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# Special Issue for the Conference on Computer Science Logic (CSL) 2017

CSL started as a series of international workshops on Computer Science Logic, and became at its sixth meeting the Annual Conference of the EACSL.

This special issue contains extended versions of selected papers presented at the 26th annual EACSL conference Computer Science Logic , CSL 2017. The conference was held in Stockholm from August 20 to August 24, 2017. It was was organised jointly by members of the Departments of Philosophy and of Mathematics and Stockholm University, and of the Department of Theoretical Computer Science at KTH Royal Institute of Technology.

There was a total of 76 full paper submissions to CSL 2017. Each paper was assigned for reviewing to at least three program committee members. The programme committee was assisted by over 100 external reviewers providing additional expertise and, after 2 months of reviewing and extensive discussion via Easychair, eventually selected 35 papers for presentation at the conference and publication in the proceedings. Out of these, we selected the top 10 and invited their authors to prepare and submit extended versions to the LMCS. These were subjected to further rigorous reviewing by at least 2 reviewers, followed by further revisions, before eventually accepted for this special issue.

We wish to thank all contributing authors for their scholarship and high quality contributions. We also thank all members of the programme committee, as well as all additional reviewers for the special issue for their hard and professional work on reviewing these papers, which essentially contributed to the high scientific quality of this special issue. Our thanks also go to the members of the organising committee of CSL 2017 and the associated workshops for organising the successful and enjoyable conference.

Mads Dam and Valentin Goranko Guest Editors of the CSL 2017 Special Issue of LMCS

### 1. Categorical structures for type theory in univalent foundations

, ; , ; ,.
In this paper, we analyze and compare three of the many algebraic structures that have been used for modeling dependent type theories: categories with families, split type-categories, and representable maps of presheaves. We study these in univalent type theory, where the comparisons between them can be given more elementarily than in set-theoretic foundations. Specifically, we construct maps between the various types of structures, and show that assuming the Univalence axiom, some of the comparisons are equivalences. We then analyze how these structures transfer along (weak and strong) equivalences of categories, and, in particular, show how they descend from a category (not assumed univalent/saturated) to its Rezk completion. To this end, we introduce relative universes, generalizing the preceding notions, and study the transfer of such relative universes along suitable structure. We work throughout in (intensional) dependent type theory; some results, but not all, assume the univalence axiom. All the material of this paper has been formalized in Coq, over the UniMath library.

### 2. On the First-Order Complexity of Induced Subgraph Isomorphism

, ; ,.
Given a graph $F$, let $I(F)$ be the class of graphs containing $F$ as an induced subgraph. Let $W[F]$ denote the minimum $k$ such that $I(F)$ is definable in $k$-variable first-order logic. The recognition problem of $I(F)$, known as Induced Subgraph Isomorphism (for the pattern graph $F$), is solvable in time $O(n^{W[F]})$. Motivated by this fact, we are interested in determining or estimating the value of $W[F]$. Using Olariu's characterization of paw-free graphs, we show that $I(K_3+e)$ is definable by a first-order sentence of quantifier depth 3, where $K_3+e$ denotes the paw graph. This provides an example of a graph $F$ with $W[F]$ strictly less than the number of vertices in $F$. On the other hand, we prove that $W[F]=4$ for all $F$ on 4 vertices except the paw graph and its complement. If $F$ is a graph on $t$ vertices, we prove a general lower bound $W[F]>(1/2-o(1))t$, where the function in the little-o notation approaches 0 as $t$ inreases. This bound holds true even for a related parameter $W^*[F]\le W[F]$, which is defined as the minimum $k$ such that $I(F)$ is definable in the infinitary logic $L^k_{\infty\omega}$. We show that $W^*[F]$ can be strictly less than $W[F]$. Specifically, $W^*[P_4]=3$ for $P_4$ being the path graph on 4 vertices. Using the lower bound for $W[F]$, we also obtain a succintness result for existential monadic second-order logic: A usage of just one monadic quantifier sometimes reduces the first-order quantifier depth at a […]

### 3. Validity and Entailment in Modal and Propositional Dependence Logics

,.
The computational properties of modal and propositional dependence logics have been extensively studied over the past few years, starting from a result by Sevenster showing NEXPTIME-completeness of the satisfiability problem for modal dependence logic. Thus far, however, the validity and entailment properties of these logics have remained mostly unaddressed. This paper provides a comprehensive classification of the complexity of validity and entailment in various modal and propositional dependence logics. The logics examined are obtained by extending the standard modal and propositional logics with notions of dependence, independence, and inclusion in the team semantics context. In particular, we address the question of the complexity of validity in modal dependence logic. By showing that it is NEXPTIME-complete we refute an earlier conjecture proposing a higher complexity for the problem.

### 4. Capturing Logarithmic Space and Polynomial Time on Chordal Claw-Free Graphs

,.
We show that the class of chordal claw-free graphs admits LREC$_=$-definable canonization. LREC$_=$ is a logic that extends first-order logic with counting by an operator that allows it to formalize a limited form of recursion. This operator can be evaluated in logarithmic space. It follows that there exists a logarithmic-space canonization algorithm, and therefore a logarithmic-space isomorphism test, for the class of chordal claw-free graphs. As a further consequence, LREC$_=$ captures logarithmic space on this graph class. Since LREC$_=$ is contained in fixed-point logic with counting, we also obtain that fixed-point logic with counting captures polynomial time on the class of chordal claw-free graphs.

### 5. Normalizing the Taylor expansion of non-deterministic {\lambda}-terms, via parallel reduction of resource vectors

,.
It has been known since Ehrhard and Regnier's seminal work on the Taylor expansion of $\lambda$-terms that this operation commutes with normalization: the expansion of a $\lambda$-term is always normalizable and its normal form is the expansion of the Böhm tree of the term. We generalize this result to the non-uniform setting of the algebraic $\lambda$-calculus, i.e. $\lambda$-calculus extended with linear combinations of terms. This requires us to tackle two difficulties: foremost is the fact that Ehrhard and Regnier's techniques rely heavily on the uniform, deterministic nature of the ordinary $\lambda$-calculus, and thus cannot be adapted; second is the absence of any satisfactory generic extension of the notion of Böhm tree in presence of quantitative non-determinism, which is reflected by the fact that the Taylor expansion of an algebraic $\lambda$-term is not always normalizable. Our solution is to provide a fine grained study of the dynamics of $\beta$-reduction under Taylor expansion, by introducing a notion of reduction on resource vectors, i.e. infinite linear combinations of resource $\lambda$-terms. The latter form the multilinear fragment of the differential $\lambda$-calculus, and resource vectors are the target of the Taylor expansion of $\lambda$-terms. We show the reduction of resource vectors contains the image of any $\beta$-reduction step, from which we deduce that Taylor expansion and normalization commute on the nose. We moreover identify a class […]

### 6. Query learning of derived $\omega$-tree languages in polynomial time

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We present the first polynomial time algorithm to learn nontrivial classes of languages of infinite trees. Specifically, our algorithm uses membership and equivalence queries to learn classes of $\omega$-tree languages derived from weak regular $\omega$-word languages in polynomial time. The method is a general polynomial time reduction of learning a class of derived $\omega$-tree languages to learning the underlying class of $\omega$-word languages, for any class of $\omega$-word languages recognized by a deterministic Büchi acceptor. Our reduction, combined with the polynomial time learning algorithm of Maler and Pnueli [1995] for the class of weak regular $\omega$-word languages yields the main result. We also show that subset queries that return counterexamples can be implemented in polynomial time using subset queries that return no counterexamples for deterministic or non-deterministic finite word acceptors, and deterministic or non-deterministic Büchi $\omega$-word acceptors. A previous claim of an algorithm to learn regular $\omega$-trees due to Jayasrirani, Begam and Thomas [2008] is unfortunately incorrect, as shown in Angluin [2016].

### 7. $\aleph_1$ and the modal $\mu$-calculus

, ; ,.
For a regular cardinal $\kappa$, a formula of the modal $\mu$-calculus is $\kappa$-continuous in a variable x if, on every model, its interpretation as a unary function of x is monotone and preserves unions of $\kappa$-directed sets. We define the fragment $C_{\aleph_1}(x)$ of the modal $\mu$-calculus and prove that all the formulas in this fragment are $\aleph_1$-continuous. For each formula $\phi(x)$ of the modal $\mu$-calculus, we construct a formula $\psi(x) \in C_{\aleph_1 }(x)$ such that $\phi(x)$ is $\kappa$-continuous, for some $\kappa$, if and only if $\phi(x)$ is equivalent to $\psi(x)$. Consequently, we prove that (i) the problem whether a formula is $\kappa$-continuous for some $\kappa$ is decidable, (ii) up to equivalence, there are only two fragments determined by continuity at some regular cardinal: the fragment $C_{\aleph_0}(x)$ studied by Fontaine and the fragment $C_{\aleph_1}(x)$. We apply our considerations to the problem of characterizing closure ordinals of formulas of the modal $\mu$-calculus. An ordinal $\alpha$ is the closure ordinal of a formula $\phi(x)$ if its interpretation on every model converges to its least fixed-point in at most $\alpha$ steps and if there is a model where the convergence occurs exactly in $\alpha$ steps. We prove that $\omega_1$, the least uncountable ordinal, is such a closure ordinal. Moreover we prove that closure ordinals are closed under ordinal sum. Thus, any formal expression built from 0, 1, $\omega$, $\omega_1$ by […]

### 8. Scalar and Vectorial mu-calculus with Atoms

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We study an extension of modal $\mu$-calculus to sets with atoms and we study its basic properties. Model checking is decidable on orbit-finite structures, and a correspondence to parity games holds. On the other hand, satisfiability becomes undecidable. We also show expressive limitations of atom-enriched $\mu$-calculi, and explain how their expressive power depends on the structure of atoms used, and on the choice between basic or vectorial syntax.

### 9. The Dynamic Geometry of Interaction Machine: A Token-Guided Graph Rewriter

, ; ,.
In implementing evaluation strategies of the lambda-calculus, both correctness and efficiency of implementation are valid concerns. While the notion of correctness is determined by the evaluation strategy, regarding efficiency there is a larger design space that can be explored, in particular the trade-off between space versus time efficiency. Aiming at a unified framework that would enable the study of this trade-off, we introduce an abstract machine, inspired by Girard's Geometry of Interaction (GoI), a machine combining token passing and graph rewriting. We show soundness and completeness of our abstract machine, called the \emph{Dynamic GoI Machine} (DGoIM), with respect to three evaluations: call-by-need, left-to-right call-by-value, and right-to-left call-by-value. Analysing time cost of its execution classifies the machine as efficient'' in Accattoli's taxonomy of abstract machines.

### 10. A Finite-Model-Theoretic View on Propositional Proof Complexity

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We establish new, and surprisingly tight, connections between propositional proof complexity and finite model theory. Specifically, we show that the power of several propositional proof systems, such as Horn resolution, bounded-width resolution, and the monomial calculus of bounded degree, can be characterised in a precise sense by variants of fixed-point logics that are of fundamental importance in descriptive complexity theory. Our main results are that Horn resolution has the same expressive power as least fixed-point logic, that bounded-width resolution captures existential least fixed-point logic, and that the polynomial calculus with bounded degree over the rationals solves precisely the problems definable in fixed-point logic with counting. We also study the bounded-degree polynomial calculus. Over the rationals, it captures fixed-point logic with counting if we restrict the bit-complexity of the coefficients. For unrestricted coefficients, we can only say that the bounded-degree polynomial calculus is at most as powerful as bounded variable infinitary counting logic, but a precise logical characterisation of its power remains an open problem. These connections between logics and proof systems allow us to establish finite-model-theoretic tools for proving lower bounds for the polynomial calculus over the rationals and also over finite fields. This is a corrected version of the paper (arXiv:1802.09377) published originally on January 23, 2019.


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# Slopes of tangent lines Find the slope of the | Ch 10.3 - 14E

ISBN: 9780321570567 2

## Solution for problem 14E Chapter 10.3

Calculus: Early Transcendentals | 1st Edition

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Problem 14E

Find the slope of the line tangent to the following polar curves at the given points. At the points where the curve intersects the origin (when this occurs), find the equation of the tangent line in polar coordinates.

$$r=2\theta;\ \ (\frac{\pi}{2},\ \frac{\pi}{4})$$

Step-by-Step Solution:

Solution 14E

Step 1:

Given that

r = 2θ;

Step 2 of 4

Step 3 of 4

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# Short answer find the quotient 6x^3-x^2-7x-9 ------------ 2x+3

143,010 questions, page 805
1. ## Math

A rectangular closet has a perimeter of about 20 feet and a width of about 3 feet. Find the diagonal.

asked by Cassie on September 14, 2010

Find the sum of each infinite series, or state that the sum doesn'nt exist 1/7+5/14+25/28+

asked by Todd on December 16, 2012
3. ## Calculus

Using the Disk Method or washer method (im not sure which one) Find The volume of y=x^3 , y=1 , x = 2 revolving around the X axis

asked by Andrew on December 16, 2008
4. ## Math

The difference of two integers is 9. Five times the smaller is 7 more than three times the larger. Find the numbers.

asked by Byran on September 13, 2011
5. ## Calculus

Find the relative extrema and absolute extrema if any. Use the first derivative test for the function f(x)=x^3-3x+2

asked by Alan on April 17, 2012
6. ## geography

Deforestation is the process whereby ( write a definition ) all i can find is the process where trees are cut down. is this right?

asked by xander on December 29, 2008
7. ## Calculus

Use logarithmic differentiation to find derivative of y=x^x^2 x > 0. Please show all steps so I can follow the steps. Thank you.!

asked by Sean on April 20, 2013
8. ## science

Number of electrons in the outermost m shell of an element is 3.find its atomic number.

asked by saurav on February 2, 2015
9. ## calculus

Find the definate integal (upper limit 1, lower limit 0) (7-4x)^2dx

asked by Roy on October 10, 2013
10. ## math

find the area between the x-axis and the graph of the given function over the given interval: y = sqrt(9-x2^2) over [-3,3] actually, it's y = sqrt(9-x^2) over [-3,3]

asked by jamay on April 18, 2007
11. ## Physics

A certain planet is located 3.54 x 10^11 m from its sun. If its orbital period is 4.9 x 10^7 s, find the mass of its sun.

asked by Anonymous on April 28, 2014
12. ## Math

A circle is divided into 20 equal parts. Find the angle measure of one of the parts

asked by Diva on March 31, 2017
13. ## math,correction

Find the y-intercept of -x - 9y= 9 so this is what i did: -9y = +x +9 i divided both sides by -9 y=(-(1)/(9)x-1 now from here i don't know what to do? y=-1/9 x -1 Yintercept is when x=0 so this is what i get: y=(-(1)/(9)x-1 y= (1(1)/(9)(0)-1 y= -1 so the y

asked by Jasmine20 on January 27, 2007
14. ## Math ~ Check Answers ~

A number cube with the numbers 1 through 6 is rolled. Find P (number is greater than or equal to 3). a.) 4/6 b.) 1/6 c.) 2/6********** d.) 5/6

asked by Chloé on April 23, 2014
15. ## Calculus BC

Find the volume if the region enclosing y=2x, y=0, x=3 is rotated a)about the y-axis b)about the line x=3 c)about the line x=4.

asked by Maya on February 12, 2011
16. ## Science

Where can I find A article on evolution? http://en.wikipedia.org/wiki/Evolution

asked by Briana on October 25, 2006
17. ## Calculus

If P'(t)= [10(t+2)]/t. What is P(t)? It is asking for the derivative, and I am so lost! Because so far we haven't covered how to find the derivative of something so complicated. Any ideas? TY

asked by Jaimie on August 15, 2010
18. ## @Steve RE: Relative Extrema: AP Calculus

Thanks so much for all your help! That was way simpler than I thought, I think I was just overthinking it. Do you think you could also help me find the concave up at x = −2, x = −1, and x = 2 and show f(x) is concave down at x = 0? (this is still for

asked by Pax on January 23, 2015
19. ## Math

the length of a rectangle is 3cm more than twice the width. the perimeter is 84cm. find the length

asked by Brett on April 7, 2010
20. ## Calc 1: Max/Min

Find values for a and b so the function f(x)= ax*e^bx has the max value f(1)=3. Give exact values for a and b, not decimals)

asked by Anonymous on October 22, 2014
21. ## Algebra

Find the center and radius of the circle. x^2+(y+7)^2=4 I know that the formula for the center at the origin x^2+y^2=r^2 Very lost. Need Help!

asked by Carrie on April 18, 2010
22. ## geometry

Given the triangle ABC and m

asked by Anonymous on June 30, 2011
23. ## pre algebra help

Find the measure of each of the interior angles and exterior angles of a regular hexagon

asked by amy on May 30, 2012

Use the unit price to find the total price. 8 gallons at $1.94 per gallon PLZ HELP! asked by Agal on November 15, 2016 25. ## Algebra Suppose we have the following identity: (px + (1-p)y)^2 = Ax^2 + Bxy + Cy^2. Find the minimum of max(A,B,C) over 0 \leq p \leq 1. asked by Oikawa on June 5, 2017 26. ## Algebra The difference of two integers is 9. Five times the smaller is 7 more than three times the larger. Find the numbers. asked by Lou on September 13, 2011 27. ## Algebra hw, coordinates!! help!! Triangle ABC has coordinates A(1, 4); B(3, –2); and C(4, 2). Find the coordinates of the image A'B'C' after a reflection over the x-axis. asked by emma C. on October 15, 2015 28. ## 5th grade math How do you find the area of an irregular shape? The shape is shaped like a cat's head. Please help me! asked by Horace on February 13, 2012 29. ## statistics In a distribution of scores, X=62 corresponds to z=+0.50, and X=52 corrsponds to z=-2.00. Find the mean and standard deviation for the distribution. asked by dray on June 20, 2010 30. ## english can you help me find facts and figures why teens should voulunteer if possible how many teens like volunteering i apreciate your help. asked by Bob on March 25, 2013 31. ## Calculus Find an equation of the tangent line to the curve at the given point. y = sin(sin x), (pi, 0) asked by Nyke on November 7, 2011 32. ## Calculus Find the linearization of f(x)= sqrtx at x=25 and use the linearization to approximate sqrt25.2 This is my math: 5+ 1/10 * (25.2-25) but it's wrong. asked by Z32 on September 27, 2009 33. ## Pre-Cal(urgent) Find u X v, where Q is the angle between u and v. magnitude of u =4 magnitude of v=10 Q=2pi/3 The Q is the part that is throwing me off. I do not know where to start. asked by Hannah on March 31, 2010 34. ## Pre-Cal I do not know how to find the angle of the function sec 4x = 2 on the unit cirlce. I know I would be looking for 2 but I don't know where since it is sec. asked by Hannah on February 17, 2010 35. ## trig Given that sin theta= 3/5 and the terminal side is in quadrant II, find tan (pi/2 - theta) asked by Ryan on October 11, 2011 36. ## geometry if ÚAOC=49, mÚBOC=2x+10, and mÚAOB=4x-15 find the degree of measure of ÚBOC and AOB asked by Ryan on November 27, 2011 37. ## math find the slope, if it exists, of the line containing the pair of points (9,5) and (10,-3) teh slope m= 1.2 m=(-3+9)/ (10-5) m=6/5 m=1.2 asked by check answer on July 22, 2009 38. ## Pre-cac Find a such that f(x)= ax^2-3x+5 has a maximum value of 15. I'm stuck because how am I suppose to complete the square? 3/2= 1.5? then square 1.5? So lost. asked by Sam on March 23, 2017 39. ## Math A floor plan has a scale of 1/4 in : 5 ft. Find the length on the drawing for an actual length of 60 ft. asked by Jennifer Roberts on April 17, 2013 40. ## Math Given TriangleABC with a = 9, b = 15, and m asked by Anonymous on November 24, 2014 41. ## Geometry How do you find the ratio if the areas of 2 similar triangles in which the ratio of a pair of corresponding sides is 4:1 asked by Jen on May 7, 2011 42. ## math translate and solve please. five times two less than a number equals seventeen. find the number asked by rosita on December 1, 2014 43. ## math find the 25th term of a geometric sequence having a common ratio r=1/4 and term a=3 asked by alex on September 17, 2009 44. ## Maths Two numbers are chosen at random from three numbers 1,3,5.Find the probability that the sum of the two is not odd? asked by Robotic on August 5, 2019 45. ## Algebra Triangle ABC has coordinates A(1, 4); B(3, –2); and C(4, 2). Find the coordinates of the image A'B'C' after a reflection over the x-axis asked by Katia on December 31, 2012 46. ## math the perimeter of a parallelogram is 88 cm and one of its adjacent sides is longer than the other by 10 cm. Find the length of each of its sides asked by suman on December 5, 2016 47. ## Calculus when trying to find the intervals of where the derivative is positive is it the intervals of where the original graph is above the x axis? asked by Anonymous on November 5, 2018 48. ## math find the area of a table top with 7/8 of a foot length and 3/5 of a foot wide. asked by tara on March 26, 2013 49. ## geometry find the measures of each side of triangle abc with vertices a(-1,5), b(6,1), c(2,6) and classify the triangle. asked by Tori on February 19, 2011 50. ## Algebra 2 Hello can you please help me to generate a number line to use in Algebra. Can you attach one for me and where can I find one on line that is printable. Thank You. asked by Lady G on October 8, 2014 51. ## Maths The first 5 terms of an arithmetic sequence are -2, 5, 12, 19, 26 Find the formula for the nth term of this sequence asked by H on May 13, 2019 52. ## maths The Hypotenuse of a right- triangle is 13 cm and the angle of the line opposite to the hypotenuse is 35 find the other two sides. asked by Kaps on November 1, 2009 53. ## math The first term of a geometric sequence is 9 and the common ratio is -1 over 3. find the fourth term asked by Anonymous on May 5, 2016 54. ## Science Find the density in g/cm3 of a 1573 cm3 piece of metal having a mass of 46 kg. asked by Angela on May 14, 2015 55. ## geometry suppose j is between h and k.use the segment addition postulate to solve for x.then find the length of each segment hj=5x jk=7x kh=96 asked by shayla on September 11, 2011 56. ## exact logarithms Use the properties of logarithms to find the exact value of the expression log[16]4*log[5]sqrt7? asked by Mc78 on January 1, 2012 57. ## Trig If the terminal side of an angle theta contains the point (-1,3), find secant theta. asked by Maxine on April 13, 2016 58. ## Math Find the probability that theven pointer will stop on an odd number or a number less than 7 asked by Anonymous on June 4, 2017 59. ## geometry A rhombus has a diagonals of length 4 and 10. 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Find the average velocity of the particle for 1 asked by Dandylove on April 8, 2012 64. ## maths A rhombus has a side 15cm and one diagonal 22cm.Find the height of the other diagonal? asked by Bismark on June 10, 2017 65. ## math for the given functions f and g, find the specified value of the following functions and state the domain of each one. f(x)=2+3/x; g(x)=3/x a.(f-g)(2)= b.(f/g)(3)= how do I do this? I need help! it throws me off because of it being in fraction form. asked by marie on March 26, 2013 66. ## math URGENT Find all values of x such that sin 2x = sin x and 0 x 2π. (List the answers in increasing order.) x= x= x= x= x= asked by Cassy on January 19, 2014 67. ## geometry Given the points K=(-2,1) and m(3,4), find coordinates for a point J that makes angle JKM a right angle. asked by zach on September 8, 2011 68. ## Organic Chem what is the pka of anhydride? i've searched the whole web and can't seem to find the pka of the functional group. asked by Sara on May 8, 2018 69. ## Math Find the inverse of the function. f(x) = -0.25x + 6 I am having trouble when there is a decimal to solve like -0.25x asked by Nathaly on October 12, 2018 70. ## Maths Two side of parallelogram are in the ratio 3:4.if its perimeter in 42cm,find its lengths of its side asked by Zen malik on September 20, 2016 71. ## Math The parallel sider of a trapezium are 25 cm and 11 cm, while its nonparallel sides are 15 cm and 13 cm. find the area of the trapezium. asked by Ahmad on August 31, 2015 72. ## Calculus f(t)=-4.9t^2+25t+3 meters what is the average velocity during the first two seconds? find the instantaneous velocity at t=2 asked by Emily on February 13, 2011 73. ## Math IN A FOUR DIGIT NUMBER THE FIRST Two DIGITS ARE BOTH 2 .the sum of the ones and tens digit is 14 find the numbers asked by Jiji on March 24, 2018 74. ## Pre Calc in a 9 digit integer, each of the digits 1,2,3,4,5,6,7,8,and 9 appears exactly once. Find the probability that the integer is divisible by 9. asked by Ryan on November 1, 2009 75. ## Math Find parametric equations and symmetric equations for the line of intersection of the given planes. x + y + z = 2, x + z = 0 X=1 y = Z= x=, y= asked by Mike on April 13, 2010 76. ## Math If y varies directly as x and y=3 when x=10, find x when y=8? Can you please show me step by step solution for this problem? asked by HELPP!!! on January 10, 2013 77. ## maths perpendicular avenue is perpendicular to tangent street and passes through D(17,8).find its equation asked by Tony on March 29, 2013 78. ## Pre-Cal I do not know how to find the angle of the function sec 4x = 2 on the unit cirlce. I know I would be looking for 2 but I don't know where since it is sec. asked by Hannah on February 17, 2010 79. ## Social Studies I need to list 10 countries in the North and 10 countries in the South during slavery times. Can't find anything. Please help. 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Find tan (A + B). asked by Molly on October 15, 2014 86. ## maths use formula v=u +at to find the velocity when the initial velocity is 3m/s, the acceleration is 1.5m/s2, the time is 7 seconds asked by kate on October 5, 2018 87. ## Math Mishelle is 7 years younger than Paul. Given that the sum of their ages is 29, find their ages asked by Stacy on September 16, 2018 88. ## maths you and your family go to a restaurant. the bill is$26.98 including \$1.53 in tax. find the percent tax. please help. thanks

asked by kjeryn on September 24, 2009
89. ## algebra

3/4 of a number is 1/2. Find the number in the lowest terms... please show work I am confused

asked by Laura on October 2, 2011
90. ## Math- Probability

A committee of five people is to be formed from six lawyers and seven teachers. Find the probability that none are lawyers.

asked by Ted on July 29, 2013

Find the equation of the circle where the center is on the line 8x + 5y - 8 =0 and passing through point (2,1) and point (3,5)

asked by Mk on February 28, 2011
92. ## math

in quadrilateral kite two equal angles are 70degrees and 80 degrees each . find other two angles.

asked by deepti on July 15, 2016
93. ## pmt coching chemistry

25 ml of 3M hno3 are mixed with 75ml of 4.0M HNO3 if volumes are additive then find molarity

asked by sanju kurmi on December 7, 2016
94. ## Math

The area of a square table is 64cm. Find the perimeter of the sqare table.

asked by Polly on March 31, 2011
95. ## Algebra

Find the center and radius of the circle. Equation: x^2 + (y+7)^2=4 Formula for center at the origin is x^2+y^2=r^2 Is this what I use and how do I go about this?

asked by Carrie on April 18, 2010
96. ## Maths

a) P is the point on the curve y= 2x^3 +kx -5 where x= 1 and gradient is -2. Find (i) The value of the constant k (ii) The equation of the normal to the curve at P

asked by Kelly on May 2, 2019
97. ## Geometry

For an ellipse, one focus is (0,0), one vertex is (2,0), and the eccentricity is 0.4. How do I find the equation for the ellipse? It seems like I have so little information!

asked by Confused on May 28, 2012
98. ## physics

Find the density in g/cm3 of a 120 cm3 piece of metal having a mass of 2 kg.

asked by Me on September 12, 2011
99. ## math

in a class of 480 students if 60% are boys then find the number of girls in the class?

asked by Niran on April 22, 2014
100. ## Delta T

In the delta T formula how do you find i = ? If its ionic like NaCl then i = 2 What if its like C6H12O6? i = 1 ? or i = 3? Since that is not ionic i would assume it will be 1

asked by Chopsticks on May 4, 2009


text:
There is no validity if we say that complex number is positive or negative. Here i = √ -1. Lecture 1 Complex Numbers Deﬁnitions. Complex Number. Nice but keep for class 9 and 10 also. It awesome buddy. CBSE Guide Complex Numbers And Quadratic Equations class 11 Notes. Inequalities in complex numbers are not defined. Complex Numbers are used in many scientific fields. Equality In Complex Number: Because of this we can think of the real numbers as being a subset of the complex numbers. Just as R is the set of real numbers, C is the set of complex numbers.Ifz is a complex number, z is of the form z = x+ iy ∈ C, for some x,y ∈ R. e.g. CBSE guide notes are the comprehensive notes which covers the latest syllabus of CBSE and NCERT. Students must free download and practice these worksheets to gain more marks in exams.CBSE Class 11 Mathematics Worksheet - Complex Numbers and Quadratic Equation Let i2 = −1. Complex numbers are often denoted by z. Complex number is of the form a +ib where a is real part and b is imaginary part. Two complex numbers are equal if: Real parts are equal; Imaginary parts are equal; E.g. Here, p and q are real numbers and $$i=\sqrt{-1}$$. Class 11 Mathematics notes on Chapter 5 Complex Numbers And Quadratic Equations class 11 Notes Mathematics are also available for download in CBSE Guide website. Keep it up. In these cases, we call the complex number a number. In real numbers if a 2 + b 2 = 0 then a = 0 = b however in complex numbers, z 1 2 + z 2 2 = 0 does not imply z 1 = z 2 = 0. CBSE Class 11 Mathematics Worksheet - Complex Numbers and Quadratic Equation - Practice worksheets for CBSE students. Complex numbers are often denoted by z. PDF download free. This PDF file for class 11 Complex Numbers subject's Mathematics topic contains brief and concise notes … Class 11 Math Chapter 5 Complex Numbers and Quadratic Equations Notes- PDF Download Complex Numbers and Quadratic Equations Class 11 Notes is prepared strictly according to the NCERT Syllabus which not only reduces the pressure on the students but also, offer them a simple way to study or revise the chapter. Let 2=−බ ∴=√−බ Just like how ℝ denotes the real number system, (the set of all real numbers) we use ℂ to denote the set of complex numbers… 3. Prepared by teachers of the best CBSE schools in India. Complex Numbers Class 11 – A number that can be represented in form p + iq is defined as a complex number. ∴ i = −1. 3 + 4i is a complex number. Complex numbers are built on the concept of being able to define the square root of negative one. E.g. : 2+ i3 ; 7+ i9 etc. Class 11 Maths Notes Chapter 5 Complex Numbers And Quadratic Equations. 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text:
# cf.Datum.equals¶

Datum.equals(other, rtol=None, atol=None, verbose=None, ignore_data_type=False, ignore_fill_value=False, ignore_type=False)[source]

Whether two instances are the same.

Equality is strict by default. This means that:

• the named parameters must be the same, with the same values and data types, and vector-valued parameters must also have same the size and be element-wise equal (see the ignore_data_type parameter).

Two real numbers x and y are considered equal if |x-y|<=atol+rtol|y|, where atol (the tolerance on absolute differences) and rtol (the tolerance on relative differences) are positive, typically very small numbers. See the atol and rtol parameters.

Any type of object may be tested but, in general, equality is only possible with another object of the same type, or a subclass of one. See the ignore_type parameter.

Parameters
other:

The object to compare for equality.

atol: float, optional

The tolerance on absolute differences between real numbers. The default value is set by the cfdm.ATOL function.

rtol: float, optional

The tolerance on relative differences between real numbers. The default value is set by the cfdm.RTOL function.

verbose: int or None, optional

If an integer from 0 to 3, corresponding to increasing verbosity (else -1 as a special case of maximal and extreme verbosity), set for the duration of the method call (only) as the minimum severity level cut-off of displayed log messages, regardless of the global configured cfdm.LOG_LEVEL.

Else, if None (the default value), log messages will be filtered out, or otherwise, according to the value of the cfdm.LOG_LEVEL setting.

Overall, the higher a non-negative integer that is set (up to a maximum of 3) the more description that is printed to convey information about differences that lead to inequality.

ignore_data_type: bool, optional

If True then ignore the data types in all numerical comparisons. By default different numerical data types imply inequality, regardless of whether the elements are within the tolerance for equality.

ignore_type: bool, optional

Any type of object may be tested but, in general, equality is only possible with another object of the same type, or a subclass of one. If ignore_type is True then equality is possible for any object with a compatible API.

Returns
bool

Whether the two instances are equal.

Examples:

>>> d.equals(d)
True
>>> d.equals(d.copy())
True
>>> d.equals(None)
False


text:
# Surface area and volume class 9 extra questions

Given below are the Class 9 MathsExtra questions and Important Questions for Surface area and volume
(a) Concepts questions
(b) Calculation problems
(c) Multiple choice questions
(e) Fill in the blank's

## Fill in the blank

Question 1
(a) A cylinder radius is doubled and height is halved, the curved surface area of the cylinder will ____ (Increase/decrease/remain same)
(b) The radius of the sphere is 7 cm, the surface area of the sphere is____ (616 cm2 / 700 cm2/ 800 cm2)
(c) Three cubes whose sides are 6 cm, 8 cm and 10 cm. They are melted and form a big cube. The volume of the big cube is ___ (1800cm3/1728 cm3)
(d) The total surface area of a hemisphere of radius 10 cm using value of π=3.14 is _____(956 cm2/942 cm2)
(e) A right circular cylinder just encloses a sphere of radius. The ratio of there surface area is ______ (1:1 / 1:2)
(f) The lateral surface area of a cube is 256 m2, the volume is ____ (456 m3/512 m3)
Solution
(a) Remain same
CSA= 2πrh
So if r is doubled, h is halved,CSA will remain same
(b) 616 cm2,  Surface area=4πr2
(c) 1728 cm3, V=V1 + V2 + V3=63+ 83 +103
(d) 956 cm2, Total surface area =3πr2
(e) 1:1 In this height of cylinder will be 2r and radius of cylinder will r. So it is equal
(f) 512 m3, Lateral surface area=4a2 and Volume=a3c

## True or False statement

Question 2
(a) A cylinder, hemisphere and cone stand on equal base and same height, the Volume ratio is 3:2:1
(b) The radius of a solid sphere is 24 cm. 8 spheres can be made from it of 12cm radius
(c) radius of the cone is doubled and height is halved, the volume will be halved
(d) A river 10m deep and 40m wide is flowing at the rate of 2m per min. 48000m3 water will flow into the sea from river
(e) A cylinder radius is halved and height is doubled, the volume will become halved
(f) The side of the cube is 4 cm, the diagonal length is  cm
Solution
(a)True.. All of them will have height as r as hemisphere height can be r only. For cylinder  =πr3
Hemisphere=(2/3) πr3
Cone=(1/3) πr3
So ratio is 3:2:1

(b) True. Sphere volume =(4/3) πr3

(c) False. Volume will be doubled. Volume is given by =(1/3)πr2h

(d) True. It is equal to the volume of the cuboid  10m,40m,and 120 m

(e) True.

(f)False

## Multiple choice Questions

Question 3
The length, breadth and height of a room are 12 m, 10 m, and 9m respectively. Find the area of  our walls of room?
(a) 636 m2
(b) 516 m2
(c) 800 m2
(d) 456m2
Solution
Area of the walls is given by =2(BH+LH) +LB=2(90+108)+120=516 m2

Question 4
The plastic paint in a Asian paint container is sufficient to paint an area equal to 93.75m2 How many blocks of dimensions 22.5 cm × 10 cm × 7.5 cm can be painted out of this container
(a) 100
(b) 800
(c) 940
(d) 1000
Solution
Total surface area of one block = 2(lb + bh + lh)
= [2(22.5 ×10 + 10 × 7.5 + 22.5 × 7.5)] cm
= 2(225 + 75 + 168.75) cm2
= (2 × 468.75) cm2
= 937.5 cm2
Let n blocks can be painted out by the paint of the container.
Area of n bricks = (n ×937.5) cm2 = 937.5n cm2
Area that can be painted by the paint of the container = 93.75 m2= 937500 cm2
937500 = 937.5n
n = 1000
Therefore, 1000 blocks can be painted out by the paint of the container

Question 5
Curved surface area of a cone is 308 cm2 and its slant height is 14 cm.
(a) Radius of the cone is 7 cm
(b) total surface area is 462 cm2
(c) Height of the cone is $\sqrt {147}$ cm
(d) None of the above
Solution
Curved surface area=πrl
So r=7 cm
Now total surface area=πr2+πrl=462 cm2

Question 6
Sita had to make a model of cylindrical kaleidoscope for her science project. She wanted to use black chart paper to make the curved surface of the kaleidoscope. What would be the area of chart paper required by her, if she wanted to make a kaleidoscope of length 30cm with a 2.7 cm radius? (Use p=22/7)
(a)1320 cm2
(b) 1400 cm2
(c) 986 cm2
(d) None of these
Solution
Curved surface of cylinder=2πrh

Question 7
The radii of two cones are in the ratio of 2:3 and their heights are in the ratio of 7:3. The ratio of their volumes is
(a) 20:27
(b) 28:27
(c) 28:27
(d) None of these
Solution
Volume =(1/3)πr2h

Question 8
Find the maximum length of the rod that can be kept in cuboidal box of sides 30cm, 20cm and 10cm.
(a)$\sqrt {1400}$ cm
(b)$2 \sqrt {400}$ cm
(c)$2 \sqrt {300}$ cm
(d) None of these
Solution
Diagonal is the longest length in the cuboid so
$D=\sqrt {L^2+B^2+H^2}$
=$\sqrt {1400}$ cm

Question 9
A box is made entirely of glass panes (including base) held together with tape. It is 3 cm long, 2.5 cm wide and 2.5 cm high. How much of tape is needed for all the 12 edges?
(a) 30cm
(b) 32cm
(c) 40 cm
(d) None of these
Solution
Length of tape=4( L+B+H)=32 cm

Question 10
The curved surface area of a right circular cylinder of height 14 cm is 88 cm2. Find the Volume of the cylinder. Assume π=22/7
(a) 144 cm3
(b) 180 cm3
(c) 176 cm3
(d) None of the above
Solution

Question 11

## Subjective Type questions

Question 12
Ramesh has build a cuboidal water tank with lid for his house with each outer edge 1.5m long. He gets the outer surface of tank excluding the base, covered with square tiles of side 25cm. Find how much he would spend for tiles, if cost of the tile is Rs 400 per dozen.
Solution
Surface Area of the cuboidal water tank where tiles to be placed= 5L2= 11.25 m2
Area of 1 Tile= 252=625 cm2=.0625 m2
No of Tiles required=11.25/.0625=180=15 dozen
Cost of tiles will be = 400 * 15=Rs 6000

Question 13
The paint in a certain container is sufficient to point an area equal to 9.375m2. How many bricks of dimensions 22.5cm x 10cm x 7.5cm can be painted out of this container.
Solution
Surface are of 1 Brick= 2(LB+ BH+ LH)=2(225 + 75 + 168.75)=937.5 cm2
Total Brick which can be painted= $\frac {9.375 \times 10^4}{937.5}= 100 Question 14 Bikanerwala sweets shop was planning an order for making cardboard boxes for packing their sweets. Two sizes of boxes were required. The bigger of dimensions 25cm x 20cm x 5cm and smaller of dimensions 15cm x 12cm x 5cm. For all overlaps, 5% of total surface area is required extra. If cost of cardboard is Rs 4 for 1000cm2, find the cost of cardboard required for supplying 250 boxes of each kind. Question 15 The diameter of a roller are 84cm and its length is 120cm. It takes 500 complete revolutions to move once more over to level a playground. Find area of playground in m2? Solution Diameter =84 cm, Therefore radius=42 cm Curved Surface area of cylinder=$2 \pi r h= 2 \times \frac {22}{7} \times 42 \times 120 $=31680 cm2= 3.1680 m2 Area of playground=500 × 3.1680=1584 m2 Question 16 In a hot water heating system, there is a cylindrical pipe of length 28m and diameter 5cm. Find the total radiating surface in system? Solution Diameter =5 cm, Therefore radius=2.5 cm Curved Surface area of cylinder is the radiating surface=$2 \pi r h= 2 \times \frac {22}{7} \times 2.5 \times 2800 =44000$cm2 Question 17 Find (a)the lateral or C.S.A of a closed cylindrical petrol storage tank that is 4.2m in diameter and 4.5m in high (b)how much steel was actually used, if 1/12 of steel actually used was wasted in making the tank. Question 18 You see the frame of lamp shade. It is to be covered with decorative cloth. The frame has a base diameter of 20cm and height of 30cm. A margin of 2.5cm is total given for folding it over top and bottom of frame How much cloth is required for covering the lamp shade? Question 19 What length of tarpaulin 3m wide will be required to make a conical tank of height 8m and base radius 6m. Assume that the extra length margins and wastage n cutting is approximately 20cm. (Use = 3.14) Question 20 A Birthday cap is in form of right circular cone of radius 7cm and height 24cm. Find area of sheet required to make 10 such caps? Question 21 A bus stop is barricaded from remaining part of road, by using 50 hollow cones made of recycles cardboard. Each cone has a base diameter of 4cm and height 1m. If outer side of earn cone is to be painted and cost of painting is Rs 12/m2. What will be the cost? ( = 3.14 and take 1.04 = 1.02) Question 22 A hemispherical dome of a building needs to be painted. If circumference of base of dome is 17.6m, find the cost of painting it, given cost of painting is Rs. 5/200 cm2. Question 23 The diameter of moon is approximately one – fourth diameter of earth. Find the ratio of their surface areas. and What fraction of the volume of the earth is the volume of the moon? Solution Let d be diameter of earth Surface area of earth=$ \pi d^2$Diameter of Moon=$\frac {1}{4}d$Surface area of Moon=$\pi D^2= \frac {1}{16} \pi d^2\frac {\text{surface area of Moon}}{\text{surface area of earth}}= \frac {1}{16}$Volume of earth=$\frac {1}{6} \pi d^3$Surface area of Moon=$\frac {1}{6} \pi D^3= \frac {1}{384} \pi d^2\frac {\text{Volume of Moon}}{\text{Volume of earth}}= \frac {1}{384}$Question 24 A right circular cone just enclosed a sphere of radius r. Find – (a)surface area of sphere (b)curved surface area of cylinder (c)ratio of areas obtained in (i) and (ii) Solution (i) Surface area of a sphere where r is the radius of the sphere =$4 \pi r^2$(ii) Height of the cylinder, =diameter of the sphere=2r And Radius of the cylinder =r CSA of cylinder formula =$2 \pi r h=4 \pi r^2$(iii) Ratio between areas =$\frac {4 \pi r^2}{4 \pi r^2}=1:1$Question 25 A wall of length 10m was to be built across an open ground. The height is 4m and thickness is 24cm. If this wall is to be build up with bricks whose dimensions are 24cm x 12cm x 8cm, how many bricks would be required? Question 26 Three cubes of each side 4cm are joining end to end. Find the surface area of resulting cuboid Solution Surface of the cuboid will be =$ 4 \times 4 \times 12 + 2 \times 4 \times 4=224$cm2 Question 27 A village having a population of 4000 requires 150 litres of water per head per day. It has a tank measuring 20m x 15m x 6m for how many days will the water of this tank last? Question 28 A godown measures 40m x 25m x 10m. Find the maximum no. of wooden crates each measuring 1.5m x 1.25m x 0.5m that can be stored in godown? Question 29 A river 3m deep and 40m wide is flowing at the rate of 2km/hr. How much water will fall into the sea in a minute? Solution Speed of water in a river=2 km/hr Length of flow of water in 1 minute=$\frac {2}{60} km =\frac {100}{3}m $Therefore Volume of water$=LBH= \frac {100}{3} \times 3 \times 40=4000 m^3 $Volume in litres= 4000 * 1000= 4 × 106 Question 30 The pillars of a temple are cylindrically shaped. If each pillar has a circular base of radius 20cm and height 10m. How much concrete mixture would be required to build 14 such pillars? Solution Volume of single pillar=$\pi r^2 h=.2 \pi$Volume of 14 Pillars=$ 14 \times .2 \pi=8.8 m^3\$

## Summary

This Surface area and volume class 9 extra questions with answers is prepared keeping in mind the latest syllabus of CBSE . This has been designed in a way to improve the academic performance of the students. If you find mistakes , please do provide the feedback on the mail.


text:
# Mach number

An F/A-18 Hornet creating a vapor cone at transonic speed just before reaching the speed of sound

In fluid dynamics, the Mach number (M or Ma) (/mɑːx/; German: [maχ]) is a dimensionless quantity representing the ratio of flow velocity past a boundary to the local speed of sound.[1][2]

${\displaystyle \mathrm {M} ={\frac {u}{c}},}$

where:

M is the Mach number,
u is the local flow velocity with respect to the boundaries (either internal, such as an object immersed in the flow, or external, like a channel), and
c is the speed of sound in the medium.

By definition, at Mach 1, the local flow velocity u is equal to the speed of sound. At Mach 0.65, u is 65% of the speed of sound (subsonic), and, at Mach 1.35, u is 35% faster than the speed of sound (supersonic).

The local speed of sound, and thereby the Mach number, depends on the condition of the surrounding medium, in particular the temperature. The Mach number is primarily used to determine the approximation with which a flow can be treated as an incompressible flow. The medium can be a gas or a liquid. The boundary can be traveling in the medium, or it can be stationary while the medium flows along it, or they can both be moving, with different velocities: what matters is their relative velocity with respect to each other. The boundary can be the boundary of an object immersed in the medium, or of a channel such as a nozzle, diffusers or wind tunnels channeling the medium. As the Mach number is defined as the ratio of two speeds, it is a dimensionless number. If M < 0.2–0.3 and the flow is quasi-steady and isothermal, compressibility effects will be small and simplified incompressible flow equations can be used.[1][2]

The Mach number is named after Austrian physicist and philosopher Ernst Mach, and is a designation proposed by aeronautical engineer Jakob Ackeret. As the Mach number is a dimensionless quantity rather than a unit of measure, with Mach, the number comes after the unit; the second Mach number is Mach 2 instead of 2 Mach (or Machs). This is somewhat reminiscent of the early modern ocean sounding unit mark (a synonym for fathom), which was also unit-first, and may have influenced the use of the term Mach. In the decade preceding faster-than-sound human flight, aeronautical engineers referred to the speed of sound as Mach's number, never Mach 1.[3]

## OverviewEdit

Mach number is useful because the fluid behaves in a similar manner at a given Mach number, regardless of other variables.[4] As modeled in the International Standard Atmosphere, dry air at mean sea level, standard temperature of 15 °C (59 °F), the speed of sound is 340.3 meters per second (1,116.5 ft/s).[5] The speed of sound is not a constant; in a gas, it increases as the absolute temperature increases, and since atmospheric temperature generally decreases with increasing altitude between sea level and 11,000 meters (36,089 ft), the speed of sound also decreases. For example, the standard atmosphere model lapses temperature to −56.5 °C (−69.7 °F) at 11,000 meters (36,089 ft) altitude, with a corresponding speed of sound (Mach 1) of 295.0 meters per second (967.8 ft/s), 86.7% of the sea level value.

## Classification of Mach regimesEdit

While the terms subsonic and supersonic, in the purest sense, refer to speeds below and above the local speed of sound respectively, aerodynamicists often use the same terms to talk about particular ranges of Mach values. This occurs because of the presence of a transonic regime around M = 1 where approximations of the Navier-Stokes equations used for subsonic design actually no longer apply; the simplest explanation is that the flow locally begins to exceed M = 1 even though the freestream Mach number is below this value.

Meanwhile, the supersonic regime is usually used to talk about the set of Mach numbers for which linearised theory may be used, where for example the (air) flow is not chemically reacting, and where heat-transfer between air and vehicle may be reasonably neglected in calculations.

In the following table, the regimes or ranges of Mach values are referred to, and not the pure meanings of the words subsonic and supersonic.

Generally, NASA defines high hypersonic as any Mach number from 10 to 25, and re-entry speeds as anything greater than Mach 25. Aircraft operating in this regime include the Space Shuttle and various space planes in development.

Regime Speed General plane characteristics
(Mach) (knots) (mph) (km/h) (m/s)
Subsonic <0.8 <530 <609 <980 <273 Most often propeller-driven and commercial turbofan aircraft with high aspect-ratio (slender) wings, and rounded features like the nose and leading edges.

The subsonic speed range is that range of speeds within all of the airflow over an aircraft is less than Mach 1. The critical Mach number (Mcrit) is lowest free stream mach number at which airflow over any part of the aircraft first reaches Mach 1. So the subsonic speed range includes all speeds that are less than Mcrit.

Transonic 0.8–1.3 530–794 609–914 980–1,470 273–409 Transonic aircraft nearly always have swept wings, causing the delay of drag-divergence, and often feature a design that adheres to the principles of the Whitcomb Area rule.

The transonic speed range is that range of speeds within which the airflow over different parts of an aircraft is between subsonic and supersonic. So the regime of flight from Mcrit up to Mach 1.3 is called the transonic range.

Supersonic 1.3–5.0 794-3,308 915-3,806 1,470–6,126 410–1,702 The supersonic speed range is that range of speeds within which all of the airflow over an aircraft is supersonic (more than Mach 1). But airflow meeting the leading edges is initially decelerated, so the free stream speed must be slightly greater than Mach 1 to ensure that all of the flow over the aircraft is supersonic. It is commonly accepted that the supersonic speed range starts at a free stream speed greater than Mach 1.3.

Aircraft designed to fly at supersonic speeds show large differences in their aerodynamic design because of the radical differences in the behaviour of flows above Mach 1. Sharp edges, thin aerofoil-sections, and all-moving tailplane/canards are common. Modern combat aircraft must compromise in order to maintain low-speed handling; "true" supersonic designs include the F-104 Starfighter, SR-71 Blackbird and BAC/Aérospatiale Concorde.

Hypersonic 5.0–10.0 3,308–6,615 3,806–7,680 6,126–12,251 1,702–3,403 The X-15, at Mach 6.72 is one of the fastest manned aircraft. Also, cooled nickel-titanium skin; highly integrated (due to domination of interference effects: non-linear behaviour means that superposition of results for separate components is invalid), small wings, such as those on the Mach 5 X-51A Waverider
High-hypersonic 10.0–25.0 6,615–16,537 7,680–19,031 12,251–30,626 3,403–8,508 The NASA X-43, at Mach 9.6 is one of the fastest aircraft. Thermal control becomes a dominant design consideration. Structure must either be designed to operate hot, or be protected by special silicate tiles or similar. Chemically reacting flow can also cause corrosion of the vehicle's skin, with free-atomic oxygen featuring in very high-speed flows. Hypersonic designs are often forced into blunt configurations because of the aerodynamic heating rising with a reduced radius of curvature.
Re-entry speeds >25.0 >16,537 >19,031 >30,626 >8,508 Ablative heat shield; small or no wings; blunt shape

## High-speed flow around objectsEdit

Flight can be roughly classified in six categories:

Regime Subsonic Transonic Sonic Supersonic Hypersonic Hypervelocity
Mach <0.8 0.8–1.2 1.0 1.2–5.0 5.0–10.0 >10.0

For comparison: the required speed for low Earth orbit is approximately 7.5 km/s = Mach 25.4 in air at high altitudes.

At transonic speeds, the flow field around the object includes both sub- and supersonic parts. The transonic period begins when first zones of M > 1 flow appear around the object. In case of an airfoil (such as an aircraft's wing), this typically happens above the wing. Supersonic flow can decelerate back to subsonic only in a normal shock; this typically happens before the trailing edge. (Fig.1a)

As the speed increases, the zone of M > 1 flow increases towards both leading and trailing edges. As M = 1 is reached and passed, the normal shock reaches the trailing edge and becomes a weak oblique shock: the flow decelerates over the shock, but remains supersonic. A normal shock is created ahead of the object, and the only subsonic zone in the flow field is a small area around the object's leading edge. (Fig.1b)

 (a) (b)

Fig. 1. Mach number in transonic airflow around an airfoil; M < 1 (a) and M > 1 (b).

When an aircraft exceeds Mach 1 (i.e. the sound barrier), a large pressure difference is created just in front of the aircraft. This abrupt pressure difference, called a shock wave, spreads backward and outward from the aircraft in a cone shape (a so-called Mach cone). It is this shock wave that causes the sonic boom heard as a fast moving aircraft travels overhead. A person inside the aircraft will not hear this. The higher the speed, the more narrow the cone; at just over M = 1 it is hardly a cone at all, but closer to a slightly concave plane.

At fully supersonic speed, the shock wave starts to take its cone shape and flow is either completely supersonic, or (in case of a blunt object), only a very small subsonic flow area remains between the object's nose and the shock wave it creates ahead of itself. (In the case of a sharp object, there is no air between the nose and the shock wave: the shock wave starts from the nose.)

As the Mach number increases, so does the strength of the shock wave and the Mach cone becomes increasingly narrow. As the fluid flow crosses the shock wave, its speed is reduced and temperature, pressure, and density increase. The stronger the shock, the greater the changes. At high enough Mach numbers the temperature increases so much over the shock that ionization and dissociation of gas molecules behind the shock wave begin. Such flows are called hypersonic.

It is clear that any object traveling at hypersonic speeds will likewise be exposed to the same extreme temperatures as the gas behind the nose shock wave, and hence choice of heat-resistant materials becomes important.

## High-speed flow in a channelEdit

As a flow in a channel becomes supersonic, one significant change takes place. The conservation of mass flow rate leads one to expect that contracting the flow channel would increase the flow speed (i.e. making the channel narrower results in faster air flow) and at subsonic speeds this holds true. However, once the flow becomes supersonic, the relationship of flow area and speed is reversed: expanding the channel actually increases the speed.

The obvious result is that in order to accelerate a flow to supersonic, one needs a convergent-divergent nozzle, where the converging section accelerates the flow to sonic speeds, and the diverging section continues the acceleration. Such nozzles are called de Laval nozzles and in extreme cases they are able to reach hypersonic speeds (Mach 13 (15,926 km/h; 9,896 mph) at 20 °C).

An aircraft Machmeter or electronic flight information system (EFIS) can display Mach number derived from stagnation pressure (pitot tube) and static pressure.

## CalculationEdit

The Mach number at which an aircraft is flying can be calculated by

${\displaystyle \mathrm {M} ={\frac {u}{c}}}$

where:

M is the Mach number
u is velocity of the moving aircraft and
c is the speed of sound at the given altitude

Note that the dynamic pressure can be found as:

${\displaystyle q={\frac {\gamma }{2}}p\,\mathrm {M} ^{2}}$

Assuming air to be an ideal gas, the formula to compute Mach number in a subsonic compressible flow is derived from Bernoulli's equation for M < 1:[6]

${\displaystyle \mathrm {M} ={\sqrt {{\frac {2}{\gamma -1}}\left[\left({\frac {q_{c}}{p}}+1\right)^{\frac {\gamma -1}{\gamma }}-1\right]}}\,}$

where:

qc is impact pressure (dynamic pressure) and
p is static pressure
${\displaystyle \gamma \,}$  is the ratio of specific heat of a gas at a constant pressure to heat at a constant volume (1.4 for air).

The formula to compute Mach number in a supersonic compressible flow is derived from the Rayleigh supersonic pitot equation:

${\displaystyle {\frac {p_{t}}{p}}=\left[{\frac {\gamma +1}{2}}\mathrm {M} ^{2}\right]^{\frac {\gamma }{\gamma -1}}\cdot \left[{\frac {\gamma +1}{1-\gamma +2\gamma \,\mathrm {M} ^{2}}}\right]^{\frac {1}{\gamma -1}}}$

### Calculating Mach number from pitot tube pressureEdit

Mach number is a function of temperature and true airspeed. Aircraft flight instruments, however, operate using pressure differential to compute Mach number, not temperature.

Assuming air to be an ideal gas, the formula to compute Mach number in a subsonic compressible flow is found from Bernoulli's equation for M < 1 (above):[6]

${\displaystyle \mathrm {M} ={\sqrt {5\left[\left({\frac {q_{c}}{p}}+1\right)^{\frac {2}{7}}-1\right]}}\,}$

The formula to compute Mach number in a supersonic compressible flow can be found from the Rayleigh supersonic pitot equation (above) using parameters for air:

${\displaystyle \mathrm {M} \approx 0.88128485{\sqrt {\left({\frac {q_{c}}{p}}+1\right)\left(1-{\frac {1}{7\,\mathrm {M} ^{2}}}\right)^{2.5}}}}$

where:

qc is the dynamic pressure measured behind a normal shock.

As can be seen, M appears on both sides of the equation, and for practical purposes a root-finding algorithm must be used for a numerical solution. It is first determined whether M is indeed greater than 1.0 by calculating M from the subsonic equation. If M is greater than 1.0 at that point, then the value of M from the subsonic equation is used as the initial condition for fixed point iteration of the supersonic equation, which usually converges in just a few iterations.[6] Alternatively, Newton's method can also be used.


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# How to use the TableText commando to create a table from a spredsheet with a variable number of rows

M.E shared this question 1 year ago

I would like to create a table from cell content in the spreadsheet with a variable number of rows by using the TableText command.

How can I do that ?

1

attached a sample

1

Thanks.

Sorry, I see I should have read this a bit more carefully in relation to how to place rows and columns in the command: https://wiki.geogebra.org/e...

Is it possible to use the command to have one column where the content is left aligned, while the rest of the columns are centered ?

Or how can I make that ?

1

see attachment

1

your first example is easier with TableText({A1:A200, B1:B200, C1:C200}, "vcrl")

1

Thank you both. :)

It's quite smart so easy it is.

"v" stands for that every element of the list (for example "A1:A200", "B1:B200" and "C1:C200") is a column.

I see the last three letters stands for the alignment of the content in each column in the listed order of the columns.

The letter "c" stands for centered, "r" for right and "l" for left alignment.

And if one would like to have the table with all borders showed, then the command will be:

TableText[ { A1:A200, B1:B200, C1:C200 }, "|_vcrl" ]

Be aware that this "|" is not a letter, but instead a symbol.

"|" stands for showed borders to the left and right of every column. And "_" stands for showed borders above and below each row.

1

Moved below.

1

No words to express my thanks. Timely help. Though posted a year ago, it is fabulous

1

Dear Colleagues,

Is there a way of controlling the column width of the columns when using either

TableText[ { A1:A200, B1:B200, C1:C200 }, "|_vcrl" ]

or the multiple table solution suggested by Rami? And, Rami, if yes, can you make the relative position to A responsive to the width of the table?

Many thanks!

1

Quote: "relative position to A responsive to the width of the table"

yes: see definition T_2, T_3 with using Corner(T_x, 3)

1

To illustrate my question better, I have created a GeoGebra file.

1

@Rami, thank you again, for all your kind and generous support! very much appreciated.

A small follow up on your response "yes: see definition T_2, T_3 with using Corner(T_x, 3)".

In the sample file you have supplied, the top of each table T_1, T_2 and T_3 are no longer vertically aligned -- is there something to be done to solve that/

2

Quote: "To illustrate my question"

use Text Tool with laTex

See text13

1

sample with T_1, T_2, T_3 "no loger vertically"

no problem in my environment (windows 10 GGB5.0.588 and GGB6.0.588 WEB)

If you transfer my sample in your file: check the sequence of objects in Construction Protokoll

1

Quote: "use Text Tool with laTex"

@Rami, thank you, thank you very much. I really and very deeply appreciate your incredible kindness in responding as generously as you do! Thank you -- thank you is all I have here -- it feels inadequate. Sorry!

1

What if I would like to have a table where all the cells in the table should be aligned centered except B2:B11 which should be aligned left. Meaning that the first cell in the B column (which contain a category) also should be centered.

So my question is: How can I get the first cell in a column to have another alignment than the rest of the cells in the same column ?

1

doing two tables and combining in one table like rami did

1

Yes, but how to handle the different width of the first cell in every column and the other cells in the column.

For example see the image below.

1

Maybe one way could be to shift from v to h so that the alignment is controlled by each row instead of each column. But then I don't know how to make the table with a variable number of rows.

1

Arh, I forgot that it isn't possible to get the cells in the range B2:B10 aligned left at the same time.

So the first row has to be created as an object for itself. And somehow combined with the others so that all the cells in each column has the same width.

1

I can't find a way to do it... :(

1

you can try spaces or phantom in 1,2 position for centering the first label onlyand clcc for each column

https://wiki.geogebra.org/e...

1

What is a "phantom" - could you insert the symbol

Sorry, I'm not sure how the script should be for this: "try spaces or phantom in 1,2 position for centering the first label only and clcc for each column." Could you write an example of a script here.

Thank you, for the link. I missed much of it earlier. But it is still very useful.

But sorry, I still can't see how to make a table according to my requirements. On the page they only describes how to make other or more simplified kind of tables. Either multi-column or tables where all cells in a column have the same alignment.

1

What is a "phantom"

más no puedo hacer

1

Thanks.

It's a low practical way to do it, but it works/does the job, so it's okay. :)

1

try in LAtex

\begin{array}{|r|l|c|}

\hline

\multicolumn{1}{|c|}{Titldfdfdfde} \multicolumn{1}{c|}{gggggklkje}{sfsdf}\\

\hline

xjhgjhgjhgnbvnbvnbjhgjh & yjhgjhgjhgjhgjhgjhgjhgjhgjhghjhmnh&jhjhgjhgjhgjhgjhgjhgjhgjhgjhgjhgjhg\\

\hline

amnbnvnbv & bhgjhgjhgj&hg\\

c & cjjgjhgjhgjhgjhgjh&gjhg\\

d & e\\ \hline

\end{array}

https://www.geogebra.org/m/RC7Pnnk5

1

Yes, that is a better way to do it. :)

Than you very much for your time and effort ! :)

And thank you for the link.


text:
# Count set bits

Count the total number of set bits of multiplication of A and B. Range of A and B is up to 1000000000. Please help someone

10^9 is not so big. If 1\le A, B \le 10^9 then definitely 1\le A\times B \le 10^{18}. The product will fit in a long int. Then counting set bits is trivial - you just keep dividing it by 2 and track the number of times the \text{LSB} is set.

CPP Snippet
long long int a = 0, b = 0;
cin >> a >> b;
long long int product = a * b;
int set_bits = 0;
while(product) {
set_bits += (product & 1);
product >>= 1;
}
cout << set_bits << '\n';

1 Like

Yes, but it is not storing.

No of set bits in 2 is 1 and in 3 is 2 if you calculate the set bit of 6 is 2 then how can you say that sum of individuals are equal to multiple of them

Sorry, I misunderstood the question.

What do you mean by not storing?

1 Like


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Finding one string inside another

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char string1[] = "Programmming"; char string2[] = "gram"; how would i find where string2 is inside string1? i know i could use the find() function but i need to write my own function to do it and i''m stumped!! can somebody please help? thanks NiNo

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Is this a do my homework for me post? Otherwise I see no reason not to use string::find () or strstr ()

Hint: Loop though each character in string1 and look for the first character in string2. If you find the first character, check the next character, etc., etc.

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no its not a "do my homewotrk for me question" its a "i''ve been trying to figure this out for hours and i''m stuck and its due tomorrow" question

i know i should loop through like you said but i just cant figure out how to do it right for some reason

NiNo

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Here is the source of Microsoft''s strstr function:

  char * __cdecl strstr ( const char * str1, const char * str2 ){ char *cp = (char *) str1; char *s1, *s2; if ( !*str2 ) return((char *)str1); while (*cp) { s1 = cp; s2 = (char *) str2; while ( *s1 && *s2 && !(*s1-*s2) ) s1++, s2++; if (!*s2) return(cp); cp++; } return(NULL);}

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quote:
Original post by nino
no its not a "do my homewotrk for me question" its a "i''ve been trying to figure this out for hours and i''m stuck and its due tomorrow" question

Hmmm... It''s not homework, but it''s DUE tomorrow?
Oops...

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Because its due tomorrow doesn''t qualify it as a homework assignment. It makes it a life and death issue

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There are several ways of doing this. The first approach is the one detailed by JY, and is a brute force approach. A more efficient approach would be to use the Knuth-Morris-Prath or Boyer-Moore algormithms. Both of these depends on first examining the pattern (the string you are looking for) for patterns. For natural language texts (e.g. english) BM is usually significantly more efficient than both other methods.

Here is some source-code for BM (searching ASCII-strings)
  int BM_Search (char P[], int plen, char T[], int tlen) { // Returns the index of the first occurence of P in T // Returns -1 if P is not found in T int i, j; int skip[255]; for (i = 0; i < 255; i++) { for (j = 0; j < plen; j++) skip[P[j]] = plen - j - 1; } i = plen-1; j = plen-1; while (j >= 0 && i < tlen) { if (T[i] == P[j]) { i--; j--; } else { if (plen - j > skip[T[i]]) i = i + plen - j; else i = i + skip[T[i]]; j = plen - 1; } } if (j == -1) return (i+1); else return -1;}

Without going in to too much detail on how this algorithm works it basically looks at the character in the text on which a mismatch occurs, and moves the pattern forward until the last occurence of that character in the pattern is lined up with it.
Oh, and it also checks from the end of the pattern backwards, rather than from the start of the pattern forwards.

Maybe a small example would help a little.
Assuming the text-string you are searcing in is
I REALLY LOVE DEBUGGING MY CODE
and the string you are searching for is
MY CODE

  First, the algorithm creates a skip-array that will look like thisskip[''E''] = 0skip[''D''] = 1skip[''O''] = 2skip[''C''] = 3skip['' ''] = 4skip[''Y''] = 5skip[''M''] = 6all other skips have the value 7Starting from the start we getI REALLY LOVE DEBUGGING MY CODEMY CODEStarting at the end of the pattern we get a mismatch on L (in the text).Since skip[''L''] is 7 we move the pattern 7 positions to the right, thus gettingI REALLY LOVE DEBUGGING MY CODE MY CODEAt this point we get a mismatch on '' '' (in the text).Since skip['' ''] is 4 we move the pattern 4 positions, gettingI REALLY LOVE DEBUGGING MY CODE MY CODENote that this skip moved the last occurence of '' '' in the pattern to match the '' '' we just found in the text. This is really the meat and bones of the algorithm, as whenever we find a character in the text we know we have in the pattern, we move the pattern so that that character matches in the pattern, and if we find a character in the text that isn''t in the pattern we move the pattern just past that point in the text.Anyway, that position above gives us a mismatch on ''U'' which doesn''t exist in our pattern, so again we move 7 positions gettingI REALLY LOVE DEBUGGING MY CODE MY CODEskip[''M''] is 6, so we move 6 spaces and getI REALLY LOVE DEBUGGING MY CODE MY CODEAnd we''re done.This algorithm is clearly a lot faster than a simple brute-force approach as the pattern is literally flying along often moving 7 spaces at the time without comparisons, while the brute-force approach would never move more than 1 before comparing again.

-Neophyte

- Death awaits you all with nasty, big, pointy teeth. -

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Anyway, the AP above was me, and boy! was that formatting ever screwed up. Sorry about that.

-Neophyte

- Death awaits you all with nasty, big, pointy teeth. -

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Looks like bullshit to me. The skip local array is never initialized, and the for i loop has no use.

Y.

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I think I would with the tried and tested method. It''s perfectly efficient enough.

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quote:
Original post by Ysaneya
Looks like bullshit to me. The skip local array is never initialized, and the for i loop has no use.

Y.

Doesn't matter what it looks like to you. It works.

But you're correct in saying that first i-loop is totally unnecessary. It doesn't do anything.

Edited by - Red Ant on November 15, 2001 6:02:49 AM

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You''re correct, that was a typo (or several) on my part. The correct loop should look like this:
  // Originally// for (i = 0; i < 255; i++) {// for (j = 0; j < plen; j++)// skip[P[j]] = plen - j - 1// }for (i = 0; i < 255; i++) skip[i] = plen;for (i = 0; i < plen; i++) skip[P[i]] = plen - i - 1;

-Neophyte

- Death awaits you all with nasty, big, pointy teeth. -

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Does wind raise air temperature?

In my undergrad physics classes, temperature was described as "the average kinetic energy of molecules". By simple, and probably naive, logic, one could then conclude that wind, which is basically just moving air, would raise the temperature of the air.

However, this seems too simple to be true. Does wind actually slightly raise air temperature?

I can see a couple potential arguments against this idea:

1. It could be the case that the macroscopic wind and the microscopic kinetic energy of the air molecules don't really "interact", for lack of a better word.
2. It could be the case that the macroscopic and the microscopic motions do "interact", but since the macroscopic wind is on such a large scale compared to the average kinetic energy of individual molecules, any system in which we use the average kinetic energy of the molecules is so small that the wind is effectively constant, and so it could be ignored.
3. Regardless, I'm guessing that the temperature only depends on the relative motion of the air molecules, so the wind just isn't even relevant

Can anybody provide some more educated input?

It could be the case that the macroscopic and the microscopic motions do "interact", but since the macroscopic wind is on such a large scale compared to the average kinetic energy of individual molecules, any system in which we use the average kinetic energy of the molecules is so small that the wind is effectively constant, and so it could be ignored.

Yes, this is -in a certain way- the right answer.

Temperature is proportional to the average kinetic energy, thus it is proportional to the average molecular speed squared. The exact formula for an ideal gas is

$$\langle v \rangle = \sqrt{\frac{2RT}{M}}$$

where $M$ is the molecular mass (the molecular mass of air is $\simeq 29$ g/mol).

If you have a volume of "still" air at temperature $T=25$°C ($298$ K), the molecules in such a volume will be moving randomly at an average speed of 467 m/s.

If you superimpose to such a motion a translation in some direction, the average speed registered by a stationary thermometer will be higher. But how fast can such a motion be?

The highest wind speed ever registered on Earth was registered on Barrow Island during the passage of the Cyclone Olivia (1996), and it was 113 m/s. Average wind speeds are far below this extreme values; you can see from this table that the order of magnitude is 10 MPH, i.e. 4.5 m/s.

So the presence of "ordinary" wind can increase the average molecular speed in one spatial direction of $\simeq 1\%$. I specified "in one spatial direction" because wind usually blows in a certain specific direction.

Let's take a reference frame such that the direction of the wind is the $x$ direction. If the other two components of the molecular speed remain unchanged and the speed in the $x$ direction increases of $1\%$, and if we assume that the three components where initially on average equal, we can roughly estimate the variation in $\langle v \rangle$ to be

$$\frac{v'-v}{v} = \frac{\sqrt{v_x'^2+v_y^2+v_z^2}-\sqrt{v_x^2+v_y^2+v_z^2}}{\sqrt{v_x^2+v_y^2+v_z^2}} = \frac{\sqrt{(1.01 u)^2 + 2 u^2}-\sqrt{3u^2}}{\sqrt{3u^2}} \simeq 0.0033 = 0.33\%$$

Since, from the previous equation,

$$T = \frac{M \langle v \rangle^2}{2R}$$

we see that the measured temperature will increase of $\simeq 0.67\%$ (i.e. $T'/T\simeq1.0067$), so if it was initially $25$°C $=298$K it will become $300$K = $27$°C.

So even in thi highly idealized situation the increase will not be very noticeable. In real life, we have at least two additional things to take into account:

1. Air is not an ideal gas. The most important factor is that air usually contains a non negligible amount of water vapor, the presence of which can greatly change its thermodynamic properties

2. In the atmosphere, wind is actually a large mass of air moving from regions of higher pressure to regions of lower pressure. Such a mass of air can be colder than the surrounding air, so the presence of wind can actually be linked to a decrease in temperature. The discussion becomes much more complicated in this cases, and a full understanding of it would require some knowledge of atmospheric science.

• If air is initially at 25C, it is at 298K, 298 degrees above absolute zero. A 2% rise in temperature is about 6K. K degrees are the same size as C degrees, they just start at absolute zero instead of freezing point of water. So a 4.5 m/s breeze will raise the temperature of room temperature air about 6 C when the wind hits a stationary barrier and the extra kinetic energy is "thermalized," distributed among all the air molecules through collisions. – mwengler May 26 '17 at 2:40
• @mwengler You are right, but I noticed I made another mistake. See updated answer. – valerio May 26 '17 at 8:47
• Besides all that, wind blowing causes forced convection. People are warmer than air generally, so by making the air flow it will feel colder by taking heat away at a higher rate. – JMac May 26 '17 at 9:47
• @JMac True, but we want to talk about the temperature measured by a thermometer. – valerio May 26 '17 at 10:52
• @valerio92 Why do we only want to talk about thermometer temperature and not the sensible temperature? At the end of the day we are people, and we feel the heat transfer. The question didn't explicitly mention it, so it seemed worth noting the way heat transfer could affect human perception of the process. – JMac May 26 '17 at 11:14

Does wind actually slightly raise air temperature?

No and yes.

First, the no part of the answer. Temperature would not be a well-defined concept if that were the case. The temperature of a gas is the random motion of the atoms / molecules that comprise the gas. What you aren't taught is that that relation between temperature and kinetic energy is made from the perspective in which the total linear momentum of the gas is zero. There is no wind in a frame that is moving with the air mass.

However, consider a solid object such as an airplane or spacecraft flying through the air. An alternative point of view is that the object is stationary and that a wind is blowing past it. (All frames of reference are equally valid.) There is no flow right at the object/air boundary; as an aside, this is why the blades of fans get so dusty. The wind stops at the boundary, and this stoppage means that the kinetic energy due to motion is converted to thermal energy. This leads to an important concept in aerodynamics, stagnation temperature. Stagnation temperature can be a serious problem for hypersonic aircraft and reentering space vehicles. Things melt, or worse.

• Interesting! I've always wondered why fan blades get dusty! – Ben Sandeen Aug 1 '16 at 4:57

If you gave every air molecule a momentum kick in one direction, then yeah $v_{rms}$ would be larger. However I don't think the temperature would actually be well-defined since the velocity distribution would no longer be a Boltzmann distribution. So yeah if someone referred to the temperature of windy air I would probably assume they meant in the air's "rest" frame.

Definitions for things like temperature tend to be a bit tricky, because they are usually only strictly speaking well defined for very particular types of systems. Usually I think you should just be more specific with what you want to know about a system.

Also keep in mind that the average kinetic energy of normal air is much larger than the velocity of wind.

• Just to clarify, when you say "it's the speed of sound", is that just a rough approximation, or is that exact? If it's exact, then that must mean that the speed of sound changes in proportion to the average kinetic energy, which is interesting – Ben Sandeen Aug 1 '16 at 1:42
• @BenSandeen: See the formula here...hyperphysics.phy-astr.gsu.edu/hbase/sound/souspe3.html#c1 – DJohnM Aug 1 '16 at 4:43
• The average molecular speed is actually different from the speed of sound. – valerio Aug 1 '16 at 10:20


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Quantity - Maple Help

# Online Help

###### All Products    Maple    MapleSim

ScientificErrorAnalysis

 Quantity
 construct a quantity-with-error

 Calling Sequence Quantity( value, uncer, opts )

Parameters

 value - central value of the quantity-with-error uncer - absolute or relative uncertainty in the quantity-with-error opts - (optional) equation(s) of the form option=value, where option is one of 'relative', 'uld'; determine behavior

Description

 • The Quantity( value, uncer, opts ) command constructs a quantity-with-error in the form Quantity().
 • The constructed quantity-with-error has two arguments, the central value and absolute uncertainty.
 • The opts argument can contain one or more of the following equations that determine the behavior.
 'relative' = true or false
 If 'relative'=true, uncer is the relative uncertainty of the quantity, and the absolute uncertainty is calculated by multiplying value by uncer. The default value of 'relative' is false.
 The equation 'relative'=true can be abbreviated to 'relative'.
 'uld' = true or false
 If 'uld'=true, uncer is the uncertainty in the units in the least digit of the quantity, and the absolute uncertainty is calculated by multiplying uncer by the appropriate power of ten.  In this case, value must be a floating-point number. The default value of 'uld' is false.
 The equation 'uld'=true can be abbreviated to 'uld'.
 It is invalid to have both 'relative'=true and 'uld'=true.
 If 'relative'=false and 'uld'=false, uncer is the absolute uncertainty of the quantity.
 • No rounding of the central value or error is performed. To round a Quantity() object, use the ApplyRule function.
 • If the absolute uncertainty of the constructed quantity-with-error is zero, only the central value is returned.

Examples

 > $\mathrm{with}\left(\mathrm{ScientificErrorAnalysis}\right):$
 > $\mathrm{Quantity}\left(20.0,0.1\right)$
 ${\mathrm{Quantity}}{}\left({20.0}{,}{0.1}\right)$ (1)
 > $\mathrm{Quantity}\left(20.0,0.005,'\mathrm{relative}'\right)$
 ${\mathrm{Quantity}}{}\left({20.0}{,}{0.1000}\right)$ (2)
 > $\mathrm{Quantity}\left(20.0,1,'\mathrm{uld}'\right)$
 ${\mathrm{Quantity}}{}\left({20.0}{,}{0.1}\right)$ (3)
 > $\mathrm{Quantity}\left(12.3,0.01,'\mathrm{relative}'\right)$
 ${\mathrm{Quantity}}{}\left({12.3}{,}{0.123}\right)$ (4)
 > $\mathrm{ApplyRule}\left(,'\mathrm{round}\left[2\right]'\right)$
 ${\mathrm{Quantity}}{}\left({12.3}{,}{0.12}\right)$ (5)

 See Also


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## 2.8 Autocorrelation

Just as correlation measures the extent of a linear relationship between two variables, autocorrelation measures the linear relationship between lagged values of a time series.

There are several autocorrelation coefficients, corresponding to each panel in the lag plot. For example, $$r_{1}$$ measures the relationship between $$y_{t}$$ and $$y_{t-1}$$, $$r_{2}$$ measures the relationship between $$y_{t}$$ and $$y_{t-2}$$, and so on.

The value of $$r_{k}$$ can be written as $r_{k} = \frac{\sum\limits_{t=k+1}^T (y_{t}-\bar{y})(y_{t-k}-\bar{y})} {\sum\limits_{t=1}^T (y_{t}-\bar{y})^2},$ where $$T$$ is the length of the time series.

The first nine autocorrelation coefficients for the beer production data are given in the following table.

$$r_1$$ $$r_2$$ $$r_3$$ $$r_4$$ $$r_5$$ $$r_6$$ $$r_7$$ $$r_8$$ $$r_9$$
-0.102 -0.657 -0.060 0.869 -0.089 -0.635 -0.054 0.832 -0.108

These correspond to the nine scatterplots in Figure 2.13. The autocorrelation coefficients are plotted to show the autocorrelation function or ACF. The plot is also known as a correlogram.

ggAcf(beer2)

In this graph:

• $$r_{4}$$ is higher than for the other lags. This is due to the seasonal pattern in the data: the peaks tend to be four quarters apart and the troughs tend to be four quarters apart.
• $$r_{2}$$ is more negative than for the other lags because troughs tend to be two quarters behind peaks.
• The dashed blue lines indicate whether the correlations are significantly different from zero. These are explained in Section 2.9.

### Trend and seasonality in ACF plots

When data have a trend, the autocorrelations for small lags tend to be large and positive because observations nearby in time are also nearby in size. So the ACF of trended time series tend to have positive values that slowly decrease as the lags increase.

When data are seasonal, the autocorrelations will be larger for the seasonal lags (at multiples of the seasonal frequency) than for other lags.

When data are both trended and seasonal, you see a combination of these effects. The monthly Australian electricity demand series plotted in Figure 2.15 shows both trend and seasonality. Its ACF is shown in Figure 2.16.

aelec <- window(elec, start=1980)
autoplot(aelec) + xlab("Year") + ylab("GWh")
ggAcf(aelec, lag=48)

The slow decrease in the ACF as the lags increase is due to the trend, while the “scalloped” shape is due the seasonality.