces) or less. What percentage of newborns are
classified as low-birthweight babies?
47
Chapter 6 Statistics
6.2 The normal distribution: Why the bell curve?
Solution:
3636.4−mean
1. − score = standard deviation =
3636.4−3332
530
=
304.4
530
= 0.6
Consulting Table 6.2, we find that this represents a percentile of
This means that about 72.6% of newborns weigh 8 pounds or less.
So, 100% − 72.6% = 27.4% of newborns weigh more than 8
pounds.
2500−mean
2. − score = standard deviation =
3332−2500
530
=
832
530
= 1.6
Table 6.2 shows a percentile of about 5.5% for a z-score of—1.6.
Hence, about 5.5% of newborns are classified as low-birthweight
babies.
48
Chapter 6 Statistics
6.2 The normal distribution: Why the bell curve?
The Central Limit Theorem
According to the Central Limit Theorem, percentages
obtained by taking many samples of the same size from a
population are approximately normally distributed.
The mean p% of the normal distribution is the mean of the
whole population.
If the sample size is n, the standard deviation of the normal
distribution is:
Standard deviation = σ =
(100 − )
percentage points
Here, p is a percentage, not a decimal.
49
Chapter 6 Statistics
6.2 The normal distribution: Why the bell curve?
Example: For a certain disease, 30% of untreated patients
can be expected to improve within a week. We observe a
population of 50 patients and record the percentage who
improve within a week. According to the Central Limit
Theorem, the results of such a study will be approximately
normally distributed.
1. Find the mean and standard deviation for this normal
distribution.
2. Find the percentage of test groups of 50 patients in which
more than 40% improve within a week.
50
Chapter 6 Statistics
6.2 The normal distribution: Why the bell curve?
Solution:
1. = 30%, = 50. A standard deviation of
=
(100−)
=
30(100−30)
50
= 6.5 percentage points
2. The z-score for 40%:
40 − mean
40 − 30 10
z − score =
=
=
= 1.5
standard deviation
6.5
6.5
Table 6.2 gives a percentile of about 93.3%.
This means that in 93.3% of test groups, we expect that
40% or fewer will improve within a week.
Only 100% − 93.3% = 6.7% of test groups will show more
than 40% improving within a week.
51
Chapter 6 Statistics
6.2 The normal distribution: Why the bell curve?
Example: Assume we know that 20% of Americans suffer
from a certain type of allergy. Suppose we take a random
sample of 100,000 Americans and record the percentage who
suffer from this allergy.
1. The Central Limit Theorem says that percentages from such
surveys will be normally distributed. What is the mean of this
distribution?
2. What is the standard deviation of the normal distribution in
part 1?
3. Suppose we find that in a town of 100,000 people, 21% suffer
from this allergy. Is this an unusual sample? What does the
52
Chapter 6 Statistics
6.2 The normal distribution: Why the bell curve?
Solution:
1. The mean is p = 20%.
2. For a sample size of 100,000,
=
(100−)
=
20(100−20)
100,000
= 0.13 percentage point.
3. Our sample of 21% is one percentage point larger than the mean
of 20%.
21 − mean
21 − 20
1
z − score =
=
=
= 7.7
standard deviation
0.13
0.13
This score is far larger than any z-score in Table 6.2. There is almost no
chance that in a randomly chosen sample of this size, 21% will suffer from this
allergy. Thus, this is a truly anomalous sample: This town is not representative
of the total population of Americans. Its allergy rate is highly unusual.
53
Chapter 6 Statistics
6.3 The statistics of polling: Can we believe the polls?
Learning Objectives:
Understand margins of error and confidence levels in
polls.
Basic terms: Margin of error, confidence interval,
and confidence level
Polls: Margin of error, confidence interval, and
confidence level
How big should the sample be?
54
Chapter 6 Statistics
6.3 The statistics of polling: Can we believe the polls?
The margin of error of a poll expresses how close to the
true result (the result for the whole population) the result of
the poll can be expected to lie.
To find the confidence interval, adjust the result of the poll
by adding and subtracting the margin of error.
The confidence level of a poll tells the percentage of such
polls in which the confidence interval includes the true result.
55
Chapter 6 Statistics
6.3 The statistics of polling: Can we believe the polls?
Polls and Margin of Error
Suppose that, based on random sampling, a poll reports the
percentage of the population having a certain property (e.g.,
planning to vote for a certain candidate) with a margin of
error m. Assuming that this margin is based on a 95%
confidence level, we can say that if we conducted this poll
100 times, then we expect about 95 of those sample results
to be within m percentage points of the true percentage
having that property.
56
Chapter 6 Statistics
6.3 The statistics of polling: Can we believe the polls?
Example: Explain the meaning of a poll that says 33% of
Americans approve of what Congress is doing, with a margin
of error of 4% and confidence level of 90%
Solution: In 90% of such polls, the reported approval of
Congress will be within four percentage points of the true
approval level.
Thus, we can be 90% confident that the true level lies in the
confidence interval between 29% and 37%.
57
Chapter 6 Statistics
6.3 The statistics of polling: Can we believe the polls?
Margin of Error
For a 95% level of confidence, we can estimate the margin
of error when we poll n people using:
100
Margin of error ≈
%
Here, the symbol ≈ means “is approximately equal to.”
58
Chapter 6 Statistics
6.3 The statistics of polling: Can we believe the polls?
Example: A recent Oricon fashion survey asked 900 people,
“Which Japanese male celebrity looks best in sneakers?” The
winner was Kimura Takuya. What is the approximate margin
of error for a 95% confidence level?
Solution:
With n = 900:
100
100
Margin of error ≈
%=
= 3.3%
900
We can be 95% confident that our poll result is within 3.3
percentage points of the true value.
59
Chapter 6 Statistics
6.3 The statistics of polling: Can we believe the polls?
1.
2.
3.
4.
Example: The Kaiser Family Foundation polled 1294 residents of
Orleans Parish in New Orleans in 2008 and found that 41% of the
residents who had lived through Hurricane Katrina in 2005 report that
their lives are still disrupted.
The poll surveyed 1294 people. What is the approximate margin of
error for a 95% confidence interval?
The poll of 1294 people found that 41% of respondents still had
disrupted lives. Can we conclude with certainty that no more than
45% of residents’ lives are still disrupted?
Suppose instead that the poll of 1294 people had found that 52% still
had disrupted lives. Explain what we could conclude from this result.
Could we assert with confidence that a majority of residents’ lives are
still disrupted by Katrina?
Suppose we wish to have a margin of error of two percentage points.
Approximately how many people should we interview?
60
Chapter 6 Statistics
6.3 The statistics of polling: Can we believe the polls?
Solution:
1. With n = 1294:
100
100
Margin of error ≈
=
= 2.8%
1294
2. Our answer to part 1 tell us that we can be 95% confident that the
poll number of 41% is within 2.8 percentage points of the true
percentage of all residents whose lives are still disrupted from
Katrina. Thus, it is very likely that the true value is:
between 41 − 2.8 = 38.2% and 41 + 2.8 = 43.8%
Because the whole interval is below 45%, we can be quite confident
(at a 95% level) that no more than 45% of residents’ lives are still
disrupted.
On the other hand, we cannot make this conclusion with
absolute certainty.
61
Chapter 6 Statistics
6.3 The statistics of polling: Can we believe the polls?
Solution (cont.):
3. We can be 95% confident that the poll number of 52% is within
2.8 percentage points of the true percentage of all residents
whose lives are still disrupted by Katrina. Thus, it is very likely that
the true value is:
between 52– 2.8 = 49.2% and 52 + 2.8 = 54.8%
Most of this interval falls above 50%, so we continue to have good
reason to think that a majority of residents’ lives are still disrupted.
But, because a portion of the interval falls below 50%, we should be
more cautious in drawing conclusions.
62
Chapter 6 Statistics
6.3 The statistics of polling: Can we believe the polls?
Solution (cont.):
4. Substitute 2% for the margin of error:
100
Margin of error = 2 =
or
= 50
Hence, = 2500.
We should interview about 2500 people.
Note that one Harris Poll with a 95% confidence level and a margin
of error of 2% surveyed 2415 people—very close to the 2500
given by the formula.
63
Chapter 6 Statistics
6.3 The statistics of polling: Can we believe the polls?
Sample Size
For a 95% level of confidence, the sample size needed to
get a margin of error of m percentage points can be
approximated using:
2
100
Sample size ≈
m
64
Chapter 6 Statistics
6.3 The statistics of polling: Can we believe the polls?
Example: What sample size is needed to give a margin of
error of 4% with a 95% confidence level?
Solution: We use the approximate formula with = 4:
2
2
100
100
Sample size ≈
=
= 625
m
4
65
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https://elsmar.com/elsmarqualityforum/threads/optimize-one-response-with-three-variables.61652/ | 1,679,947,137,000,000,000 | text/html | crawl-data/CC-MAIN-2023-14/segments/1679296948684.19/warc/CC-MAIN-20230327185741-20230327215741-00680.warc.gz | 272,594,728 | 28,988 | # Optimize One Response with Three Variables
C
#### Cenderze
Hello!
I'm intending to do a response optimization of one response, y, having three variables x1, x2, x3. These variables are coded in the following manor:
A B C y
-1,00000 -1,00000 -1,00000 66
1,00000 -1,00000 -1,00000 80
-1,00000 1,00000 -1,00000 78
1,00000 1,00000 -1,00000 100
-1,00000 -1,00000 1,00000 70
1,00000 -1,00000 1,00000 100
-1,00000 1,00000 1,00000 60
1,00000 1,00000 1,00000 75
-1,68179 0,00000 0,00000 100
1,68179 0,00000 0,00000 80
0,00000 -1,68179 0,00000 68
0,00000 1,68179 0,00000 63
0,00000 0,00000 -1,68179 65
0,00000 0,00000 1,68179 82
0,00000 0,00000 0,00000 113
0,00000 0,00000 0,00000 100
0,00000 0,00000 0,00000 118
0,00000 0,00000 0,00000 88
0,00000 0,00000 0,00000 100
0,00000 0,00000 0,00000 85
I dont know how to perform this optimization. I've tried to "Define Custom Surface Response Design" and choosing the Optimize Response option under Stat -> DOE -> Response Surface
I receive a value though which has only desirability = 0,7 (hence not an optimum -the highest value in my set of y i 118) and using x1 = -0,5 which is not in the range of my valid values for it.
When I've googled all I see is people who merely makes surface plots and conclude in what path the response is optimized - however using three variables I do not know how to perform this.
Any help really appreciated!
Regards,
Cenderze
Last edited by a moderator:
#### Miner
##### Forum Moderator
Include the actual response values and I'll take a look at it.
C
#### Cenderze
Sure I can add the response values. I believe adding a column after the x1, x2, x3 columns in my question makes it too difficult to copy everything as intended, so I merely write out the response column as a whole.
It is the following
y
66
70
78
60
80
70
100
75
100
75
100
80
68
63
65
82
113
100
118
88
100
85
It is desirable to have high values on y.
C
#### Cenderze
Include the actual response values and I'll take a look at it.
Thanks again for writing. Just wanted to make sure you noticed that the whole data set was edited. I saw a 1 in the wrong place.
Doing the exercise as
Stat -> DOE -> Define Custom Response Surface
And choosing 3 responses, 6 center points, 1 replicate and alpha 1,682 and subsequently choosing to
Stat -> DOE -> Optimize Response variable
and choosing Maximize and lowest 100 and target 118 I got that the max y is
101,711
and D=0,41
but A=0,6 B = -0,11 and C=0,15
which, if I've understood correct is invalid for this design.
So, basicially, I stil need help even though I've found an error in my original data set.
Hmm.. guess this became more of an update than an actual response
#### Miner
##### Forum Moderator
The data in post 3 has two more entries than in post 1.
C
#### Cenderze
Thank you for replying. The "main" post has been edited now so it should be correct size and correct data.
#### Miner
##### Forum Moderator
I'll take a look on Monday. You might want to bump it again Sunday night.
#### Miner
##### Forum Moderator
I see your problem. Response Optimizer is only as good as the predictive model that you develop. If the predictive model is poor, the Optimizer results will also be poor.
I analyzed your experimental data as a response surface design and found the two quadratic terms BB and CC to be significant. The R^2(adj) was low (34.7%) and R^2(pred) was very low (15.7%). Since Minitab's DOE analysis is heirarchical (meaning you must include the non-significant B and C terms), I reanalyzed the data using General Regression and included only the quadratic terms. This improved the model slightly to a R^2(pred) of 33.4%. However, this is still far from a good model.
The error partitioning did not indicate a Lack of Fit issue. All of the error was in the Pure Error category. This would indicate that you potentially have an issue with your measurement variation or your experimental setup variation.
Can you tell us any more about the factors and responses as well as the process? That would enable us to offer better advice.
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F Response to ISO 13485 Audit Minor Nonconformances ISO 13485:2016 - Medical Device Quality Management Systems 6
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D Showing the Response Surface Experimental Point in MiniTab Using Minitab Software 5 | 2,353 | 9,201 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.609375 | 3 | CC-MAIN-2023-14 | latest | en | 0.749906 |
https://stackoverflow.com/questions/2411392/double-epsilon-for-equality-greater-than-less-than-less-than-or-equal-to-gre | 1,553,636,962,000,000,000 | text/html | crawl-data/CC-MAIN-2019-13/segments/1552912206016.98/warc/CC-MAIN-20190326200359-20190326222359-00171.warc.gz | 617,680,599 | 33,833 | # Double.Epsilon for equality, greater than, less than, less than or equal to, greater than or equal to
http://msdn.microsoft.com/en-us/library/system.double.epsilon.aspx
If you create a custom algorithm that determines whether two floating-point numbers can be considered equal, you must use a value that is greater than the Epsilon constant to establish the acceptable absolute margin of difference for the two values to be considered equal. (Typically, that margin of difference is many times greater than Epsilon.)
So is this not really an epsilon that could be used for comparisons? I don't really understand the MSDN wording.
Can it be used as the epsilon in the examples here? - What is the most effective way for float and double comparison?
And finally this seems really important so I'd like to make sure I have a solid implementation for equality, greater than, less than, less than or equal to, and greater than or equal to.
I don't know what they were smoking when they wrote that. `Double.Epsilon` is the smallest representable non-denormal floating point value that isn't 0. All you know is that, if there's a truncation error, it will always be larger than this value. Much larger.
The `System.Double` type can represent values accurate to up to 15 digits. So a simple first order estimate if a double value `x` is equal to some constant is to use an epsilon of constant * 1E-15
``````public static bool AboutEqual(double x, double y) {
double epsilon = Math.Max(Math.Abs(x), Math.Abs(y)) * 1E-15;
return Math.Abs(x - y) <= epsilon;
}
``````
You have to watch out though, truncation errors can accumulate. If both `x` and `y` are computed values then you have to increase the epsilon.
• Microsoft says `Represents the smallest positive Double value that is greater than zero.` See msdn.microsoft.com/en-us/library/…. – david.pfx Mar 22 '14 at 6:00
• @Ian That's what `Double.Epsilon` should be, but david.pfx defines it as it is :-( – Mark Hurd Jun 21 '14 at 13:59
• 1.0 + double.Epsilon = 1.0 – David Sykes Sep 5 '14 at 14:14
• 1.0 + 1E-16 = 1.0, plenty more of those :) – Hans Passant Sep 5 '14 at 15:28
• Tsk, tsk, experienced programmers making statements like that does explain why we get so many questions about the topic. Base 2 != base 10. – Hans Passant Jul 5 '16 at 22:55
I'd like to make sure I have a solid implementation for equality, greater than, less than, less than or equal to, and greater than or equal to.
You are using binary floating point arithmetic.
Binary floating point arithmetic was designed to represent physical quantities like length, mass, charge, time, and so on.
Presumably then you are using binary floating point arithmetic as it was intended to be used: to do arithmetic on physical quantities.
Measurements of physical quantities always have a particular precision, depending on the precision of the device used to measure them.
Since you are the one providing the values for the quantities you are manipulating, you are the one who knows what the "error bars" are on that quantity. For example, if you are providing the quantity "the height of the building is 123.56 metres" then you know that this is accurate to the centimetre, but not to the micrometer.
Therefore, when comparing two quantities for equality, the desired semantics is to say "are these two quantities equal within the error bars specified by each measurement?"
So now we have an answer to your question. What you must do is keep track of what the error is on each quantity; for example, the height of the building is "within 0.01 of 123.56 meters" because you know that is how precise the measurement is. If you then get another measurement which is 123.5587 and want to know whether the two measurements are "equal" within error tolerances, then do the subtraction and see if it falls into the error tolerance. In this case it does. If the measurements were in fact precise to the micrometre, then they are not equal.
In short: you are the only person here who knows what sensible error tolerances are, because you are the only person who knows where the figures you are manipulating came from in the first place. Use whatever error tolerance makes sense for your measurements given the precision of the equipment you used to produce it.
• While certainly correct to point out that tolerance is a more practical and straightforward measure to define I was looking into the inaccuracies of the encoded representation as a general first pass rule and tolerance would be an optional second pass depending on the specific situation. – ss2k Mar 10 '10 at 13:41
• Eric, if you create somekind of Data Structure (eg. for game) and want to allow other games to use it, everyone will have different tolerance as they all use different coordinate systems. So, deciding what your epsilon will be, is not just a matter of personal conditions, I think. – Tom May 25 '10 at 7:33
• You did not answer the question. It is not a question of taste or context, it's related the the definition of a double in memory (IEEE 754). – Eric Ouellet Feb 15 '16 at 19:39
• Floating point values are used quite often for synthetic calculations that have no error bars based on a physical measurement technique. Understanding the precision limits of a data type is important for many problems. – tukra Jan 23 '18 at 2:27
If you have two double values that are close to 1.0, but they differ in only their least significant bits, then the difference between them will be many orders of magnitude greater than Double.Epsilon. In fact, the difference is 324 decimal orders of magnitude. This is because of the effect of the exponent portion. Double.Epsilon has a huge negative exponent on it, while 1.0 has an exponent of zero (after the biases are removed, of course).
If you want to compare two similar values for equality, then you will need to choose a custom epsilon value that is appropriate for the orders-of-magnitude size of the values to be compared.
If the double values that you are comparing are near 1.0. Then the value of the least siginificant bit would be near 0.0000000000000001. If the double values that you are comparing are in the quadrillions, then the value of the least significant bit could be as much as a thousand. No single value for epsilon could be used for equality comparisons in both of those circumstances.
I just did this - using Kent Bogarts idea.
``````private bool IsApproximatelyEqual(double x, double y, double acceptableVariance)
{
double variance = x > y ? x - y : y - x;
return variance < acceptableVariance;
//or
//return Math.Abs(x - y) < acceptableVariance;
}
``````
It could be used for comparisons, assuming you want to ensure the two values are either exactly equal, or have the smallest representable difference for the double type. Generally speaking, you would want to use a number greater than `double.Epsilon` to check whether two doubles are approximately equal.
Why the .NET framework doesn't define something like
``````bool IsApproximatelyEqual(double value, double permittedVariance);
``````
is beyond me.
I think the pertinent bits in the MSDN link you posted are these:
However, the Epsilon property is not a general measure of precision of the Double type; it applies only to Double instances that have a value of zero.
Note: The value of the Epsilon property is not equivalent to machine epsilon, which represents the upper bound of the relative error due to rounding in floating-point arithmetic.
This value is not defined as smallest positive number x, such that x + 1.0 is not equal to 1.0, so Double.Epsilon cannot be used for "almost equality". There exists no constant in the framework whose value is smallest positive number x, such that x + 1.0 is not equal to 1.0.
I have to say, that surprises me. I too had assumed that Double.Epsilon was the equivalent of DBL_EPSILON in c/c++ - clearly not!
From what I can read of that link it seems to be saying 'you need to figure out a decent value yourself for comparisons' which is rather surprising to say the least.
Perhaps someone more knowledgable can clarify :)
I use the following
``````public static class MathUtil {
/// <summary>
/// smallest such that 1.0+EpsilonF != 1.0
/// </summary>
public const float EpsilonF = 1.192092896e-07F;
/// <summary>
/// smallest such that 1.0+EpsilonD != 1.0
/// </summary>
public const double EpsilonD = 2.2204460492503131e-016;
[MethodImpl( MethodImplOptions.AggressiveInlining )]
public static bool IsZero( this double value ) {
return value < EpsilonD && value > -EpsilonD;
}
[MethodImpl( MethodImplOptions.AggressiveInlining )]
public static int Sign( this double value ) {
if ( value < -EpsilonD ) {
return -1;
}
if ( value > EpsilonD )
return 1;
return 0;
}
``````
and if you want to check for equality of two doubles 'a' and 'b', you can use
``````(a-b).IsZero();
``````
and if you want to get the comparison result, use
``````(a-b).Sign();
``````
The problem with comparing doubles is that when you do a comparison between two different math results that are equal but which, due to rounding errors, aren't evaluating to the same value, they will have some difference...which is larger than epsilon, except on edge cases. And using a reliable epsilon value is also difficult. Some people consider two doubles equal if the difference between them is less than some percentage value, since using a static minimum difference epsilon may mean your differences are too small or large when the double itself is high or low.
Here's some code that included twice within the Silverlight Control Toolkit:
`````` public static bool AreClose(double value1, double value2)
{
//in case they are Infinities (then epsilon check does not work)
if(value1 == value2) return true;
// This computes (|value1-value2| / (|value1| + |value2| + 10.0)) < DBL_EPSILON
double eps = (Math.Abs(value1) + Math.Abs(value2) + 10.0) * DBL_EPSILON;
double delta = value1 - value2;
return(-eps < delta) && (eps > delta);
}
``````
In one place they use `1e-6` for epsilon; in another they use `1.192093E-07`. You will want to choose your own epsilon.
• This one looks promising. Looks like they take care of the problem with diminishing decimal precision, when comparing high values. Of course, you must consider whether you want this kind of precision scaling. It's just as applicable as the version with a static epsilon. – Ultroman the Tacoman Jun 12 '17 at 21:17
There is no choice you have to calculate it yourself or define own constant.
``````double calculateMachineEpsilon() {
double result = 1.0;
double one = 1.0/256;
while(one + result/2.0 != 1.0) {
result/=2.0;
}
return result;
}
`````` | 2,529 | 10,676 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.375 | 3 | CC-MAIN-2019-13 | latest | en | 0.896545 |
https://studymoose.com/slfglkj-essay | 1,487,703,417,000,000,000 | text/html | crawl-data/CC-MAIN-2017-09/segments/1487501170823.55/warc/CC-MAIN-20170219104610-00147-ip-10-171-10-108.ec2.internal.warc.gz | 760,233,601 | 9,473 | # The purpose of this study is to investigate the use of internet activities that may affect the sleeping hours of a person or else cause of sleep deprivation
Custom Student Mr. Teacher ENG 1001-04 20 March 2016
## The purpose of this study is to investigate the use of internet activities that may affect the sleeping hours of a person or else cause of sleep deprivation
Please read all questions and instructions carefully. Note that you only need to enter answers in terms of numbers and without any symbols (including \$, %, commas, etc.). Enter all dollars without decimals and all interest rates in percentage with up to two decimals. Read the syllabus for examples.The points for each question are listed in parentheses at the start of the question, and the total points for the entire assignment adds up to 100. In accordance with the Coursera Honor Code, I (Shravan Vepa) certify that the answers here are my own work. Thank you! Question 1
(5 points) In a world with no frictions (i.e., taxes, etc.), having debt is always better because it increases the value of the firm/project. False.
True.
Question 2
(5 points) The return on equity is equal to the return on assets of a project/firm. Always true.
Never true.
Sometimes true.
Question 3
(10 points) Moogle, Inc. is in the same business as Google, Inc., but has recently retired all its debt to become an all-equity firm. Its return on equity has dropped from 12.25% to 10.60% as a result of this. Google, Inc. continues to have debt in its capital structure, and its debt-to-equity ratio is 30%. What is the return on assets of Google, Inc.(No more than two decimals in the percentage interest rate, but do not enter the % sign.) Answer for Question 3
Question 4
(10 points) Suppose CAPM holds, and the beta of the equity of your company is 2.00. The expected market risk premium (the difference between the expected market return and the risk-free rate) is 4.5% and the risk-free rate is 3.00%. Suppose the debt-to-equity ratio of your company is 20% and the market believes that the beta of your debt is 0.20. What is return on assets of your business? (No more than two decimals in the percentage interest rate, but do not enter the % sign.) Answer for Question 4
Question 5
(10 points) You are planning on opening a consulting firm. You have projected yearly cash flows of \$2 million starting next year (t = 1) with a growth rate of 3% over the foreseeable future thereafter. This endeavor will require a substantial investment and you will have to convince investors to provide you the capital to do so. You will invest some of your own money, convincing other investors will of course be useful for your valuing your own investment decision. A critical piece of your analysis is figuring out the present value of the cash flows of the business. Your research has
revealed the following information: similar consulting businesses equity has an average beta of 2.40 and the average debt-to-equity ratio in this industry is 10%. The risk-free rate is 3.25% and the expected market risk premium (the average difference between between the market return and the risk-free rate) is 4.50%. What is the value of the cash flows of your business? 20060181.
Do not have enough information to value.
18099548.
Question 6
(10 points) Your firm has been plodding along without much attention from the stock market; both analysts and investors are not showing much interest in your company. Your boss insists that (a) he can increase the return on equity of the company by simply taking on more debt and (b) that will attract new investors to the firm because of the higher returns. False.
Partly true; partly false.
True.
Question 7
(10 points) Two firms, Alpha, Inc., and Beta, Inc., are in the same business. Alpha, Inc., has debt that is viewed by the market as risk-less with a market value of \$500 million. Beta, Inc., has no debt. Both firms are expected to generate cash flows of \$100 million per year for the foreseeable future and the market value of the equity of Beta, Inc is \$1 billion. Estimate the return on equity of Alpha, Inc. Assume there are no taxes, and the risk-free rate is 5%. (No more than two decimals in the percentage interest rate, but do not enter the % sign.) Answer for Question 7
Question 8
(10 points) Mango, Inc. has had debt with market value of \$1 million that has paid a 6% coupon and has had an expiration date that is far, far away. The expected annual earnings before interest and taxes for the firm are \$2 million and the firm has not grown, nor does it have plans for any growth. The firm however has just raised more equity to retire all its debt. If the required rate of return to equity-holders (after the capital structure
change) is now 20%, what is the market value of the firm? Assume there are no taxes. (Enter just the number without the \$ sign or a comma; round to the nearest whole dollar.) Answer for Question 8
Question 9
(15 points) Suppose all investors are risk-averse and hold diversified portfolios. You are evaluating a new drug company that is going to have two divisions: an R&D unit and a Sales unit. Your CEO and you are arguing about whether the two units should have the same cost of capital (WACC), or whether the discount rates should be different. If different, what should be the relative magnitudes of the discount rates, that is, which unit R&D or Sales should have the higher discount rate. Assume the discount rates of the two units are labeled as R (for R&D) and S (for Sales), respectively. What do you think? S>R.
The same discount rates for both divisions/units (R=S).
R>S.
Question 10
(15 points) NorthSouth Airlines has been granted permission to fly passengers between major U.S. cities. The new company faces competition from four airlines that operate between the major cities. The betas of the equity of the four major competitors (A, B, C, D) are 2.25, 2.50, 2.75, and 3.00; and the debt-to-equity ratios of these four companies (in the same order: A, B, C, D) are 0.21, 0.42, 0.63, 0.83. Although these D/E ratios vary, all airline debt is rated the same. Suppose the yield on airline debt is 7%, the risk-free rate is 3% and the expected market risk premium (the average difference between the market return and the risk-free rate) is 5%. What is the cost of capital (or discount rate) that you should use in valuing NorthSouth Airlines? (No more than two decimals in the percentage interest rate, but do not enter the % sign.) Answer for Question 10
In accordance with the Coursera Honor Code, I (Shravan Vepa) certify that the answers here are my own work. Thank you!
A+
• Subject:
• University/College: University of Chicago
• Type of paper: Thesis/Dissertation Chapter
• Date: 20 March 2016
• Words:
• Pages:
We will write a custom essay sample on The purpose of this study is to investigate the use of internet activities that may affect the sleeping hours of a person or else cause of sleep deprivation
for only \$16.38 \$12.9/page | 1,640 | 7,011 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.96875 | 3 | CC-MAIN-2017-09 | longest | en | 0.938559 |
https://money-zine.com/definitions/investing-dictionary/index-arbitrage/ | 1,628,034,057,000,000,000 | text/html | crawl-data/CC-MAIN-2021-31/segments/1627046154486.47/warc/CC-MAIN-20210803222541-20210804012541-00034.warc.gz | 419,386,218 | 44,876 | HomeDefinitionsInvesting DictionaryIndex Arbitrage
# Index Arbitrage
## Definition
The investing term index arbitrage refers to a trading strategy that takes advantage of the difference between the futures price of an index and its cash price. Program traders can take advantage of these anomalies by buying or selling the index future, while simultaneously assuming the opposite position for the stocks that make up the index.
### Explanation
Index arbitrage is typically accomplished using program trading techniques. Computer systems are instructed to monitor the difference between the spot price of a stock index (such as the S&P 500) and its futures contracts. When the difference between a futures contract and its corresponding cash price moves through pre-programmed thresholds, simultaneous buy and sell orders will be placed for stocks, index futures, or exchange traded funds (ETF).
The difference between a futures contract and its spot price is referred to as basis or spread. Generally, there are sets of rules which index arbitrage follows:
• If a futures contract is deemed high relative to the cash price of the index, the index future is sold and the stocks making up the index are purchased.
• If a futures contract is deemed low relative to the cash price of the index, the index future is purchased and the stocks making up the index are sold.
In either of the above scenarios, the arbitrageur is fully hedged against upward or downward movement in the value of the index. Profits are made when the spread between the futures contract and its spot price returns to its “normal” or expected value.
### Example
In this example the futures price is deemed high relative to the cash price of the index, so simultaneous orders are placed to sell (short) the futures contract and buy the index. This position is now fully hedged against movements in the index itself. If the index goes up, the value of the cash position increases, while the value of the futures contract decreases by the same amount. If the index goes down, the value of the cash position decreases, while the value of the futures contract increases by the same amount.
If the difference between the futures contract and the spot price narrows, the arbitrageur will profit from their short position in the futures contract. | 445 | 2,318 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.53125 | 3 | CC-MAIN-2021-31 | latest | en | 0.925405 |
https://www.stata.com/statalist/archive/2008-04/msg00613.html | 1,723,252,920,000,000,000 | text/html | crawl-data/CC-MAIN-2024-33/segments/1722640782288.54/warc/CC-MAIN-20240809235615-20240810025615-00087.warc.gz | 776,457,013 | 4,067 | # Re: st: Weighted Least Squares
From Maarten buis <[email protected]> To [email protected] Subject Re: st: Weighted Least Squares Date Tue, 15 Apr 2008 15:57:43 +0100 (BST)
```I read your question as follows:
- you have two (or more) variables: x1 = % in zone1, and x2 = % in
zone 2, which must add up to 100%
- x1 and x2 are explanatory variables
- you want these variables to have only one effect instead two (or
more) because you see them as one variable.
Is that correct?
-- Maarten
--- Tam Phan <[email protected]> wrote:
> Hello Stata Users:
>
> I have a quick question on weighted Least squares: Below is the
> following econometric equation:
>
> Q1+Q2 = alpha + [ (Q1/(Q1+Q2))*X1 + (Q2/(Q1+Q2))*X2 ]b + e
>
> Where X1 and X2 are the same explanatory variables, however with
> different numerical values. For example, X1 would be the % in one
> zone, and X2 is the percent in another zone.
>
> Is there anything wrong w/ this logic?
>
>
> y1=alpha1+beta1(P)+e
> y2= alpha2 +beta2(P)+v
>
> To obtain the aggregate effect one would sum:
>
> y1+y2=(alpha1+alpha2) +(beta1+beta2)P+(e+v)
>
> Is there anything wrong with the above logic?
>
> Tam
>
> Tam
> *
> * For searches and help try:
> * http://www.stata.com/support/faqs/res/findit.html
> * http://www.stata.com/support/statalist/faq
> * http://www.ats.ucla.edu/stat/stata/
>
-----------------------------------------
Maarten L. Buis
Department of Social Research Methodology
Vrije Universiteit Amsterdam
Boelelaan 1081
1081 HV Amsterdam
The Netherlands
Buitenveldertselaan 3 (Metropolitan), room Z434
+31 20 5986715
http://home.fsw.vu.nl/m.buis/
-----------------------------------------
___________________________________________________________
Yahoo! For Good helps you make a difference
http://uk.promotions.yahoo.com/forgood/
*
* For searches and help try:
* http://www.stata.com/support/faqs/res/findit.html
* http://www.stata.com/support/statalist/faq
* http://www.ats.ucla.edu/stat/stata/
``` | 593 | 1,993 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.75 | 3 | CC-MAIN-2024-33 | latest | en | 0.739934 |
https://www.indiabix.com/logical-reasoning/cause-and-effect/ | 1,716,443,765,000,000,000 | text/html | crawl-data/CC-MAIN-2024-22/segments/1715971058611.55/warc/CC-MAIN-20240523050122-20240523080122-00044.warc.gz | 713,274,772 | 9,670 | # Logical Reasoning - Cause and Effect
Exercise : Cause and Effect - Cause and Effect
• Cause and Effect - Cause and Effect
Directions to Solve
In each of the following questions, two statements numbered I and II are given. There may be cause and effect relationship between the two statements. These two statements may be the effect of the same cause or independent causes. These statements may be independent causes without having any relationship. Read both the statements in each question and mark your answer as
• (A) If statement I is the cause and statement II is its effect;
• (B) If statement II is the cause and statement I is its effect;
• (C) If both the statements I and II are independent causes;
• (D) If both the statements I and II are effects of independent causes; and
• (E) If both the statements I and II are effects of some common cause.
1.
Statements:
1. The prices of petrol and diesel in the domestic market have remained unchanged for the past few months.
2. The crude oil prices in the international market have gone up substantially in the last few months.
Statement I is the cause and statement II is its effect
Statement II is the cause and statement I is its effect
Both the statements I and II are independent causes
Both the statements I and II are effects of independent causes
Both the statements I and II are effects of some common cause
Explanation:
The prices of petrol and diesel being stagnant in the domestic market and the increase in the same in the international market must be backed by independent causes.
2.
Statements:
1. The government has recently fixed the fees for professional courses offered by the unaided institutions which are much lower than the fees charged last year.
2. The parents of the aspiring students launched a severe agitation last year protesting against the high fees charged by the unaided institutions.
Statement I is the cause and statement II is its effect
Statement II is the cause and statement I is its effect
Both the statements I and II are independent causes
Both the statements I and II are effects of independent causes
Both the statements I and II are effects of some common cause
Explanation:
The parents' protest against high fees being charged by the institutions led the government to interfere and fix the fees at a more affordable level.
3.
Statements:
1. The Reserve Bank of India has recently put restrictions on few small banks in the country.
2. The small banks in the private and co-operative sector in India are not in a position to withstand the competitions of the bigger in the public sector.
Statement I is the cause and statement II is its effect
Statement II is the cause and statement I is its effect
Both the statements I and II are independent causes
Both the statements I and II are effects of independent causes
Both the statements I and II are effects of some common cause
Explanation:
The inability of the small banks to compete with the bigger ones shall not ensure security and good service to the customers, which is an essential concomitant that has to be looked into by the Reserve Bank. I seems to be a remedial step for the same.
4.
Statements:
1. All the schools in the area had to be kept closed for most part of the week.
2. Many parents have withdrawn their children from the local schools.
Statement I is the cause and statement II is its effect
Statement II is the cause and statement I is its effect
Both the statements I and II are independent causes
Both the statements I and II are effects of independent causes
Both the statements I and II are effects of some common cause
Explanation:
Closing the schools for a week and the parents withdrawing their wards from the local schools are independent issues, which must have been triggered by different individual causes.
5.
Statements:
1. India has surpassed the value of tea exports this year over all the earlier years due to an increase in demand for quality tea in the European market.
2. There is an increase in demand of coffee in the domestic market during the last two years.
Statement I is the cause and statement II is its effect
Statement II is the cause and statement I is its effect
Both the statements I and II are independent causes
Both the statements I and II are effects of independent causes
Both the statements I and II are effects of some common cause | 879 | 4,368 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.9375 | 3 | CC-MAIN-2024-22 | latest | en | 0.960546 |
https://www.studypool.com/discuss/448542/how-do-you-find-the-length-of-an-apothem-of-a-hexagon-1?free | 1,508,826,030,000,000,000 | text/html | crawl-data/CC-MAIN-2017-43/segments/1508187828178.96/warc/CC-MAIN-20171024052836-20171024072836-00611.warc.gz | 987,373,927 | 14,923 | Time remaining:
##### How do you find the length of an apothem of a hexagon
label Mathematics
account_circle Unassigned
schedule 1 Day
account_balance_wallet \$5
I was only given the length of one side of the regular hexagon. I would like step by step on how to complete this. the side length was 22
Oct 23rd, 2017
The apothem is the line from the center of the hexagon to the middle of any side.
Since each side of a hexagon is the base of an equilateral triangle, then the apothem is really just the ALTITUDE of the equilateral triangle.
If the side is 22, the the altitude divide the base into two equal segments, both 11.
So, then the apothem would be found by
tan 60 = apothem / 11
apothem = 11tan60
= 19.05
Mar 27th, 2015
How did you get that final answer? do you divide those two numbers?
Mar 27th, 2015
draw an equilateral triangle. The altitude is the apothem.
notice that it divide the triangle into 2 RIGHT TRIANGLES.
The base angle is 60 because the whole triangle is an equilateral.
Then just use the relationship of tan = opposite side / adjacent side.
That gives you tan60 = apothem / 11
solve from there.
Does that clear it up?
Mar 27th, 2015
yes. thank you very much.
Mar 27th, 2015
you're welcome :)
Mar 27th, 2015
...
Oct 23rd, 2017
...
Oct 23rd, 2017
Oct 24th, 2017
check_circle | 379 | 1,325 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.0625 | 4 | CC-MAIN-2017-43 | latest | en | 0.924652 |
https://www.instructables.com/id/Propeller-Clock/ | 1,571,509,506,000,000,000 | text/html | crawl-data/CC-MAIN-2019-43/segments/1570986697439.41/warc/CC-MAIN-20191019164943-20191019192443-00399.warc.gz | 936,803,720 | 23,498 | # Propeller Clock
160,083
101
160
Hi Friends,
now the propeller clock available make using 8051 controller family.
The 8051 controller is generally used in college project so u can make your own propeller using 8051,
for this propeller u required 1200RPM motor for drive it other thing for that power supply for running circuit u just make it by concept of any DC MOTOR brush supply.
### Teacher Notes
Teachers! Did you use this instructable in your classroom?
Add a Teacher Note to share how you incorporated it into your lesson.
## Step 1: 8051
The 8051 is the 40 pin microcontroller IC which is used for this project
## Step 2: Propeller Clock Circuit on Universal PCB
I have made it on universal PCB and other connection direct drag and connect with the LED's. This project suggest me Anupam Dubey to make this amazing project.
## Step 3: Circuit Diagrame
The circuit diagram of this circuit is given here in Propeller Clock PDF file so download it for schematic diagram.
## Step 4: PCB Layout
I have made it PCB layout in Dip Trace PCB designer software. And it given here my propeller clock PCB layout, but for my project i made circuit on universal PCB, i just upload it PCB if any one want to make it on designed PCB then can make it easily fron that PCB design. And this design for 8051 only
## Step 5: Programming
Programming and its hex file given here so you can directly burn your 8051controller ic using suitable ISP/loader. I used for programming Keil Programmer and for burn ic I used Flash Magic loader, for that you can use the
mikroC PRO for 8051 for programming and its flash burner also available with its setup file.
## Step 6: POV Propeller Massage Display Using Atmega16
Now I've made my new POV propeller display using atmega16, you can also make it using atmega8 for more cheaper and easy and compact.
## Step 8: Method for Codding Your Massage
In POV Massage display generally two type of circuit can build, one was Active High and other was Active Low.
As we know about this concept so active high circuit make LED on when that gives logic "1", and active Low circuits make LED on when logic "0" is applied. So as far that concept here figure given for make your own code for your massage.
## Step 9: Source Code of All Word for All Active High Condition
This source file providing for writing any name any sign in POV propeller display.
NOTE:-
This code for ACTIVE HIGH circuit only if your POV display you made in active high mode then simply past this code for all word and any provided sign in PDF.
## Recommendations
• ### Internet of Things Class
22,802 Enrolled
## 160 Discussions
i have implemented the ckt but as output i am getting only 8 concentric circles ( 8 LED's) not the time . I am not able to find the faults . The ckt not showing the time . Now what I have to do? Plz tell me. Plz reply me fast as fast as u can.
i have connected every connection in proper manner but my 8th led is not glowing why?
Hi,
have you check this one??
actually we make/coding for only 7 LED's if you want to glow you 8th LED you just make little bit of changes in massage code any you can glow it also. :)
Bro.. I have made this project.. It's really cool and interesting. However I am having a problem. While I am connecting the vcc of the microcontroller and led to the 9volt battery with the help of connectors.. Everytime I am getting different output. Sometime all leds are glowing, while sometime the output is coming properly.. It's really frustrating. Few times, I have to connect the connector to 9 volt battery like 20-30 times just to get the proper output.. And then it comes.. Can you please help me.. what should I do..?
how will it detect the blade postion ? in one of the circuit diagrams you mentioned ir led photo diode ? how does it work ?
sir plz tell me about the list of all components and there exact values
3 replies
you need
1ps-atmega16 or atmega8
8ps-LEDs
8ps-Resistors
2ps-ceramic capacitors
1ps-crystal
1ps-battery
1ps-pc coller fan
and a PCB.
and for its value please check circuits.
:)
the project is very well explained with proper pictures and documents attached. but, i was confused about the role of 8051 MC in the circuit.
why is 8051 used? what's the difference between 8051 and atmega6?
both are controller IC just its from different family 8051 is another family controller and Atmega16 is AVR family controller and program hex file saving ROM is different in both of this. :)
sir, can u suggest any 32 pin microcontroller ic to run this project. | 1,077 | 4,555 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.625 | 3 | CC-MAIN-2019-43 | latest | en | 0.87805 |
https://www.splashlearn.com/s/math-worksheets/divide-hundredths-by-2-digit-numbers-with-remainder-horizontal-division | 1,725,738,400,000,000,000 | text/html | crawl-data/CC-MAIN-2024-38/segments/1725700650920.0/warc/CC-MAIN-20240907193650-20240907223650-00692.warc.gz | 976,186,964 | 25,880 | Divide Hundredths by 2-Digit Numbers with Remainder: Horizontal Division - Math Worksheets - SplashLearn
Math > Divide Hundredths by 2-Digit Numbers with Remainder: Horizontal Division Worksheet
# Divide Hundredths by 2-Digit Numbers with Remainder: Horizontal Division Worksheet
## Know more about Divide Hundredths by 2-Digit Numbers with Remainder: Horizontal Division Worksheet
Does your child know how to divide hundredths by 2-digit numbers with remainder? Invite them to practice this concept here. Students divide decimals by 2-digit numbers in the same manner as they would divide whole numbers first. Depending on how many decimal digits are in the dividend, they add a decimal point at the appropriate place in the quotient. Their accuracy with this concept is improved by using the divide hundredths by 2-digit numbers with remainder worksheet. | 173 | 859 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.984375 | 3 | CC-MAIN-2024-38 | latest | en | 0.872592 |
https://ask.sagemath.org/questions/26630/revisions/ | 1,606,365,804,000,000,000 | text/html | crawl-data/CC-MAIN-2020-50/segments/1606141186414.7/warc/CC-MAIN-20201126030729-20201126060729-00105.warc.gz | 195,188,006 | 6,147 | # Revision history [back]
### How can I search and count all induced subgraphs in a directed graph with edge labels?
Hello all,
I can extract all induced subgraphs and count them in a graph. My question is how I can search and count all induced subgraphs in a labeled graph (a graph with edge labels)? In documentation of g.subgraph() and g.subgraph_search_count() is mentioned that these functions should work on labeled graphs as well. but it does not work. Here is an example:
sage: s=DiGraph() sage: s.add_vertex(0) sage: s.add_vertex(1) sage: s.add_edge(0,1,label="a") sage: s.plot(color_by_label=true)
sage: g.subgraph_search_count(s) 2
While it should return 1 not 2.
### How can I search and count all induced subgraphs in a directed graph with edge labels?
Hello all,
I can extract all induced subgraphs and count them in a graph. My question is how I can search and count all induced subgraphs in a labeled graph (a graph with edge labels)? In documentation of g.subgraph() and g.subgraph_search_count() is mentioned that these functions should work on labeled graphs as well. but it does not work. Here is an example:
sage: g.subgraph_search_count(s) 2
While it should return 1 not 2.
### How can I search and count all induced subgraphs in a directed graph with edge labels?
Hello all,
I can extract all induced subgraphs and count them in a graph. My question is how I can search and count all induced subgraphs in a labeled graph (a graph with edge labels)? In documentation of g.subgraph() and g.subgraph_search_count() is mentioned that these functions should work on labeled graphs as well. but it does not work. Here is an example:
sage: g=DiGraph()
sage: s=DiGraph() sage: s.add_vertex(0) sage: s.add_vertex(1) sage: s.add_edge(0,1,label="a") sage: s.plot(color_by_label=true)
sage: g.subgraph_search_count(s) 2
While it should return 1 not 2.
### How can I search and count all induced subgraphs in a directed graph with edge labels?
Hello all,
I can extract all induced subgraphs and count them in a graph. My question is how I can search and count all induced subgraphs in a labeled graph (a graph with edge labels)? In documentation of g.subgraph() and g.subgraph_search_count() is mentioned that these functions should work on labeled graphs as well. but it does not work. Here is an example:
sage: g=DiGraph()
g=DiGraph() | 574 | 2,370 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.875 | 3 | CC-MAIN-2020-50 | longest | en | 0.888162 |
https://www.hpmuseum.org/forum/thread-11176-post-101937.html | 1,638,213,611,000,000,000 | text/html | crawl-data/CC-MAIN-2021-49/segments/1637964358786.67/warc/CC-MAIN-20211129164711-20211129194711-00091.warc.gz | 903,554,300 | 6,730 | FYI: WarpPi calculator
08-06-2018, 05:52 AM
Post: #1
Luigi Vampa Member Posts: 257 Joined: Dec 2015
FYI: WarpPi calculator
Saludos Saluti Cordialement Cumprimentos MfG BR + + + + +
Luigi Vampa +
Free42 HuaweiP10 '<3' I + + +
08-06-2018, 07:28 AM
Post: #2
Dan Member Posts: 162 Joined: Jan 2017
RE: FYI: WarpPi calculator
Anyone know the model of the LCD on that calculator?
08-07-2018, 06:37 PM (This post was last modified: 08-09-2018 05:52 PM by compsystems.)
Post: #3
compsystems Senior Member Posts: 1,329 Joined: Dec 2013
RE: FYI: WarpPi calculator
WarpPi Is an Impressive DIY Raspberry Pi Calculator, and is the firts with Step-By-Step Algebra
https://www.neoteo.com/construye-una-cal...y-pi-zero/
mechanical binary calculator https://www.youtube.com/watch?v=NQX6irk11qA
08-09-2018, 01:04 PM
Post: #4
Eddie W. Shore Senior Member Posts: 1,234 Joined: Dec 2013
RE: FYI: WarpPi calculator
Several things I like about it:
I like how you can designate how the variables act: blue for CAS variables, black for numerical constants, etc
I like how big the screen is. The screen is clear, well done on the display.
I can't wait to see what the finished keyboard looks like.
08-10-2018, 03:51 PM (This post was last modified: 08-10-2018 03:59 PM by compsystems.)
Post: #5
compsystems Senior Member Posts: 1,329 Joined: Dec 2013
RE: FYI: WarpPi calculator
If it is a calculator with CAS, the 6 algebraic operators should be together
[+], [-], [·](*), [÷](/), [^], [√]
PHP Code:
[f1][f2][f3][f4][f5][←][↑][→] [↓][ ][ ][ ][ ][↤][,][ ][ ][ ][ ][()][±][√][^][÷][EEX][7][8][9][*][α][4][5][6][-][⇧][1][2][3][+][esc][0][.][spc][enter]
mechanical binary calculator https://www.youtube.com/watch?v=NQX6irk11qA
08-11-2018, 04:39 AM
Post: #6
Jlouis Senior Member Posts: 668 Joined: Nov 2014
RE: FYI: WarpPi calculator
Doesn't the HP 50G have step by step CAS?
Even the 28c has step by step, well, not all the steps, but anyway, the 28c is from 1987, isnt it?
I think I'm missing something...
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