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# If a polynomial has only rational roots does that automatically mean it is solvable?
Note that I am talking about rational roots not rational coefficients. I know that Galois theory can tell you but I want to know if knowing whether all the roots of a polynomial are rational can also tell you. Note also that the roots and their properties are usually unknown, but I'm talking about when you only know that they are rational but don't know their values.
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## 2 Answers
By definition, a polynomial is solvable if and only if its roots can be expressed using rational numbers, addition/subtraction, multiplication/division, and radicals (squareroot symbols, cuberoot symbols, etc.) If the roots are rational, they certainly can be expressed in the above form.
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Thank you, that is exactly what I wanted to know. So if you know the roots are rational, then Galois theory can help you find them, right? You are talking about THAT "solvable" aren't you? – Kenny Mar 7 '12 at 17:15
@Kenny: Well, Galois theory doesn't really give you any information in this case, since the Galois group is trivial. But if the roots are rational, you don't need such powerful tools. See Tib's post. – Brett Frankel Mar 7 '12 at 17:18
What do you mean by "the Galois group would be trivial?" So even if you don't know the roots but do know they are rational, Galois theory doesn't tell you anything about the roots? – Kenny Mar 7 '12 at 17:34
Yes. Galois theory also explains, in abstract terms, why some equations are solvable by radical and others aren't. And of course there are myriad applications to number theory, algebraic geometry, group theory, etc. – Brett Frankel Mar 7 '12 at 18:12
@BlueRaja-DannyPflughoeft: It is solvable over $\mathbb{Q}(\pi)$, but this post is about polynomials with rational coefficients. – Brett Frankel Mar 7 '12 at 22:40
You can always find all the rational roots of a polynomial using the rational root theorem. See http://en.wikipedia.org/wiki/Rational_root_theorem
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(1) This does not answer OP's question about whether solvability equivalent to having all roots rational. (2) OP is interested in polynomials whose all of its roots are rational rather than finding all the rational roots of a polynomial. – user2468 Mar 7 '12 at 17:21
@J.D. True, but this does answer OP's follow-up comment. Perhaps it would be better as a comment than a solution, but I don't think it deserves a downvote. – Brett Frankel Mar 7 '12 at 17:24
@BrettFrankel Well, then this is more of a comment. Answers should address the question title & body. – user2468 Mar 7 '12 at 17:26
BTW I did not downvote! I only posted my remarks. – user2468 Mar 7 '12 at 17:27
He didn't ask whether it's equivalent to being solvable; he asked whether it implies that it's solvable. Clearly there are some cases where it's solvable and none of the roots are rational. – Michael Hardy Mar 7 '12 at 18:08
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# Numerically efficient way to compute sparse-matrix arithmetic on GPU?
Can anyone tell me some very good/efficient numerial algorthims for GPU/CUDA to compute multiplication/ between sparse matrices (its good if you can recommend me some research papers)?
I googled some papers on sparse vectors, but it looks like they are more interested in the operations involves sparse matrix and dense vectors, but what I am dealing with is some math operations involves only sparse matrices and sparse vectors.
Thanks!
• In practice sparse matrix times dense vector multiplication is heavily used in iterative methods for the solution of sparse systems of equations. In these algorithms the vector almost always becomes fully dense within a few iterations, so there's no point in worrying about multiplying by sparse vectors. – Brian Borchers Nov 4 '13 at 22:49
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# Package framed
Default LaTeX style in knitr
By default, knitr uses a LaTeX package named framed for typesetting; the most obvious feature is the light gray shading. In this page, we introduce some tricks and known problems.
As listed in FAQ’s, you may see the output overflow the shading box, and you can set options('width') to a smaller value in this case.
## Overflow of elements
Besides text overflow, figures may also exceed the margin of the shading. If a figure is too wide, LaTeX may complain there is something wrong with the kframe environment, which is what knitr uses to wrap up the chunk output. A known case is about PNG graphics in #154. To make sure your figures do not exceed the page margin, knitr uses the following command in the LaTeX preamble:
The chunk option out.width is set to '\\maxwidth' by default if the output format is LaTeX.
It does not work well with the figure* environment in two-column documents; see knitr-twocolumn.pdf for one approach to deal with this situation.
If you use the Tufte handout/book classes, the fullwidth environment does not work well with the framed package either; see discussion in #222 for possible solutions.
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# Pre algebra questions and answers
Recent questions in Pre-Algebra
York 2021-02-05
### The Manitou Incline, which leads to the top of Pike's Peak in Colorado, climbs from an elevation of 6574 feet to 8585 feet above sea level. The fastest reported time for running up the incline is 16 minutes and 42 seconds, by Mike Fretta. How many vertical feet per second did Mark climb during his record-breaking performance? (Round to 1 decimal digit.)
Wierzycaz 2021-02-05
### Find the negation of the following statement: "There exists a negative real number z such that -3
Burhan Hopper 2021-02-05
### 2.37 - (-1.55) - 2.48
opatovaL 2021-02-05
### An elevator ride down Steven stories
Chesley 2021-02-04
### A certain scale has an uncertainty of 3 g and a bias of 2 g. a) A single measurement is made on this scale. What are the bias and uncertainty in this measurement? b) Four independent measurements are made on this scale. What are the bias and uncertainty in the average of these measurements? c) Four hundred independent measurements are made on this scale. What are the bias and uncertainty in the average of these measurements? d) As more measurements are made, does the uncertainty get smaller, get larger, or stay the same? e) As more measurements are made, does the bias get smaller, get larger, or stay the same?
Nannie Mack 2021-02-04
### Among 25- to 30-years-olds, $28\mathrm{%}$ say they have operated heavy machinery while under the influence of alcohol Suppose three 25- to 30-years-olds are selected at random. Complete psrts (a) through (d) below: a) What is the probability that all three have operated heavy machinery while under the influence of alcohol? b) What is the probability that at least one has not haavy operated heavy machinery while under the influence of alcohol? c) What is the probability that none of three have operated heavy machinery while under the influence of alcohol? d) What is the probability that at least one has operated heavy machinery while under the influence of alcohol?
Brennan Flores 2021-02-04
### To solve:
foass77W 2021-02-04
### Basic facts and techniques of Boats and Streams of Quantitative Aptitude Boats and Streams is a part of the Quantitative aptitude section. This is just a logical extension of motion in a straight line. One or two questions are asked from this chapter in almost every exam. Today I will tell you some important facts and terminologies which will help you to make better understanding about this topic.
preprekomW 2021-02-03
### An electrical light circuit in John's welding shop had a load of 2800 watts. He changed the circuit to ten 150-watt bulbs and six 100-watt bulbs. What fraction represents a comparison of the new load with the old load?
Lewis Harvey 2021-02-03
### The total amount to stage an event was 1300.00. Person A spent 175.00 Person B spent 440.00 Person C spent 700.00 and Person D 0.00. What is the fair equal distribution of money
texelaare 2021-02-03
### Use a calculator to evaluate the expression. Round your answer to the nearest hundredth. arcsin $\left(0.25\right)\approx ?$ radians
Chesley 2021-02-02
### $\frac{\frac{5}{7}}{\frac{5}{6}}$
nicekikah 2021-02-02
### Simon is making wreaths to sell. He has 60 bows, 36 silk roses, and 48 silk carnations. He wants to put the same number of items on each wreath. All the items on a wreath will be the same type. How many items can Simon put on each wreath?
Nannie Mack 2021-02-02
### Find the equation $5+6$
postillan4 2021-02-01
amanf 2021-02-01
### Which expression below is equivalent to $\left(9x-4\right)+13\left(3x-2\right)$? (A) $9x+26$ (B) $48x-30$ (C) $39x-26$ (D) $48x+30$
mattgondek4 2021-01-31
### For the given fraction and decimals we have to write its equivalent percent. Given fractions are $a\right)\frac{3}{25}b\right)\frac{1}{5}c\right)\frac{2}{5}$ And the decimals are, $d\right)0.01,e\right)4.06,f\right)0.6$ We have to find its equivalent percent.
Marvin Mccormick 2021-01-31
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An almost exponential improvement in bounds against ACC
Source from previous paper
Cody Murray is a PhD student of Ryan Williams at MIT. He and Ryan have a new paper that greatly improves Ryan’s separation of nonuniform ${\mathsf{ACC}}$ circuits from uniform nondeterministic time classes. The previous best separation was from ${\mathsf{NEXP}}$, that is, nondeterministic time ${2^{n^{O(1)}}}$. The new one is from ${\mathsf{NQP}}$, which is nondeterministic time ${2^{(\log n)^{O(1)}}}$. The ‘Q’ here means “quasi-polynomial,” not “quantum.”
Today we discuss the new ideas that led to this breakthrough on a previous breakthrough.
Since I live in Buffalo and my hometown Buffalo Bills are named for the frontiersman and showman William Cody, I can’t help making Wild West associations. The complexity frontier has now existed for over 60 years, a fourth of United States history, longer than the time from the Civil War to Arizona becoming the last contiguous state in 1912. So much is still unknown about it that any new avenue of progress commands attention.
The main idea calls to mind the classic movie The Searchers. In the movie the two principals are searching for an abducted small child. Here we are searching for a witness string ${y}$ to an ${\mathsf{NP}}$-type predicate ${V(x,y)}$ that is small in the following sense: it is the truth table of a small circuit ${C_y}$. In the first applications using ${\mathsf{NEXP}}$, the length ${m}$ of ${y}$ was exponential in the length ${n}$ of the input ${x}$ and the size ${\ell}$ of ${C_y}$, while polynomial in ${n}$, was (poly-)logarithmic in ${m}$. In the new paper, all of the auxiliary size functions are related by operations under which polynomials are closed.
The small nature of the witness depends on strong hypotheses that the proof is ultimately trying to contradict. Ryan’s previous results used the powerful hypothesis ${\mathsf{NEXP \subset ACC}}$. The present result starts by supposing ${\mathsf{NQP \subset ACC}}$. What we wish to emphasize first are the clever alterations to previous techniques that enable sweeping over the vast expanses of strings in a new way to reveal a new unconditional lower bound.
## The “Almost Almost Everywhere” Hardness Condition
Let ${s(n)}$ be a complexity function. We will think of ${s(n)}$ as polynomial or quasi-polynomial but the reasoning works clear up to ${s(n) < 2^n/(2n)}$. With ${s(n)}$ in mind, say that a language ${L}$ is “easy at length ${n}$'' if there exists an ${n}$-input circuit ${C}$ of size ${s(n)}$ such that for all ${x \in \{0,1\}^n}$, ${C(x) = L(x)}$. Say ${L}$ is “hard at ${n}$'' otherwise. Consider the following conditions:
1. There are infinitely many ${n}$ such that ${L}$ is hard at ${n}$.
2. For some polynomial ${p(m)}$ and all ${m}$, there is an ${n}$ such that ${m \leq n \leq p(m)}$ and ${L}$ is hard at ${n}$.
3. For all but finitely many ${n}$, ${L}$ is hard at ${n}$.
When ${L}$ encodes a natural problem like ${\mathsf{SAT}}$, we might expect its hardness not to fluctuate with ${n}$, so that these conditions have equal force. When ${L}$ (like ${\mathsf{SAT}}$) is downward self-reducible, easiness at lengths just below ${n}$ implies easiness at ${n}$. The easiness does slip to ${C\cdot s(n)}$ in place of ${s(n)}$, where ${C}$ is the number of queries in the self-reduction, but this still supports the intuition that easiness and hardness should be evenly spread.
To get a language ${L_d \in \mathsf{DSPACE}[s(n)^2]}$ that meet the third criterion, we can loop over Boolean functions ${f}$ with ${u_n = \lfloor 2\log_2(s(n))\rfloor}$ inputs until we find ${f_n}$ that has no circuit of size ${s(n)}$. Then define:
$\displaystyle L_d = \{1^{n - 1 - u_n}0v: |v| = u_n \wedge f_n(v) = 1\}.$
Almost-everywhere hardness may fail technically if, say, the encoding of ${\mathsf{SAT}}$ as ${L}$ uses no odd-length strings, so that ${L}$ is trivially easy at odd ${n}$. We could fill in the gap by using ${L' = L \cup \{x1: x \in L\}}$ as the encoding, but this is ad hoc and might not work for other kinds of gaps. We could instead craft notions of ${L}$ being hard on a polynomially dense set ${D}$, strengthening condition 2 by making ${D}$ easy to decide and denser. Against this backdrop the new “a.a.e.” condition used in the paper is short and neat:
Definition 1 ${L}$ is almost-almost-everywhere hard if there is a polynomial ${q}$ such that for all ${n}$, either ${L}$ is hard at ${n}$ or ${L}$ is hard at ${q(n)}$.
We may relax ${q}$ to be a more general function, such as a polynomial composed with other bounds. We assume all bounds in question are increasing functions that are time-constructible or space-constructible as required.
Besides the diagonal language ${L_d}$, the paper uses a special ${\mathsf{PSPACE}}$-complete set ${L_0}$ credited to Rahul Santhanam drawing on work by Luca Trevisan and Salil Vadhan. ${L_0}$ is downward self-reducible, paddable in that ${1^a 0x \in L_0 \iff x \in L_0}$ for all ${a}$, and autoreducible in the following sense: there is a probabilistic polynomial-time oracle TM ${M_0}$ that on inputs ${x}$ makes only queries of length ${|x|}$, always accepts ${x}$ when ${x \in L_0}$ and ${L_0}$ is the oracle, and when ${x \notin L_0}$ rejects with probability at least 2/3 given any oracle. (The last clause prevents simply having ${M_0(x)}$ query ${x}$.)
## The First Lower Bound
The new lower bound really comes from a new kind of upper bound using “Merlin-Arthur with advice.” A predicate ${R(x,y,z)}$ related to a language ${L}$ has the Merlin-Arthur property if (informally speaking):
• If ${x \in L}$, then there exists a ${y}$ such that for most ${z}$, ${R(x,y,z)}$ holds.
• If ${x \notin L}$, then there are no ${y}$ or ${z}$ such that ${R(x,y,z)}$ holds.
One reason this double-barreled quantification enters the scene is that we still don’t know how to separate ${\mathsf{NEXP}}$ from ${\mathsf{P/poly}}$, but we do know that the exponential-time Merlin-Arthur class is not in ${\mathsf{P/poly}}$. The new tweak involves adding a quantifier but will allow dropping the time. It comes from exempting a small piece ${\alpha_n}$ of ${y}$ from the second (“soundness”) condition, where ${\alpha_n}$ depends only on the length ${n}$ of ${x}$:
Definition 2 ${L}$ belongs to ${\mathsf{MA}[t(n)]/a(n)}$ if there is a predicate ${R(x,y,z)}$ decidable in time ${t(|x|)}$ such that for all ${n}$ there is a string ${\alpha_n}$ of length ${a(n)}$ such that for all ${x \in \{0,1\}^n}$:
$\displaystyle \begin{array}{rcl} x \in L &\implies& (\exists y)\Pr_z[R(x,\alpha_n y,z)] > 2/3;\\ x \notin L &\implies& (\forall y) \Pr_z[R(x,\alpha_n y,z)] < 1/3. \end{array}$
The point is that the condition for ${x \notin L}$ is allowed to fail for other strings ${\alpha'_n}$. The string ${\alpha_n}$ will in fact either be ${q(n)}$ from the a.a.e. definition or will give the maximum length ${\ell = \ell(n)}$ at which the language ${L_0}$ mentioned above has circuits of some size ${r = r(n)}$ depending only on ${n}$ (so ${\ell}$ exists). Now we can state the lower bound, importing what we’ve said about ${s(n)}$ and ${q(n)}$ previously and ${r(n)}$ just now:
Theorem 3 There are constants ${b,c,d \geq 1}$ such that for all ${s(n)}$ as above, and any auxiliary functions ${q(n),r(n)}$ such that ${q(n) > s(n)^{2c}}$, ${r(n) \geq s(q(n))}$, and ${r(n) \geq s(n)^{2b+1}}$, we can construct a language ${L \in \mathsf{MA}[r(n)^2 \cdot q(n)^d]/2\log(q(n))}$ that is a.a.e.-hard.
The proof blends the constructions of ${L_d}$ and ${L_0}$ mentioned in the previous section. In particular, it uses the manner in which ${L_d}$ reduces to ${L_0}$ under appropriate padding. The reduction ${f}$ maps any string ${x}$ of length ${n}$ to a padded string ${x'}$ of length ${n' = s(n)^{2c}}$ in ${O(n')}$ time. Note that the oracle TM ${M_0}$ that comes with ${L_0}$ obeys a kind of ${\mathsf{MA}}$ condition. We have all the pieces we need to carry out the diagonalization needed for a.a.e.-hardness while obtaining a Merlin-Arthur protocol with advice that works as follows on any input ${x}$, ${n = |x|}$:
1. Advice ${\alpha_n = \ell(n)}$ unless ${q(n)}$ is smaller.
2. Merlin guesses a circuit ${C}$ of size ${r(n)}$ with ${\alpha_n}$ inputs.
3. Arthur runs ${M_0^C(w)}$ for a selected string ${w}$:
• In the case ${\alpha_n = q(n)}$, let ${x' = f(x)}$, ${n' = |x'|}$, and take ${w = 1^{q(n)-1-n'} 0 x'}$. By the way ${L_d}$ is defined above and ${n' = s(n)^{2c} < q(n)}$, this slice has no ${s(n)}$-sized circuits.
• In the case ${\alpha_n = \ell(n) < q(n)}$ we parse ${x}$ as ${1^a 0 x'}$ and put ${n' = |x'|}$ as before. If ${\alpha_n < n' + 1}$ then reject, else we can take ${w = 1^{\alpha_n - 1 - n'}0x'}$.
First note that we saved up to ${2^{q(n)}}$ time by not computing ${\ell(n)}$. The latter case uses the magnitude not the length of ${\alpha_n}$ in the padding. The proof analyzes the cases.
In the former case, because of how ${L_0}$ is defined with padding and ${q(n) \leq \ell}$, there is a circuit ${C}$ of size ${r(n)}$ that decides ${L_0}$ at length ${q(n)}$, so Merlin can guess it. In the latter case, Merlin guesses ${C}$ for the length-${\ell(n)}$ slice of ${L_0}$ directly. The property of ${M_0}$ ensures that for all ${x}$ and the appropriate ${\alpha_n}$, either there is a ${C}$ leading Arthur to accept always or all ${C}$ make Arthur reject with probability at least 2/3, so the set of ${x}$ giving the former case defines a language in ${\mathsf{MA}}$ with the stated time and advice bounds.
The a.a.e. hardness follows from how the protocol either yields the length-${n}$ slice of ${L_d}$ or implicitly maps the length-${q(n)}$ slice of it under the reduction to ${L_0}$. In fact, the argument is more delicate and starts by negating both sides of the “a.a.e.” hardness definition for sake of contradiction. For that we refer to the paper.
## The Easy-Witness Theorem
Let ${\mathsf{EW}_{t(n)}[w(n)]}$ denote the “easy-witness” class of languages ${L}$ such that for all witness predicates ${V(x,y)}$ for ${L}$ that are decidable in time ${t(|x|)}$, and all ${x \in L}$, there is a circuit ${C}$ of size ${w(|x|)}$ whose graph is a string ${y}$ such that ${V(x,y)}$ holds. It doesn’t much matter whether we restrict witnesses ${y}$ to have length a power of 2 or let the graph be ${1^a 0 y}$ for some ${a}$. Let ${\mathsf{SIZE}[s(n)]}$ denote the class of languages with (nonuniform) circuits of size ${s(n)}$.
Theorem 4 There are universal constants ${e,g \geq 1}$ and ${h > 0}$ such that for every ${s(n) < \frac{1}{n}2^{n/e}}$ and time function ${t(n) \geq q(q(q(n)))^h}$, where ${q(n) = s(en)^e}$:
$\displaystyle \mathsf{NTIME}[t(n)^e] \subset \mathsf{SIZE}[s(n)] \implies \mathsf{NTIME}[t(n)] \subseteq \mathsf{EW}_{t(n)}[q(q(q(n)))^{2g}].$
The two triple compositions of ${q(n)}$ (which is called ${s_2(n)}$ in the paper) foreshadow the proof being a three-day ride. The proof again works by contradiction, and it helps that the negation of the “easy-witness” condition is concrete and helpful: it gives a verifier ${V}$ and an ${x \in L}$ such that there are ${y}$ of length at most ${t(|x|)}$ giving ${V(x,y)}$ but none with small circuit complexity. In fact, we get this for infinitely many ${x \in L}$. The proof finally applies a theorem of Chris Umans that constructs a polynomial-time pseudorandom generator ${G}$ such that whenever ${y}$ has circuit complexity not less than ${s^g}$ and ${n' = \lceil g\log|y|\rceil}$, all circuits ${C}$ of size ${s}$ give:
$\displaystyle \left|\Pr_{v \in \{0,1\}^{n'}}[C(G(y,v)) = 1] - \Pr_{x \in \{0,1\}^m}[C(m) = 1]\right| < \frac{1}{s},$
where ${m \leq s}$ is the output length of ${G}$ in terms of ${n'}$ and the length of ${y}$. This is where the constant ${g}$ comes in, while ${h}$ can be taken as ${2}$ divided by the exponent in the running time of ${G}$. The generator is used to de-randomize the ${\mathsf{MA}}$ protocol. This yields a nondeterministic TM whose running time violates an application of the nondeterministic time hierarchy theorem, producing the desired contradiction.
## The Roundup
The horses driving the final results come from various families ${{\cal C}}$ of circuits, which we may suppose obey some simple closure properties. The nub is the speed with which a nondeterministic TMs ${N}$ can distinguish ${n}$-input circuits ${H \in {\cal C}}$ in the following sense:
• If ${H}$ computes the all-zero function then some paths of ${N(H)}$ say ${H}$ is “white” while all others say “another color.”
• If ${H}$ gives output ${1}$ for at least ${\frac{1}{4}2^n}$ arguments then some paths say “black” while all others say “another color.”
• ${N(H)}$ never has some paths that say “white” and others that say “black.”
Note that distinguishing the all-zero and quarter-dense cases is easy to do randomly in basically ${O(n + |H|)}$ time, which converts to deterministic time under certain de-randomizing hypotheses. We only need to achieve this nondeterministically with a little advantage over the brute-force time (which cycles through ${2^n}$ assignments). The main theorem works for any ${\epsilon}$ such that ${0 < \epsilon < 1}$:
Theorem 5
• If ${n}$-input ${{\cal C}}$-circuits of size ${2^{\epsilon n}}$ can be nondeterministically distinguished in ${O(2^{n - \epsilon n})}$ time, then there is a ${c \geq 1}$ such that for all ${k}$, ${\mathsf{NTIME}[n^{ck^4/\epsilon}]}$ does not have size-${n^k}$ circuits in ${{\cal C}}$.
• If ${n}$-input ${{\cal C}}$-circuits of size ${2^{n^\epsilon}}$ can be nondeterministically distinguished in ${O(2^{n - n^\epsilon})}$ time, then for all ${k}$ there is a ${c \geq 1}$ such that ${\mathsf{NTIME}[2^{(\log n)^{ck^4/\epsilon}}]}$ does not have size-${2^{(\log n)^k}}$ circuits in ${{\cal C}}$.
The proof applies the easy-witness theorem to a particular verifier constructed by Eli Ben-Sasson and Emanuele Viola, and its easy witnesses lead the charge. By distinguishing the adversaries’ horse colors they lift their cover of darkness and drive them to a final contradiction shootout in the gulch of the nondeterministic time hierarchy theorem. In terms of ${h = 2^{\epsilon n}}$ the first statement’s target time happens to be ${O(h^{1/\epsilon})}$, which is polynomial in ${h}$, while the second statement’s time in terms of ${h = 2^{n^{\epsilon}}}$ is ${O(h^{(\log h)^{(1-\epsilon)/\epsilon}})}$, which is quasi-polynomial in ${h}$.
Note that the second statement pulls a fast one: the order of quantifying ${c}$ and ${k}$ is switched. The gun it draws, however, was already in plain sight from earlier work by Ryan. Let ${\mathsf{ACC}^+}$ denote ${\mathsf{ACC}}$ circuits plus one layer of threshold gates at the inputs:
Theorem 6 For all ${m,d}$ there is an ${\epsilon > 0}$ such that given ${\mathsf{ACC}^+}$ circuits of modulus ${m}$, depth ${d}$, and size ${2^{n\epsilon}}$, whether they compute the all-zero function can be decided deterministically in time ${O(2^{n - n^\epsilon})}$.
The sheriff holding this gun rides all the ${\mathsf{ACC}^+}$ circuits out of the highlands of ${\mathsf{NQP}}$. And if a gunslinger NTM ${N}$ can be found to enforce the first clause in Theorem 5, a trail may open up for showing ${\mathsf{NP} \not\subset \mathsf{ACC}}$.
## Open Problems
What other consequences follow from this march into new lower-bound territory? Already these movies are serials.
[fixed a subscript, m-vs-n in condition 2, and last sentence before “Open Problems”; fixed LaTeX in algorithm case; fixed Theorem 6.]
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6 Comments leave one →
1. January 24, 2018 10:57 am
In condition 2 the two occurrences of $p(n)$ should be $p(m)$.
• January 24, 2018 11:02 am
Done—thanks!
2. Pinchas permalink
January 25, 2018 7:39 am
Thanks for an accessible explanation of the result. The paper’s introduction also does a nice job at explaining the result, and its building blocks.
3. January 26, 2018 3:51 pm
Wow! NQP seems amazingly close to NP (but for obvious reasons I’m not expecting that we’ll see a proof of ACC not contained in NP anytime soon). Question: Does NQP have any relations to more usual complexity classes? I’m not seeing any obvious ones. (The complexity zoo doesn’t even list NQP! Or rather, it lists something called “NQP”, but it’s not this.)
• January 29, 2018 2:28 pm
Er, obviously I meant “NP not contained in ACC”, not “ACC not contained in NP”…
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## Plasma Optics (II)
Two limits for $\epsilon(\omega,K)$: $\epsilon(\omega,0)$ and $\epsilon(0,K)$. The first one refers to the collective excitations of the Fermi sea, which is related to the plasma, and the latter describes the electrostatic screening of the electron with electron, lattice and impurity interactions in crystals.
To obtain plasma frequency $\omega_p$, we can proceed with the equation of motion of a free electron in an electric field:
x is the position of electron. Assuming electron moves according to the behaviour of simple planar wave, we need to introduce $e^{-i\omega t}$ on both side,
Further, we have polarization: the dipole moment per unit volume,
Since we are dealing with polarization, we can’t forget electric field, E, which both of them expressed in the electric displacement, D,
That is, we find the characteristic of each material depicted by specific mass, m. Thus, this is the plasma frequency $\omega_p$,
The dielectric function can be re-written as follows
In the next section, we will deal with the implication of this equation.
The main concept from Kittel’s book: Introduction to solid state physics, 8th edition.
## The Wave Equation
This is what I want to write since a long time ago: The wave equation, derived from Maxwell’s equations in free space. Finally! I can write now :)
To start with, we need to realize that an electromagnetic field is described by two vector fields, both are functions of position and time:
1. The electric field, $\vec{\varepsilon}(\vec{r},t)$
2. The magnetic field, $\vec{H}(\vec{r},t)$
The famous Maxwell’s equations in free space are defined by
the $\nabla \times$ and $\nabla \cdot$ are the curl and divergence. the constants of $\epsilon_0$ and $latex \mu_0$ represents the electric permitivity and the magnetic permeability in the free space.
Basically, we can express the wave equation based on the described Maxwell’s equation above. So, here we go
Next, we need to use the vector identitiy (curl of the curl) of $\nabla \times \left( \nabla \times \vec{\varepsilon} \right)=\nabla \left( \nabla \cdot \vec{\varepsilon} \right)-\nabla^2 \vec{\varepsilon}$. From (3), we arrive to the equation of $\nabla \times \left( \nabla \times \vec{\varepsilon} \right)=\nabla \left( 0 \right)-\nabla^2 \vec{\varepsilon} \longrightarrow \nabla \times \left( \nabla \times \vec{\varepsilon} \right)=-\nabla^2 \vec{\varepsilon}$. Therefore, we will have
By applying the speed of light equation in free space $c_0=\frac{1}{\sqrt{\varepsilon_0 \mu_0}}$, we arrive to the wave equation in the free space according to the Maxwell’s equation
If we start with (1) i.e $\nabla \times \left( \nabla \times \vec{H} = \epsilon_0 \frac{\partial \vec{\varepsilon}}{\partial t} \right)$ and follow the same step, the final result for the wave equation will be
When we speak about scalar wavefunction, the electric ($\vec{\varepsilon}(\vec{r},t)$) and magnetic field ($\vec{H}(\vec{r},t)$) can be represented in the scalar wavefunction ($u(\vec(r),t)$), and we will have the wave equation as it is explained below,
Great! Now I am satisfied enough :D
## Excursion to the Col Dei Rossi
Last week, between 15 – 19 March I went to Italy, attending international conference in Canazei, about 180 km north of Venice. The name of the conference is The 18th Molecular Beam Epitaxy (http://web.nano.cnr.it/eurombe2015/)
Canazei is located in the upper part of the Val di Fassa, and this region is part of the Dolomites, a mountain range located in the northeastern of Italy. This is my first time of my life seeing Alps as it is just standing before my eyes, so huge! According to the wikipedia, The Dolomites form a part of Southern Limestone Alps and extend from the River Adige in the west to the Plave Valley in the east. A huge range of lining mountain in the north of Italy.
The Dolomites is acknowledged as The Natural World Heritage Sites by UNESCO. Basically this place is heaven for those who can ski, both down hill and cross country. It has almost 260 km range to do ski activity: Ski until you drop :)
It has hundred of squared kilometres of walls, towers, pinnacles and valleys. A well known phenomenon, called Enrosadira, takes place at dawn and dusk when the sun light reflects on the walls of the Dolomites. Unfortunately, I was not able to see this phenomenon. The break I had was unfortunately insufficient to find out. It is supposed look like this:
In winter, the Enrosadira will appear like this
This is the map for the Val di Fassa:
And here I was, standing on the Col Dei Rossi, after taking some breathtaking cablecar from Canazei. The transportation was almost made me crazy, the wind blew and shaked our cable car! Who is not afraid of those disturbance?? Anyway, I relieved I could made to the one of the top peak in here.
The first two days were cloudy, but the last two days were awesome. Unfortunately, I did not have good schedule on the last two days. I wish I could climbed the mountain on that days :(
Well, eventhough I was not able to see Enrosadira, I was able to see the sun rays were splitted by the mountain behind hotel I stayed and I guess it was Col Dei Rossi.
That was wonderful 4-day stay in Canazei. The last photo was taken before I left Canazei to Venice, where my flight back to Trondheim departed. I enjoyed and surprised with the great hospitality I received there. I met some of the people who were hardly speaking English, but they tried their best, and the Italian (or Ladin?) language was used eventhough I did not understand what they wanted to say. I wonder why they continue talking in Italian to me? Maybe, they wanted to make me welcomed in Canazei.
Grazie!
## Plasma Optics (I)
When we talk about optics, we always relate it with the interaction between light and matters. The interaction will give varying result as it depends on what kind of material are being interacted with. One important properties of material is called dielectric function $\epsilon(\omega,K)$, a function whose frequency and wavevector has impact on the physical interaction between light and matters.
We have two fascinating interaction probabilities in here: the light can be reflected or propagate from/through matter. Before we start with everything, it is better to have understanding of what plasma is. Basically, plasma, one of the fundamental state of matters (others are solid, liquid and gas), is medium with equal concentration of positive and negative charges, of which at least one charge type is mobile. Plasma takes form in gas which composed of free electrons and ions. Plasma has high energy.
Plasma has frequency, called plasma frequency $\omega_{p}$ which becomes a guideline for deciding whether the light will be reflected or propagate. I will try to explain this later. All of us know the relation between energy E and wavelength $\lambda$: $E=h\frac{c}{\lambda}$. It is also obvious that metal, in the visible light, is reflecting incoming light. But how does it can be explained by plasma optics?
The answer lies on the value of incoming wavelength light $\lambda$. When $\lambda>\lambda_{p}$, incident light will be reflected. In the other hand, as $\lambda<\lambda_{p}$, incident light will propagate through matter. Illustration in figure \ref{fig:vis} will give a brief explanation of where should the light get reflected or propagate through material. The example is group of alkali metal, with wavelength ranging from 155 – 362 nm.
We will come back later to this concept in the next section.
# 3. Fiber optic: application of total internal reflection
The first and second part should be enough to provide illustration of how the fiber optic works: first, it depends heavily on the Snell’s law in governing how the light behaves after passing the interface of two materials with different refractive index number. Second, total internal reflection phenomenon is established owing to the fact that light coming at certain angle, defined as critical angle, is able to have reflection instead of refraction, as the light comes from the denser medium to less denser medium ($n_{1}>n_{2}$). Mathematical expression for the first and second part are described in here and here, respectively.
Now, we can go back to the figure 1. First, we need to realize that there are three different refractive index, as depicted below:
Refractive index for air (where, later, the source of light-laser- comes from), core and cladding of fiber optic are designed as it is so that light with certain range of angle, can be guided (controlled) from one point to another. To get an idea of the value of critical angle need to be fulfilled, we can start first inside the fiber optic, in the interface between core and cladding. We can start describe it as it in the figure 4 in the right picture. To have total internal reflection means that refractive index of core must be larger than refractive index of cladding ($n_{2}>n_{3}$).
Then we follow the same procedure as in the second part,
Next, we can determine the correlation between air and core of the fiber optic (figure 7). Since total internal reflection in this boundary is not needed (or precisely, avoided), we can proceed with the refractive index of air smaller than core of the fiber optic ($n_{1}).
Applying Snell’s law, we can calculate the incident angle required to achieve total internal reflection,
$n_1 \sin \theta_1 = n_2 \sin \theta_4$
We do not know $\sin \theta_4$ but we do know the value of $\sin \theta_2$ or $\sin \theta_{c(i)}$. Based on the figure 7, we can figure the value of $\theta_4$ from ordinary trigonometry case,
Mathematically, $\sin \theta_{4}$ is equal to $\sin \left( \frac{\pi}{2}-\theta_2 \right)$. To re-define $\sin \theta_{4}$, trigonometry identity of $\sin (\frac{\pi}{2}-\theta)=\cos \theta$ can be utilized. Then the continuation of the last equation is,
There it is, we find already the incident angle required to conduct total internal reflection in the fiber optic. {\textit{Numerical aperture (NA)} is defined as the qualitative limitation of receiving angle of the fiber optic with respect to the incident angle of light. Therefore, the numerical aperture for fiber optic can be defined as follows:
# 2. Total Internal Reflection
As it has been discussed in the first part that, the incident light coming from one medium to another as it strikes the boundary with certain angle, will be refracted in definite manner according to the $n_{1} \sin \theta_{1}=n_{2} \sin \theta_{2}$. Let’s just start this section by increasing incident angle of light in figure 2, first part. This is what will happen if the incident angle is increased:
As we know from the behavior described in the first part, we observed the light being refracted in increasing manner when the angle of incident light is increased as well. It is less likely that we can notice something intriguing in here.
How about if medium one has larger value than medium two ($n_{1}>n_{2}$)? We should have light will have lower velocity in medium 1 than in medium 2 ($v_{1} < v_{2}$) and thus angle of light with respect to the normal in medium 1 is smaller than in medium 2 ($\theta_{1}<\theta_{2}$). As we increase the angle of incident in medium 1, we will this result:
Total internal reflection is depicted in the right illustration of figure 2. In this phenomenon, the medium boundary acts as a reflector, instead of refractor under two conditions:
• Light propagates from medium with higher refractive index to the medium with lower refractive index ($n_{1}>n_{2}$).
• The angle of incident light has to satisfy minimum of certain angle, i.e. critical angle $\theta_{c}$.
Total internal reflection can be considered when the refracted light is in parallel direction with the interface (figure 2, middle illustration), i.e. $\theta_{2}=90^{\circ}$. Using Snell’s law equation described earlier in the first part and two conditions stated above, we can quantify the required incident angle so that it can trigger total internal reflection:
$n_{1} \sin \theta_{1}=n_{2} \sin \theta_{2}$
$n_{1} \sin \theta_{c}=n_{2} \sin 90^{\circ}$
$n_{1} \sin \theta_{c}=n_{2}$
$\sin \theta_c=\frac{n_2}{n_1}$
$\theta_c=\sin^{-1} \frac{n_2}{n_1}$
Total internal reflection is occured when the incident angle fulfills the critical angle whose value is determined by the medium 1 and 2, as it is expressed in the last formula of the above equations, called the critical angle equation:
$\theta_c=\sin^{-1} \frac{n_2}{n_1}$
This equation establishes the requirement for total internal reflection to take place: medium 2 must have refractive index smaller than medium 1. If it happened in opposite way, there will be no solution for this equation, meaning that total internal reflection does not take place.
## Let the density and its respective radius being known, then a critical beam angle shall be found!
It was happened when I read a paper from V. Consonni et al published in Physical Review B (or you can find it in here) about the shadowing effect part, in the page 4. V. Consonni et al gave a mathematical description (modelling) for the growth rate of self-induced GaN nanowires, and extracted several important parameters (for instance effective diffusion length on the sidewalls and substrate surface, desorption rate, driving forces for the diffusion of gallium adatoms to the nanowire top) based on the finding of equations from the fitting with the experimental results, which are as a function of growth time, gallium rate and growth temperature.
So, I found this “magic” at page 4, on the sub-part of “Theoretical modelling of the NW axial growth rate for the self-induced approach”, namely “shadowing effects”. Beforehand, I agreed with them regarding the geometrical considerations in the molecular beam epitaxy chamber determining the final form of nanowires which are heavily influenced by the incident angle of the gallium effusion cell and nitrogen plasma source with respect to the substrate (read here). It is important also to realize that the incident angle of these sources will be having an impact in the shadowing of the grown nanowires after they reach certain height.
To my surprise, V. Consonni et al provided three examples, just straight answer, of how shadowing effects took a role in affecting structural morphology of the nanowire (density, radius, spacing, height) and the respective angle of gallium beam. In all cases, they assumed nanowire to have regular square area. The first case is nanowire having density, radius, spacing of 100 ${\mu}m^{-2}$, 30 nm and 51 nm, respectively. They found out that with an angle of 21 degree, the gallium impinged on a nanowire with height of 133 nm. The second example is nanowire with density and radius of 100 ${\mu}m^{-2}$ and 30 nm, a critical gallium beam angle of 49 degree was found. What… The last example, nanowire with radius and gallium beam angle of 30 nm and 21 degree, the density was deduced to be 200 ${\mu}m^{-2}$. I just don’t understand that straight answer.. It is just BOOM! Just let the radius and density to be known, then gallium beam angle will be discovered by some calculations which I do not understand.
To elucidate this matter, I make simple self-explanation. I used plain figure, based on the first case, which I think the most reasonable example I want to approach due to the more data in it. The objective is simple, before shadowing plays role, I want to find the maximum gallium beam angle and density, while radius, spacing and height are known. The figure has same scale measurement as with nm:
Let’s consider that gallium beam angle comes to the nanowires in a single line (of course in reality, the nanowires are received bunch of gallium flux on them). The consideration of “just before” shadowing effect means that this effect does not occur, i.e. maximum height of nanowires are found to be around 133 nm. The figure below gives an illustrative idea of how the maximum gallium beam angle “just before” the shadowing effect.
If I made a larger angle, than shadowing effect would take place. This is the maximum angle before shadowing. We can calculate that angle by doing a little hack. We can consider those figure with this:
I doubt the angle will be 21 degree. With $\tan \theta = \frac{111 nm}{133 nm}$, the maximum gallium beam angle is 39.84 degree.
The density of the nanowire itself is lower with my calculation. Since 1 ${\mu}m^{-2}$ is equal with $10^{-6}$ $nm^{-2}$. We have 1 nanowire every 111 nm, meaning that in 2000 x 500 $nm^{-2}$, we have only 81 nanowire for 1 ${\mu}m^{-2}$.
It might be that V. Consonni et al have lower maximum limit to ensure that shadowing effect really will not be occurred, while my illustration really push the maximum angle of gallium beam angle. Using my approximation, with gallium beam angle of 21 degree, the maximum height of nanowire can be around 192 nm.
Ok, I will leave the shadowing part here.
|
UNKNOWN
## Project description
Easily gather measurements from your multimeter using the Fortune Semiconductors FS9721_LP3 protocol.
## Installation
Install from Github directly:
git clone https://github.com/coddingtonbear/python-fs9721.git
cd python-fs9721
python setup.py install
or, install from PyPI using pip:
pip install fs9721
## Use as a Library
Create the client you’ll use for gathering measurements first. Using the path to the serial device, create an instance of fs9721.Client:
from fs9721 import Client
my_multimeter = Client('/dev/tty.usbserial')
Then, you can gather measurements from your multimeter using:
print(my_multimeter.getMeasurement())
## Command-Line Use
For basic use, just run:
fs9721 /path/to/serial/port
For example, on my computer the device is connected via the serial port at /dev/tty.usbserial, for me to gather measurements directly from the multimeter, I would run:
fs9721 /dev/tty.usbserial
### Command-Line Options
• --timeout=3.0: Number of seconds to wait before timing out when communicating with the multimeter. Default: 3 seconds.
• --retries=COUNT: Number of times to retry after failing to communicate with the multimeter.
• --format=FORMAT: One of json, csv, or text (defaulting to text) corresponding with the format in which you would like the data formatted.
• --file=PATH: Rather than writing the output to the console via stdout, write file to the specified file.
• --raise: Due to the relative commonness of errors in communication with the multimeter, communication errors are suppressed by default. Use this option to raise exceptions for errors that occur.
• -show-null: Null measurements from the multimeter are suppressed by default, use this option to display null measurements when they are returned.
## Does this support my multimeter?
This library should support any multimeter using the Fortune Semiconductors FS9721_LP3 chip. Common multimeters using this chip are often low-end and include the following:
• TekPower TP4000ZC
• UNI-T_UT60E
• V&A V18b
• Voltcraft VC-820 and VC-840
If your multimeter is not on the above list, do not despair! This specific IC is very common, and it may very use this chip. Sigrok has a nice reference of which chips various multimeters use; search for your multimeter on their wiki to see if yours also uses this DMM IC.
## Project details
### Source Distribution
fs9721-1.1.tar.gz (8.2 kB view hashes)
Uploaded source
|
# Complex number problem
If z= CosA+iSinA, express 2/1+z in the form I-Tan(kA).
dextercioby
Homework Helper
Well,what is that number (2/(1+z)) equal to?
And what do you mean by "I-Tan(kA)"...?
Daniel.
where k is a constant and A is the angle from above, it says to express the answer in that form- z is a complex number in polar form.
I tried multiplying the 2/(1+z) by the conjugate (1-z) on the top and bottom but I can't get it into the form 1-Tan(kA).
I think this involves trignometric identities and de Moivre's theorem.
you write z as (cosA + i SinA) and multilply the denominator by (1+ CosA --iSinA)...Simplify it...
|
Thread: I have no idea how to do this one
1. I have no idea how to do this one
Use the Definition to find an expression for the area under the graph of f as a limit. Do not evaluate the limit.f(x) = x2 +
1 + 2x
, 4x6
lim n
n i = 1
2. Re: I have no idea how to do this one
Again, we cannot help you identify your error or your confusion if you do not show us what you have done (or at the VERY least thought about).
3. Re: I have no idea how to do this one
Originally Posted by brendasants
Use the Definition to find an expression for the area under the graph of f as a limit. Do not evaluate the limit.f(x) = x2 +
1 + 2x
, 4 ≤ x ≤ 6
lim n → ∞
n i = 1
If you really want our help then learn to post using LaTeX.
I personally have no interest in trying to guess at what a question really means.
What you posted above is total gibberish.
4. Re: I have no idea how to do this one
Originally Posted by brendasants
Use the Definition to find an expression for the area under the graph of f as a limit. Do not evaluate the limit.f(x) = x2 +
1 + 2x
, 4x6
lim n
n i = 1
I agree with Plato. You need to find another way to write your post. Do you have access to a scanner? Then you could write out your problem and post it.
For now, is $f(x) = x^2 + \sqrt{1 + 2x}$?
-Dan
|
## Precalculus (6th Edition) Blitzer
a. We can graph $y=e^x$ (red) with $y=1+x+\frac{x^2}{2}$ (blue) as shown in the figure. b. We can add $y=1+x+\frac{x^2}{2}+\frac{x^3}{6}$ (green) to the graph. c. We can add $y=1+x+\frac{x^2}{2}+\frac{x^3}{6}+\frac{x^4}{24}$ (purple) to the graph. d. We can see that with an increasing number of terms in the polynomial, we get a better approximation to the function $y=e^x$ Extra: as a matter of fact, the polynomials used in the graph are the first few terms of the Taylor expansion of the function $y=e^x$
|
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# Consulting: Johnson Cornell (fin aid) x Michigan (no aid)
Author Message
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Consulting: Johnson Cornell (fin aid) x Michigan (no aid) [#permalink] 08 May 2013, 03:39
Hello guys,
first of all, I’ve not been admitted to Ross yet, so I might be stressing over something that doesn’t actually happen.
Nonetheless, if I do get an admission offer from Ross, I have to decide very quickly (usually within 24 hours).
The problem is that I’m an international applicant and I have no cosigner and therefore, no loan.
I have saved some cash for the course, but I will definitely be in a very tight financial situation (I probably will have the opportunity to go out with current students for dinners and beers, but I would not have enough money for several travels)
My main objective is consulting. I know that Michigan is pretty good in consulting, but is it worth choosing it over Cornell, considering that I will have a considerable lower standard of living?
Thanks!
Current Student
Status: Now or never
Joined: 07 Aug 2010
Posts: 328
Location: India
Concentration: Strategy, Technology
GPA: 3.5
WE: Consulting (Consulting)
Followers: 7
Kudos [?]: 160 [0], given: 27
Re: Consulting: Johnson Cornell (fin aid) x Michigan (no aid) [#permalink] 08 May 2013, 04:52
rainfall wrote:
Hello guys,
first of all, I’ve not been admitted to Ross yet, so I might be stressing over something that doesn’t actually happen.
Nonetheless, if I do get an admission offer from Ross, I have to decide very quickly (usually within 24 hours).
The problem is that I’m an international applicant and I have no cosigner and therefore, no loan.
I have saved some cash for the course, but I will definitely be in a very tight financial situation (I probably will have the opportunity to go out with current students for dinners and beers, but I would not have enough money for several travels)
My main objective is consulting. I know that Michigan is pretty good in consulting, but is it worth choosing it over Cornell, considering that I will have a considerable lower standard of living?
Thanks!
Do you have offer from Cornell? I assume you do. As of now the only offer you have should not create any problems. When you have problem of choosing Ross vs Cornell get back with your specific queries. I can possibly offer some insights
_________________
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Re: Consulting: Johnson Cornell (fin aid) x Michigan (no aid) [#permalink] 08 May 2013, 11:58
crackHSW wrote:
rainfall wrote:
Hello guys,
first of all, I’ve not been admitted to Ross yet, so I might be stressing over something that doesn’t actually happen.
Nonetheless, if I do get an admission offer from Ross, I have to decide very quickly (usually within 24 hours).
The problem is that I’m an international applicant and I have no cosigner and therefore, no loan.
I have saved some cash for the course, but I will definitely be in a very tight financial situation (I probably will have the opportunity to go out with current students for dinners and beers, but I would not have enough money for several travels)
My main objective is consulting. I know that Michigan is pretty good in consulting, but is it worth choosing it over Cornell, considering that I will have a considerable lower standard of living?
Thanks!
Do you have offer from Cornell? I assume you do. As of now the only offer you have should not create any problems. When you have problem of choosing Ross vs Cornell get back with your specific queries. I can possibly offer some insights
What’s the cost of giving your opinion without knowing that I’m going to be admitted after all?
Nevermind…
Yes, I do have an offer from Johnson.
Current Student
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Posts: 147
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Re: Consulting: Johnson Cornell (fin aid) x Michigan (no aid) [#permalink] 08 May 2013, 12:14
to clarify, whats the financial situation w/ cornell? have they given you a scholarship? (how much $) or were you able to get some loans to go there for the full price? Current Student Joined: 15 Nov 2009 Posts: 124 Concentration: Finance GMAT 1: 680 Q48 V35 GMAT 2: 700 Q48 V37 Followers: 1 Kudos [?]: 46 [0], given: 15 Re: Consulting: Johnson Cornell (fin aid) x Michigan (no aid) [#permalink] 08 May 2013, 18:17 old1442 wrote: to clarify, whats the financial situation w/ cornell? have they given you a scholarship? (how much$) or were you able to get some loans to go there for the full price?
The financial aid at cornell is not about scholarship, it's just the loan for the full tuition
Current Student
Status: Now or never
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Re: Consulting: Johnson Cornell (fin aid) x Michigan (no aid) [#permalink] 08 May 2013, 20:18
rainfall wrote:
crackHSW wrote:
rainfall wrote:
Hello guys,
first of all, I’ve not been admitted to Ross yet, so I might be stressing over something that doesn’t actually happen.
Nonetheless, if I do get an admission offer from Ross, I have to decide very quickly (usually within 24 hours).
The problem is that I’m an international applicant and I have no cosigner and therefore, no loan.
I have saved some cash for the course, but I will definitely be in a very tight financial situation (I probably will have the opportunity to go out with current students for dinners and beers, but I would not have enough money for several travels)
My main objective is consulting. I know that Michigan is pretty good in consulting, but is it worth choosing it over Cornell, considering that I will have a considerable lower standard of living?
Thanks!
Do you have offer from Cornell? I assume you do. As of now the only offer you have should not create any problems. When you have problem of choosing Ross vs Cornell get back with your specific queries. I can possibly offer some insights
What’s the cost of giving your opinion without knowing that I’m going to be admitted after all?
Nevermind…
Yes, I do have an offer from Johnson.
Well its not about the cost in any form (time or money), but if you don't have an offer from Ross what's the point of comparing. Anyways here are my two cents on the hypothetical situation :
Considering Ross doesn't offer loan for internationals its could be mighty tough to arrange for the cost of attending the program. Again on the assumption that you need loan.
On the other hand with Cornell your problem is sorted out. So here is the thing if you want to make money as the criteria for the decision then its simple - Cornell. Otherwise I know you cant go wrong at any of the schools. Both are top notch schools with Cornell having a slight advantage in fin and Ross having a slight advantage in Consulting . The location of both the schools is something that cannot help you make a decision. The class size is something you might want to look into. Are you comfortable in a larger group or a smaller group. Also you might want to check with the career service the kind of companies you are interested in. One final thing talk to as many alums and 2nd year students at both the schools to get a feel of the culture at both the schools. This would help immensely and give you a sense of belongingness. I hope this helps.
Cheers
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Re: Consulting: Johnson Cornell (fin aid) x Michigan (no aid) [#permalink] 08 May 2013, 22:21
rainfall wrote:
Hello guys,
first of all, I’ve not been admitted to Ross yet, so I might be stressing over something that doesn’t actually happen.
Nonetheless, if I do get an admission offer from Ross, I have to decide very quickly (usually within 24 hours).
The problem is that I’m an international applicant and I have no cosigner and therefore, no loan.
I have saved some cash for the course, but I will definitely be in a very tight financial situation (I probably will have the opportunity to go out with current students for dinners and beers, but I would not have enough money for several travels)
My main objective is consulting. I know that Michigan is pretty good in consulting, but is it worth choosing it over Cornell, considering that I will have a considerable lower standard of living?
Thanks!
i think ross would be better for consulting. sounds like you can manage to pay for school although you won't be living it up and might miss some travel opportunities. at the end of the day ur still paying full price for both programs and perhaps more for cornell with interest on your loans. ross is a better ranked school and better for consulting. also, remember the internship should add 8-10k/month to your pocket. if you had some kind of scholarship i think it'd be another story. but if you can go to ross and forgo some travels i think you'll be fine. or find some part time work until august to make a few extra $to help you. if you dont get in to ross, cornell is a great option as well. but between the two, id go to ross. and see if you can apply for scholarships while you're there or be a t.a. or something (not sure how it works for international students) Manager Joined: 05 Aug 2011 Posts: 66 Location: United States Concentration: General Management, Sustainability GMAT 1: Q V0 Followers: 0 Kudos [?]: 8 [0], given: 14 Re: Consulting: Johnson Cornell (fin aid) x Michigan (no aid) [#permalink] 08 May 2013, 22:41 rainfall wrote: old1442 wrote: to clarify, whats the financial situation w/ cornell? have they given you a scholarship? (how much$) or were you able to get some loans to go there for the full price?
The financial aid at cornell is not about scholarship, it's just the loan for the full tuition
Rainfall, What's the APR on the student loan Cornell offers to internationals without cosigner? Is it reducing balance and if so how much do you need to pay per month after graduation over say a 10 year period? Have you worked those numbers? Could you share some insight, much appreciated. thanks
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Re: Consulting: Johnson Cornell (fin aid) x Michigan (no aid) [#permalink] 08 May 2013, 22:48
IMHO, both schools are top notch and have the ability to get you into any top consulting firm.. But as far as getting you prepared for consulting is concerned, I think Ross leads the way.. They have a really beautiful well structured action learning program (arguably the best action learning program I've seen among the top 15) and they also have some amazing consulting related clubs...
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Thanks To The Almighty - My GMAT Debrief
GMAT Reading Comprehension: 7 Most Common Passage Types
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Re: Consulting: Johnson Cornell (fin aid) x Michigan (no aid) [#permalink] 09 May 2013, 13:27
ace312 wrote:
rainfall wrote:
old1442 wrote:
to clarify, whats the financial situation w/ cornell? have they given you a scholarship? (how much \$) or were you able to get some loans to go there for the full price?
The financial aid at cornell is not about scholarship, it's just the loan for the full tuition
Rainfall, What's the APR on the student loan Cornell offers to internationals without cosigner? Is it reducing balance and if so how much do you need to pay per month after graduation over say a 10 year period? Have you worked those numbers? Could you share some insight, much appreciated. thanks
That's a good question
In Johnson you pay a high interest rate for internationals (something about 6.75 per year), but that’s my only choice. I could ask funds from my family, but I only would ask something if I was in a really bad situation.
After graduation, the student can choose to make amortization over the period (up to 25 years depending on the amount borrowed) or just pay interest for 2 years and then start to amortize in the remaining years.
It is an expensive credit, I know (even though in my country, the credit would be at least twice as much as expensive).
Current Student
Joined: 15 Nov 2009
Posts: 124
Concentration: Finance
GMAT 1: 680 Q48 V35
GMAT 2: 700 Q48 V37
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Re: Consulting: Johnson Cornell (fin aid) x Michigan (no aid) [#permalink] 09 May 2013, 13:30
And thanks for the opinions on both schools. It's clear that Johnson is good enough for consulting, and Ross is slightly better for this role.
Current Student
Joined: 22 Dec 2012
Posts: 12
Concentration: General Management, Entrepreneurship
GMAT 1: 750 Q50 V41
GPA: 3.45
WE: Consulting (Pharmaceuticals and Biotech)
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Re: Consulting: Johnson Cornell (fin aid) x Michigan (no aid) [#permalink] 10 May 2013, 13:46
rainfall wrote:
My main objective is consulting. I know that Michigan is pretty good in consulting, but is it worth choosing it over Cornell, considering that I will have a considerable lower standard of living?
I went to Cornell for undergrad. The cost of living was not that low..
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Re: Consulting: Johnson Cornell (fin aid) x Michigan (no aid) [#permalink] 12 May 2013, 07:24
If you're thinking about going into consulting back in your home country there is no big difference in any of the top 15 or so schools - you can generally get a first round interview in any of them.
Things are different if you want to do consulting in the US - your chances are better the bigger of a consulting school you go to but by no means assured.
Re: Consulting: Johnson Cornell (fin aid) x Michigan (no aid) [#permalink] 12 May 2013, 07:24
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# Consulting: Johnson Cornell (fin aid) x Michigan (no aid)
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# Lesson 7Slacker’s SimulationSolidify Understanding
## Jump Start
Find the following:
### 1.
The probability that a fair coin is tossed twice and lands on heads both times.
### 2.
The probability that a spinner with colors (red, blue, green, yellow, pink, and orange) lands on red, and then a coin is flipped and it lands on tails.
## Learning Focus
Perform a simulation to determine if an event can occur.
Is there a way to test a claim without performing a study on actual subjects?
## Open Up the Math: Launch, Explore, Discuss
I know a student who forgot about the upcoming history test and did not study at all. To protect his identity, I’ll just call him Slacker. When I reminded Slacker that we had a test in the next class, he said that he wasn’t worried because the test has true/false questions. Slacker said that he would totally guess on every question, and since he’s always lucky, he thinks he will get at least out of . That’s what he did on the last quiz and it worked great.
I’m skeptical, but Slacker said, “Hey, sometimes you flip a coin and it seems like you just keep getting heads. You may only have a chance of getting heads, but you still might get heads several times in a row. I think this is just about the same thing. I could get lucky.”
### 1.
What do you think of Slacker’s claim? Is it possible for him to get out of questions right? Explain.
I thought about it for a minute and said, “Slacker, I think you’re on to something. I’m not sure that you will get on the test, but I agree that the situation is just like a coin flip. It’s either one way or the other and they are both equally likely if you’re just guessing.” My idea is to use a coin flip to simulate the T/F test situation. We can try it many times and see how often we get out of questions right. I’m going to say that if the coin lands on heads, then you guessed the problem correctly. If it lands on tails, then you got it wrong.
### 2.
Try it a few times yourself. To save a little time, just flip coins at once and count up the number of heads for each test.
# Incorrect (Tails)
% Correct
Test 1
Test 2
Test 3
Test 4
Test 5
Did you get out of correct in any of your trials?
### 3.
Based on your trials, do you think Slacker is likely to get correct?
### 4.
Check out the histogram that represents the data from the whole class. Now what do you think of Slacker’s chances of getting correct? Explain why.
Pause and Reflect
### 5.
What would you expect the graph to look like if you continued to collect samples? Why?
### 6.
Based upon your understanding of this distribution, what would you estimate the likelihood of Slacker getting on the test without studying?
Padma has been playing a board game with her friends where the moves are determined by the sum of the numbers on three number cubes. In the last game, one of Padma’s friends needed to roll a sum of on the cubes to win and it took her a lot of turns to get it. One of the players remarked, “I don’t know why it took so long, since is no more or less likely than any of the other numbers between and .” Padma decided to investigate this claim with a simulation. She rolled number cubes times and got this distribution:
#### a.
Which of these statements are true about rolling in this game?
#### A.
It is not possible to roll 5 in this game.
#### B.
It is not very likely to roll in this game.
#### C.
Rolling is less likely than rolling in this game.
#### D.
Rolling is less likely than rolling .
#### E.
Rolling and rolling have the same likelihood.
Simulation:
## Lesson Summary
In this lesson, we used simulation to model the outcome of a random event. Using many trials, we created a distribution and used it to predict the likelihood of an event.
## Retrieval
### 1.
• What kind of numbers are and ?
• What kind of number is ?
• What kind of number is the product of and ?
### 2.
In a group of students, are taking algebra, are taking biology, and are taking both algebra and biology.
#### a.
Draw a Venn diagram to represent this information.
#### b.
If a student, chosen at random, is taking algebra, what is the probability that he or she is taking biology? (Let be algebra and biology.)
#### c.
Which notation means the same thing as the question in part b?
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## Calculus (3rd Edition)
$$\theta=54.73^o.$$
By making use of the result from problem 36 (d), we get $n_1=\langle 1,1,1\rangle$ and we have $n_2=\langle 1,0,0\rangle$; then we get \begin{align*} \cos \theta &=\frac{n_1 \cdot n_2}{\|n_1\|\|n_2\|}\\ &=\frac{1+0+0}{\sqrt{1+1+1}\sqrt{1+0+0}}\\ &=\frac{1}{\sqrt{3} }. \end{align*} That is $$\theta=54.73^o.$$
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# Grothendieck and Non-commutative Geometry?
When Grothendieck and his followers were working on their profound progress of algebraic geometry, did they ever consider non-commutative rings? Is there anyway evidence that Grothendieck foresaw the developments that would later come in non-commutative geometry or quantum group theory?
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Ok. Edited accordingly. – Abtan Massini Nov 30 '10 at 18:11
A friend of mine postulated the following: in English language calculus course and books, exponential processes, population growth or radioactivity, are often introduced under the heading "Growth and Decay" problems, see for example math.dartmouth.edu/~klbooksite/3.02/302.html The suggestion was that this is the reason for the choice of letter in Grothendieck K-theory. I'm here all week. Don't forget to tip your waiter or waitress. – Will Jagy Nov 30 '10 at 19:02
The bigger the groan, the better they are! – Todd Trimble Nov 30 '10 at 19:15
Will, that's awesome! Keep 'em coming :) – Philip Brooker Nov 30 '10 at 22:15
No and yes, depending on the level of understanding. The consideration of noncommutative rings telling about geometry is almost nonexistent in Grothendieck's published opus. One of the exceptions is that he considered cohomologies for the possibly noncommutative sheaves of $\mathcal{O}$-algebras for commutative $\mathcal{O}$ (the latter is used in Semiquantum geometry). On the other hand, Grothendieck has been pioneer on abandoning the points of spaces as primary objects and promoting the category of sheaves over the space as defining the space. This is the point of view of topos theory which he invented; he noticed that the topological properties do not depend on a site but only on the associated topos of sheaves, and proposed a topos as a natural generalization of a topological space. Manin took Grothendieck's advice that one should consider the topos of sheaves as replacing the space, together with Serre's theorem that the category of quasicoherent modules determines a projective variety, as a motivation to his approach to noncommutative geometry and quantum groups. The modern view of noncommutative geometry is that it is about the presentation of space via the structures consisting of all possible objects of some kind living on a space (algebra of functions, some structures consisting of cocycles, like category of vector bundles, category of sheaves, higher category of higher stacks).
In late 1960s W. Lawvere, with help from Tierney, extended the Grothendieck topoi to the theory of elementary topoi. This was not the only contribution of Lawvere in the 1960s. Lawvere promoted also the duality between spaces and dual objects which he calls quantity (cf. space and quantity). While Lawvere's impact has been deep, I object to the terminology: in physics a quantity is normally a single observable; physicist do not consider the algebra of all observables a quantity, but rather a field of quantities, or algebra of quantities. But never mind the terminology, Lawvere went on very deeply in presenting this point of view, which is really generalized noncommutative geometry. Of course, neither Grothendieck nor Lawvere did not pay that particular attention to reconstructing the differential geometry and measure theory from the study of operator algebras, what is the huge contribution of Connes, or from the study of noncommutative rings, which was implicit in Gabriel 1961 and more explicit with works of J. S. Golan, van Oystaeyen (and P. M. Cohn with his affine spectrum) and others in mid 1970s, working with spectra of noncommutative rings and noncommutative localization theory as a noncommutative analogue of Zariski topology. One should mention that sporadic appearance of operator algebras from the noncommutative geometry point of view is present to some extent in 1970s book of Semadeni on Banach spaces of continuous functions (MR296671), where he studies, among other topics, the noncommutative analogues of many topological properties of topological spaces; in less explicit form there are also works of Irving Segal which had a similar motivation.
Grothendieck says in his memoirs that the concept of abelian category as he promoted it in Tohoku is part of the same philosophy -- abelian categories, possibly with additional axioms like AB5 are sort of categories of sheaves of modules, and should be viewed as an idea which is sort of abelian/stable version of Grothendieck topoi. More precisely, in this line, there is a recent Nikolai Durov's concept of a vectoid. Pierre Gabriel, who was close to Grothendieck's school in his early days, had in his prophetic work of 1961 reconstruction theorem for schemes and study of subcategories and localizations in abelian categories which represent open or closed subschemes and so on. Gabriel's work is in fact the first big work in noncommutative algebraic geometry and his reconstruction theorem is really the basic motivation in algebraic flavour of the theorem. In a sense, Gabriel's work is an abelian version of some Grothendieck's basic ideas of topos theory (cf. noncommutative scheme for one of the modern ideas along that line of thought) and Grothendieck was well aware of the abelian direction of this thinking from the Tohoku times.
For a general vista, I recommend
• Pierre Cartier, A mad day's work: from Grothendieck to Connes and Kontsevich The evolution of concepts of space and symmetry, Bull. Amer. Math. Soc. 38 (2001), 389-408, pdf.
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# View Antimicrobial Pharmacodynamics In Theory And Clinical Practice, 1St Edition (Infectious Disease And Therapy) 2002
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# zbMATH — the first resource for mathematics
Characters, asymptotic and n-homology of Harish-Chandra modules. (English) Zbl 0523.22013
##### MSC:
22E46 Semisimple Lie groups and their representations 20G10 Cohomology theory for linear algebraic groups 57T10 Homology and cohomology of Lie groups
Full Text:
##### References:
[1] Atiyah, M. F. &Schmid, W., A geometric construction of the discrete series for semisimple Lie groups.Invent. Math., 42 (1977), 1–62. · Zbl 0373.22001 [2] Borel, A.,Introduction aux groupes arithmétiques. Hermann, Paris 1969. · Zbl 0186.33202 [3] Borel, A. Wallach, N.,Continuous chomology, discrete subgroups, and representations of reductive groups. Annals of Mathematical Studies 94. · Zbl 0980.22015 [4] Bruhat, F., Sur les représentations induites des groupes de Lie.Bull. Soc. Math. France, 84 (1956), 97–205. · Zbl 0074.10303 [5] Cartan, H. &Eilenberg, S.,Homological Algebra. Princeton University Press, Princeton 1956. [6] Casselman, W., Characters and Jacquet modules.Math. Ann., 230 (1977), 101–105. · Zbl 0337.22019 [7] Casselman, W., Jacquet modules for real reductive groups. InProceedings of the International Congress of Mathematicians, Helsinki 1978, pp. 557–563. [8] Casselman, W. & Miličić, D., Asymptotic behavior of matrix coefficients of admissible representations. Preprint. [9] Casselman, W. &Osborne, M. S., The n-cohomology of representations with an infinitesimal character.Compositio Math., 31 (1975), 219–227. · Zbl 0343.17006 [10] –, The restriction of admissible representations to n.Math. Ann., 233 (1978), 193–198. · Zbl 0355.20041 [11] Dixmier, J., Algèbres enveloppantes. Gauthier-Villars, Paris 1974. · Zbl 0308.17007 [12] Fomin, A. I. &Shapovalov, N. N., A property of the characters of real semisimple Lie groups.Functional Anal. Appl., 8 (1974), 270–271. · Zbl 0302.22017 [13] Harish-Chandra, Representations of semisimple Lie groups I.Trans. Amer. Math. Soc., 75 (1953), 185–243. · Zbl 0051.34002 [14] –, Representations of semisimple Lie groups II.Trans. Amer. Math. Soc., 76 (1954), 26–65. · Zbl 0055.34002 [15] –, Representations of semisimple Lie groups III.Trans. Amer. Math. Soc., 76 (1954), 234–253. · Zbl 0055.34002 [16] –, The characters of semisimple Lie groups.Trans. Amer. Math. Soc., 83 (1956), 98–163. · Zbl 0072.01801 [17] –, Spherical functions on a semisimple Lie group I.Amer. J. Math., 80 (1958), 241–310. · Zbl 0093.12801 [18] –, Invariant eigendistributions on a semisimple Lie group.Trans. Amer. Math. Soc., 119 (1965), 457–508. · Zbl 0199.46402 [19] –, Discrete series for semisimple Lie groups II.Acta Math., 116 (1966), 1–111. · Zbl 0199.20102 [20] Harish-Chandra, On the theory of the Eisenstein integral. InConference on Harmonic Analysis, Springer Lecture Notes in Mathematics 266 (1972), pp. 123–149. · Zbl 0245.22019 [21] –, Harmonic analysis on real reductive groups I.J. Funct. Anal., 19 (1975), 104–204. · Zbl 0315.43002 [22] –, Harmonic analysis on real reductive groups III.Ann. of Math., 104 (1976), 117–201. · Zbl 0331.22007 [23] Hecht, H., On characters and asymptotics of representations of a real reductive Lie group.Math. Ann., 242 (1979), 103–126. · Zbl 0405.22010 [24] Hecht, H. &Schmid, W., A proof of Blattner’s conjecture.Invent. Math., 31 (1975), 129–154. · Zbl 0319.22012 [25] Hirai, T., The characters of some induced representations of semisimple Lie groups.J. Math. Kyoto Univ., 8 (1968), 313–363. · Zbl 0185.21503 [26] Hirai, T., Explicit form of the characters of discrete series representations of semisimple Lie groups. InProceedings of Symposia in Pure Mathematics 26 (1973), pp. 281–287. · Zbl 0287.22013 [27] Knapp, A. W. & Zuckerman, G., Classification of irreducible tempered representations of semisimple Lie groups. Preprint. · Zbl 0329.22013 [28] Langlands, R. P.,On the classification of irreducible representations of real algebraic groups. Mimeographed notes, Institute for Advanced Study 1973. [29] Lepowsky, J., Algebraic results on representations of semisimple Lie groups.Trans. Amer. Math. Soc., 176 (1973), 1–44. · Zbl 0264.22012 [30] McConnel, J., The intersection theorem for a class of noncommutative rings.Proc. London Math. Soc., 17 (1967), 487–498. · Zbl 0148.26501 [31] Miličić, D., Asymptotic behavior of matrix coefficients of the discrete series.Duke Math. J., 44 (1977), 59–88. · Zbl 0398.22022 [32] Nouazé, Y. &Gabriel, P., Idéaux premiers d’algèbre enveloppante d’une algèbre de Lie nilpotente.J. Algebra, 6 (1967), 77–99. · Zbl 0159.04101 [33] Osborne, M. S., Lefschetz formulas on non-elliptic complexes. Thesis, Yale University 1972. [34] Schmid, W.,L 2-cohomology and the discrete series.Ann. of Math., 103 (1976), 375–394. · Zbl 0333.22009 [35] Schmid, W., Two character identities for semisimple Lie groups. InNoncommutative Harmonic Analysis, Springer Lecture Notes in Mathematics 587 (1977), pp. 196–225. [36] –, Vanishing theorems for Lie algebra cohomology and the cohomology of discrete subgroups of semisimple Lie groups.Adv. in Math., 41 (1981), 78–113. · Zbl 0472.22003 [37] Speh, B. &Vogan, D., Reducibility of generalized principal series representations.Acta Math., 145 (1980), 227–299. · Zbl 0457.22011 [38] Trombi, P. C., The tempered spectrum of a real semisimple Lie group.Amer. J. Math., 99 (1977), 57–75. · Zbl 0373.22007 [39] Trombi, P. C. &Varadarajan, V. S., Asymptotic behavior of eigenfunctions on a semisimple Lie group; the discrete spectrum.Acta Math., 129 (1972), 237–280. · Zbl 0244.43006 [40] Vogan, D., Lie algebra cohomology and the representations of semisimple Lie groups. Thesis, M.I.T. 1976. [41] Vogan, D., Complex geometry and representations of reductive groups. Preprint. · Zbl 0681.22013 [42] Wolf, J. A.,Unitary representations on partially holomorphic cohomology spaces. Amer. Math. Soc. Memoir 138 (1974). · Zbl 0288.22022 [43] Zuckerman, G., Tensor products of finite and infinite dimensional representations of semisimple Lie groups.Ann. of Math., 106 (1977), 295–308. · Zbl 0384.22004
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# How to integrate $\int_{\gamma_1} \frac{dz}{z(z-i)}$ with $\gamma_1 = Re^{it}$, $R>1$?
I am stuck calculating the integral $$\int_{\gamma_1} \frac{dz}{z(z-i)}$$ over $\gamma_1 = Re^{it}, R>1$.
If I had to integrate over $\gamma_2 = re^{it}, r < 1$, I could just expand the integrand into a power series (using the geometric series) around $z=0$, but with $R>1$ this approach won't work. I quite frankly don't have any other idea on how to approach this.
Any hints?
2
2022-07-25 20:46:07
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# magnitude of acceleration
• February 19th 2008, 04:11 AM
galactus
magnitude of acceleration
Hello All:
Here's a problem I am ashamed to admit has me stymied. I am sure someone here has an idea.
"Professor lives S miles from the university and it takes him T minutes to drive from home to work. Prove that at some instant the magnitude of the acceleration of his car is at least $\frac{4S}{T^{2}}$.
I think we can use the MVT.
Wouldn't he be accelerating to the midpoint and then decelerating?.
I was thinking perhaps the max acceleration would be at S/4.
• February 19th 2008, 07:58 AM
topsquark
Quote:
Originally Posted by galactus
Hello All:
Here's a problem I am ashamed to admit has me stymied. I am sure someone here has an idea.
"Professor lives S miles from the university and it takes him T minutes to drive from home to work. Prove that at some instant the magnitude of the acceleration of his car is at least $\frac{4S}{T^{2}}$.
I think we can use the MVT.
Wouldn't he be accelerating to the midpoint and then decelerating?.
I was thinking perhaps the max acceleration would be at S/4.
Assuming that the professor ends his/her trip at 0 m/s. (I suppose he could crash his car at the end of the problem, but then we can calculate the deceleration at the end and find out it is enormously larger than the given answer. :) ) So the least possible acceleration will be given by a constant acceleration to the half-way point and the same magnitude constant deceleration to the destination. (That this gives the "minimum possible maximum" acceleration is guaranteed by the MVT.)
So the acceleration may be calcuated by
$d = \frac{1}{2}at^2$
The professor travels S/2 m to get to the half-way point, in a time T/2.
Thus
$\frac{S}{2} = \frac{1}{2}a \left ( \frac{T}{2} \right )^2$
Solve for a. This is the least possible maximum acceleration to get to the destination on time.
-Dan
• February 19th 2008, 08:04 AM
wingless
Quote:
Originally Posted by galactus
Wouldn't he be accelerating to the midpoint and then decelerating?.
You're right about that. As we're looking for the least acceleration, we will take the movement uniformly increasing and decreasing. (You can easily prove this)
Here's the V-t graph:
http://img513.imageshack.us/img513/6215/gphza6.png
Area under the V-t graph gives the distance travelled.
Then, the area from 0 to T under the graph is S.
$Area = \frac{1}{2} T \cdot V_{max} = S$
$S = \frac{T\cdot V_{max}}{2}$
$V_max = \frac{2S}{T}$
Slope of the lines (which also means the derivative) gives the acceleration.
$Slope = a = \frac{\Delta y}{\Delta x} = \frac{V_{max}-0}{\frac{T}{2}-0}$
$a = \frac{2\cdot V_{max}}{T}$
Plug the $V_{max}$ we found before,
$a = \frac{2\cdot \frac{2S}{T}}{T}$
$a = \frac{4S}{T^2}$
Edit: topsquark was faster :P
• February 19th 2008, 08:42 AM
galactus
I knew I'd feel stupid. I forgot all about the $\frac{1}{2}at^{2}$
I was also trying to use the MVT which wasn't necessary.
Thanks.
• February 19th 2008, 07:19 PM
topsquark
Quote:
Originally Posted by galactus
I knew I'd feel stupid. I forgot all about the $\frac{1}{2}at^{2}$
I was also trying to use the MVT which wasn't necessary.
Thanks.
The MVT is needed to prove that this is the minimum necessary acceleration, so I wouldn't say that it wasn't necessary. I'd say it's rather critical. (Though it admittedly doesn't help the actual calculation.)
-Dan
• February 20th 2008, 04:33 AM
galactus
That's pretty close to what wingless showed, isn't it?.
• February 20th 2008, 07:51 AM
topsquark
Quote:
Originally Posted by galactus
That's pretty close to what wingless showed, isn't it?.
Yeah, I guess you are right about that. (Nod)
-Dan
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## Relaxed energies for harmonic maps.(English)Zbl 0793.58011
Variational methods, Proc. Conf., Paris/Fr. 1988, Prog. Nonlinear Differ. Equ. Appl. 4, 37-52 (1990).
[For the entire collection see Zbl 0713.00009.]
Let $$\Omega \subset \mathbb{R}^ 3$$ be an open bounded set with smooth boundary $$\partial \Omega$$. Set $H^ 1(\Omega;S^ 2) = \{u \in H^ 1(\Omega;\mathbb{R}^ 3): | u(x)| = 1 \text{a.e.\} and}$
$H^ 1_ \varphi(\Omega;S^ 2) = \{u\in H^ 1(\Omega;S^ 2): u = \varphi \text{ on }\partial \Omega\},$ where $$\varphi: \partial \Omega \to S^ 2$$ is a given boundary datum.
The aim of this article is to establish some basic properties of the functionals $$L(u)$$ and $$F(u)$$ defined below. The interest of this approach lies on its applications to the theory of liquid crystals and to the study of singularities of harmonic maps into spheres. $L(u) = {1\over 4\pi}\displaystyle{\sup_{\substack{ \xi: \Omega\to \mathbb{R}\\ \| \nabla\xi \|_ \infty \leq 1}} }\left \{\int_ \Omega D(u)\cdot \nabla \xi - \int_{\partial \Omega} D(u) \cdot \nu \xi d\sigma\right\},$ where $$\nu$$ denotes the outward normal to $$\partial \Omega$$ and the vector field $$D(u)$$ is given by $$D(u) = (u.u_ y \wedge u_ z,\;u.u_ z \wedge u_ x,\;u.u_ x \wedge u_ y)$$.
$$F(u) = E(u) + 8 \pi L(u)$$, where $$E(u)$$ is the usual energy for harmonic maps.
By way of example, let us quote one result obtained in this paper:
Theorem: $$F$$ is sequentially lower semi-continuous on $$H^ 1_ \varphi$$ for the weak $$H^ 1$$ topology.
Reviewer: A.Ratto (Brest)
### MSC:
58E20 Harmonic maps, etc. 35J65 Nonlinear boundary value problems for linear elliptic equations
### Keywords:
regularity; liquid crystals; singularities; harmonic maps; spheres
### Citations:
Zbl 0793.58012; Zbl 0713.00009
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# Vibrational and rotational spectroscopy
I don't understand why vibrational spectroscopy only has 1 intense absorption peak whereas the rotational spectroscopy has many separate peaks and the distance between the peaks is equal. $$\Delta E\text{(vib)}$$ is independent of quantum number so vibrational spectroscopy should instead have a graph of many separate peaks and the distance between which is the same. Is it due to the selection rule? But then both vibrational- and rotational spectroscopy share the same selection rule. $$\Delta E\text{(rot)}$$ depends on the quantum number $$J$$ which means that the rotational energy levels are not equally spaced in energy so its spectroscopy should not have equally spaced absorption peaks should it?
Summary of my question:
• Explanation for the the shape of vib- and rotational spectroscopy
• How does $$\Delta E$$ of vibration and rotation affect the number of peaks in the respective spectra, even though they share the same selection rule?
There are several different issues conflated together here: selection rules, separation between energy levels, and energy level population (which you didn't mention).
For vibrational spectroscopy, in the approximation that a vibrational mode behaves like a quantum harmonic oscillator, the energy levels are equally spaced and the selection rule is $$\Delta n=\pm 1$$, where $$n$$ is the quantum number. The reason for this is explained here. So you expect to see (and do see) an absorption transition from $$n=0$$ to $$n=1$$. You might also expect to see a transition from $$n=1$$ to $$n=2$$ etc. This would occur at the same frequency since the gap between successive energy levels is the same. However, it relies on there being a thermal equilibrium population of molecules already in the $$n=1$$ state. For most molecules, at normal temperatures, the population of $$n=1$$ and higher levels (determined by the Boltzmann factor) is rather low. At elevated temperatures, you might see such transitions; also the frequency won't be exactly at the same frequency as the $$n=0\rightarrow 1$$ transition, because of anharmonicity effects.
For a linear rotor, the quantum levels are at $$BJ(J+1)$$ where $$B$$ is a constant and $$J$$ is the quantum number. These are not evenly spaced. The selection rule is $$\Delta J=\pm 1$$ (angular momentum conservation). So you expect to see (and do see) transitions between successive levels: $$J=0\rightarrow 1$$, $$J=1\rightarrow 2$$ etc. In this case, at normal temperatures, the spacing between rotational levels is typically small compared with the available thermal energy. So those higher states are populated, at least for $$J$$ not too high. The frequencies are not all the same, but the energy level spacings change linearly with $$J$$: $$\Delta E_{J\rightarrow J+1}=B(J+1)(J+2)-BJ(J+1)=2B(J+1).$$ So you see a spectrum with equally spaced lines for $$J=0,1,2\ldots$$ (in this rigid rotor approximation).
EDIT following OP comment.
• The first line, $$J=0\rightarrow 1$$ is at a frequency $$\nu$$ given by $$h\nu=\Delta E=2B$$ where $$h$$ is Planck's constant.
• The second line, $$J=1\rightarrow 2$$ is at $$h\nu=\Delta E=4B$$.
• The third line, $$J=2\rightarrow 3$$ is at $$h\nu=\Delta E=6B$$.
And so on. Hence the lines in the spectrum are equally spaced, $$2B$$ apart (in energy units) or $$2B/h$$ in frequency units. You can also see a diagram of this in the Linear Molecules section of the Rotational Spectroscopy Wikipedia page (reproduced below under the terms of the CC BY-SA 3.0 licence). The diagram shows the link between the energy levels and the lines in the spectrum (the only difference is that the transitions on the energy level diagram on that page are drawn for emission lines, $$J\leftarrow J+1$$, but exactly the same frequencies occur for the corresponding absorption lines $$J\rightarrow J+1$$).
• Hi, like you say, the spacings (harmonic potential energy of a rigid rotor) are dependent of J which means that the spacing in the spectrum should not be equal should it? Thanks for the answer – Jung Dec 17 '18 at 15:52
• No, the linear dependence on $J$ means that the lines in the spectrum are equally spaced. I've edited my answer to clarify. – LonelyProf Dec 17 '18 at 16:27
• ah yes, i forgot the absorbed energy is not E of the energy level itself but instead delta E. Delta (delta E) is 2 hcB, which is a constant which explains the equal spacing. Thanks for the clarification. – Jung Dec 17 '18 at 19:30
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# Relaxion from particle production
Recently a new solution to the hierarchy problem was proposed which makes use of the cosmological evolution of a light scalar field, a scanner, instead of symmetry or anthropic arguments to select a small Higgs mass. In the original proposal this scanner field could be the QCD axion and thus such class of solution became known as relaxion’’. Two central features required of the relaxion are a reason for why a small Higgs mass is special from the scanner’s viewpoint and a mechanism fo the scanner to dissipate its energy in order to stop in a value associated with small Higgs mass. In this talk we propose a novel mechanism that achieves both goals and opens new possibilities for the relaxion. We show that particle production can create an effective friction force for the relaxion and show how it can be used to select’’ the TeV scale as a special energy scale from reasonable initial conditions. This allows for the scanning to happen much faster than in previous relaxion models, which usually require very large amounts of inflation, and also allows for the scanning to take place after inflation has ended.
Collection/Series:
Event Type:
Seminar
Scientific Area(s):
Event Date:
Tuesday, March 7, 2017 - 13:00 to 14:30
Location:
Space Room
Room #:
400
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# Precalculus Examples
Determine if A and B are Mutually Exclusive
,
Set up the intersection notation of set and .
The intersection of two sets is the set of elements which are in both sets. The intersection in this case is an empty set.
The two sets share no members, therefore the sets are mutually exclusive.
Mutually Exclusive
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Step-by-step work + explanations
• Step-by-step work
• Detailed explanations
• Access anywhere
Access the steps on both the Mathway website and mobile apps
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# Swift Overview¶
The Swift scripting language provides a simple, compact way to write parallel scripts that run many copies of ordinary programs concurrently in various workflow patterns, reducing the need for complex parallel programming or arcane scripting. Swift is very general, and is in use in domains ranging from earth systems to bioinformatics to molecular modeling.
# Access¶
Type the following commands to run a simple Swift script:
% module load swift
% swift -config swift.conf myscript.swift
# Swift Tutorial on NERSC Systems¶
This site has a set of excellent introductory tutorials for Swift that run on NERSC systems. On Cori, just remember to type "module load swift" before starting the tutorials.
An introductory talk by Michael Wilde was given at NERSC in December 2015. The slides from this seminar can be found here.
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Wave antenna 5/8 pro VKV FM
Wave antenna 5/8 consists of a vertical radiator which is fed at the base of the antenna. A suitable device of some sort should be added between the antenna and feedline if you want to eat with coax. Adding a coil in series with the antenna on the base is one of these methods are suitable.
So why would anyone use an antenna 5/8 wave if they have to go through all that extra work? After all, a ground plane antenna provides a good match. There are several answers. The first is GAIN. The computer shows that the antenna (mounted 1 foot above the ground) has a margin of about 1.5 dBd higher than a dipole (also installed 1 foot above the ground.) The second reason you might want to use the wave 5/8 vertical is to get a lower angle of radiation. Peak radiation angle A half-wave antenna is 20 degrees. You will find that the angle 5/8 wave antenna radiation is only 16 degrees so it is better dx antenna.
You may have noticed a pattern developing here. A quarter-wave ground plane antenna has a radiation pattern that produces the maximum gain at about 25 degrees and half-wave antenna drops to 20-degree angle, and wave antenna 5/8 further drops to 16 degrees angle. So why not just keep extending the antenna to one full wave? Well it would be nice if it worked, but unfortunately the wave patterns begin to create a very high angle of radiation waves exceed 5/8. So we've reached the maximum gain at this point and extend the antenna further reduce profits only where we want it (low angle).
Of course if you are interested in a very short jump, extend the antenna will produce a nice profit on the dipole. All the length of the antenna depends on various factors. Some of these factors are: height above ground, the diameter of the wire, nearby structures, the effects of other antennas in the area and even the conductivity of the soil. This page allows you to calculate the wavelength for the antenna 5/8. It uses the standard formula, 585 / f (178.308 / f for metric) MHz to calculate the length of the element. If you have experimented with 5/8 wave antenna before and know a better formula for your QTH, feel free to change the formula accordingly. This formula is for the antenna wire.
Of course if you build your antenna out of the tube, total length of the antenna will be shorter, for example I have found that 21.5 feet seems to provide maximum benefit to the frequency of 28.5 MHz when using a 1 "tube, and 22.5. Foot seems be the best long-wire at the same frequency. Since the formula to calculate the antenna to be about 2 feet shorter, be sure to experiment and maybe add a little for your final term.
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Checkout JEE MAINS 2022 Question Paper Analysis : Checkout JEE MAINS 2022 Question Paper Analysis :
# Poisson's Ratio - Longitudinal Strain and Lateral Strain
In mechanics, Poisson’s ratio is the negative of the ratio of transverse strain to lateral or axial strain. It is named after Siméon Poisson and denoted by the Greek letter ‘nu’, It is the ratio of the amount of transversal expansion to the amount of axial compression for small values of these changes.
## What is Poisson’s Ratio?
Poisson’s ratio is “the ratio of transverse contraction strain to longitudinal extension strain in the direction of the stretching force.” Here,
Symbol Greek letter ‘nu’,ν Formula Poisson’s ratio = – Lateral strain / Longitudinal strain Range -1.0 to +0.5 Units Unitless quantity Scalar / Vector Scalar quantity
## Poisson’s Ratio Formula
Imagine a piece of rubber, in the usual shape of a cuboid. Then imagine pulling it along the sides. What happens now?
It will compress in the middle. If the original length and breadth of the rubber are taken as L and B respectively, then when pulled longitudinally, it tends to get compressed laterally. In simple words, length has increased by an amount dL and the breadth has increased by an amount dB.
In this case,
$$\begin{array}{l}\varepsilon _{t}=-\frac{dB}{B}\end{array}$$
$$\begin{array}{l}\varepsilon _{l}=-\frac{dL}{L}\end{array}$$
The formula for Poisson’s ratio is,
$$\begin{array}{l}Poisson’s\;ratio=\frac{Transverse\;strain}{Longitudinal\;strain}\end{array}$$
$$\begin{array}{l}\Rightarrow \nu =-\frac{\varepsilon _{t}}{\varepsilon _{l}}\end{array}$$
where,
εt is the Lateral or Transverse Strain
εl is the Longitudinal or Axial Strain
$$\begin{array}{l}\nu \end{array}$$
is the Poisson’s Ratio
The strain on its own is defined as the change in dimension (length, breadth, area…) divided by the original dimension.
### Poisson’s Effect
When a material is stretched in one direction, it tends to compress in the direction perpendicular to that of force application and vice versa. The measure of this phenomenon is given in terms of Poisson’s ratio. For example, a rubber band tends to become thinner when stretched.
## Poisson’s ratio values for different material
It is the ratio of transverse contraction strain to longitudinal extension strain, in the direction of the stretching force. There can be a stress and strain relation that is generated with the application of force on a body.
• For tensile deformation, Poisson’s ratio is positive.
• For compressive deformation, it is negative.
Here, the negative Poisson ratio suggests that the material will exhibit a positive strain in the transverse direction, even though the longitudinal strain is positive as well.
For most materials, the value of Poisson’s ratio lies in the range, 0 to 0.5.
A few examples of poisson’s ratio are given below for different materials.
Material Values Concrete 0.1 – 0.2 Cast iron 0.21 – 0.26 Steel 0.27 – 0.30 Rubber 0.4999 Gold 0.42 – 0.44 Glass 0.18 – 0.3 Cork 0.0 Copper 0.33 Clay 0.30 – 0.45 Stainless steel 0.30 – 0.31 Foam 0.10 – 0.50
Watch the video below to understand the Poisson’s ratio in detail.
Physics Related Topics:
## Frequently Asked Questions – FAQs
### Does Poisson’s ratio depend on temperature?
In general, Colder temperature decreases both strains and high-temperature increases both horizontal and vertical strain. Thus, the net effect on Poisson’s Ratio is small since the change in both horizontal and vertical strain is by a similar amount.
### Is Poisson’s ratio constant?
Poisson’s ratio for material remains approximately constant within elastic limits.
### Define Poisson’s ratio.
The ratio of transverse strain to longitudinal strain in the direction of the stretching force.
### Write the Poisson’s ratio formula.
$$\begin{array}{l}Poisson’s\;ratio=\frac{Transverse\;strain}{Longitudinal\;strain}\end{array}$$
### State true or False: Poisson’s ratio is negative for Tensile deformation.
False. Poisson’s ratio is Positive for Tensile deformation
### What is Poisson’s ratio of concrete?
The Poisson’s ratio of concrete is 0.1 to 0.2.
### What does the Poisson’s ratio 0.5 mean?
Poisson’s ratio 0.5 means a perfectly in-compressible material is deformed elastically at small strains.
### What are the units of Poisson’s ratio?
Poisson’s ratio is the unitless scalar quantity.
### What is Poisson’s ratio of cork?
Poisson’s ratio of cork is 0.0.
Hope you have understood Poisson’s ratio, how it is defined, its symbol, units, formula, terms and values for various materials. Stay tuned with BYJU’S for more such interesting articles. Also, register to “BYJU’S-The Learning App” for loads of interactive, engaging physics-related videos and an unlimited academic assist.
Test your Knowledge on Poissons Ratio
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# Choose $n$ out of $2n-1$ boxes containing at least half of all white balls and half of all black balls
We are given $$2n - 1$$ boxes with a total of $$B$$ black and $$W$$ white balls. In the $$i$$-th, box there are $$w_i$$ white and $$b_i$$ black balls. It is required to choose $$n$$ boxes so that, the sum of the white balls is at least $$W/2$$, and the sum of black balls is at least $$B/2$$. Solve for $$O(n\log{n})$$.
What I am currently thinking is this:
Generate some hashtable to be able to track the boxes that we chose.
Sort the boxes by the quantity of the white (in increasing order) balls and chose the last $$n$$ balls every time keeping them in our hashtable.
Sort by the quantity of the black balls and do the same. Check every time to see if we already chose the box or not. Here comes the problem: Suppose we didn't. Then we can face a situation where we already have $$n$$ boxes that have in total at least $$W/2$$ white balls but at the same time, they have in total less than $$B/2$$ black balls. How can we overcome this problem?
We can't just switch the chosen boxes that have the least number of white balls with the one that have the maximum available number of black balls since the box with white balls can contain significant amount of black balls on its own.
• maybe sorting them by the total number of balls could prove useful? – nir shahar Jun 3 '20 at 7:35
• Can you show that there always exists a solution? That would be a good start. – Yuval Filmus Jun 3 '20 at 7:48
• You haven't specified what $W$ and $B$ are. – Yuval Filmus Jun 3 '20 at 7:49
Let $$B^*, B_1, B_2, \dots, B_{2n-2}$$ be the the bins sorted by the number of white balls, in non-increasing order (break ties arbitrarily). Notice that the first bin is called $$B^*$$.
Consider the two groups $$G_E = \{ B^* \} \cup \{B_i \mid i \text{ is even} \}$$ and $$G_O = \{ B^* \} \cup \{B_i \mid i \text{ is odd} \}$$. Clearly $$|G_E| = |G_O| = 1 + \frac{2n-2}{2} = n$$.
Moreover, each of $$G_E$$ and $$G_O$$ contains at least $$W/2$$ white balls. This is true since since for each bin not in $$G_E$$ (resp. $$G_O$$) the previous bin in the sorted order must be in $$G_E$$ (resp. $$G_O$$). That is, the total number of white balls in $$G_E$$ (resp. $$G_O$$) is at least as large as the number of white balls not in $$G_E$$ (resp. $$G_O$$).
Now, total the number of black balls in at least one group $$G^* \in \{G_E, G_O\}$$ must be at least $$B/2$$. This is easy to see since $$\{G_E, G_O \setminus \{B^*\}\}$$ is a partition of the set of bins, and hence the bins in at least one set of the partition (which is either $$G_E$$ or a subset of $$G_O$$) must contain at least half the black balls.
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# Variable voltage/frequency power supply.
#### arthur92710
Joined Jun 25, 2007
307
I want to add a Variable voltage/frequency option to my power supply.
I read that a Multivibrater can do this. But I cant find a datasheet that explains the pins. I have a CD74HC221EE4 , SN74LV4046ANE4 (cmos logic w/voc) , HCF4098BEY also i have a UC252AN its a pulse width mod?.
Which would be the best for a Variable voltage/frequency power supply.
#### beenthere
Joined Apr 20, 2004
15,819
You might want to expand on the variable nature of the PS.
Most power supplies have the ability to use 50/60 Hz AC inputs, at either 115 VAC or 230 VAC.
Most power supplies output DC, so the variable frequency doesn't apply.
If it's a switcher, the frequency is governed by the primary transformer. Running a frequency above of below the requirement will lead to inefficiency, resulting in wasted power as heat and poor regulation of the outputs.
#### arthur92710
Joined Jun 25, 2007
307
No... what i meant was :
I have a power supply (450w pc power supply in a custom case) I works with both 50/60hz and 110/220. But it only outputs fixed dc voltages. (+12 -12 +5 -5 +3.3 0) But I also want it to output a variable(frequency and voltage) square, triangle and sine wave.
We have one at school. It has 3 outputs.(square, sine, and triangle)
then it has 2 adjustable knobs. One for the multiplier(1-10,10-100,100-1k...) and one for fine adjustment.
I cant figure out how it works but i want one for my psu.
#### SgtWookie
Joined Jul 17, 2007
22,221
mrmeval guessed what you wanted first Good kit he pointed you to, as well.
I suggest that you build that kit as-is, and not try to add it to your PSU power supply.
The signal generator is an analog device; your PSU is a "switched" power supply designed to power digital devices. Switching-type power supplies are very efficient, but they are electrically "noisy", and so are not really suitable for powering analog devices. If you DID try to add it to your PSU, you would find that the output signals were not very stable in either frequency or shape.
Once you completed the kit as it was designed and have it working, you might power it from a linear 9v power supply. You could build one using a small 12v "wall wart", a 7809 voltage regulator and a couple of capacitors. Remove the batteries before using the "wall wart" powered supply.
#### arthur92710
Joined Jun 25, 2007
307
Does any one have a circuit for that? I would buy it but the shipping cost is extremely high. $11 for ground shipping from California to New York. When i bought a Pc Case (22lb) it cost me 12$ and it was 3 day shipping not ground. Also from California.
#### Ron H
Joined Apr 14, 2005
7,014
Google "function generator". You can find schematics, kits, and commercial units.
#### SgtWookie
Joined Jul 17, 2007
22,221
Try this search, too:
The first one that pops up is this one:
www.aldinc.com/pdf/wf_47003.0.pdf
Of course they specify several models of op amps that they make, but there is no reason you couldn't adapt that schematic to using LM741's, LM1458's or LM324's (single, dual and quad op amps respectively) - you'd just have to make adjustments per the individual op amp's specified voltage ranges.
You'd probably be better off to stick with a pair of LM1458's. You could pick up a couple of those and a general purpose IC PC board (catalog # 276-150A) at your local Radio Shack.
#### nanovate
Joined May 7, 2007
666
You could also look at building something around the XR2206 from EXAR if you can find them.
#### SgtWookie
Joined Jul 17, 2007
22,221
I found this one http://www.edn.com/contents/images/236418f1.pdf
I tryed to build the first part with the 74ls123 but it only outputted a 5.5v signal.
That's because you're working with TTL-level logic at that point. TTL needs to be supplied between 4.75 and 5.25 volts, or it will have a very short life span.
You could use a general-purpose op amp to amplify it's output.
But then you'll basically just have a square- or rectangular-wave generator.
How much output do you need?
If you wish to have the capability of generating sine and/or triangle waves, you need to keep looking.
You could even build a signal generator out of some 555 or 556 timer chips. It'll be for low frequencies, but it can be made to work.
#### arthur92710
Joined Jun 25, 2007
307
I just need a simple square triangle and sine wave signal generator. with variable frequencies.
#### Ron H
Joined Apr 14, 2005
7,014
I just need a simple square triangle and sine wave signal generator. with variable frequencies.
Google ICL8038. It's a function generator chip designed to do exactly what you want. You can find the datasheet, an application note, and several DIY projects.
#### arthur92710
Joined Jun 25, 2007
307
I looked at the 8038 at both maxim and intersil. (for a sample) but they are both inactive.
I looked at the data sheet and i see that it is a good part for what i want.
they sent me an email
MAXIM IC said:
This part was discontinued March 2006. We can no longer quote or accept orders.
At this time there is no replacement.
Thank you
Maxim Direct! -JF
#### Ron H
Joined Apr 14, 2005
7,014
You can probably get the 8038 at an online hobbyist or surplus shop, but you might also look at XR2206. Jameco sells them, and there are probably other sources as well.
#### thingmaker3
Joined May 16, 2005
5,084
There's on on eBay.
#### arthur92710
Joined Jun 25, 2007
307
So there all of these are discontinued -.-
Why did they do that?
Ill try to get one on ebay of somewhere.
cool the ICL8038 is $20 on ebay +5 for shipping. the XR2206 is$7 +5 for shipping
#### beenthere
Joined Apr 20, 2004
15,819
You can go to Mouser and get an NTE864. It is an exact replacement for the ICL8038. It's under $5.00. #### SgtWookie Joined Jul 17, 2007 22,221 Or you COULD just do it the old-fashioned way - and build it out of a few op-amps, using schematics that've already been posted. The trouble with using obsolete IC's is that they'll keep getting harder (read: more expensive) to obtain - so if you go to the effort of building something, and your precious IC gets fried, you're going to have to either throw it out and start over, or do the painful search/pay-through-the-nose-and-wait-for-it game. Thread Starter #### arthur92710 Joined Jun 25, 2007 307 but why did they stop making them? the NTE864 at mouser is$30? not 5.
Sgt. you got me... i found some op amps lm224n (quad) and a comparator lm319n (dual) can i use the LM224N?
i cant figure out if i have to connect to the inverting or non inverting part of the op amp? also how do i get -2.5 from a 9v?
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# On the mass of the Higgs Boson
Thomson, in Modern Particle Physics, chapter 17, says
The SM Higgs boson H is a neutral scalar particle. Its mass is a free parameter of the SM that is given by $m_H=2\lambda v^2.$
In the next section he adds
Prior to the turn-on of the LHC at CERN, the window for the SM Higgs was relatively narrow.The absence of a signal from direct searches at LEP implied that $m_H>114GeV$. At the same time, the limits on the size of the quantum loop corrections from the precision electroweak measurements at LEP and Tevatron suggested that $m_H <\approx 150 GeV$ and that $m_H$ was unlikely to be greater than 200GeV.
I would like some help with
1. Why do people sometimes refer to the mass of the Higgs boson and sometimes to the mass parameter of the Higgs boson?
2. Why is it a free parameter?
3. If I get it write the mass of the Higgs boson can be calculated by the equation above, right? Then why did the experimentalists had to narrow the possibilities of where (in the "mass spectrum") they would find it instead of going straight to 125GeV?
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# zbMATH — the first resource for mathematics
## Li, Shi
Compute Distance To:
Author ID: li.shi Published as: Li, S.; Li, Sh.; Li, Shi
Documents Indexed: 90 Publications since 1992
all top 5
#### Co-Authors
9 single-authored 4 Xiang, Zhengrong 3 Chuzhoy, Julia 3 Im, Sungjin 3 Krishnaswamy, Ravishankar 3 Luo, Dang 3 Moseley, Benjamin 3 Xu, Jinhui 2 Cai, Yunpeng 2 Chakrabarty, Deeparnab 2 Charikar, Moses S. 2 Chen, Jiang 2 Hajiaghayi, Mohammad Taghi 2 Hu, Wei 2 Kulkarni, Janardhan 2 Li, Hong 2 Li, Jian 2 Li, Sujian 2 Saha, Barna 2 Svensson, Ola 2 Yao, Jinyi 2 Ye, Minwei 1 Ahmed, Shabbir 1 Ahn, Choon Ki 1 Bansal, Nikhil 1 Batterman, Stuart 1 Chalermsook, Parinya 1 Chattopadhyay, Arkadev 1 Chekuri, Chandra S. 1 Chen, Guang 1 Demirci, Gökalp 1 Duan, Guangren 1 Ene, Alina 1 Feige, Uriel 1 Fu, Yanming 1 Garg, Shashwat 1 Ghosh, Malay 1 Grandoni, Fabrizio 1 Guo, Xiangyu 1 Harris, David G. 1 He, Qie 1 He, Wenmin 1 Jia, Peifa 1 Khanna, Sanjeev 1 Kim, David Hong Kyun 1 Laekhanukit, Bundit 1 Langberg, Michael 1 Li, Ran 1 Li, Yanling 1 Luo, Chengxin 1 Mao, Yong 1 Mukherjee, Bhramar 1 Narayanan, Srivatsan 1 Nemhauser, George L. 1 Pensyl, Thomas W. 1 Qiu, Zhen 1 Ren, Jirong 1 Rudra, Atri 1 Sandeep, Sai 1 Srinivasan, Aravind 1 Torng, Eric K. 1 Trinh, Khoa 1 Wang, Di 1 Wang, Jiaxin 1 Wang, Yigang 1 Wang, Ying 1 Yang, Wenbin 1 Yang, Zehong 1 Zhang, Jing 1 Zhong, Xueling
all top 5
#### Serials
2 SIAM Journal on Computing 2 ACM Transactions on Algorithms 1 Classical and Quantum Gravity 1 Journal of the Franklin Institute 1 Applied Mathematics and Computation 1 Biometrics 1 Fuzzy Sets and Systems 1 Mathematics of Operations Research 1 Mathematics in Practice and Theory 1 Algorithmica 1 Information and Computation 1 SIAM Journal on Discrete Mathematics 1 Journal of Tsinghua University. Science and Technology 1 Machine Learning 1 International Journal of Robust and Nonlinear Control 1 SIAM Journal on Optimization 1 Mathematical Problems in Engineering 1 Journal of the ACM 1 Discrete Dynamics in Nature and Society 1 Journal of Shaanxi Normal University. Natural Science Edition 1 Journal of Machine Learning Research (JMLR) 1 Control and Decision 1 Journal of Shenyang Normal University. Natural Science Edition 1 Pacific Journal of Optimization 1 Journal of Inner Mongolia Normal University. Natural Science Edition 1 Theory of Computing 1 International Journal of Systems Science. Principles and Applications of Systems and Integration
all top 5
#### Fields
36 Computer science (68-XX) 31 Operations research, mathematical programming (90-XX) 5 Combinatorics (05-XX) 5 Systems theory; control (93-XX) 3 Statistics (62-XX) 2 Mathematical logic and foundations (03-XX) 1 Partial differential equations (35-XX) 1 Numerical analysis (65-XX) 1 Mechanics of particles and systems (70-XX) 1 Classical thermodynamics, heat transfer (80-XX) 1 Quantum theory (81-XX) 1 Relativity and gravitational theory (83-XX) 1 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 1 Biology and other natural sciences (92-XX)
#### Citations contained in zbMATH
51 Publications have been cited 403 times in 366 Documents Cited by Year
A 1.488 approximation algorithm for the uncapacitated facility location problem. Zbl 1281.68236
Li, Shi
2013
Symmetric weak-form integral equation method for three-dimensional fracture analysis. Zbl 0906.73074
Li, S.; Mear, M. E.; Xiao, L.
1998
A numerical method for singularly perturbed turning point problems with an interior layer. Zbl 1291.65231
Geng, F. Z.; Qian, S. P.; Li, S.
2014
Finite-time stability of cascaded time-varying systems. Zbl 1117.93004
Li, S.; Tian, Y.-P.
2007
A 1.488 approximation algorithm for the uncapacitated facility location problem. Zbl 1334.68301
Li, Shi
2011
Approximating $$k$$-median via pseudo-approximation. Zbl 1293.90061
Li, Shi; Svensson, Ola
2013
Numerical manifold method based on the method of weighted residuals. Zbl 1109.74373
Li, S.; Cheng, Y.; Wu, Y.-F.
2005
On $$(1,\varepsilon)$$-restricted assignment makespan minimization. Zbl 1372.68044
Chakrabarty, Deeparnab; Khanna, Sanjeev; Li, Shi
2015
Approximating $$k$$-median via pseudo-approximation. Zbl 1338.90346
Li, Shi; Svensson, Ola
2016
On saturation-strip model of a permeable crack in a piezoelectric ceramic. Zbl 1064.74155
Li, S.
2003
A dependent LP-rounding approach for the $$k$$-median problem. Zbl 1272.90020
Charikar, Moses; Li, Shi
2012
The effects of shear on delamination in layered materials. Zbl 1045.74595
Li, S.; Wang, J.; Thouless, M. D.
2004
A two-stage hybrid flowshop with uniform machines and setup times. Zbl 1185.90078
Huang, W.; Li, S.
1998
Li, S.; Miskioglu, I.; Altan, B. S.
2004
Error of partitioned Runge-Kutta methods for multiple stiff singular perturbation problems. Zbl 0956.65057
Xiao, A.; Li, S.
2000
Lie-Poisson integration for rigid body dynamics. Zbl 0834.70006
Li, S.; Qin, Mengzhao
1995
Kronrod extension of Turán formula. Zbl 0724.65015
Li, S.
1994
Adaptive prescribed performance control for switched nonlinear systems with input saturation. Zbl 1385.93040
Li, Shi; Xiang, Zhengrong
2018
A polylogarithmic approximation algorithm for edge-disjoint paths with congestion 2. Zbl 1426.68302
Chuzhoy, Julia; Li, Shi
2016
Improved approximation for node-disjoint paths in planar graphs. Zbl 1376.68170
Chuzhoy, Julia; Kim, David H. K.; Li, Shi
2016
On uniform capacitated $$k$$-median beyond the natural LP relaxation. Zbl 1371.90077
Li, Shi
2015
Capacitated network design on undirected graphs. Zbl 1405.68439
Chakrabarty, Deeparnab; Krishnaswamy, Ravishankar; Li, Shi; Narayanan, Srivatsan
2013
Approximation of the solution and its derivative for the singularly perturbed Black-Scholes equation with nonsmooth initial data. Zbl 1210.91134
Li, S.; Shishkin, G. I.; Shishkina, L. P.
2007
Estimation of coefficients in a hyperbolic equation with impulsive inputs. Zbl 1111.35123
Li, S.
2006
On global energy release rate of a permeable crack in a piezoelectric ceramic. Zbl 1110.74554
Li, S.
2003
On the micromechanics theory of Reissner-Mindlin plates. Zbl 0993.74034
Li, S.
2000
Determination of domains of attraction based on a sequence of maxima. Zbl 0884.62051
Wang, J. Z.; Cooke, P.; Li, S.
1996
On uniform capacitated $$k$$-median beyond the natural LP relaxation. Zbl 06972815
Li, Shi
2017
A constant factor approximation algorithm for fault-tolerant $$k$$-median. Zbl 1421.68228
Hajiaghayi, Mohammadtaghi; Hu, Wei; Li, Jian; Li, Shi; Saha, Barna
2016
Approximating capacitated $$k$$-median with $$(1 + \epsilon)k$$ open facilities. Zbl 1418.90141
Li, Shi
2016
Better algorithms and hardness for broadcast scheduling via a discrepancy approach. Zbl 1422.68288
Bansal, Nikhil; Charikar, Moses; Krishnaswamy, Ravishankar; Li, Shi
2014
Hawking radiation of the fermionic field and anomaly in $$(2+1)$$-dimensional black holes. Zbl 1195.83059
Li, Ran; Li, Shi; Ren, Ji-Rong
2010
A polychromatic sets approach to the conceptual design of machine tools. Zbl 1068.90076
Xu, L.; Li, Z.; Li, S.; Tang, F.
2005
A polynomial time constant approximation for minimizing total weighted flow-time. Zbl 1431.68153
Feige, Uriel; Kulkarni, Janardhan; Li, Shi
2019
On the computational complexity of minimum-concave-cost flow in a two-dimensional grid. Zbl 1349.90153
Ahmed, Shabbir; He, Qie; Li, Shi; Nemhauser, George L.
2016
A dynamic programming framework for non-preemptive scheduling problems on multiple machines. Zbl 1371.90056
Im, Sungjin; Li, Shi; Moseley, Benjamin; Torng, Eric
2015
A constant factor approximation algorithm for fault-tolerant $$k$$-median. Zbl 1421.68229
Hajiaghayi, Mohammadtaghi; Hu, Wei; Li, Jian; Li, Shi; Saha, Barna
2014
Prediction of mechanical gear mesh efficiency of hypoid gear pairs. Zbl 1203.70014
Kolivand, M.; Li, S.; Kahraman, A.
2010
Essential norm of weighted composition operators from analytic Besov spaces into Zygmund type spaces. Zbl 1426.30041
Hu, Q.; Li, S.; Zhang, Y.
2019
A stable node-based smoothed finite element method for metal forming analysis. Zbl 07053715
Yang, H.; Cui, X. Y.; Li, S.; Bie, Y. H.
2019
Constant approximation for $$k$$-median and $$k$$-means with outliers via iterative rounding. Zbl 1428.68393
Krishnaswamy, Ravishankar; Li, Shi; Sandeep, Sai
2018
Extreme equitable block colorings of $$C_4$$-decompositions of $$K_v-F$$. Zbl 1404.05055
Li, S.; Matson, E. B.; Rodger, C. A.
2018
Constant approximation algorithm for non-uniform capacitated multi-item lot-sizing via strong covering inequalities. Zbl 1418.90018
Li, Shi
2017
Simulation on the interaction between multiple bubbles and free surface with viscous effects. Zbl 1403.76089
Li, S.; Ni, Bao-yu
2016
Constant approximation for capacitated $$k$$-median with $$(1+\epsilon)$$-capacity violation. Zbl 1388.68306
Demirci, Gökalp; Li, Shi
2016
Construct the stable vendor managed inventory partnership through a profit-sharing approach. Zbl 1317.90027
Li, S.; Yu, Z.; Dong, M.
2015
Approximation algorithms and hardness of integral concurrent flow. Zbl 1286.05060
Chalermsook, Parinya; Chuzhoy, Julia; Ene, Alina; Li, Shi
2012
Architecture of TsinghuAeolus. Zbl 1050.68958
Yao, Jinyi; Chen, Jiang; Cai, Yunpeng; Li, Shi
2002
Estimation for type II domain of attraction based on the $$W$$ statistic. Zbl 0962.62043
Hasofer, A. M.; Li, S.
1999
Tool condition monitoring in machining by fuzzy neural networks. Zbl 0875.93387
Li, S.; Elbestawi, M. A.
1996
$$h$$- and $$p$$-adaptive boundary element methods. Zbl 0770.65076
Ye, T. Q.; Zhang, Daming; Li, S.; Cheng, J. L.
1992
A polynomial time constant approximation for minimizing total weighted flow-time. Zbl 1431.68153
Feige, Uriel; Kulkarni, Janardhan; Li, Shi
2019
Essential norm of weighted composition operators from analytic Besov spaces into Zygmund type spaces. Zbl 1426.30041
Hu, Q.; Li, S.; Zhang, Y.
2019
A stable node-based smoothed finite element method for metal forming analysis. Zbl 07053715
Yang, H.; Cui, X. Y.; Li, S.; Bie, Y. H.
2019
Adaptive prescribed performance control for switched nonlinear systems with input saturation. Zbl 1385.93040
Li, Shi; Xiang, Zhengrong
2018
Constant approximation for $$k$$-median and $$k$$-means with outliers via iterative rounding. Zbl 1428.68393
Krishnaswamy, Ravishankar; Li, Shi; Sandeep, Sai
2018
Extreme equitable block colorings of $$C_4$$-decompositions of $$K_v-F$$. Zbl 1404.05055
Li, S.; Matson, E. B.; Rodger, C. A.
2018
On uniform capacitated $$k$$-median beyond the natural LP relaxation. Zbl 06972815
Li, Shi
2017
Constant approximation algorithm for non-uniform capacitated multi-item lot-sizing via strong covering inequalities. Zbl 1418.90018
Li, Shi
2017
Approximating $$k$$-median via pseudo-approximation. Zbl 1338.90346
Li, Shi; Svensson, Ola
2016
A polylogarithmic approximation algorithm for edge-disjoint paths with congestion 2. Zbl 1426.68302
Chuzhoy, Julia; Li, Shi
2016
Improved approximation for node-disjoint paths in planar graphs. Zbl 1376.68170
Chuzhoy, Julia; Kim, David H. K.; Li, Shi
2016
A constant factor approximation algorithm for fault-tolerant $$k$$-median. Zbl 1421.68228
Hajiaghayi, Mohammadtaghi; Hu, Wei; Li, Jian; Li, Shi; Saha, Barna
2016
Approximating capacitated $$k$$-median with $$(1 + \epsilon)k$$ open facilities. Zbl 1418.90141
Li, Shi
2016
On the computational complexity of minimum-concave-cost flow in a two-dimensional grid. Zbl 1349.90153
Ahmed, Shabbir; He, Qie; Li, Shi; Nemhauser, George L.
2016
Simulation on the interaction between multiple bubbles and free surface with viscous effects. Zbl 1403.76089
Li, S.; Ni, Bao-yu
2016
Constant approximation for capacitated $$k$$-median with $$(1+\epsilon)$$-capacity violation. Zbl 1388.68306
Demirci, Gökalp; Li, Shi
2016
On $$(1,\varepsilon)$$-restricted assignment makespan minimization. Zbl 1372.68044
Chakrabarty, Deeparnab; Khanna, Sanjeev; Li, Shi
2015
On uniform capacitated $$k$$-median beyond the natural LP relaxation. Zbl 1371.90077
Li, Shi
2015
A dynamic programming framework for non-preemptive scheduling problems on multiple machines. Zbl 1371.90056
Im, Sungjin; Li, Shi; Moseley, Benjamin; Torng, Eric
2015
Construct the stable vendor managed inventory partnership through a profit-sharing approach. Zbl 1317.90027
Li, S.; Yu, Z.; Dong, M.
2015
A numerical method for singularly perturbed turning point problems with an interior layer. Zbl 1291.65231
Geng, F. Z.; Qian, S. P.; Li, S.
2014
Better algorithms and hardness for broadcast scheduling via a discrepancy approach. Zbl 1422.68288
Bansal, Nikhil; Charikar, Moses; Krishnaswamy, Ravishankar; Li, Shi
2014
A constant factor approximation algorithm for fault-tolerant $$k$$-median. Zbl 1421.68229
Hajiaghayi, Mohammadtaghi; Hu, Wei; Li, Jian; Li, Shi; Saha, Barna
2014
A 1.488 approximation algorithm for the uncapacitated facility location problem. Zbl 1281.68236
Li, Shi
2013
Approximating $$k$$-median via pseudo-approximation. Zbl 1293.90061
Li, Shi; Svensson, Ola
2013
Capacitated network design on undirected graphs. Zbl 1405.68439
Chakrabarty, Deeparnab; Krishnaswamy, Ravishankar; Li, Shi; Narayanan, Srivatsan
2013
A dependent LP-rounding approach for the $$k$$-median problem. Zbl 1272.90020
Charikar, Moses; Li, Shi
2012
Approximation algorithms and hardness of integral concurrent flow. Zbl 1286.05060
Chalermsook, Parinya; Chuzhoy, Julia; Ene, Alina; Li, Shi
2012
A 1.488 approximation algorithm for the uncapacitated facility location problem. Zbl 1334.68301
Li, Shi
2011
Hawking radiation of the fermionic field and anomaly in $$(2+1)$$-dimensional black holes. Zbl 1195.83059
Li, Ran; Li, Shi; Ren, Ji-Rong
2010
Prediction of mechanical gear mesh efficiency of hypoid gear pairs. Zbl 1203.70014
Kolivand, M.; Li, S.; Kahraman, A.
2010
Finite-time stability of cascaded time-varying systems. Zbl 1117.93004
Li, S.; Tian, Y.-P.
2007
Approximation of the solution and its derivative for the singularly perturbed Black-Scholes equation with nonsmooth initial data. Zbl 1210.91134
Li, S.; Shishkin, G. I.; Shishkina, L. P.
2007
Estimation of coefficients in a hyperbolic equation with impulsive inputs. Zbl 1111.35123
Li, S.
2006
Numerical manifold method based on the method of weighted residuals. Zbl 1109.74373
Li, S.; Cheng, Y.; Wu, Y.-F.
2005
A polychromatic sets approach to the conceptual design of machine tools. Zbl 1068.90076
Xu, L.; Li, Z.; Li, S.; Tang, F.
2005
The effects of shear on delamination in layered materials. Zbl 1045.74595
Li, S.; Wang, J.; Thouless, M. D.
2004
Li, S.; Miskioglu, I.; Altan, B. S.
2004
On saturation-strip model of a permeable crack in a piezoelectric ceramic. Zbl 1064.74155
Li, S.
2003
On global energy release rate of a permeable crack in a piezoelectric ceramic. Zbl 1110.74554
Li, S.
2003
Architecture of TsinghuAeolus. Zbl 1050.68958
Yao, Jinyi; Chen, Jiang; Cai, Yunpeng; Li, Shi
2002
Error of partitioned Runge-Kutta methods for multiple stiff singular perturbation problems. Zbl 0956.65057
Xiao, A.; Li, S.
2000
On the micromechanics theory of Reissner-Mindlin plates. Zbl 0993.74034
Li, S.
2000
Estimation for type II domain of attraction based on the $$W$$ statistic. Zbl 0962.62043
Hasofer, A. M.; Li, S.
1999
Symmetric weak-form integral equation method for three-dimensional fracture analysis. Zbl 0906.73074
Li, S.; Mear, M. E.; Xiao, L.
1998
A two-stage hybrid flowshop with uniform machines and setup times. Zbl 1185.90078
Huang, W.; Li, S.
1998
Determination of domains of attraction based on a sequence of maxima. Zbl 0884.62051
Wang, J. Z.; Cooke, P.; Li, S.
1996
Tool condition monitoring in machining by fuzzy neural networks. Zbl 0875.93387
Li, S.; Elbestawi, M. A.
1996
Lie-Poisson integration for rigid body dynamics. Zbl 0834.70006
Li, S.; Qin, Mengzhao
1995
Kronrod extension of Turán formula. Zbl 0724.65015
Li, S.
1994
$$h$$- and $$p$$-adaptive boundary element methods. Zbl 0770.65076
Ye, T. Q.; Zhang, Daming; Li, S.; Cheng, J. L.
1992
all top 5
#### Cited by 707 Authors
36 Xu, Dachuan 18 Du, Donglei 18 Wu, Chenchen 14 Arqub, Omar Abu 14 Li, Shihua 10 Rungamornrat, Jaroon 9 Mear, Mark E. 9 Zhang, Dongmei 7 Al-Smadi, Mohammed H. 6 Byrka, Jarosław 6 Ding, Shihong 6 Wang, Yishui 5 Du, Haibo 5 Li, Gaidi 5 Li, Yu 5 Rybicki, Bartosz 5 Solis-Oba, Roberto 5 Svensson, Ola 4 Abbasbandy, Saeid 4 Bonnet, Marc 4 Duarte, C. A. M. 4 Jansen, Klaus 4 Lin, Xiangze 4 Page, Daniel R. 4 Pedrosa, Lehilton L. C. 4 Salavatipour, Mohammad R. 4 Tran, Han D. 4 Xiu, Naihua 3 An, Hyung-Chan 3 An, Xinmei 3 Behsaz, Babak 3 Chazallon, Cyrille 3 Chekuri, Chandra S. 3 Chen, Xiaohong 3 Fang, Xing 3 Friggstad, Zachary 3 Geng, Fazhan 3 Gray, Leonard J. 3 Im, Sungjin 3 Krishnaswamy, Ravishankar 3 Lapusta, Y. M. 3 Li, Shi 3 Li, Xiuying 3 Liu, Fei 3 Loboda, Volodymyr V. 3 Ma, Guowei 3 Maayah, Banan 3 Momani, Shaher M. 3 Mouhoubi, Saïda 3 Nagarajan, Viswanath 3 Nguyen, Bao-Hoang 3 Ou, Meiying 3 Spalević, Miodrag M. 3 Spoerhase, Joachim 3 Srinivasan, Aravind 3 Wang, Chaoli 3 Williamson, David P. 3 Wu, Boying 3 Xiao, Aiguo 3 Xu, Yicheng 3 Zhang, Zhenning 3 Zhuang, Xiaoying 2 Aardal, Karen I. 2 Babolian, Esmail 2 Bakhtiari, Parisa 2 Bley, Andreas 2 Boyanov, Borislav Dechev 2 Chakrabarty, Deeparnab 2 Chen, Ting-Yu 2 Chen, Zengtao 2 Cheng, Yung-Ming 2 Cheung, Sin-Shuen 2 Chuzhoy, Julia 2 Dang, HuaYang 2 Ding, Hu 2 Elmachtoub, Adam N. 2 Fan, CuiYing 2 Fraga Alves, M. Isabel 2 Gan, Siqing 2 Gao, Fangzheng 2 Han, Lu 2 Hayat, Tasawar 2 Hu, Keqiang 2 Ji, Sai 2 Jiang, Yanjun 2 Karasözen, Bülent 2 Khuller, Samir 2 Kovalyov, Mikhail Yakovlevich 2 Kumar, Amit 2 Lee, Orlando 2 Lei, Jing 2 Levi, Retsef 2 Li, Haixia 2 Li, Min 2 Li, Qi 2 Li, Shanfei 2 Li, Shumin 2 Liu, Li 2 Maack, Marten 2 Meira, Luís A. A. ...and 607 more Authors
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#### Cited in 118 Serials
22 Engineering Analysis with Boundary Elements 16 SIAM Journal on Computing 15 Applied Mathematics and Computation 13 Journal of Computational and Applied Mathematics 13 Algorithmica 11 Computer Methods in Applied Mechanics and Engineering 11 Theoretical Computer Science 10 Journal of Combinatorial Optimization 9 Mathematical Problems in Engineering 8 International Journal of Control 8 Journal of the Franklin Institute 7 Acta Mechanica 7 International Journal of Solids and Structures 7 International Journal for Numerical Methods in Engineering 6 Applied Mathematical Modelling 6 Mathematical Programming. Series A. Series B 6 International Journal of Fracture 5 Operations Research Letters 5 European Journal of Operational Research 4 Computers & Mathematics with Applications 4 Mathematics of Operations Research 4 Computers & Operations Research 4 Applied Mathematics Letters 4 Nonlinear Dynamics 4 European Journal of Mechanics. A. Solids 4 Asian Journal of Control 3 Discrete Applied Mathematics 3 Chaos, Solitons and Fractals 3 Automatica 3 Applied Numerical Mathematics 3 Acta Mathematicae Applicatae Sinica. English Series 3 Computational Mechanics 3 SIAM Journal on Discrete Mathematics 3 Journal of Global Optimization 3 International Journal of Robust and Nonlinear Control 3 Abstract and Applied Analysis 3 Soft Computing 3 Optimization Letters 2 International Journal of Engineering Science 2 Information Processing Letters 2 Journal of the Mechanics and Physics of Solids 2 ZAMP. Zeitschrift für angewandte Mathematik und Physik 2 Calcolo 2 Information Sciences 2 Kybernetika 2 Meccanica 2 Operations Research 2 Applied Mathematics and Mechanics. (English Edition) 2 Optimization 2 INFORMS Journal on Computing 2 Theory of Computing Systems 2 Science China. Mathematics 2 Journal of the Operations Research Society of China 1 International Journal of Theoretical Physics 1 Journal of Mathematical Analysis and Applications 1 Mathematical Methods in the Applied Sciences 1 Metrika 1 Wave Motion 1 Mathematics of Computation 1 The Annals of Statistics 1 BIT 1 Journal of the American Statistical Association 1 Journal of Computer and System Sciences 1 Journal of Optimization Theory and Applications 1 Journal of Statistical Planning and Inference 1 Mathematics and Computers in Simulation 1 Mechanics Research Communications 1 Optimal Control Applications & Methods 1 Circuits, Systems, and Signal Processing 1 International Journal of Production Research 1 Discrete & Computational Geometry 1 Information and Computation 1 Asia-Pacific Journal of Operational Research 1 Mathematical and Computer Modelling 1 International Journal of Foundations of Computer Science 1 Numerical Algorithms 1 International Journal of Computer Mathematics 1 Journal of Elasticity 1 SIAM Journal on Applied Mathematics 1 Distributed Computing 1 Archive of Applied Mechanics 1 Computational Optimization and Applications 1 Test 1 International Journal of Modern Physics D 1 Applied Mathematics. Series B (English Edition) 1 Journal of Inverse and Ill-Posed Problems 1 Journal of Multi-Criteria Decision Analysis 1 Complexity 1 Journal of Difference Equations and Applications 1 Mathematics and Mechanics of Solids 1 Neural Computing and Applications 1 European Journal of Mechanics. B. Fluids 1 Extremes 1 Computational Geosciences 1 Journal of High Energy Physics 1 Optimization and Engineering 1 Lobachevskii Journal of Mathematics 1 Qualitative Theory of Dynamical Systems 1 Journal of Systems Science and Complexity 1 Journal of Applied Mathematics ...and 18 more Serials
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#### Cited in 28 Fields
123 Operations research, mathematical programming (90-XX) 98 Computer science (68-XX) 88 Mechanics of deformable solids (74-XX) 86 Numerical analysis (65-XX) 41 Systems theory; control (93-XX) 36 Ordinary differential equations (34-XX) 22 Partial differential equations (35-XX) 16 Combinatorics (05-XX) 11 Integral equations (45-XX) 10 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 9 Statistics (62-XX) 7 Approximations and expansions (41-XX) 6 Dynamical systems and ergodic theory (37-XX) 6 Operator theory (47-XX) 6 Mechanics of particles and systems (70-XX) 6 Fluid mechanics (76-XX) 5 Functional analysis (46-XX) 4 Calculus of variations and optimal control; optimization (49-XX) 3 Relativity and gravitational theory (83-XX) 2 Potential theory (31-XX) 2 Statistical mechanics, structure of matter (82-XX) 2 Information and communication theory, circuits (94-XX) 1 Mathematical logic and foundations (03-XX) 1 Topological groups, Lie groups (22-XX) 1 Functions of a complex variable (30-XX) 1 Differential geometry (53-XX) 1 Optics, electromagnetic theory (78-XX) 1 Quantum theory (81-XX)
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anonymous one year ago determine whether the series is convergent or divergent..... equation below
$\sum_{n=1}^{\infty} 1/n ^{2}+6n+13$
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### 11th Standard CBSE Physics Study material & Free Online Practice Tests - View Model Question Papers with Solutions for Class 11 Session 2020 - 2021 CBSE [ Chapter , Marks , Book Back, Creative & Term Based Questions Papers - Syllabus, Study Materials, MCQ's Practice Tests etc..]
#### Class 11th Physics - Waves Case Study Questions and Answers 2022 - 2023 - by Study Materials - View & Read
• 1)
A wave motion is a means of transferring energy and momentum from one point to another without any actual transportation of matter between these points. In wave motion, disturbance travels through some medium but medium does not travel along with the disturbance. For propagation of wave, medium must possess two essential property viz. inertia and elasticity. Disturbance produced at one point is communicated to the adjoining particle which also start vibrating simple harmonically about their mean positions. Hence the wave motion travels on and on. Wave motion is categorised as longitudinal and transverse on the basis of mode of vibration of particles of medium.
(i) What are elastic waves?
(ii) Explain two types of wave motion.
(iii) Mention the characteristic of medium in which longitudinal and transverse waves propagate.
(iv) What are compression and rarefaction in a longitudinal wave propagation?
(v) Why are longitudinal waves called pressure waves?
(vi) What type of waves are the sound and the light waves?
(vii) An explosion occurs inside a lake. What type of waves are produced inside the water?
• 2)
Sound travels through a gas in the form of compressions and rarefactions. Newton assumed that the changes in pressure and volume of gas, when sound waves are propagated through it, are isothermal. The amount of heat produced during compression, is lost to the surrounding and similarly the amount of heat lost during rarefaction is gained from the surroundings. So as to keep the isothermal elasticity. Laplace, a french mathematician pointed out that Newton's assumption was wrong. According to Laplace, the changes in pressure and volume of a gas, when sound waves propagated through it, are not isothermal but adiabatic.
(i) Write the Newton's formula for velocity of sound in gases and Laplace correction in it.
(ii) What is the effect of pressure on velocity of sound in gases?
(iii) Find the ratio of velocity of sound in Hydrogen and Oxygen.
(iv) Define temperature coefficient of velocity of sound in air.
(v) What is effect of humidity on the speed of sound in air?
(vi) Explain why propagation of sound in air is an adiabatic process?
(vii) If tension of a wire is increased to four times, how is the wave speed changed?
• 3)
The principle of super position of waves enables us to determine the net waveform when any number of individual waveforms overlap. The net displacement at a given time is the algebraic sum of the displacements due to each wave at that time. When two sets of progressive wave trains of the same type having the same amplitude and same time period travelling with the same speed along the same straight line in opposite directions superimpose a new set of waves are formed. These are called strationary waves or standing waves. The resultant waves do not propagate in any direction, nor there is any transfer of energy in the medium. In stationary waves, there are nodes and anti nodes point where particles are at rest and have largest amplitude respectively.
(i) How amplitude of vibration vary in stationary wave?
(ii) What is energy of stationary wave?
(iii) What is distance between consecutive node, antinode and between node and antinode?
(iv) What is phase difference between particles vibrating in a segment of stationary wave and between adjoining segments?
(v) Why is a stationary wave so named?
(vi) Where will a person hear maximum sound, at node or antinode?
(vii) Name the type of stationary wave produced by an organ pipe, open at both ends.
• 4)
When two sound waves of-nearly same frequency and amplitudes travelling in a medium along the same direction, super-impose on each other, then the intensity of the resultant sound at a particular position rises and falls alternately with time. This phenomenon is known as beat. if intensity of sound is maximum at time t = 0, one beat is said to be formed when intensity becomes maximum again, after becoming minimum once in between. The time interval between two successive beats is called beat period. The number of beats produced per second is called beat frequency.
(i) Two sound waves of frequency v1 and v2 superimpose to form beats. What is the beat frequency?
(ii) What should be the difference in frequency of two sound waves to form beats? Give reason
(iii) Write two applications of the phenomenon of beats.
(iv) Two sounds of very close fequencies, say 256 Hz and 260 Hz are produced simultaneously. What is the frequency of resultant sound and also write the number of beats heard in one second?
(v) A sitar wire and a tabla, when sounded together, produce 5 beats per second. What can be concluded from this? If the tabla membrane is tightened will the beat rate increase or decrease?
(vi) A tuning fork of unknown frequency gives 4 beats with a tuning fork of frequency 310 Hz. It gives the same number of beats on filing. Find the unknown frequency.
• 5)
Whenever there is a relative motion between the source of sound, the observer and the medium, the frequency of sound as received by the observer is different from the frequency of sound emitted by the source. For example to a man standing on a railway platform, when a train blowing its whistle, approaches him, the pitch of the whistle appears to rise and it suddenly appears to drop as the engine moves away from him. Similar effect is observed when the source is at rest and observer moves towards or away from the source. This phenomenon is noticeable only when the relative velocity between the source and the observer is an appreciable fraction of the wave velocity.
(i) Name the phenomenon observed in the passage. Define it.
(ii) On what factors does the apparent frequency of sound depends?
(iii) What physical change occurs when a source of sound moves and the listener is stationary?
(iv) What physical change occurs when the source of sound is stationary but the listener moves?
(v) A particle travelling with a speed of 0.9 of the speed of sound and is emitting radiations of frquency of 1 KHz and moving towards the observer. What is the apparent frequency of radiation?
(vi) Write the condition in doppler effect when apparent frequency of sound increases.
(vii) Write two applications of Doppler effect.
#### Class 11th Physics - Oscillation Case Study Questions and Answers 2022 - 2023 - by Study Materials - View & Read
• 1)
In our daily life, we come across various types of motions: such as periodic, non-periodic, oscillatory and non oscillatory. The study of oscillatory motion is of great importance as its concepts are required for the understanding of many physical phenomena, for example
(i) vibrating strings produces pleasant. sounds in musical instruments like the sitar, guitar or the violin
(ii) the oscillations of the atoms in solid about their mean positions to convey the sensation of temperature.
All oscillatory motions are periodic motions but all periodic motion may not be oscillatory. An oscillatory motion is further classified as harmonic and non-harmonic oscillation.
(i) What are oscillatory motions? Give example.
(ii) Give example of periodic motion which are not oscillatory.
(iii) Differentiate between harmonic and non-harmonic oscillations.
(iv) What causes a system to oscillate?
(v) On what factor does the time period of a simple harmonic oscillator depends?
(vi) What are periodic functions? Express one graphically.
• 2)
Simple harmonic motion is a special type of periodic motion, in which a particle moves to and fro repeatedly about a mean position under a restoring force, which is always directed towards the mean position and whose magnitude at any instant is directly proportional to the displacement of the particle from the mean position at that instant i.e., Fresloring =-Kx, where K is force constant and x - the displacement of the particle executing S.H.M. It can be expressed in terms of one simple harmonic function x = a sin $\left(\omega t+\phi_{0}\right)$ 0)' where a - is the amplitude of oscillation and $\left(\omega t+\phi_{0}\right)$ is the phase of vibrating particle at the instant t, $\phi_{0}$ is initial phase and co is the angular frequency of the vibrating particle in S.H.M.
(i) Give geometrical interpretation of S.H.M.
(ii) Express the velocity and acceleration of S.H.M in terms of displacement and angular frequency $(\omega)$.
(iii) State the condition when S.H.M particle velocity is in phase and opposite in phase with the acceleration of particle.
(iv) What is the average value of total energy of a particle in S.H.M in one complete oscillation?
(v) What is the frequency of P.E, K.E and total energy in S.H.M?
(vi) What will be the time period of second's pendulum if its length is doubled?
(vii) The girl sitting on a swing stands up. What will be the effect on the periodic time of the swing?
• 3)
Consider two springs of spring constants KJ and K. Let there be three spring combinations as shown in fig (a), (b), and (c). A body of mass rn oscillates about its mean position under influence of restoring force (F) produced when displaced from the equilibrium position of the body.
(i) Determine the net restoring force and spring constant of the combination of spring in fig (a) for a displacement of y in mass from mean position.
(ii) Determine the spring constant of combination of spring in fig (b) and also the restoring force produced when displaced from mean position.
(iii) Determine the spring constant of combination of spring in fig (c) an,d also the restoring force produced when displaced from mean position.
(iv) Determine the frequency of oscillation in each of combination.
• 4)
A body capable of oscillating S.H.M when displaced from mean position start oscillating. During oscillation amplitude of oscillating body may either vary or remain constant. It is categorised as undamped, damped and free or forced vibraton depending on whether there is some resistive force present in the medium where body is oscillating or not and also if the body under the influence of some external force or not. Amplitude of oscillation is function of resistive force of the medium, external periodic force and phase difference between external force and oscillating body.
(i) What are undamped simple harmonic oscillations?
(ii) Define damped oscillations. Give example.
(iii) What are free oscillations? On what factor does the natural frequency of an oscillating system depends?
(iv) What are forced oscillations?
(v) Give an example of forced oscillation.
(vi) What are resonant oscillations? Give example.
(vii) Is the damping force, constant on a system executing S.H.M?
• 5)
A 2 kg block hangs without vibrating at the bottom end of a spring with a force constant of 800 N/m. The top end of the spring is attached to the ceiling of an elevator car. The car is rising with an upward acceleration of 10 ms-2. When the acceleration suddenly ceases at time t = 0, the car moves upward with constant speed. (g = 10 ms-2 )
(i) What is the angular frequency of oscillation of the block if their acceleration' ceases?
(ii) Determine the amplitude of the oscillation.
(iii) Determine the initial phase angle observed by a rider in the elevator, taking downward direction to be positive.
(iv) A particle starts oscillating from half the amplitude position. What is its initial phase?
(v) A simple harmonic motion of amplitude A, has a time period T. What will be the acceleration of the oscillator when its displacement is half of the amplitude?
#### Class 11th Physics - Kinetic Theory Case Study Questions and Answers 2022 - 2023 - by Study Materials - View & Read
• 1)
The Kinetic theory was developed by Maxwell and Boltzmann to explain the behaviour of gases based on the idea that gases consist of rapidly moving atoms or molecules. The inter atomic forces binding the atoms are negligible and their size is. negligible. The theory is consistent with various gas laws and Avogadro's hypothesis. It gives molecular interpretation oftemperature pressure. Specific heat capacities. Compared to solids and liquids, it is easier to understand the properties of gases. This is because molecules in gases are far apart.
(i) Write the characteristics of ideal gas. Write the conditions at which real gases acquire ideal gas behaviour.
(iii) Write the perfect gas equation by explaining Boyle's law and Charles's law.
(iv) What is meant by Boltzmann constant? Calculate its value in S.I units.
(v) The graph shows the variation of the product PV w.r.t the pressure P of given masses of three gases, A, Band C. The temperature is kept constant. State with proper arguments which of these gases is ideal.
• 2)
The molecules of a gas move in all directions with various speeds. The speeds of the molecules of a gas increase with rise in temperature. During its random motion, a fast molecule of ten strikes against the walls of the container of the gas. The collisions are assumed to be prefectly elastic i.e., the molecule bounces back with the same speed with which it strikes the wall. Since the number of molecules is very large, billions of molecules strike against the walls of the container every second. These molecules exert force on the wall. The force exerted per unit area is the pressure exerted by the gas on the walls. According to the kinetic theory, the pressure of a gas of density p at absolute temperature T is given by $P=\frac{1}{3} \rho$ $v_{r m s}^{2},$ where vrms is the root mean square speed of the gas molecule and is given by $v_{r m s}=\sqrt{\frac{3 K_{B} T}{M}}$ where M is the mass of a molecule and KB is Boltzmann constant.
(i) State the relation between pressure and kinetic energy of the gas.
(ii) On what factors does the average kinetic energy of translation per molecule of the gas depends?
(iii) State absolute zero temperature in terms of root mean square velocity of gas.
(iv) Define root mean square speed and establish its relation with temperature.
(v) The absolute temperature of a gas is made four times. How many times will its total kinetic energy pressure and r.m.s velocity become?
(vi) Two different gases have exactly the same temperature. Does this mean that their molecules have the same r.ms. speed?
(vii) On reducing the volume of the gas at constant temperature, the pressure of the gas increases. Explain on the basis of kinetic theory.
• 3)
The number of degrees of freedom of a dynamical system is the total number of co-ordinates or independent quantites required to describe completely the position and configuration of the system. For a dynamical system, the number of degrees of freedom is obtained by subtracting the number of independent relations from the total number of co-ordinates required to specify the position of constituebnt particles of the system.
N = 3A - R, where N - number of degrees of freedom of the system.
A - number of particles in the system
R - number of independent relations among the particles.
Each degree of freedom contributes equally In the distribution of the energy associated with each molecule. In thermal equilibrium the energy associated with each molecule per degree of freedom is $\frac{1}{2} K_{B} T$ .
(i) Determine the number of degrees of freedom of a non-linear triatomic molecule.
(ii) State law of equipartition of energy.
(iii) If a gas has n degrees of freedom, determine the ratio of principal specific heat of the gas.
(iv) Determine the energy contributed by a vibrational mode in total energy.
(v) Determine the ratio of specific heat for monoatomic gas molecule.
• 4)
During their random motion, the molecules of a gas often come close to each other. Molecules are perfect elastic spheres and their size is very small compared to the distance between them. Gas molecules undergo elastic collision. Therefore, they cannot move straight unhindered. The paths of molecules keep on getting deflected incessantly. Path of a single gas molecule consists of a series of short zig zag paths of different lengths. These paths of different lengths are called free paths of the molecules and their mean is called mean free path. Mean free path of gas molecules depends on diameter (d) of gas molecule and molecular density (n) as follows $\lambda=\frac{1}{n \pi d^{2}}$
(i) Define mean free path of gas molecules.
(ii) On what factors do the mean free path of gas molecules depends?
(iii) What is significance of mean free path?
(iv) How many collisions per second does each molecule of a gas make, when the average speed of a molecule is 500 ms-1 and mean free path is 2.66 x 10- 7 m?
(v) Calculate the mean free path of molecules, if number of molecules per cm3 is 3 x 1019 and diameter of each molecule is 2A.
#### Class 11th Physics - Thermodynamics Case Study Questions and Answers 2022 - 2023 - by Study Materials - View & Read
• 1)
A change in pressure and volume of a gas without any change in its temperature is called an isothermal change. In such a change, there is free exchange of heat between the gas and its surrounding. These changes are governed by Boyle's law i.e., PV = Constant. The change in pressure and volume of a gas when temperature also changes is called an adiabatic change. In such a change, no heat is allowed to enter into or escape from the gas. The equation of adiabatic changes $P V^{\gamma}$ = constant or $T V^{\gamma-1}$ constant or $P^{1-\gamma} T^{\gamma}$ = constant, where $\gamma$ - ratio of two principal specific heats of the gas = $\frac{C_{P}}{C_{V}}$ .
(i) What are conditions for the changes to be isothermal and for adiabatic?
(ii) A gas is compressed isothermally to half its volume. By what factor does the pressure of the gas increase?
Given $\gamma$ = 1.4.
(iii) A gas is compressed adiabatically to half its volume. By what factor does the pressure of gas increases?
Given $\gamma$ = 1.4.
(iv) A gas is suddenly compressed to $\frac{1}{4}$th of its original 4 volume. Calculate the rise in temperature, when the original temperature is 27°C and $\gamma$ = 1.5.
(v) Draw PV variation diagram representing isothermal and adiabatic processes respectively.
(vi) Ice at 0 °C is converted into steam at 100°C. State the isothermal changes in this process
• 2)
Thermodynamics is the study of transformation of heat into other forms of energy and vice-versa. Thermodynamical system is said to be in equilibrium when macroscopic variables like pressure, volume, temperature, mass, composition, etc. that characterise the system variables which are not necessarily independent. Some important thermodynamic processes are: isothermal, adiabatic, isobaric, and isochoric. A thermodynamic state is represented by equation of state that represents the connection between the state variables of a system.
(i) Which physical quantity determine the thermal equilibrium of a system? State zeroth law of thermodynamics.
(ii) What is quasi static process? What does it represent?
(iii) What is equation of state? Write equation of state for adiabatic operation.
(iv) Determine the specific heat of gas during isothermal and adiabatic process.
(v) The volume of an ideal gas is Vat a pressure P. On increasing the pressure by $\Delta \boldsymbol{P}$ , the change in volume of the gas is $\Delta \boldsymbol{V}_{\mathbf{1}}$ under isothermal conditions and $\Delta \boldsymbol{V}_{2}$ under adiabatic condition. Is $\Delta V_{1}>\Delta V_{2}$ or vice versa and why?
(vi) An ideal gas at temperature T1, undergoes expansion under adiabatic conditions, to attain temperature T2. Write expression for workdone.
• 3)
Every system (solid, liquid or gas) possess a certain amount of energy. This energy is called the internal energy and is consists of two parts
(i) Kinetic energy due to the motion (translation, rotational, and vibrational) of the molecules and
(ii) potential energy due to the configuration (separation) of the molecules. The internal energy of a homogeneous system depends on its thermodynamic state i.e., on its thermodynamic coordinates P, V, and T. Each definite state of the system possesses a definite quantity of internal energy. A change in the internal energy can occur only if a transfer of energy between the system and surrounding is permitted. This can take place if some work is performed on or by the system and some heat is absorbed or given out by the system.
(i) How temperature, pressure and volume of gas change in an isothermal process?
(ii) 200 J of work is done on a gas to reduce its volume by compressing it. If this change is done under adiabatic conditions. Find the change in internal energy of the gas and also the amount of heat absorbed by the gas?
(iii) In a given process on an ideal gas dW = 0 and dQ >0, then what change would occur in temperature and internal energy?
(iv) What are two different modes of changing state of a thermodynamic system?
(v) Write the sign convention regarding to change in heat content, internal energy change and work done over a thermodynamic system during a thermodynamic process.
(vi) What will be the change in internal energy during isothermal, adiabatic and cyclic process?
• 4)
The first law of thermodynamics establishes the essential equivalence between work and heat, as according to this law, internal energy (and hence temperature) of a system can be increased either by supplying heat to it or by doing work on the system or both. However, this law has limitations as it does not indicate the direction in which the change can occur and gives no idea about the extent of change. Limitations of first law of thermodynamics is overcome by second law of thermodynamics expressed as Kelvin Planck statement and Clausius statement based on working of heat engine and refrigerator respectively.
(i) Write the relation between two principal specific heats of an ideal gas based on first law of thermodynamics.
(ii) Which fundamental law of conservation is represented by first law of thermodynamics?
(iii) State Kelvin Planck statement and Claus statement of second law of thermodynamics.
(iv) Define thermal efficiency of a heat engine.
(v) Define coefficient of performance of a refrigerator $\text { ( } \beta \text { ) }$.State the relation between $\eta \text { and } {\beta}$ .
(vi) Can a heat engine of 100% efficiency be designed? Explain.
• 5)
A sample of2 kg of mono atomic helium is taken through the process ABC and another sample of 2 kg of the same gas is taken through the process ADC as shown in diagram. The molecular mass of Helium = 4 and R (gas constant) = 8.3 JK-1 mol-1.
(i) If number of moles of helium being 500, then Determine the temperature of State A.
(ii) What is isochoric process? Determine the temperature at state B.
(iii) What is isobaric process? Determine the temperature at state C and D.
(iv) Determine the work done in process ABC.
#### Class 11th Physics - Thermal Properties of Matter Case Study Questions and Answers 2022 - 2023 - by Study Materials - View & Read
• 1)
Heat is a form of energy which produces the sensation of warmth. It is total thermal energy of the body which is sum of kinetic energies of all the individual molecules of the body. It is transferred from one body to the other on account of temperature difference between two bodies. Joule found that when' mechanical work (W) is converted into heat (Q), then the ratio of Wand Q is always constant, represented by J i.e., Joule's mechanical equivalent of heat.
$J=\frac{W}{Q} \text { or } W=J Q$
A measure of temperature is obtained using a thermometer that use Physical Properties which change uniformly with temperature, for example, in common liquid in glass thermometers, mercury, alcohol etc are used whose volume varies linearly with temperature over a wide range.
(i) Define temperature and its significance.
(ii) Define One calorie.
(iii) How does a thermometer work?
(iv) Name the two convenient fixed reference point for measuring temperature.
(v) How celsius temperature and Fahrenheit scale are related to each other?
(vi) What is absolute zero temperature?
(vii) What is Kelvin scale? How it is related with celsius scale?
• 2)
When solid is heated, the amplitude of vibration of atoms and molecules increases. Therefore effective interatomic separation increases and cause thermal expansion. Thermal expansion of solids are of three types:
(i) Linear expansion
(ii) Area expansion
(iii) Volume expansion. Thermal expansion of liquid are of two kinds. Real expansion and apparent expansion that occurs in form of volume expansion. Thermal expansion of water is anomalous i.e., volume of given amount of water first decreases with increase in temperature from 0 °C to 4 °C and beyond 4 °C volume of water increases with rise of temperature. In thermal expansion of gases is more than solid and liquid but coefficient of volume expansion is dependent on temperature for gases.
(i) On what factors does the coefficient of thermal expansion depend? Write its S.I. unit.
(ii) Write the relation between the three coefficients of expansion $\alpha, \beta$ and $\gamma$ for solid.
(iii) A body at higher temperature contains more heat. Comment.
(iv) Why are clock pendulums usually made of invar?
(v) Draw graphical variation of volume and temperature for water.
(vi) How is coefficient of thermal expansion of gases related to temperature?
(vii) State water equivalent.
• 3)
States of matter viz: solid, liquid and gas are function of temperature and heat content. During the change of state of a substance, the exchange of heat takes place between the substance and surrounding. In this process temperature of substance remains constant. At certain temperature known as melting point. Both the solid and liquid states of the substance coexist in thermal equilibrium. Similarly, at boiling point both the liquid and vapour states of the substance co-exist in the thermal equilibrium. There are certain substance which on heating directly pass from solid to vapour state without passing through the liquid state. This is sublimation process in which solid changes to vapour state of the substance. Process of change of state depends on pressure and temperature.
(i) Define triple point.
(ii) Define latent heat of a substance.
(iii) What is principle of calorimetry?
(iv) What is effect of pressure on the melting point of a substance?
(v) State phenomenon of relegation.
(vi) What is boiling point? What is effect of pressure on the boiling point?
(vii) What is sublimation?
• 4)
Three cylindrical rods A, Band C of equal lengths and equal diameters are joined in series as shown in the figure. Their thermal conductivities are 2K, K and 0.5 K respectively.
100°C $\begin{array}{|l|l|l|} \hline \mathrm{A} & \mathrm{B} & \mathrm{C} \\ \hline \end{array}$ 0°C
In the steady-state, the free ends of rods A and C are at 100 °C and a 0°C respectively. Neglecting loss of heat from the curved surfaces of rods.
(i) Determine the temperature of the junction between rods A and B.
(ii) Determine the temperature of the junction between rods Band C.
(iii) Determine the equivalent thermal conductivity of the combination.
(iv) Define coefficient of thermal conductivity of a solid. Write its S.I unit.
Two rods A and B are of equal length. Each rod has the ends at temperature T1 and T2 What is the condition that will ensure equal rates of flow of heat through the rods A and B?
• 5)
All bodies emit heat energy from their surface by virtue of their temperature. This heat energy is called radiant energy or thermal radiation. The heat that we receive from the sun is transferred to us by a process which, unlike conduction and convection, does not require the help of a medium in intervening space which is almost free of particles. Radiant energy travels in space as electromagnetic waves in the infra-red region of electromagnetic spectrum. They exhibit the phenomenon of interference, diffraction, and polarization as light does.
The emission of radiation from a hot body is expressed in terms of that emitted from a reference body (called the black body) at the same temperature. A black body absorbs and emits radiations of all wavelengths. The total energy E emitted by a unit area of a black body per second is given by E = $\sigma T^{4}$
Where T - is absolute temperature of the body and a is Stefan's constant, if the body is not the perfect black body, then E = $\varepsilon \sigma T^{4}$ , where $\varepsilon$ is the emissivity of the body.
(i) Determine the dimensions of Stefan's constant a from Stefan - Boltzman law.
(ii) What is S.I unit of Stefan's constant?
(iii) In which region of the electromagnetic spectrum do thermal radiation lie?
(iv) Which device is used to detect thermal radiation?
(v) When a body A at a higher temperature T1 is surrounded by another body B at a lower temperature T2. Write the relation between the rate of loss of heat from body A and temperature.
(vi) On what factor does the rate at which energy is radiated by a body depends?
(vii) On which parameter does the colour of a star depends upon?
#### Class 11th Physics - Mechanical Properties of Fluids Case Study Questions and Answers 2022 - 2023 - by Study Materials - View & Read
• 1)
Fluid is the name given to a substance which begins to flow when external force is applied on it. Shape and volume of the fluid changes by the application of very small shear stress. The branch of physics which deals the study of fluids at rest is called hydrostatics and that branch of physics which. deals with study of fluid in motion is called hydrodynamics when liquid is at rest in a container, it exerts a force on the surface of object in contact with liquid, which is always normal to the surface of object. The total normal force exerted by liquid at rest on a given surface in contact with it is called thrust of liquid on that surface.
(i) What is the cause of thrust of liquid on surface in contact?
(ii) Define hydrostatic pressure. Write its S.I. unit
(iii) Is hydrostatic pressure a vector quantity or scalar quantity? Give reason.
(iv) Define relative density of a substance
(v) What will be the measure of total preassure at a depth h below the liquid surface?
(vi) What is gauge pressure? On what factors does it depend?
• 2)
It is observed that if gravity effect is neglected the pressure at every point of liquid in equilibrium of rest is same and the increase in pressure at one point of the enclosed liquid in equilibrium of rest is transmitted equally to all other points of the liquid. This is accounted to Pascal's law. Hydraulic lift and hydraulic brakes working is based on the Pascal's law, in which a small force applied on the smaller piston will appear as a very large force on the large piston.
(i) State Pascal's law
(ii) A bottle full of a liquid is fitted with a tight cork. Explain why a slight blow on the cork may be sufficient to break the bottle?
(iii) Two Pistons of hydraulic press have diameter of 30.0 cm and 2.5 cm. Find the force exerted by longer piston when 50.0 kg wt is placed on smaller Piston.
(iv) In the above question, find the distance through which the longer piston would move after 10 strokes if the stroke of the smaller piston is 40 cm.
(v) In a car lift, compressed air exerts a force F1 on a small piston having a radius of 5.0 cm. This pressure is transmitted to a second piston of radius 10.0 cm, If the mass of the car to be lifted is 1300 kg. Calculate F1. What is pressure necessary to accomplish the task?
• 3)
It has been found that a liquid in small quantity at rest, free from external force like gravity, always tends to have a spherical shape. Since for a given volume, a sphere has the least surface area, hence it shows that the free surface of every liquid at rest has a tendency to have a least surface area. The free surface of liquid behaves as if covered by a stretched membrane, having tension in all directions parallel to the surface. This tension in the free surface of liquid at rest is called the surface tension. It arises due to the fact that the free surface of liquid at rest has some additional potential energy.
(i) What is surface tension and its origin?
(ii) Why does oil spread over the surface of water?
(iii) At what temperature the surface tension of a liquid is zero?
(iv) Surface tension of all lubricating oils and paints is kept low. Why?
(v) What is the effect of impurities on the surface tension of liquid?
(vi) What is work done in blowing a soap bubble of radius r and surface tension S?
(vii) Define surface energy of liquid.
• 4)
Stokes' Law: A body falling through a viscous medium experiences a retarding force resulting in absorption of energy by the medium in the form of heat .:The motion of the body produces a relative motion between the different layers of the fluid. Consequently, it experiences a force which tends to retard its motion. When a small spherical body is dropped in a viscous liquid such as glycerine, it accelerates first, but soon begins to experience a retarding force. When the retarding force becomes equal to the effective weight of the body in the fluid, the body experiences no net force and falls with a constant velocity known as the terminal velocity. George stokes found that a small spherical body of radius r moving with a uniform velocity v in a fluid of coefficient of viscosity $\eta$ experiences a retarding force F given by F = $6 \pi \eta r v$
(i) Define viscosity.
(ii) Define S.I unit of coefficient of viscosity of a liquid.
(iii) Name the forces which act on the small spherical body falling freely through a viscous medium.
(iv) Write the dimenslonal formula of coefficient of viscosity $\eta$ .
(v) What would be the terminal velocity of the body if the upthrust on the body is negligible to its weight?
(vi) Write the expression for terminal velocity of spherical body of radius r and density $\rho$ falling freely through a liquid of density $\sigma$ and coefficient of viscosity $\eta$.
• 5)
Equation of continuity is a fundamental equation of liquid flow and is a special case of the general law of conservation of mass. Consider an incompressible and non viscous liquid flowing slowly and steadily through a pipe of non-uniform cross-section. Let A and B be two different sections of a pipe having cross-sectional area Q 1 and Q 2 respectively. Let v1 and v 2 be the respecstive velocities of the liquid flow through these cross sections. According to the equation of continuity of flow Q 1 v1 = Q2V2 or QV = constant i.e., the velocity of liquid flow at any section of the pipe is inversely proportional to area of cross-section of the pipe at that section.
(i) Water flows through a horizontal pipe of nonuniform cross-section at the rate of 31.4 litre per minute. Determien the velocity of flow of water at the section of the pipe where diameter is 2 cm.
(ii) Water flows through a horizontal pipe of diameter 2 cm at a speed of 3 cm s-1. The pipe has a nozzle of diameter 5 mm at its end. Determine the speed of water emerging from the nozzle.
(iii) What is meant by streamline flow?
(iv) Distinguish between laminar flow and turbulent flow.
(v) Still water runs deep, why?
#### Class 11th Physics - Mechanical Properties of Solids Case Study Questions and Answers 2022 - 2023 - by Study Materials - View & Read
• 1)
The atoms in solids are held together by interatomic forces. the average locations of the atoms in a lattice do not change with time and lack mobility. This makes a solid rigid and becomes a cause of elasticity in solids. In some solids such as steel, the atoms are bound together by larger inter-atomic forces than in others. Thus elastic behaviour varies from solid to solid. Even fluids exhibit elasticity. All material bodies get deformed when subjected to a suitable force. The ability of a body to regain its original shape and size is called elasticity. The deforming force per unit area is called stress. The change in dimension divided by the original dimension is called strain. The three kinds of stresses are tensile stress, shearing stress and volumetric stress similarly strains too. According to Hooke's law, within the elastic limit stress is proportional to strain.
(i) Which state of matter has volume elasticity?
(ii) When we stretch a wire, we have to perform work. Why? What happens to the energy given to the wire in this process?
(iii) Define elastic limit.
(iv) Define modulus of elastcity on what factors does it depend?
(v) Why solids are more elastic and gases are least?
(vi) The ratio of radii of two wires of same material is 2 : 1. If these wires are stretched by equal force, find the ratio of stresses Produced in them.
• 2)
A wire of uniform area of cross-section is suspended vertically from a rigid support through one end with the help of an attached hanger by putting different known weights in the hanger Plot a graph between stress and strain for the stretched wire is as shown.
(i) What does the portion OA of graph represent?
(ii) Which point does the elastic limit represent?
(iii) Which region represents permanent set?
(iv) Which point does represent the yield point?
(v) Which point corresponds to breaking point or breaking stress?
(vi) Draw stress-strain variation graph for elastomers.
• 3)
One end of a string of length L and cross-sectional area A is fixed to a support and the other end is fixed to a bob of mass m. The bob is revolved in a horizontal circle of radius r with an angular velocity co such that the string makes an angle $\theta$ with the vertical.
(i) Determine the angular velocity $\omega$ of the bob.
(ii) Determine the tension T in the string.
(iii) Determine the increase in length of the string when bob is freely suspended from rigid support.
(iv) Determine the stress in the string.
(v) A metallic wire is suspended by attaching some weight to it. If $\alpha$ is the longitudinal strain and Y is Young's modulus, find the ratio between elastic potential energy and the energy density.
• 4)
When an elastic body is subjected repeatedly to the action of alternating deforming forces, its behaviour corresponds to that of less elastic bodies due to elastic fatigue. In our daily life, elastic properties are considered while designing a structure of the material. For example, the metallic parts of the machinery are never subjected to a stress beyonds elastic limit otherwise they will get permanently deformed. The thickness of the metallic rope used in the crane in order to lift a given load is decided from the knowledge of elastic limit of the material of the rope and the factor of safety. Similarly the bridges are designed in such a way that they do not bend much or break under the load of heavy traffic, force of strongly blowing wind and its own weight.
(i) What does it mean by elastic after effect?
(ii) Define elastic fatigue
(iii) Why are bridges and girders given I shape?
(iv) A hollow shaft is found to be stronger than a solid shaft made of some equal material against twisting. Explain why?
(v) Define Poisson's ratio
(vi) An elastic wire is cut to half its original length. How would it affect the maximum load that the wire can support?
(vii) Why is a spring made of steei, not of copper?
(viii) Why are the bridges declared unsafe after long use?
• 5)
A thin rod of negligible mass and cross-sectional area 4 x 10-6m-2, suspended vertically from one end, has a length of 0.5 m at 100° C, The rod is cooled to 0° C, Young's modulus is 1011 Nm-2, Coefficient of linear expansion = 10- 5 K-1 and g = 10 ms-2.
(i) Determine the decrease in length of the rod on cooling.
(ii) What mass must be attached at the lower end of the rod so that the rod is prevented from contracting on cooling?
(iii) Determine the total energy stored in the rod.
(iv) What is origin of elastic potential energy in a stretched wire? Give its relation with Young's modulus and strain.
#### Class 11th Physics - Gravitation Case Study Questions and Answers 2022 - 2023 - by Study Materials - View & Read
• 1)
Galileo was the first to recognise the fact that irrespective of their masses, bodies fall towards the earth with a constant acceleration. Later on gravity and its laws were given by Newton in his universal law of gravitation and motion of planets around the sun was explained by Kepler in his laws of planetary motion. Newton's universal law of gravitation and third law of motion have a similarity that two bodies exert equal and opposite force on each other.
(i) What does the word 'Gravity' mean?
(ii) Define acceleration due to gravity
(iii) State Newton's law of gravitation.
(iv) What would be the gravitational force on a point mass body situated insde the hollow spherical shell of uniform density?
(v) State Kepler's law of period for planetary motion.
(vi) Give relationship between angular momentum and areal velocity of a planet around sun.
(vii) At what distance force of gravitation between two bodies would be zero?
• 2)
There are three identical point mass bodies each of mass m located at the vertices of an equilateral triangle with side r. They are exerting gravitational force of attraction on each other, which can be given by newton's law of gravitation. Each mass body produces its gravitational field in the surrounding region. The magnitude of gravitational field at a point due to a point mass body is the measure of gravitational intensity at that point. The gravitational potential at a point in a gravitational field is the amount of work done in bringing a unit mass body from infinity to the given point without acceleration.
(i) What is the magnitude of the gravitational force on one body due to other two bodies in the given arrangement.
(ii) What is the work done in taking one body far away from the other two bodies?
(iii) Determine the gravitational potential at the centroid point O.
(iv) When a body falls towards earth, earth also moves towards the body. Why is earth's motion not noticed?
• 3)
A rocket is fired vertically upwards with speed v = 5 kms-1 from the surface of earth. It goes up to a height h before returning to earth. At height h a body is thrown from the rocket with speed vo in such a way so that the body becomes satellite of earth. Let the mass of the earth, M = 6 x 1024kg, mean radius of the earth, R = 6.4 x 10 6 m, G = 6.67 x 10- 11 Nm2 kg-2, g = 9.8 ms-2.
(i) Determine the value of height h above the surface of earth from which body is thrown out from the, rocket.
(ii) Determine the orbital velocity of the satellite.
(iii) Calculate the time period of revolution of satellite around the earth.
(iv) If this satellite is to be taken at double of the present height from the surface of the earth, then find the new time period of revolution of satellite.
(v) What is the value of gravitational field on the surface of earth?
(vi) Is the potential energy of a system of bodies positive or negative? Give reason.
(vii) What is the maximum value of gravitational potential energy and where?
• 4)
Geostationary satellite is a particular type of satellite used in communication. A number of communication satellites are launched which remain in fixed position at a specified height above the equator. They are called synchronous satellites. These appear fixed at a position above a certain place on the earth, it must corotate with the earth so that its orbital period around the earth is exactly equal to the rotational period of the earth about its axis of rotation.
(i) What is the height of geostationary satellite above the surface of the earth?
(ii) What is the time period and direction of revolution of a geostationary satellite respectively?
(iii) What is the orbital-speed of geostationary satellite?
(iv) Which electromagnetic wave is used in satellite communication?
(v) Can a Pendulum vibrate in an artificial satellite?
(vi) From where does a satellite revolving around planet get the required centripetal Force?
• 5)
A body moving in an orbit around the earth is called earth satellite. The First artificial satellite was put into earth's orbit in 1956. Artificial satellites are put into orbit at an altitude of a few hundred kilometers. The satellite is carried in a rocket which is launched from the earth with a velocity greater than the escape velocity. The escape velocity is the velocity with which a body must be projected in order that is may escape the gravitational pull of the earth. When the rocket has achieved the desired height, the satellite is released horizontally by imparting to it a very high speed so that it remains moving in a nearly circular orbit around the earth. This velocity is called the orbital velocity which is about 8 kms-1 for a satellite at a few hundred kilometeres above the earth.
(i) What are the factors on which escape velocity of a rocket fired from the earth depends upon?
(ii) What provides the necessary centripetal force to keep a satellite in a circular orbit around the earth?
(iii) An artificial satellite is orbiting the earth at an altitude of 500 km. A bomb is released from the satellite. How will the bomb move after its release?
(iv) How escape velocity is related to orbital velocity of satellite?
(v) A satellite is orbiting around the earth with a speed v. To make the satellite escape, what is the minimum percentage increase in its speed?
(vi) A satellite is launched into a circular orbit of radius R around the earth. A second satellite is launched into an orbit of radius 1.01 R. By what percentage the period of second satellite is larger than that of first satellite?
(vii) Does the speed of a satellite remain constant in a particular orbit?
#### Class 11th Physics - System of Particles and Rotational Motion Case Study Questions and Answers 2022 - 2023 - by Study Materials - View & Read
• 1)
The centre of mass of a body is a point at which the entire mass of the body is supposed to be concentrated. The $\vec{r}$ of c.m of the system of two particles of masses m1 and m2 with position vector $\vec{r}_{1} \text { and } \vec{r}_{2}$ is given by
$\vec{r}=\frac{m_{1} \vec{r}_{1}+m_{2} \vec{r}_{2}}{m_{1}+m_{2}}$
for an isolated system, where no external force is acting $\overrightarrow{v_{c m}}$ = constant
Under no circumstance, the velocity of centre of mass of an isolated system can undergo a change.
(i) What should be the position of the centre of mass of a system of two particles of unequal masses?
(ii) An electron and a proton move towards each other with velocities v1 and v2 respectively. What is the velocity of their centre of mass?
(iii) Two bodies of masses 1kg and 2 kg are located at (1, 2) and (-1, 3) respesctively, determine the coordinates of the centre of mass.
(iv) A bomb dropped from an aeroplane in level flight explodes in the middle. How would be the motion of centre of mass of the fragments?
(v) Two blocks of masses 5 kg and 2 kg are placed on a frictionless surface and connected by a spring. An external kick gives a velocity of 14 ms-1 to the heavier block in the direction of the lighter one. Determine the velocity gained by the centre of mass.
(vi) Can centre of mass of a body lie where there is absolutely no mass? Give example.
(vii) Can centre of mass of a body coincide with the geometrical centre of the body?
• 2)
Moment of inertia of a body about a given axis is the rotational inertia of the body about that axis. It is represented by 1= MK2, where M is mass of body and K is radius of gy ration of the body about that axis. it is a scalar quantity, which is measured in kg m2.
When a body rotates about a given axis and the axis of rotation also moves, then total K.E of body = K.E of translation + kinetic energy of rotation.
$K=\frac{1}{2} m v^{2}+\frac{1}{2} I \omega^{2}$
(i) Is the M.I of a body about a given axis is vector or scalar quantity?
(ii) On what factors does M.I of a body depend?
(iii) Determine the moment of inertia of circular disc and circular ring of same mass and radius about an axis perpendicular to plane.
(iv) A 40 kg flywheel in the form of a uniform circular disc of diameter 1 m is making 120 rpm. What is the M.I about a transverse axis through its centre?
(v) Determine kinetic of rotation of the flywheel in the above case.
(vi) Calculate radius of gyration of a cylindrical rod of mass m and length L about an axis of rotation perpendicular to its length and passing through its centre,
(vii) Determine the ratio of the radii of gyration of a circular disc about a tangential axis in the plane of the disc and of a circular ring of the same radius about a tangential axis in the plane of the ring.
• 3)
A hollow sphere of mass M and Radius R is initially at rest on a horizontal rough surface. It moves under the action of constant horizontal force F as shown in Fig.
(i) Determine the direction of frictional force between the sphere and the surface.
(ii) Determine the linear acceleration of the sphere.
(iii) What is the frictional force between the sphere and the surface?
(iv) State the relationship between angular momentum and torque of rotating body about an axis.
(v) Solid cylinder of mass m and radius r rolling down an inclined plane of $\theta$ inclination e without slipping. Determine the acceleration of the cylinder down the inclined plane.
• 4)
Angular momentum of a rotating body is measure of quantity of motion in rotational motion about an axis which is measured by product of moment of inertia and angular velocity i.e., L = $I \omega$ . It is moment of linear momentum about axis of rotation. Being a vector quantity its direction is along the axis of rotation. In the absencse of an external torque, angular momentum vector remains constant.
(i) If angular momentum is conserved in a system whose moment of inertia is decreased, will its rotational kinetic energy be also conserved? Explain.
(ii) Why spin angular velocity of a star is greatly enhanced when it collapses under gravitational pull and becomes a neutron star?
(iii) A Person sits near the edge of a circular platform revolving with a uniform angular speed. What will be the change in the motion of the platform?
(iv) What would happen if the person starts moving from the edge toward the centre of the platform?
(v) Why are there two propellers in a helicopter?
(vi) A thin uniform circular disc of mass M and Radius R is rotating in a horizontal plane about an axis passing through its centre and perpendicular to its plane with an angular velocity $\omega \text { . }$ Another disc of same dimensions but of mass $\frac{\boldsymbol{M}}{\mathbf{4}}$ is placed gentally 4 on the first disc coaxially show that angular velocity of the system is $\frac{4}{5} \omega.$
#### Class 11th Physics - Work, Energy and Power Case Study Questions and Answers 2022 - 2023 - by Study Materials - View & Read
• 1)
Mechanical energy exists in two forms: Kinetic energy and Potential energy. Kinetic energy is the energy possesed by a body by virtue of motion. Potential energy is the energy possessed by the body by virtue of its position or configuration. These two forms of energy are interconvertible. If no other form of energy is involved in a process, the sum of kinetic energy and potential energy always remains constant.
(i) State law of conservation of mechanical energy.
(ii) State two particles having mass m1 and m2 , both have equal linear momenta. What is the ratio of their kinetic energies?
(iii) Two particles of masses m1 and m2 have equal kinetic energies. What is the ratio of their linear momenta?
(iv) A particle of mass m has half the kinetic energy of another particle of mass m/2. If the speed of the heavier particle is increased by 2 ms-1, its new kinetic energy equals the original kinetic energy of the lighter particle. What is the ratio of the original speeds of the lighter and heavier Particle?
(v) A uniform rod of mass m and length I is made to stand vertically on one end. What is the potential energy of the rod in this position?
(vi) Give an example where a force does work on a body but fails to change its K-E.
(vii) Does K-E depend upon the direction of motion involved? Can it be negative? Does its value depend on frame of refrence?
• 2)
Work is said to be done by a force acting on a body, provided the body is displaced actually in any direction except in a direction perpendicular to the direction of the force-mathematically, $W=\bar{F} \cdot \bar{s}=F s \cos \theta$ whereas energy is the capacity of a body to do the work and Power is the rate at which the body do the work.
$P=\frac{\mathrm{W}}{t}=\frac{\overline{\mathrm{F}} \cdot \bar{s}}{t}=\overline{\mathrm{F}} \cdot \bar{v}$
Both, work and energy are measured in Joule while power is measured in watt.
(i) A box is pushed through 4.0 m across a floor offering 100 N resistance. Determine the work done by the applied force.
(ii) In the above question, determine the work done by the resistive force and by the gravity.
(iii) A truck draws a tractor of mass 1000 kg at a steady rate of 20 ms-1 on a level road. The tension in the coupling is 2000 N. What is the power spent on the tractor?
(iv) Determine the work done on the tractor in one minute?
• 3)
The work done by a constant force acting on a body is given by $W=\bar{F} \cdot \bar{r}, \text { where } \bar{F}$ is the force vector and $\bar{r}$ is displacement vector. the displacement vector $\bar{r}=\bar{r}_{2}-\bar{r}_{1}$ where r 1 is the initial position vector and $\bar{r}_{2}$ is the final position vector. If the force is variable, the work done in moving a body from a Position $\bar{r}$ to a position $\bar{r}_{2}$ is given by $W=\int_{v}^{r_{2}} \bar{F} \cdot d \bar{r}, \text { where } d \bar{r}$ is an infinitesimally small displacement.
(i) A particle is moved from a position $\bar{r}_{1}=(3 \hat{i}+$$2 \hat{j}-4 \hat{k})$ metre to position $\bar{r}_{2}=(5 \hat{i}+6 \hat{j}+9$$\hat{\boldsymbol{k}})$ metre under the action of a force $\overline{\boldsymbol{F}}=\mathbf{(} \hat{\boldsymbol{i}}+$ $\mathbf{3} \hat{\boldsymbol{j}}+\hat{\boldsymbol{k}})$ newton. Determine the net work done.
(ii) A body of mass m is projected from a tower of height h at angle $\theta$ above the horizontal. Determine the workdone by the gravitational force during the time it takes to hit the ground.
(iii) A body of mass m is projected from the ground with a velocity u at an angle $\theta$ above the horizontal. Determine the work done by the gravitational force in time $t=\frac{u \sin \theta}{g}$
(iv) In the above question if $\theta$ = 45°, then determine the gravitational force in time t = $\frac{2 u \sin \theta}{g}$
(v) A force F acting on an object varies with distance x as shown in Fig. Force is in Nand x is m. Determine the work done by the force in moving the object from x = 0 to x = 6 m.
(vi) The Potential energy of a 1kg particle free to move along the x-axis is given by $V(x)=\left(\frac{x^{4}}{4}-\frac{x^{2}}{2}\right) \mathrm{J}$ .The total mechanical energy of the particle is 2J. Determine the maximum speed of the particle in m/s.
• 4)
In a conservative force field, we can find the component of force from the potential energy at a point in the field. A positive force means repulsion and a negative force means attraction. From the given potential energy function U (r) we can find the equilibrium position where force is zero. Suppose the potential energy at a distance r from centre of the field is given as
$\mathrm{U}(r)=\frac{A}{r^{2}}-\frac{B}{r}$Where A and B are positive constants.
(i) What should be the nature of the field as per conclusion drawn from the form of given potential energy?
(ii) Determine the work done to move the particle from equilibrium to infinity.
(iii) If K.E of a body becomes 4 times of its initial value, then what would be new linear momentum?
(iv) Determine the percentage change in K.E of a body if momentum of a body increases by 0.01 %
(v) What does it meant by unstable equilibrium of a particle? Write condition for unstable equilibrium.
(vi) What are conservative forces?
• 5)
A ball P moving with a velocity u strikes an identical stationary ball Q such that after the collision, the direction of motion of balls P and Q makes an angle 30° with the original direction of motion of ball P as shown.
(i) Determine the speed V1 of ball P after collision.
(ii) Determine the ratio of the total kinetic energy of the balls after collision to that before collision.
(iii) Determine the ratio of velocity V1 and v 2 after the collision in terms of coefficient of restitution e.
(iv) A ball hits a floor and rebounds after an inelastic collision. What change would occur in total energy, kinetic energy and momentum of ball?
#### Class 11th Physics - Motion in a Plane Case Study Questions and Answers 2022 - 2023 - by Study Materials - View & Read
• 1)
Physical quantities are broadly classified into vectors and scalars. Physical quantities which have a sense of direction are known as vectors, these are categorised as polar and axial. Vectors cannot be added by simple laws of algebra but by adopting vector algebra. Resultant of two or more vectors could be zero depending upon their magnitude and direction. When resultant of three vectors are zero or null then Lami's theorem is applicable. Vector substraction is similar to addition where vector addition of one vector and negative vector of other is done. In our daily life vector addition and vector substraction is very important as far as application of mechanical force is concerned.
(i) Define vector physical quantity. Give example.
(ii) Distinguish between Polar and Axial vectors. Give example
(iii) What are unit vector and null vector?
(iv) State Lami's theorem of equilibrium of three vectors.
(v) State triangle law of vector addition.
• 2)
An aeroplane is flying with velocity $\vec{v}_{p}$ (= 100 ms-1) towards east, with respect to ground through motionless air and $\vec{v}_{w}$ is the wind velocity with respect to ground. The total velocity of aeroplane is $\vec{v}=\overrightarrow{v_{p}}+\overrightarrow{v_{w}}$. The magnitude of the velocity is often called speed. The heading of the plane is the direction in which the nose of the plane points. Infact, it is the direction in which the engine propels the plane. Resultant velocity and direction is the vector sum of different components of velocity, while relative velocity is the velocity of one with respect to another obtained by vector substraction.
(i) What is resultant velocity of plane if the wind velocity is towards south with respect to ground at 25 ms-1?
(ii) Define relative velocity? How it is determined?
(iii) If the wind blows with velocity 25 ms-1 northwards, then by which angle the plane velocity is deflected from east?
(iv) Vectors cannot be added algebraically. Why?
(v) Mention the condition when the magnitude of vectors $\overline{\mathbf{A}}-\overline{\mathbf{B}}$and $\overline{\mathbf{A}}+\overline{\mathbf{B}}$ is same.
• 3)
Projectile is the name given to a body thrown with some initial velocity with the horizontal direction and then allowed to move in two dimensions under the action of gravity alone, without being propelled by any engine or fuel. A projectile moves under the combined effect of two velocities: one a uniform velocity in the horizontal direction and other a uniformly changing velocity. It is observed that path of projectile which is known as trajectory is parabolic in shape.
(i) Does the time of flight, horizontal range and maximum height depend on the mass of the projectile?
(ii) State the relation between the maximum height attained by the projectile and the maximum range.
(iii) What is the angle between velocity and acceleration at the highest point of projectile path?
(iv) A body is projected with velocity u at angle e with the horizontal, what would be the angle and speed of projectile when it strike at the ground at same horizontal plane from which it is projected?
(v) What are the angles of projection for same initial velocity of projection at which horizontal range of the projectile is same?
(vi) Write the equation of the path of projectile, projected at angle e with initial velocity u, from a horizontal plane.
• 4)
If a body moves in circle with a constant speed, then the magnitude of the velocity is constant but the direction of the velocity vector is changing all the time. Thusvelocity is changing with time. Therefore the motion of the body is accelerated. The acceleration is directed towards the centre of the circle and is called centripetal acceleration ac. If m is the mass of the body then the centripetal force is Fc = mac·
(i) What is effect on the linear momentum, angular momentum and kinetic energy of body, moving uniformly in circular path?
(ii) What is the magnitude and direction of ac of body moving in circular path of radius r with speed v?
(iii) A stone of mass m is tied to a string is whirled in circular path of radius r in horizontal plane at a constant speed v. What is the angle between velocity and acceleration? Also determine the tension in the string.
(iv) State the relation between linear velocity (v) and angular velocity (ω) of body moving in circular path of radius r,
(v) What would be the direction of motion of stone if at certain time its string get broken?
(vi) In a non-uniform circular motion with tangential acceleration at and radial acceleration ar. What would be the resultant acceleration?
#### Class 11th Physics - Motion in a Straight Line Case Study Questions and Answers 2022 - 2023 - by Study Materials - View & Read
• 1)
It must be clearly understood that distance is not the same as displacement. Distance is a scalar quantity and is given by the total length of the path travelled by the body in a certain interval of time. Displacement is a vector quantity and is given by the shortest distance (in a specified direction) between the initial and the final positions of the body. The direction of the displacement vector is from the initial position to the final position of the motion. Speed IS a scalar quantity. The average speed and average velocity are different in many respect. The direction of the velocity vector is the same as that of the displacement vector. Acceleration is defined as the rate of change of velocity and it is a vector quantity.
(i) Mention a condition when displacement and distance are both equal.
(ii) Define average speed and average velocity.
(iii) Draw position-time graph of uniform accelerated motion.
(iv) What does the area under velocity-time graph and time axis signifies?
(v) What does the slope of position-time graph and velocity-time graph represent at any instant?
(vi) Mention a condition when body is at rest but still it has acceleration.
(vii) A body is moving in circular parth with uniform speed. What is the acceleration and average velocity during one complete revolution?
• 2)
A sports car passing a police checkpost at 60 km h-1 immediately started slowing down uniformly until its speed was 40 km h-1. It continued to move at the same speed until it was passed by a Police car 1 km from the check post. This police car had started from rest at the check post at the same instant as the sport car had passed the check post. The police car had moved with a constant acceleration until it had passed sports car. Assuming that the time taken by the sports car in slowing down from 60 km h-' to 40 km h-' was equal to the time that it travelled at constant speed before passed by the Police car.
(i) Determine the average velocity of sports car.
(ii) Determine the speed of the police car at the instant when it passed the sports car.
(iii) Determine the acceleration of the police car.
(iv) Draw the velocity-time graph for police car and sport car.
(v) How much time taken by police car to pass the sports car?
• 3)
Kinematics is the branch of mechanics which deals with the study of motion of material objects without taking into account the factors affecting the motion. Rest and motion are relative concept and nothing is absolute. The Position of the object at a given instant of time is described in terms of position coordinates. The coordinate system along with a clock constitutes a frame of reference. Frame of reference can be of two types viz: inertial frame of reference and non-inertial frame of reference. When position of body change in a frame of reference, it is said to be in motion which is categorised as uniform and non-uniform. Motion of a body is studied in terms of position-time graph and velocity-time graph.
(i) "Rest and motion are relative not absolute." Comment.
(ii) What are different types of frames of reference? Explain.
(iii) Draw position-time graph for uniform and Non-uniform motion.
(iv) Draw velocity-time graph for uniform and non-uniform motion.
(v) Relative veolcity of two bodies is zero. What is nature of Position-time graph for it?
#### Class 11th Physics - Units and Measurements Case Study Questions and Answers 2022 - 2023 - by Study Materials - View & Read
• 1)
The dimensional method is a very convenient way of finding the dependence of physical quantity on other physical quantities of a given system. This method has its own limitations. In a complicated situation, it is often not easy to guess the factors on which a Physical quantity will depend. Secondly, this method gives no information about the dimensionless proportionality constant. Thirdly this method is used only if a Physical quantity depends on the product of other physical quantities. Fourthly this method will not work if a physical quantity depends on another quantity being a trignometric or exponential function. Finally this method does not give complete information in cases where a Physical quantity depends on more than three quantities in problems in mechanics.
(i) What are Physical quantity? Write their types.
(ii) What are dimensional formula of a physical quantity?
(iii) Which principle does the method of dimensional analysis use?
(iv) What is principle of homogeneity?
(v) Write two limitations of dimensional analysis.
(vi) Write one use of dimensional analysis.
• 2)
In the study of Physics, we often have to measure the physical quantities. The numerical value of a measured quantity can only be approximate as it depends upon the least count of the measuring instrument used. The number of significant figures in any measurement indicates the degree of precision of that measurement. The importance of significant figures lies in calculation. A mathematical calculation cannot increase the precision of a physical measurement. Therefore, the number of significant figures in the sum or product of a group of numbers cannot be greater than the number that has the least number of significant figures because a chain cannot be stronger than its weakest link. The difference in the true value and the measured value of a quantity is the measure of error in measurement.
(i) What are significant figures?
(ii) Define least count
(iii) Is there any relation between precision and accuracy?
(iv) State the relationship between significant figures and precision.
(v) Determine the maximum error in the addition of two physical quantities.
(vi) How Systematic errors can be minimised?
• 3)
To measure distance or length between two points, various methods are adopted which are categorised as direct and indirect methods. Direct methods are used to measure shorter but accessible distances while indirect methods are used for inaccessible distance. Large distances, such as the distance of a planet or star from earth can be measured by the Parallax method. Parallax is the change in position of an object with respect to fixed background when object is seen from two different positions. Parallex method is used for measuring distances of stars which are less than 100 light years away.
(i) Name the instruments that are used to measure distance of the order 10- 5 m.
(ii) Define the terms 'basis' and parallactic angle used in Parallax method for measuring distance.
(iii) What is the measure of basis in measuring distance of a nearby star by Parallax method?
(iv) Why Parallax method is used for measuring distances of nearby stars only?
(v) What is meant by angular diameter of moon?
#### Class 11th Physics - Physical World Case Study Questions and Answers 2022 - 2023 - by Study Materials - View & Read
• 1)
Physics has revealed that all the forces occuring in different contexts arise from a small number of fundamental forces in nature. Though the origin of derived forces is complex, yet they can be understood in terms of the fundamental forces which govern the macroscopic as well as microscopic world. These fundamental forces are gravitational force, weak nuclear forces, electromagnetic forces, strong nuclear forces. Among these gravitational and electromagnetic forces have some common characteristics like being central, conservative, long range and obeying inverse square law while weak nuclear forces and strong nuclear forces are of nuclear origin, microscopic and very short range attractive forces. All these forces govern the diverse phenomena of physical world.
(i) What is the relative strengths of four types of basic forces in nature?
(ii) In which phenomenon does weak nuclear forces involve? Explain.
(iii) What are graviton and photon?
(iv) What does the inverse square law represent? Show graphically.
(v) What is range of weak nuclear force and strong nuclear force?
• 2)
Physics is a basic discipline in the category of natural sciences. It refers to the study of the physical world and it is devoted to the study of nature and natural phenomenon. In the study of physics, there are two principal thrusts: unification and Reductionism. Unification means attempt to explain the diverse physical phenomenon in terms of a few concepts and laws, while reductionism attempt to derive the properties of a bigger, more complex system from the properties of its constituent simple parts.
(i) Mention the origin of the word 'Physics' and write its meaning.
(ii) Name some phenomena which are unified by Newton's law of gravitation.
(iii) Name the equation that unified basic laws of electromagnetism.
(iv) What reductionism has done in thermodynamics to explain some macroscopic quantities?
(v) What are main five branches of Physics?
(vi) What is Physics?
• 3)
The two main domains of interest in Physics are macroscopic and microscopic. The macroscopic domain includes the study of Phenomena involving objects offinite size, this makes up classical Physics, developed upto the year 1900. The microscopic domain includes the study of Phenomenon involving molecules, atoms, nuclei, electron and other. elementary particles. This makes up modern physics developed after the year 1900. Overall Physics has played a key role in the development of many other branches of science.
(i) Mention the different branches of classical Physics.
(ii) What are limitations of classical Physics?
(iii) Which theory explains properly the microscopic domain.
(iv) What does the branch of Physics, Thermodynamics deals with?
(v) Explain how Physics is in relation to mathematics?
#### 11th Standard CBSE Physics Annual Exam Model Question 2020 - by Girish - Chennai - View & Read
• 1)
The range of strong nuclear force is about
• 2)
The density of a cube is measured by measuring its mass and the length of its sides. If the maximum errors in the measurement of mass and length are 3% and 2% respectively, then the maximum error in the measurement of density is
• 3)
In case of a moving body
• 4)
A particle moves on a given line with a constant speed $\upsilon$. At a certain time it is at a point P on its straight line path. O is fixed point. The value of $\overrightarrow { OP } \times \overrightarrow { \upsilon }$ is (where y is perpendicular distance from O to given line)
• 5)
An insect is crawling up on the concave surface of a fixed hemispherical bowl of radius R. If the coefficient of friction is ${1\over3}$ then the height up to which the insect can crawl is nearly,
#### 11th Standard CBSE Physics Public Exam Sample Question 2020 - by Girish - Chennai - View & Read
• 1)
Who proposed the wave theory?
• 2)
The dimensions of entropy are
• 3)
The displacement x of a particle varies with time according to the relation x=$\frac { a }{ b }$(1-e-bt). Then
• 4)
A particle moves on a given line with a constant speed $\upsilon$. At a certain time it is at a point P on its straight line path. O is fixed point. The value of $\overrightarrow { OP } \times \overrightarrow { \upsilon }$ is (where y is perpendicular distance from O to given line)
• 5)
A particle of mass 5 kg is pulled along a smooth horizontal surface by a horizontal string. The acceleration of the particle is 10 ms-2. The tension in the string is
#### 11th Standard CBSE Physics Public Exam Important Question 2019-2020 - by Girish - Chennai - View & Read
• 1)
The range of strong nuclear force is about
• 2)
A wire has a mass 0.3 ± 0.003 g, radius 0.5 ± 0.005 mill and length 6 ± 0.06 cm. The maximum percentage error in the measurement of its density is
• 3)
Distance-time graph of a body at rest is
• 4)
A plane is indined at an angle of 30° with horizontal. The magnitude of component of a vector $\overset\rightarrow{A}$=-10$\hat{k}$ perpendicular to this plane is (here z-direction is vertically upwards
• 5)
If the tension in the cable supporting an elevator is equal to the weight of the elevator, the elevator may
#### 11th Standard Physics Board Exam Sample Question 2020 - by Girish - Chennai - View & Read
• 1)
The range of strong nuclear force is about
• 2)
The velocity of a body moving in viscous medium is given by v =$\frac { A }{ B } \left[ 1-{ e }^{ \frac { -t }{ b } } \right]$where t is time, A and B are constants .Then the dimensions ot A are
• 3)
In case of a moving body
• 4)
If $\overrightarrow { { a }_{ 1 } }$ and $\overrightarrow { { a }_{ 2 } }$ are two non collinear unit vectors and if $\left| \overrightarrow { { a }_{ 1 } } +\overrightarrow { { a }_{ 2 } } \right|$ =$\sqrt{3}$, then the value of $\left( \overrightarrow { { a }_{ 1 } } -\overrightarrow { { a }_{ 2 } } \right) .\left( 2\overrightarrow { { a }_{ 1 } } +\overrightarrow { { a }_{ 2 } } \right)$ is
• 5)
A particle of mass 5 kg is pulled along a smooth horizontal surface by a horizontal string. The acceleration of the particle is 10 ms-2. The tension in the string is
#### 11th Standard Physics Board Exam Model Question 2019-2020 - by Girish - Chennai - View & Read
• 1)
Who proposed the wave theory?
• 2)
The SI units of the universal gravitational constant G are
• 3)
The displacement of an object at any instant is given by x = 30 + 20 t2, where x is in metres and t in seconds. The acceleration of the object will be
• 4)
If the resultant of three forces $\overrightarrow { F } _{ 1 }=p\hat { i } +3\hat { j } -\hat { k } ,\overrightarrow { F } _{ 2 }$and $\overrightarrow { F } _{ 3 }=6\hat { i } -\hat { k }$ acting on a particle has a magnitude equal to 5 units, then the value of p is
• 5)
A rectangular body is held at rest by pressing it against a vertical wall. Which of the following is generally true?
#### CBSE 11th Physics - Public Model Question Paper 2019 - 2020 - by Girish - Chennai - View & Read
• 1)
Who proposed the wave theory of light?
• 2)
A cylindrical solid of mass M has raidus R and length L. Its moment of inertia about a generator is:
• 3)
A black body is at 727°C. It emits energy at a rate which is proportional to
• 4)
Which of the following statements is true?
#### CBSE 11th Physics - Waves Model Question Paper - by Girish - Chennai - View & Read
• 1)
The time period ofmass suspended from a spring is T. Ifthe spring is cut into four equal parts and the same mass is suspended from one of the parts, then the new time period will be
• 2)
Two pulses in a stretched string whose centres are initially 8 cm apart are moving towards each other as shown in figure. The speed of each pulse is 2 cms-1. After 2 second, the total energy of the pulses will be
• 3)
A transverse wave propagating along X-axis is represented by y(x, t) = 8.0 sin(0.5 πx-4πt-π/4) where x is in metre and t is in seconds. The speed of the wave is
• 4)
Which of the following statements is true?
• 5)
A source X of unknown frequency produces 8 beats per second with a source of 250 Hz and 12 beats per second with a source of 270 Hz. The frequency of the source X is
#### CBSE 11th Physics - Oscillation Model Question Paper - by Girish - Chennai - View & Read
• 1)
A simple pendulum of frequency n is taken upto a certain height above the ground and then dropped along with its support so that it falls freely under gravity. The frequency of oscillations of the falling pendulum will
• 2)
The length of a simple pendulum is increased by 44%. What is the percentage increase in its time period?
• 3)
A particle executing simple harmonic motion along y-axis has its motion described by the equation y = A sin (ωt) + B. The amplitude of the simple harmonic motion is
• 4)
Masses in and 3m are attached to the two ends of a spring of constant k. If the system vibrates freely, the period of oscillation will be
• 5)
The following are the quantities associated with a body performing SHM.
1. The velocity of the body.
2. The accelerating of the body.
3. The accelerating force acting on the body.
Which of these quantities are exactly in phase with each other?
#### CBSE 11th Physics - Kinetic Theory Model Question Paper - by Girish - Chennai - View & Read
• 1)
According to kinetic theory of gases the r.m.s. velocity of the gas molecules is directly proportional to
• 2)
The speed of sound in a gas is v. The rms speed of molecules of this gas is C. If $\Upsilon =\frac { { C }_{ P } }{ C_{ V } }$ then the ratio of v and C is
• 3)
A sealed container with negligible thermal coefficient of expansion contains helium (a monoatomic gas). When it is heated from 300 to 600 K, the average kinetic energy of the helium atom is
• 4)
During an adiabatic process, the pressure of a gas is proportional to the cube of its absolute temperature. The value of Cp/ C v for that gas is
• 5)
Two vessels having equal volume contain molecular hydrogen at one atmosphere and helium at two atmosphere pressure respectively.If both samples are at the same temperature the mean velocity of hydrogen molecule is
#### CBSE 11th Physics - Thermodynamics Model Question Paper - by Girish - Chennai - View & Read
• 1)
The internal energy of an ideal gas depends on:
• 2)
In an adiabatic change, the pressure P and temperature T of a diatomic gas are related by the relation P ∝ TC where C equals
• 3)
An ideal heat engine exhosting heat at 27°C is to have 25%efficiency. It must take heat at:
• 4)
For a gas, r = 1.4 then atomicity, CP, and CV of the gas are
• 5)
The given quantity of an ideal gas is at pressure P and absolute Temperature T. The isothermal bulk Modulus of the gas is
#### CBSE 11th Physics - Thermal Properties of Matter Model Question Paper - by Girish - Chennai - View & Read
• 1)
Black body radiation is white. Comment.
• 2)
Give the relation between celsis, fahrenheit and reaumur scale temperature.
• 3)
Van temperature on celsius scale and kelvin scale related?
• 4)
Two bodies at different temperatures T1 and T2, if brought in thermal contact do not necessarily settle at the mean temperature $\frac { ({ T }_{ 1 }+{ T }_{ 2 }) }{ 2 }$ . Why?
• 5)
Usually a good conductor of heat is a good conductor of electricity also. Give reason.
#### CBSE 11th Physics - Mechanical Properties of Fluids Model Question Paper - by Girish - Chennai - View & Read
• 1)
The Bernauli's Theorem is based on the conservation of:
• 2)
The mass of water rises in capillary tube of radius R is M. The mass of water that rises in tube of radius 2R is
• 3)
Two small drops of mercury, each of radius R, coalesce to form a single large drop. The ratio of the total surface energies before and after the change is:
• 4)
For a ball falling in a liquid with constant velocity, ratio of resistance force due to the liquid to that due to gravity is
• 5)
A cylindrical vessel is filled with water upto height H. A hole is bored in the wall at a depth h from the free surface of water. For maximum range, h is equal to
#### CBSE 11th Physics - Mechanical Properties of Solids Model Question Paper - by Girish - Chennai - View & Read
• 1)
Dimensional formula of stress is same as that of
• 2)
Young's modulus of a material has the same unit as
• 3)
Elastic limit is equal to
• 4)
Which of the following is not a unit of Young's modulus?
• 5)
A wire suspended vertically from one end, is stretched by attaching a weight 200 N to the lower end. The weight stretches the wire by 1 mm. The energy gained by the wire is
#### CBSE 11th Physics - Gravitation Model Question Paper - by Girish - Chennai - View & Read
• 1)
A satellite is orbiting the earth. If its distance from the earth is increased, its
• 2)
If g is the acceleration due to gravity on the earth's surface, the gain in the potential energy of an object of mass m raised from the earth's surface to a height equal to the radius R of the earth,is
• 3)
If three uniform spheres, each having mass M and radius r, are kept in such a way that each touches the other two, the magnitude of the gravitational force on any sphere due to the other two is
• 4)
A satellite of mass m revolves around the earth of radius R at a height x from its surface. If g is the acceleration due to gravity on the surface of the earth, the orbital speed of the satellite is
• 5)
If a particle is fired vertically upwards from the surface of earth and reaches a height of 6400 km, the initial velocity of the particle is (assume R = 6400 km and g = 10 ms-2)
#### CBSE 11th Physics - System of Particles and Rotational Motion Model Question Paper - by Girish - Chennai - View & Read
• 1)
A couple produces a:
• 2)
A cylindrical solid of mass M has raidus R and length L. Its moment of inertia about a generator is:
• 3)
A man of mass M is standing at the centre of a rotating turn table rotating with an angular velocity w. The man holds two 'dumb bells' of mass M/4 each in each of his two hands. If he stretches his arms to a horizontal position, the turn table acquires a new angular velocity w' where
• 4)
A particle performing uniform circular motion has angular momentum L. If its angular frequency is doubled and its kinetic energy halved, then the new angular momentum is
• 5)
A loaded spring gun of mass M fires a 'shot' of mass m with a velocity $\vartheta$ at an angle of elevation $\theta$. The gun is initially at rest on a horizontal frictionless surface. After firing, the centre of mass of the gun-shot system
#### CBSE 11th Physics - Work, Energy and Power Model Question Paper - by Girish - Chennai - View & Read
• 1)
Equal masses (m each) are attached at the two ends of a string passing over two pulleys. Another mass is attached at the centre of the string. In order that there is no sag in the string, this mass should be
• 2)
The work done by all the forces (external and internal) on a system equals the change in
• 3)
A heavy stone is thrown from a cliff of height h with a speed v. The stone will hit the ground with maximum speed if it is thrown
• 4)
Two bodies of masses m and 4 m are moving with equal kinetic energy. The ratio of their linear momenta is
• 5)
Two bodies of masses m and 4 m are moving with equal linear momentum. The ratio of their kinetic energies is
#### CBSE 11th Physics - Laws of Motion Model Question Paper - by Girish - Chennai - View & Read
• 1)
If the tension in the cable supporting an elevator is equal to the weight of the elevator, the elevator may
• 2)
The dimension of Impulse is
• 3)
A force of 200 N is required to push a car of mass 500 kg slowly at constant speed on a level road. If a force of 500 N is applied, the acceleration of the car (in m S-2) will be
• 4)
A block of mass m is placed on a smooth inclined plane of inclination $\theta$ with the horizontal. The force exerted by the plane on the block has a magnitude
• 5)
An insect is crawling up on the concave surface of a fixed hemispherical bowl of radius R. If the coefficient of friction is ${1\over3}$ then the height up to which the insect can crawl is nearly,
#### CBSE 11th Physics - Motion in a Plane Model Question Paper - by Girish - Chennai - View & Read
• 1)
If $\overrightarrow { { a }_{ 1 } }$ and $\overrightarrow { { a }_{ 2 } }$ are two non collinear unit vectors and if $\left| \overrightarrow { { a }_{ 1 } } +\overrightarrow { { a }_{ 2 } } \right|$ =$\sqrt{3}$, then the value of $\left( \overrightarrow { { a }_{ 1 } } -\overrightarrow { { a }_{ 2 } } \right) .\left( 2\overrightarrow { { a }_{ 1 } } +\overrightarrow { { a }_{ 2 } } \right)$ is
• 2)
The sum of magnitudes of two forces acting at a point is 18 units and the magnitude of their resultant is 12 units. The resultant is at 90° with the force of the smaller magnitude. The magnitude of the individual forces is
• 3)
If the resultant of three forces $\overrightarrow { F } _{ 1 }=p\hat { i } +3\hat { j } -\hat { k } ,\overrightarrow { F } _{ 2 }$and $\overrightarrow { F } _{ 3 }=6\hat { i } -\hat { k }$ acting on a particle has a magnitude equal to 5 units, then the value of p is
• 4)
A particle moves on a given line with a constant speed $\upsilon$. At a certain time it is at a point P on its straight line path. O is fixed point. The value of $\overrightarrow { OP } \times \overrightarrow { \upsilon }$ is (where y is perpendicular distance from O to given line)
• 5)
From the top of a tower of height 40 m, a ball is projected upwards with a speed of 20 m/ s at an angle of elevation of 30°. The ratio of the total time taken by the ball to hit the ground to its time of flight (time taken to come back to the same elevation) is (Take g = 10 m/s2)
#### CBSE 11th Physics - Motion in a Straight Line Model Question Paper - by Girish - Chennai - View & Read
• 1)
The displacement x of a particle varies with time according to the relation x=$\frac { a }{ b }$(1-e-bt). Then
• 2)
The displacement of an object at any instant is given by x = 30 + 20 t2, where x is in metres and t in seconds. The acceleration of the object will be
• 3)
The area under the velocity time graph between any two instants t = t1 and t = t2 gives the distance covered in a time $\delta$ t = t2 - t1.
• 4)
Which of the following is not a vector quantity?
#### CBSE 11th Physics - Units and Measurements Model Question Paper - by Girish - Chennai - View & Read
• 1)
The SI units of magnetic field is
• 2)
The dimensions of energy per unit volume are the same as those of
• 3)
A physical quantity is represented by X = Ma Lb T-c . If percentage error in the measurement of M, Land Tare $\alpha$%, $\beta$% and $\gamma$% respectively, then total percentage error is
• 4)
'Parsec' is the unit of:
• 5)
A wire has a mass 0.3 ± 0.003 g, radius 0.5 ± 0.005 mill and length 6 ± 0.06 cm. The maximum percentage error in the measurement of its density is
#### CBSE 11th Physics - Physical World Model Question Paper - by Girish - Chennai - View & Read
• 1)
Who gave Universal Law of Gravitation?
• 2)
Who discovered scattering of light?
• 3)
Which scientific principle is steam engine based on:
• 4)
Who gave theory of relativity?
• 5)
Who proposed the wave theory of light?
#### CBSE 11th Physics - Full Syllabus One Mark Question Paper with Answer Key - by Girish - Chennai - View & Read
• 1)
Which scientific principle is steam engine based on:
• 2)
Who proposed the wave theory of light?
• 3)
Who proposed the wave theory?
• 4)
• 5)
Who discovers famous theory of relativity?
#### CBSE 11th Physics - Full Syllabus Five Marks Questions - by Girish - Chennai - View & Read
• 1)
One mole of an ideal gas at standard temperature and pressure occupies 22.4 L(molar volume). What is the ratio of molar volume to the atomic volume of a mole of hydrogen? (Take the size of hydrogen molecule to be about 1$\mathring{A}$). Why is the ratio so large?
• 2)
Find an expression for viscous force F acting on a tiny steel ball of radius r moving in a viscous liquid of viscosity $\eta$ with a constant speed v by the method of dimensional analysis.
• 3)
Briefly explain how you will estimate the molecular diameter of oleic acid.
• 4)
The speed-time graph of a particle moving along a fixed direction is shown in Fig. Obtain the distance traversed by the particle between
(a) t = 0 s to 10 s.
(b) t = 2 s to 6 s.
What is the average speed of the particle over the intervals in (a) and (b)?
• 5)
Derive the three basic kinematic equations by calculus method.
#### CBSE 11th Physics - Full Syllabus Four Marks Questions - by Girish - Chennai - View & Read
• 1)
Write down the number of significant figure in the following.
12.000
• 2)
In successive measurements, the reading of the period of oscillation of a simple pendulum were found to be 2.63 s,2.56 s,2.42 s,2.71 s and 2.80 s in an experiment.Calculate express the result in proper form.
• 3)
Check whether the given equation is dimensionally correct $\frac { 1 }{ 2 } mv^{ 2 }=mgh$
• 4)
Shruti goes to school with his brother Alok in their own car.The school is about 10km apart from their home.They drive on alternate days.Alok is a very careful driver but Shruti drives rashly.She takes 3 min less than Alok in reaching the school.Alok advices Shruti to drive safely but she hardly listens.
What is the difference between average speeds of Shruthi and Alok if later takes 15 min to drive to the school?
• 5)
The two thigh bones(femurs), each of cross-sectional area 10 cm2 support the upper part of a human body of mass 40 kg. Estimate the average pressure sustained by the femurs.
#### CBSE 11th Physics - Full Syllabus Three Marks Questions - by Girish - Chennai - View & Read
• 1)
Write in about 1000 words, a fiction piece based on your speculation on the science and technology of the twenty second century.
• 2)
Write in about 100 words a fiction piece based on your speculation on the science and technology of the twenty-second century.
• 3)
Compute the following with regards to significant figures.
$4.6\times 0.128$.
• 4)
A train 500m long crosses a bridge of 1000 m in 10s. Find the average speed of the train when it just crosses the bridge.
#### CBSE 11th Physics - Full Syllabus Two Marks Questions - by Girish - Chennai - View & Read
• 1)
Name that branch of science which deals with the study of stars.
• 2)
A jeweller put a diamond weighing 5.42 g in a box weighing 1.2 Kg. Find the total weight of the box and the diamond to correct number of significant figures.
• 3)
The farthest objects in our universe discovered by modern astronomers are so distant that light emitted by them takes billions of years to reach the earth.These objects(known as quasars) have many puzzling features which have not yet been satisfactorily explained. What is the distance in km of a quasar from which light takes 3.0 billion years to reach us?
• 4)
What are uses of a velocity - time graph ?
#### 11th CBSE Physics - Waves Five Marks Model Question Paper - by Jemi Bhouseya - Indore - View & Read
• 1)
Two tuning forks A and B give 5 beats. A resounds with a closed column of air 15 cm long and B with an open column of ar 30.5 cm long. Caluculate their frequencies. Negelct and correction.
• 2)
A SONAR system fixed in a submarine operates at a frequency 40.0 KHz. An enemy submarine of 360 km/h. What is the frequency of sound reflected by the submarine? Take the speed of sound in water to be 1450 ${ ms }^{ -1 }$
• 3)
For the harmonic travelling wave $y=2\cos { 2\pi } \left( 10t-0.0080x+3.5 \right)$ , where, x and y are in cm and t is in second. What is the phase difference between the oscillatory motion at two points separated by a distance of $\frac { \lambda }{2 }$.
• 4)
For the harmonic travelling wave $y=2\cos { 2\pi } \left( 10t-0.0080x+3.5 \right)$, where, x and y are in cm and t is in second. What is the phase difference between the oscillatory motion at two points separated by a distance of What is the phase difference between the oscillation of a particle located at x=100 cm, at t=Ts and t = 5s?
• 5)
Explain why (or how) bats can ascertain distance, directions, nature and sizes of the obstacles without any eyes?
#### 11th Standard CBSE Physics - Oscillation Five Marks Model Question Paper - by Jemi Bhouseya - Indore - View & Read
• 1)
A particle is subjected to two simple harmonic otions in the same direction having equal magnitude and equal frequency. If the resultant amplitude is equal to the amplitude of the individual motion. Fnd the phase difference between two individual motions.
• 2)
A particle moving with SHM ina stright line has a speed of 6 m/s when 4 m/s from the centre of oscillation and a speed of 8 m/s when 3 m from the oscillation and the shortest time taken by the partile in moving from the extreme position to a point mid way between the extreme position and the centre.
• 3)
A spring of force constant ${ 1200\ Nm }^{ -1 }$is mounted on a horiontal table as a mass of 3 kg is attached to the free end of the spring, pulled sideways to adistance of 2.0 cm and released. What is
the speed of the mass, if the spring in compressed?
• 4)
A spring of force constant ${ 1200\ Nm }^{ -1 }$is mounted on a horiontal table as a mass of 3 kg is attached to the free end of the spring, pulled sideways to adistance of 2.0 cm and released. What is
total energy of the mass during oscillation?
• 5)
The motion of a particle executing simple harmonic motion is described by the displacement function.
x(t) = A cos (ωt + $\phi$)
If the initial (t = 0) position of the particle is I cm and its initial velocity is ω cm/s, what are its amplitude and initial phase angle? The angular frequency of the particle is $\pi$s-1. If instead of the cosine junction, we choose the sine junction to describe the SHM: x = B sin (ພt + α), what are the amplitude and initial phase of the particle with the above initial conditions?
#### 11th Standard CBSE Physics - Kinetic Theory Five Marks Model Question Paper - by Jemi Bhouseya - Indore - View & Read
• 1)
A box of 1.00 m3 is filled with nitrogen at 1.5 atm at 300 K. The box has a hole of an area 0.010 mm2 . How much time is required for the pressure to reduce by 0.10 atm, if the pressure outside is 1 atm.
• 2)
Given below are densities of some solids and liquids.Give rough estimate of the size of their atoms.
Substance Atomic Mass(u) Density (10-3 kgm-3) (i) Carbon(diamond) 12.01 2.22 (ii) Gold 197.00 19.32 (iii) Nitrogrn(liquid) 14.01 1.00 (iv) Lithium 6.94 0.53 (v) Fluorine(liquid) 19.00 1.14
[Hint : Assume the atoms to be ‘tightly packed’ in a solid or liquid phase, and use the known value of Avogadro’s number. You should, however, not take the actual numbers you obtain for various atomic sizes too literally. Because of the crudeness of the tight packing approximation, the results only indicate that atomic sizes are in the range of a few Å].
#### 11th CBSE Physics - Thermodynamics Five Marks Model Question Paper - by Jemi Bhouseya - Indore - View & Read
• 1)
A Carnot cycle is performed by 1 mole of air (r = 1.4) initially at 327o C. Each stage represents a compression or expansion in the ratio 1:6 Calculate the lowest temperature
Take R = 8.31 J/ mol-K
• 2)
A Carnot cycle is performed by 1 mole of air (r = 1.4) initially at 327o C. Each stage represents a compression or expansion in the ratio 1:6 Calculate network done during each side
Take R = 8.31 J/ mol-K
• 3)
A Carnot cycle is performed by 1 mole of air (r = 1.4) initially at 327o C. Each stage represents a compression or expansion in the ratio 1:6 Calculate efficiency of the engine
Take R = 8.31 J/ mol-K
• 4)
Two cylinders A and B of equal capacity are connected to each other via a stopcock. A contains a gas at standard temperature and pressure. B is completely evacuated. The entire system in thermally insulated. The stopcock is suddenly opened. Answer the following
What is the change in internal energy of the gas?
• 5)
Two cylinders A and B of equal capacity are connected to each other via a stopcock. A contains a gas at standard temperature and pressure. B is completely evacuated. The entire system in thermally insulated. The stopcock is suddenly opened.
What is the change in temperature of the gas?
#### 11th Standard CBSE Physics - Thermal Properties of Matter Five Marks Model Question Paper - by Jemi Bhouseya - Indore - View & Read
• 1)
A circular disc made by iron is rotating about its axis of rotation with a uniform angular speed $\omega$
Determine the change in the linear speed of particle at the rim in percentage. The disc of rim is slowly heated from 20o C to 50o C keeping the angular speed uniform. Give that coefficient of linear expansion for the material of iron is $1.2\times 10^{ -5\quad }$$^{0}$C-1
• 2)
A copper cube of mass 200 g slides down on a rough inclined plane having inclination 37o at a constant speed. If any loss in mechanical energy goes into the copper block as thermal energy. Find the increase in the temperature of the block as it slides down through 60 cm. Given, specific heat of copper is 420 J Kg-1K-1 .
• 3)
A brass wire 1.8 m long at 27o C is held taut with little tension between two rigid support. If the wire is cooled to a temperature of -39o C, what is the tension developed in the wire if its diameter is 2mm?
• 4)
(a) The triple point of water is a standard fixed point in modern thermometry. Why? What is wrong in taking the melting point of ice and the boiling point of water as standard fixed points (as was originally done in the Celsius scale)?
(b) There were two fixed points in the original Celsius scale as mentioned above which were assigned the number 0 °C and 100 °C respectively. On the absolute scale, one of the fixed points is the triple-point of water, which on the Kelvin absolute scale is assigned the number 273.16 K. What is the other fixed point on this (Kelvin) scale?
(c) The absolute temperature (Kelvin scale) T is related to the temperature tc on the Celsius scale by tc = T – 273.15 Why do we have 273.15 in this relation, and not 273.16?
(d) What is the temperature of the triple-point of water on an absolute scale whose unit interval size is equal to that of the Fahrenheit scale?
• 5)
What is the temperature of the triple point of water on an absolute scale whose unit interval size is equal to that of the fahrenheit scale?
#### CBSE 11th Physics - Mechanical Properties of Fluids Four and Five Marks Questions - by Girish - Chennai - View & Read
• 1)
The flow of blood in a larger artery of an anesthetised dog is diverted through a venturimeter. The wider part of the meter has a cross-sectional area equal to that of the artery, a1 = 8mm2. The narrow part has an area a2 = 4mm2. The pressure drop in the artery is 24 Pa. What is the speed of the blood in the artery?
• 2)
A fully loaded Boeing aircraft has a mass of 3.3$\times$105 kg. Its total wing area is 500 m2. It is in level flight with a speed of 960 km/h
Estimate the fractional increase in the speed of the air on the upper surface of the wing relative to the lower surface.[The density of air is $\rho$= 1.2 kgm-3
• 3)
Near the surface of the river, the velocity of water is 160 kmh -1 .Find the shearing stress between horizontal layers of waters, if the river is 6 m deep and the coefficient of viscosity of water is 10-2 poise.
• 4)
If 27 drops of rain were to be combine to form one new large spherical drop, then what should be the velocity of this large spherical drop? Consider the terminal velocity of 27 drops of equal size falling through the air is 0.20 ms-1
• 5)
The flow rate of water is 0.58 L/mm from a tap of diameter of 1.30 cm. After some times, the flow rate is increased to 4 L/min. Determine the nature of the flow for both the flow rates. The coefficient of visocity of water is 10-3 Pa-s and the density of water is 103 kg/m3.
#### CBSE 11th Physics - Thermal Properties of Matter Four and Five Marks Questions - by Girish - Chennai - View & Read
• 1)
Perfect Black Body
Calculate the temperature (in K) at which perfect black body radiates energy at the rate of 5.67 W/cm.Given, $\sigma$ = 5.67 x 10- 8 Wm-2 K-4
• 2)
A hot body having the surface temperature 13270C. Determine the wavelength at which it radiates maximum energy. Given wien's constant = 2.9 x 10-3 mK.
• 3)
Aman went for a weekend trip with his parents and grandparents to a remote village. His grandfather showed him the fields and the crops they grow. As they moved forward, they saw that a bullock cart got struck in wet mud and the driver was not able to push it out by himself. Seeing him in distress, Aman ran to his help and together they pushed it out, but iron rim of the wheel came out.
They tried to put it on the wheel but it was smaller than diameter of wheel. Suddenly, he got an idea. He collected some wood and set them on fire and heated the rim and then rim easily slipped on the wheel. Cartman thanked Aman and moved away.
If the diameter of the rim and ring were 5.243 m and 5.231 m respectively at 27 $^{0}$C. To what temperature had Aman heated the ring so as to fit the rim of the wheel? Coefficient of linear expansion of iron = 1.20 x 10-5 K-1
• 4)
Radha was studying when she noticed her 2 years old younger brother was learning towards kitchen where her mother had kept the boiling water.
Radha ran towards him and caught him just as he was about to touch the boiling water. She moved him to the side and fell on the floor but she was happy that she saved her brother from getting burnt.
At atmospheric pressure, 2g of water having volume of 2.00 cm3 becomes 3342 cm3 of steam when boiled. The latent heat of vaporisation of water is 539 cal/g at 1 atm. What is the amount of heat added tot he system?
• 5)
What do you understand by thermal resistance?
closed cubical box is made of perfectly insulating material and the only way for heat to enter or leave the box is through two solid cylindrical metal plugs, each of cross-sectional area 12 cmand length 8 em fixed in the opposite walls of the box. The outer surface of one plug is kept at a temperature of 100°C while the outer surface of other plug is maintained at a temperature of 4°C. The thermal conductivity of the material of the plug is 2.0 W/m-°C. A source of energy generating 13 W is enclosed inside the box. Find the equilibrium temperature of the inner surface of the box assuming that it is the same at all points on the inner surface
#### CBSE 11th Physics -Thermodynamics Four and Six Marks Questions - by Girish - Chennai - View & Read
• 1)
Give an example of each of given below Isochoric process
• 2)
Find the ratio of $\frac { \triangle Q }{ \triangle U } \quad and\quad \frac { \triangle Q }{ \triangle W }$ in an isobaric process. The ratio of molar specific heats, $\frac { C_{ p } }{ { C }_{ v } } =\gamma$ .
• 3)
1g of water at 1000C is converted into steam of the same temperature.If the volume of steam is 1551 cm3, find out the change in internal energy of the water.Latent heat of steam= $2256\times { 10 }^{ 3 }J/kg$ . Consider atm pressure.
• 4)
Consider a Carnot cycle operating between T= 500K and T= 300K producing 1KJ of mechanical work per cycle. Find the heat transferred to/by the engine by/to the reservoir.
• 5)
A refrigerator has to transfer an average of 263J of heat per second from temperature -100C to 250C.Calculate the average power consumed, assuming no energy losses in the process.
#### CBSE 11th Physics - Kinetic Theory Four and Five Marks Questions - by Girish - Chennai - View & Read
• 1)
The density of water is 1000 kg m–3. The density of water vapour at 100 °C and 1 atm pressure is 0.6 kg m–3.The volume of a molecule multiplied by the total number gives ,what is called, molecular volume. Estimate the ratio (or fraction) of the molecular volume to the total volume occupied by the water vapour under the above conditions of temperature and pressure. Estimate the volume of a water molecule.What is the average distance between atoms in water?
• 2)
Two non-reactive gases are kept in a container. The ratio of their partial pressures is given 5:3. Find the ratio of number of molecules.
• 3)
A gas at 270C in a cylinder has a volume of 4L and pressure 100 N/m2. If the gas is first compressed at constant temperature so that the pressure is 150 N/m2. Estimate the change in volume.
• 4)
A container is filled with a gas at a pressure of 76 cm of mercury at a certain temperature. The mass of a gas is increased by 50% by introducing more gas in the container at same temperature. Calculate the final pressure of the gas.
• 5)
If the mass of each molecule of a gas is halved and speed is doubled. Find the ratio of initial and final pressure.
#### CBSE 11th Physics - Waves Four and Five Marks Questions - by Girish - Chennai - View & Read
• 1)
One morning, sameer went to the washroom for taking bath. He noticed that the pitch of sound produced went on increasing as he noticed that the pitch of sound produced went on increasing as he opened the tap to fill an empty bucket with water. He was surprise to observe that as the bucket started filling with water, the pitch of sound become higher. He shared this with his Physics teacher in the physics period. On getting a solution of problem from the teacher, sameer felt satisfied and expressed gratitude to the teacher.
(i) When we start filling an empty bucket with water, the pitch of sound goes on increasing why?
• 2)
One day kapil had to go to his friend's house. He took hids mobile and headphone to listen to music while walking. On the way , he had to cross a railway line. He was so engrossed in the music that he did not hear the approaching train. Train driver was blowing the horn but Kapil did not seeing this ran towards kapil and pushed him away just as train reached there. Kapil realised his mistake and thanked that person.
(i) If a train emitting sound of frequency 500 Hz, is moving towards an observer with a speed of 30ms-1 then what is the frequency as heard by the listener?
• 3)
Shubham was good football player. but since last few dasy, he is suffering from stomach ache, could not attend and play matches. his parents took him to a doctor who examined and asked him to get an ultrasound done to detect the exact cause. Shubham was afraid of ultrasound scanner and refused to get it done. His parents made him understand that the scanner uses ultrasonic rays which go inside and detect any problem inside the body . So he got it done became fine again.
(i) On which principle does the ultrasonic scanner work? If the ultrasound uses the operating frequency of 4.2 MHz, the speed of sound in the tissue is 1.7 kms-1. What is the wavelength of the sound in tissue?
• 4)
An aeroplane is going towards East at speed of ${ 510kmh }^{ -1 }$at a height of 2000 m. At a certain instant, the sound of the plane heard by a ground observer appears to come from a point vertically above him.
What values do you learn from this problem?
• 5)
Given below are some examples of wave motion. State in each case if the wave motion is transverse, longitudinal or a combination of both
(i) Motion of a kink in a longitudinal spring produced by displacing one end of the spring sideways
(ii) Waves produced in a cylinder containing a liquid by moving its piston back and forth
(iii) Waves produced by a motorboat sailing in water
(iv) Ultrasonic waves in air produced by a vibrating quartz crystal
#### CBSE 11th Physics - Mechanical Properties of Solids Four and Five Marks Questions - by Girish - Chennai - View & Read
• 1)
A Cubical Body Gets Deformed
If the angle of shear is 30$^0$ for a cubical body and the change in length is 250 cm, then what must be the volume of this cubical body ?
• 2)
An Elongated Wire
If a wire of length 4 m and crosssectional area of 2m2 is stretched by a force of 3 KN, then determine the change in length due to this force. Given Young's modulus of material of wire 110 x 109 N/m2.
• 3)
Elongation of Copper Wire
A copper wire is streched by 10 N force. If radius of wire decreases by 2%. How will Young's modulus of wire be affected?
• 4)
Volumetric Analysis
What will be the decrease in vloume of 100 cm3 of water under pressure of 100 atm if the compressibility of water is 4 x 10-5 per unit atmospheric pressure?
• 5)
Shear Modulus is Less than Young's Modulus
The shear modulus of a material is always considerably smaller than the Young's modulus for it. What does it signify?
#### CBSE 11th Physics - Oscillation Four and Five Marks Questions - by Girish - Chennai - View & Read
• 1)
The physics teacher of class XI has assigned the work of finding the percentage change in the periodic time of a simple pendulum. Two students Rahul and Rohit made a theoretical study as well as verified experimentally whereas Rahul could not complete the work. When the teacher enquired the next day, Rohit could given the answer whereas Rahul could not
Calculate the percentage change in the periodic time of a simple pendulum. If the mass of the bob be increased by 30%
• 2)
The potential energy of a particle of mass 1 kg in motion along the x-axis is given by U = 4 (1 - cos 2x) J Here x is in metres . Find the period of small oscillations.
• 3)
A body of mass m falls from a height h on to the pan of a spring balance. The masses of the pan and spring are negligible. The spring constant of the spring is k. Having stuck to the pan the body starts performing harmonic oscillations in the vertical direction. Find the amplitude and energy of oscillation.
• 4)
Two linear simple harmonic motions of equal amplitudes and frequencies ω and 2ω are impressed on a particle along the axes of X and Y respective/yo If the initial phase difference between them is π/2 find the resultant path followed by the particle.
• 5)
A particle is vibrating in SHM when the displacements of the particle from its equilibrium position are x1 and x2 it has velocities v1 and v2 respectively. Show that its time period is given by $T=2\pi\sqrt{x_1^2-x_2^2\over v_2^2-v_1^2}$
#### CBSE 11th Physics -Gravitation Four and Five Marks Questions - by Girish - Chennai - View & Read
• 1)
What will be the value of g at the bottom of sea 7 km deep?Diameter of the earth is 12800 km and g on the surface of the earth is 9.8 ms-2 .
• 2)
As you will learn in the text, a geostationary satellite orbits the earth at a height of nearby 36000 km from the surface of the earth.What is the potential due to the earth's gravity at the site of this satellite?(take the potential energy at infinity to be zero).Mass of the earth = $6.0\times { 10 }^{ 24 }$kg, radius = 6400 km.
• 3)
An earth's satellite has a period of 90 min.Assuming the orbit to be circular, calculate its height.Take radius of the earth equal to 6380 km and g at the surface of the earth equal 9.8m/s2 .
• 4)
Shweta was reading a book on the biography of Issac Newton. She read that Newton was sitting under an apple tree when a falling apple led him to develop a whole new science of gravity. After reading the book, shweta realised that every phenomenon in universe has some scientific fact associated with it, it depends on us whether we look for a scientific fact or associate a superstition with it.
How can you find the mass of the earth using law of gravitation?
• 5)
Arjun was a student of class IX. He was sitting in a garden along with his grandmother, who was a retired physics teacher. Suddenly he saw an orange falling from the tree. Immediately he asked his grandmother that both of the orange and earth experience equal and opposite forces of gravitation, then why it is the orange that falls towards the earth and not the earth towards the orange. His grandmother explained him the reason in a simple way.
(i) What are the values being displayed by Arjun?
(ii) What in your opinion may be the reason for this observation?
#### CBSE 11th Physics -System of Particles and Rotational Motion Four and Five Marks Questions - by Girish - Chennai - View & Read
• 1)
Two bodies of masses 1 kg and 2 kg are located at (1, 2) and (-1, 3), respectively. Calculate the coordinates of the centre of mass.
• 2)
Centre of gravity of a body on the earth coincides with its centre of mass for a small object and for a large object, it may not. What is the qualiative meaning of small and large in this regards? For which following two of them coincides, a building, a pond, a lake, a mountain.
• 3)
The angular speed of a motor wheel is increased from 1200 rpm to 3120 rpm in 16s.
(i) What is its angular acceleration, assuming the acceleration to be uniform?
(ii) How many revolutions does the engine make during this time?
• 4)
To maintain a rotor at a uniform angular speed of 200 rad s-1, an engine needs to transmit a torque of 180 N m. What is the power required by the engine ? (Note: uniform angular velocity in the absence of friction implies zero torque. In practice, applied torque is needed to counter frictional torque). Assume that the engine is 100% efficient
• 5)
Since his childhood Sanjay had always seen his mother grinding flour in the grindstone. He had observed that his mother had to do a lot of hand work in order to get flour from wheat. He felt very helpless at that time. As he grew older he thought of an idea to connect an electric motor to the wheel of grindstone . Now, it become very easy to get flour with help of grindstone and now his mother is very happy and felt proud of his intelligence.
A grinding stone of diameter 4 m revolving at 120 rpm accelerates to 660 rpm in 9s. Calculate the angular acceleration and linear acceleration.
#### CBSE 11th Physics - Work, Energy and Power Four and Five Marks Questions - by Girish - Chennai - View & Read
• 1)
The nucleus Fe27 emits a $\gamma$-ray of energy 14.4 keV. If the mass of the nucleus is 56.935 amu, calculate the recoil energy of the nucleus.
• 2)
One day Pawan went to super market to purchase some groceries. There he saw an old lady struggling with her shopping. He immediately showed her the lift and explained to her how she can carry her goods from one floor to the other. Even then the old lady showed hesitation to use the lift. On seeing this, Pawan took the lady into the lift and showed her how to operate the lift. The old lady was very happy and easily finished her shopping.
(i) An elevator which can carry a maximum load of 1800 of 2m/s. The frictional force opposing he motion is 4000 N.Determine the maximum power delivered by the motor to the elevator in horse power.
• 3)
Rocket Propulsion
A toy rocket of mass 0.1 kg has a small fuel of mass 0.02 kg which it burns out in 3 s. Starting from rest on horizontal smooth track it gets a speed of 20 ms-1 after the fuel is burnt out. What is the approximate thrust of the rocket ? What is the energy content per unit mass of the fuel ? (Ignore the small mass variation of the rocket of the rocket during fuel burning).
• 4)
When a pebble Hits the Ground
Consider a drop of small pebble of mass 1.00 g falling from a cliff of height 1.00 km. It hits the ground with a speed of 50.0 ms-1. What is the work done by the unknown resistive force ?
• 5)
A railway carriage of mass 9000kg moving with a speed of 36km/h collides with stationary carriage of the same mass. After the collision, the carriage get coupled of the same mass. After the collision, the carriage get coupled and move together. What is their common speed after collision? what type of collision is this?
#### 11th Standard CBSE Physics - Laws of Motion Four and Five Marks Questions - by Girish - Chennai - View & Read
• 1)
A cyclist speeding at 18km/h on a level road takes a sharp circular turn of radius 3m without reducing the speed. The coefficient of static friction between the tyres and the road is 0.1. will the cyclist slip while taking the turn?
• 2)
A circular race track of radius 300m is banked at angle of 15o . IF the coefficient of friction between the wheels of a race car and the road is 0.2, what is the (i)optimum speed of the race to avoid wear and tear on its tyres and (ii) maximum permissible speed to avoid slipping?
$\left[ tan{ 15 }^{ o }=0.2679 \right]$
• 3)
The radius of curvature of a railway track at a place, where the train is moving at a speed of 72kmh-1 is 625m. The distance between the rails is 1.5m. Find the angle and the elevation of the out rails so that there may be no side pressure on the rails. Take, g = 9.8 ${ m }/{ { { s }^{ 2 } } }$
$\left[ { tan }^{ -1 }(0.00653)=3.7{ 4 }^{ o },sin3.7{ 4 }^{ o }=0.06522 \right]$
• 4)
A girl riding a bicycle along a straight road with a speed of 5 m/ s throws a stone of mass 0.5 kg which has a speed of 15 m /s with respect to the ground along her direction of motion. The mass of the girl and bicycle is 50 kg. Does the speed of the bicycle change after the stone is thrown ? What is the change in speed, if so ?
• 5)
Vipul was driving on the road with his old grandmother. She was sitting on the front seat with him. When vipul was about to reach his destination, he stopped the engine and did not apply the brakes. Even then the car was running on the road for sometimes.
His grandmother surprised and asked her grandson the reason the car running without the engine on. Vipul was the student of science studying in class XIth. He explained his grandmother that it is only the momentum due to which the car is going on.
(i) What values Vipul exhibit here ?
(ii) What is momentum and on which factor it depends ?
#### CBSE 11th Standard Physics - Mechanical Properties of Fluids Five Mark Model Question Paper - by Jemi Bhouseya - Indore - View & Read
• 1)
The reading of pressure meter attached with a closed pipe is $3.5\times { 10 }^{ 5 }Nm^{ -2 }$ .On opening the valve of the pipe, the reading of the pressure meter is reduced to $3.0\times { 10 }^{ 5 }Nm^{ -2 }$ .Calculate the speed of the water flowing in pipe.
• 2)
A liquid is kept in cylindrical vessel which is rotated along its axis. The liquid rises at the sides. If the radius of vessel is 0.05 m and the speed of rotation is 2 rev/s, find the difference in height of the liquid at the centre of the vessel and its sides.
• 3)
Explain why
(a) To keep a piece of paper horizontal, you should blow over, not under, it
(b) When we try to close a water tap with our fingers, fast jets of water gush through the openings between our fingers
(c) The size of the needle of a syringe controls flow rate better than the thumb pressure exerted by a doctor while administering an injection
(d) A fluid flowing out of a small hole in a vessel results in a backward thrust on the vessel
(e) A spinning cricket ball in air does not follow a parabolic trajectory
• 4)
Explain why?
A fluid flowing out of small hole in vessel results in a backward thrust on the vessel. According to Bernoulli's theorem, for horizontal flow of fluids, $\left( p+\frac { 1 }{ 2 } \rho { v }^{ 2 }=constant \right)$ Therefore, when velocity of fluid increases, its pressure decreases and vice-versa.
• 5)
Mercury has an angle of contact equal to 1400 with soda lime glass. A narrow tube of radius 1.00 mm made of this glass is dipped in a trough containing mercury. By what amount does the mercury dip down in the tube relative to the liquid surface outside? Surface tension of mercury at the temperature of th experiment is 0.456 N/m .Density of mercury? =13.6$\times$103 kg/m3
#### 11th CBSE Physics - Mechanical Properties of Solids Five Mark Model Question Paper - by Jemi Bhouseya - Indore - View & Read
• 1)
What is the length of a wire that breaks under its own weight when suspended vertically? Breaking stress = 5 x 107 Nm-2 and density of the material of the wire = 3 x 103 kg/m3
• 2)
Two wires of equal cross-section but one made of steel and the other copper are joined end to end. When the combination is kept under tension, the elongation in the two wires is found to be equal. Given Young's moduli of steel and copper are 2.0 x 1011 Nm-2 and 1.1 x 1011Nm-2 Find the ratio between the lengths of steel and copper wires
• 3)
The edge of an aluminium cube is 10 cm long. One face of the cube is firmly fixed to a vertical wall. A mass of 100 kg is then attached to the opposite face of the cube. The shear modulus of aluminium is 25 GPa.What is the vertical deflection of this face?
• 4)
A14.5kg mass, fastened to one end of a steel wire of unstretched length 1m is whirled in a vertical circle with an angular frequency of 2 rev/s at the bottom ofthe circle. The cross-sectional area of the wire is 0.065 cm2.Calculate the elongation of the wire when the mass is at the lowest point of its path.
• 5)
Four identical hollow cylindrical columns of mild steel support a big structure of mass 50,000 kg. The inner and outer radii of each column are 30 cm and 60 cm respectively. Assuming the load distribution to be uniform, calculate the compressional strain of each column. Young's modulus, $\Upsilon$ = 2.0 x 1011 Pa.
#### CBSE 11th Standard Physics - Gravitation Five Mark Model Question Paper - by Jemi Bhouseya - Indore - View & Read
• 1)
A spaceship is launched into a circular orbit close to the surface of the earth. What additional velocity has now to be imparted to the spaceship in the orbit to overcome the gravitational pull.
• 2)
Consider two solid uniform spherical object of the same density $\rho$ . One has a radius R and the other a radius 2R. They are in outer space where the gravitational field from other objects are negligible. If they are at rest with their surfaces touching, then what is the contact force between the objects due to their gravitational attraction?
• 3)
Choose the correct alternatives.
Acceleration due to gravity increases/decreases with increasing depth(assume the earth to be a sphere of uniform density).
• 4)
The planet Neptune travels around the sun with a period of 165 yr. Show that the radius of its orbit is approximately thirty times that of the earth's orbit, both being considered as circular.
• 5)
A particle is fired vertically upwards with a speed of 15 km/s. Find the speed of particle when it goes out of the earth's gravitational pull.
#### 11th CBSE Physics - System of Particles and Rotational Motion Five Mark Model Question Paper - by Jemi Bhouseya - Indore - View & Read
• 1)
In given pulley mass system, mass m1 = 500 g, m2 = 460 g and the pulley has a radius of 5 cm. When released from rest, heavier mass falls through 7.50 cm in 5 s. There is no slippage between pulley and string.
What is magnitude of acceleration of mass?
• 2)
Find the centre of mass for a solid cone of base radius r and height h.
• 3)
A man stands on a rotating platform with his arms stretched horizontally holding a 5 kg weight in each hand. The angular speed of the platform in 30 rpm. The man then brings his arms close to his body with the distance of each weight from the axis changing from 90 cm to 20 cm. The moment of inertia of the man together with the platform may be taken to be constant and equal to 7.6 kg-m2.
Is kinetic energy conserved in the process? If not, from where does the change come about?
• 4)
A disc of radius R is rotating with an angular speed $\omega$ , about a horizontal axis. It is placed on a horizontal table. The coefficient of kinetic friction is ${ \mu }_{ k }$.
What happens to the linear velocity of a point on its rim when placed in contact with the table?
• 5)
A disc of radius R is rotating with an angular speed $\omega$ , about a horizontal axis. It is placed on a horizontal table. The coefficient of kinetic friction is ${ \mu }_{ k }$. Calculate the time taken for the rolling to begin.
#### CBSE 11th Physics - Motion in a Plane Four Marks and Five Marks Questions - by Girish - Chennai - View & Read
• 1)
Resultant velocity of a Boat
A motor boat is racing towards North at 25km/h and the water current in that region is 10 km/h in the direction of 600 East to South. Find the resultant velocity of the boat.
• 2)
A soccer player kicks a ball at an angle of ${ 30 }^{ \circ }$with an initial speed of 20m/s. Assuming that the ball travels in a vertical plane. Calculate the maximum height reached$g=10m/s^{ 2 }$
• 3)
A cricket ball us thrown at a speed of 28${ ms }^{ -1 }$in a direction ${ 30 }^{ \circ }$above the horizontal.
(i) the maximum height
(ii) the time taken by the ball to return to the same level and
(iii) the distance from the thrower to the point where the baU returns to the same level.
• 4)
A body of mass 10 kg revolves in a circle of diameter 0.4m making 1000 revolutions per minute. Calculate its linear velocity and centripetal acceleration.
• 5)
From a school, a group of boys went for a picnic in a village. They went through fields and enjoyed the beauty of nature. While walking, they saw a well which they had never seen in the city. They were very excited and started drawing water from well. They planned to have a competition in which they decide that the who would draw more water would become winner . A villager who was listening to them, went to them and told them about the importance of water. He also explained that they use the water of this for irrigating their fields and also for drinking.
If the two boys raising the bucket, pull it an angle $\theta$ to each other and each exerts a force of 20N, their effective pull is 30N. What is the angle between their arms?
#### CBSE 11th Physics - Motion in a Straight Line Four Marks and Five Marks Questions - by Girish - Chennai - View & Read
• 1)
A train 100 m long is moving with a speed of 60 km/h. In what time shall it cross a bridge of 1 km long?
• 2)
During a hard sneeze, your eyes might shut for 0.5 s. If you are driving a car at 90 km/h during such a sneeze, how far does the car move during that time?
• 3)
If the average speed of the particle is [ ${ 2t }^{ 2 }\hat { i } +3t\hat { j }$ ], then find out the instantaneous speed of the particle.
• 4)
Prove Galileo's Law of Odd Numbers
• 5)
Shruti goes to school with his brother Alok in their own car.The school is about 10km apart from their home.They drive on alternate days.Alok is a very careful driver but Shruti drives rashly.She takes 3 min less than Alok in reaching the school.Alok advices Shruti to drive safely but she hardly listens.
What is the difference between average speeds of Shruthi and Alok if later takes 15 min to drive to the school?
#### 11th Standard CBSE Physics - Units and Measurements Four Marks and Five Marks Questions - by Girish - Chennai - View & Read
• 1)
The sun's angular diameter is measured to be 1920. The distance r of the sun from the earth is $1.496\times { 10 }^{ 11 }$ m. What is the diameter of the sun?
• 2)
Write down the number of significant figure in the following.
0.060
• 3)
A drop of olive oil of radius 0.25 mm spreads into a circular film of radius 10 cm on the water surface. Estimate the molecular size of olive oil.
• 4)
The refractive index of water is found to have the values 1.29,1.33,1.34,1.35,1.32,1.36,1.30 and 1.33.Calculate mean absolute error
• 5)
The refractive index of water is found to have the values 1.29,1.33,1.34,1.35,1.32,1.36,1.30 and 1.33.Calculate fractional error
#### 11th Standard CBSE Physics - Waves Three Marks Questions - by Girish - Chennai - View & Read
• 1)
A guitar string is 90 cm long and has a fundamental frequency of 124 H. Where should it be pressed to produce a fundamental frequency of 180 Hz?
• 2)
A standing wave is formed by two harmonic waves, ${ Y }_{ 1 }=Asin(kx-\varpi t)$and ${ Y }_{ 2 }=Asin(kx-\varpi t)$travelling on a string in opposite directioons. Mass density of the string is $\rho$ and area of cross-section is s. Find the total mechanical energy between two adjacent nodes on the string.
• 3)
A train, standing at the outer signal of a railway station blows a whistle of frequency 400 Hz in still air. What is the speed of sound in each case? The speed of sound in still air can be taken as 340ms-1.
• 4)
One end of a long string of linear mass density 8.0 x 10-3 kg m-1 is connected to an electrically driven tuning fork of frequency 256 Hz. The other end passes over a pulley and is tied to a pan containing a mass of 90kg. The pulley end absorbs all the incoming energy so that reflected waves at the end have negligible amplitude. At t=0, the left and (fork end) of the string x = 0 has zero transverse displacement (y=0) and is moving along positive y-direction. The amplitude of the wave is 5.0 cm. Write down the transverse displacement y as function of x and t that describes the wave on the string.
• 5)
A transverse harmonic wave on a string is described by y(x, t) = 3.0sin (36t + 0.018x + $\pi /4$) Where, x and y are in cm and t in seconds. The positive direction of x is from left to right
What is the initial phase at the origin?
#### 11th Standard CBSE Physics - Oscillation Three Marks Questions - by Girish - Chennai - View & Read
• 1)
A body oscillates with SHM according to the equation x(t) $=5cos\left( 2\pi t+\frac { \pi }{ 4 } \right)$ , where x is in metres and t is in seconds.
Calculate the following magnitude of maximum velocity
• 2)
When the mass is displaced a little to one side, one spring gets compressed and another is elongated.Due to which the combination of sp[rings. Here, effective spring factor k will be given by $k={ k }_{ 1 }+{ k }_{ 2 }=600+600=1200{ Nm }^{ -1 }$
• 3)
The motion of a simple pendulum is approximately simple harmonic for small angle oscillations. For larger angles of oscillation, a more involved analysis sjows that T is greater than $2\pi \sqrt { \frac { l }{ g } }$ .Think of a qualitative argument to appreciate this result.
• 4)
Let us take the position of mass when the spring is unstretched as x = 0 and the direction from left to right as the positive direction of the x-axis. Given x as a function of time t for the oscillating mass, if at the moment we start the stopwatch (t = 0), the mass is at the maximum stretched position.
In what way do these functions for SHM differ from each other, in frequency, in amplitude or the initial phase?
• 5)
A body weighing 10 g has a velocity of 6 cms-1 after one second of its starting from mean position. If the time period is 6 s, then find the kinetic energy, potential energy and the total energy.
#### 11th Standard CBSE Physics - Kinetic Theory Three Marks Questions - by Girish - Chennai - View & Read
• 1)
Write the difference between ideal gas and real gas.
• 2)
If one mole of a monoatomic gas is mixed with three moles of a diatomic gas.What is the molar specific heat of mixture at constant value?[Take, R = 8.31 J mol-1K-1]
• 3)
A gaseous mixture contain 16 g of helium and 16g of oxygen, then calculate the ratio of Cp/CV of the mixture.
• 4)
What will be the mean free path of nitrogen gas at STP of given diameter of nitrogen molecule = 2$\overset { 0 }{ A }$ ?
#### 11th Standard CBSE Physics - Thermodynamics Three Marks Questions - by Girish - Chennai - View & Read
• 1)
An ideal Carnot engine takes heat from a sources at 317o C, does some external work and delivers the remaining energy to a heat sink at 117o C. If 500 Kcal of heat is taken from the sources.
How much heat is delivered to the sink?
• 2)
A monoatomic ideal gas$(\gamma =\frac { 5 }{ 3 } )$initialy at 170C is suddenly compressed to one-eight of its original volume. Find the final temperature after compression.
• 3)
Explain why
(a) Two bodies at different temperatures T1 and T2 if brought in thermal contact do not necessarily settle to the mean temperature (T1 + T2 )/2.
(b) The coolant in a chemical or a nuclear plant (i.e., the liquid used to prevent the different parts of a plant from getting too hot) should have high specific heat.
(c) Air pressure in a car tyre increases during driving.
(d) The climate of a harbour town is more temperate than that of a town in a desert at the same latitude.
• 4)
Explain, why?
The climate of a harbor town is more temperate than that of a town in a desert at the same latitude
• 5)
In changing the state of a gas adiabatically from an equilibrium state A to another equilibrium state B, an amount of work equal to 22.3 J is done on the system. If the gas is taken from state A to B via a process in which the net heat absorbed by the system is 9.35 cal, how much is the net work done by the system in the latter case? (Take 1 cal = 4.19 J).
#### 11th Standard CBSE Physics - Thermal Properties of Matter Three Marks Questions - by Girish - Chennai - View & Read
• 1)
At what temperature, if any, do the following pairs of scales gives the same reading?
Fahrenheit and Kelvin.
• 2)
A box having total surface area 0.05 m2 and of 6 mm thick side walls is filled with melting ice and kept in room. Calculate the thermal conductivity of the box material if 0.5 kg of ice melts in 1 h. The room temperature is 40o C and latent heat of fusion of ice = $3.33\times { 10 }^{ 5 }J{ kg }^{ -1 }$
• 3)
Explain the following
(i) Hot tea cools rapidly when poured into the saucer from the cup.
(ii) Temperature of a hot liquid falls rapidly in the beginning but slowly afterward.
(iii) A hot liquid cools faster if outer surface of the container is blackened.
• 4)
A fat man is used to consuming about 3000 kcal worth of food every day. His food contains 50g of butter plus a plate of sweets every day, besides items which provide him with other nutrients (proteins, vitamins, minerals, etc) in addition to fats and carbohydrates. The calorific value of 10g of butter is 60kcal and that of a plate of sweets is of average 700kcal per day? Assume the man cannot resist eating the full plate of sweets once it is offered to him.
• 5)
A metallic ball has a radius of 9.0 cm at 00C. Calculate the change in its volume when it is heated to 900 C. Given the coefficient of linear expansion of metal of ball is 1.2 x 10-5 K.
#### 11th Standard CBSE Physics - Mechanical Properties of Fluids Three Marks Questions - by Girish - Chennai - View & Read
• 1)
27 identical drops of water are falling down vertically in air each with a terminal velocity 0.15$ms^{ -1 }$>if they combine to form a single bigger drop, what will be its terminal velocity?
• 2)
Two soap bubbles have radii in the ratio2:3.Compare the excess of pressure inside these bubbles.
• 3)
(a) What is the largest average velocity of blood flow in an artery of radius 2 $\times$ 10- 3m, if the flow must remain laminar?
(b) What is the corresponding flow rate ? (Take viscosity of blood to be 2.084 x 10–3 Pa -s)
• 4)
What is the corresponding flow rate? (Take viscosity of blood to be 2.084$\times$ 10- 3Pa-s)
• 5)
Show that ifn equal rain droplets falling through air with equal steady velocity ono cms-1 coalesce, the resultant drop attains a new terminal velocity ono n2/3cms- 1.
#### 11th Standard CBSE Physics - Mechanical Properties of Solids Three Marks Questions - by Girish - Chennai - View & Read
• 1)
Two parallel steel wires A and B are fixed to rigid support at the upper ends and subjected to the same load at the lower ands. The lengths of wires are in the ratio 4:5 and their radii are in the ratio 4:3. The increase in the length of the wire A is 1mm. Calculate the increase in the length of the wire B.
• 2)
Determine the volume contraction of a solid copper cube, 10cm on an edge, when subjected to a hydraulic pressure of 7 x 106 Pa. Bulk modulus for copper = 140 x 109 Pa.
• 3)
Read the following two statements below carefully and state, with reasons, if it is true or false.
(a) The Young's modulus of rubber is greater than that of steel;
(b) The stretching of a coil is determined by its shear modulus.
• 4)
A steel wire of length 4 m is stretched through 2 mm. The cross-section area of the wire is 2.0 mm2. If Young's modulus of steel is 2.0 x 1011 N/m2, find
(i) the energy density of the wire and
(ii) the elastic potential energy stored in the wire.
• 5)
Explain why steel is more elastic than rubber.
#### 11th Standard CBSE Physics - Gravitation Three Marks Questions - by Girish - Chennai - View & Read
• 1)
An artificial satellite is going round the earth, close to the surface. What is the time taken by it to complete one round?
• 2)
The distances of two planets from the sun are 1013 m and 1012 m, respectively. Calculate the ratio of time period and the speeds of the two planets
• 3)
If the earth is 1/4 of its present distance from the sun, then what is the duration of he year?
• 4)
The mass of a spaceship is 1000 kg. It is to be launched from the earth's surface out into free space. The value of g and R(radius of earth) are 10${ m }/{ { s }^{ 2 } }$and 6400km, respectively.What is the required energy for this work done?
• 5)
What will be the potential energy of a body of mass 67kg at a distance of $6.6{ \times 10 }^{ 10 }m$from the centre of the earth? Find gravitational potential at this distance.
#### 11th Standard CBSE Physics - System of Particles and Rotational Motion Three Marks Questions - by Girish - Chennai - View & Read
• 1)
A solid cylinder of mass 20 kg rotates about its axis with angular speed 100 rad s-1. The radius of the cylinder is 0.25 m. What is the kinetic energy associated with the rotation of the cylinder? What is the magnitude of angular momentum of the cylinder about its axis?
• 2)
A bullet of mass 10 g and speed 500 m/s is fired into a door and gets embedded exactly at the centre of the door. The door is 1.0 m wide and weight 12 kg. It is hinged at one end and rotates about a vertical axis practically without friction. Find the angular speed of the door just after the bullet embeds into it.
(Hint: The moment of inertia of the door about the vertical axis at one end is ML2/3.)
• 3)
A cylinder of mass 10 kg and radius 15 cm is rolling perfectly on a plane of inclination ${ 30 }^{ \circ }$. The coefficient of static friction, ${ \mu }_{ s }=0.25$.
(a) How much is the force of friction acting on the cylinder?
(b) What is the work done against friction during rolling?
(c) If the inclination θ of the plane is increased, at what value of θ does the cylinder begin to skid, and not roll perfectly?
• 4)
A particle on a rotating disc have initial and final angular position are -2rad, +6rad. In which case, particle undergoes a negative displacement.
• 5)
A particle on a rotating disc have initial and final angular position are -4rad, -8rad. In which case, particle undergoes a negative displacement.
#### 11th Standard CBSE Physics - Work, Energy and Power Three Marks Questions - by Girish - Chennai - View & Read
• 1)
The blades of windmill sweep out a circle of area A.
(i) If the wind flows at a velocity v perpendicular to the circle, what is the mass of the air passing through it in time t?
• 2)
The blades of windmill sweep out a circle of area A.
(i) What is the kinetic energy of the air?
• 3)
Under the correct alternative.
Whan a conservation force does positive work on a body, the potential energy of the body increase decrease/remains unaltered.
• 4)
Work done by a body against friction always results in a loss of its kinetic/potential energy
• 5)
The rate of change of total momentum of a many particle systems is proportional to the external force/sum of the internal forces on the system.
#### 11th CBSE Physics - Laws of Motion Five Mark Question Paper - by Jemi Bhouseya - Indore - View & Read
• 1)
A hammer weighing 1 kg moving with the speed of 20 m / s strikes the head of a nail driving it 20 cm into a wall. Neglecting the mass of the nail, calculate
(i) the acceleration during the impact
(ii) the time interval during the impact
(iii) the impulse.
• 2)
A hammer weighing 1 kg moving with the speed of 20 m / s strikes the head of a nail driving it 20 cm into a wall. Neglecting the mass of the nail, calculate
(iii) the impulse.
• 3)
Give the magnitude and direction of the net force acting on a cork of mass 10 g floating on water.
• 4)
Give the magnitude and direction of the net force acting on a kite skillfully held stationary in the sky.
• 5)
A circular motion addict of mass 80 kg rides a Ferris wheel around in a vertical circle of radius 10m at a constant speed of 6.1 m/s the highest point of the circular path
#### 11th CBSE Physics - Work, Energy and Power Five Mark Question Paper - by Jemi Bhouseya - Indore - View & Read
• 1)
A stone of mass 0.4 kg is thrown vertically upward wit a speed of 9.4 m/s/ Find the potential and kinetic energies after half second?
• 2)
A running man has half the kinetic energy that a boy of his mass has. the man speeds up by 1.0 m/s and then same energy as the boy. what were the original speeds of the man and the boy?
• 3)
State if each of the following staement is true or false. Give reasons for your answer.
Total energy of a system is always conserved, no matter what internal and external forces on the body are present?
• 4)
(a) In an elastic collision of two billiard balls, is the total kinetic energy conserved during the short time of collision of the balls i.e. when they are in contact?
(b) Is the total linear momentum conserved during the short time of an elastic collision of two balls ?
(c) What are the answers to (a) and (b) for an inelastic collision ?
(d) If the potential energy of two billiard balls depends only on the separation distance between their centres, is the collision elastic or inelastic ? (Note, we are talking here of potential energy corresponding to the force during collision, not gravitational potential energy).
• 5)
If the potential energy of two billiard balls depends only on the separation distance between their centres, is the collison elastic or inelastic?
#### 11th CBSE Physics - Motion in a Plane Five Mark Question Paper - by Jemi Bhouseya - Indore - View & Read
• 1)
State the reason , whether the following algebraic operations with scalar and vector physical quantities are meaningful
Adding a scalar to a vector of dimensions
• 2)
State the reason , whether the following algebraic operations with scalar and vector physical quantities are meaningful
• 3)
State the reason , whether the following algebraic operations with scalar and vector physical quantities are meaningful.
adding a component of a vector to the same vector.
• 4)
If A and B are two vectors such that $\left| A\times B \right| =\sqrt { 3 } A.B$ Then,
Also, find the value of $\left| A\times B \right|$
• 5)
A ball rolls of the top of a stairway with horizontal velocity of 1.8 m/s. The steps are 0.24 m high and 0.2 m wide. Which step will the ball hit first? Take g=9.8 m/s2
#### CBSE 11th Standard Physics - Motion in a Straight Line Five Mark Question Paper - by Jemi Bhouseya - Indore - View & Read
• 1)
Two parallel rail tracks run North-South.Train A moves North with a speed of 54 kmh-1. and train B moves South with a speed of 90 kmh-1. What is the relative velocity of ground with respect to B?
• 2)
(i) Draw position-time graph for
(a) Accelerated motion
(b) Retarded motion
(ii) A juggler throws balls into air. He throws one whenever the previous one is at its highest point.How high do the balls rise if he throws n balls in each second? Take acceleration due to gravity as g.
• 3)
A juggler throws balls into air. He throws one whenever the previous one is at its highest point.How high do the balls rise if he throws n balls in each second? Take acceleration due to gravity as g.
• 4)
A motor boat covers the distance between two spots on the river in t1= 8 h and t= 12 h, downstream and upstream, respectively.What is the time required for the boat to cover this distance in still water?
• 5)
A train passes a station A at 40 kmh -I and maintains its speed for 7 km an? is then uniformly retarded, stopping at B which is 8.5 km from A. A second train starts from A at the instant the first train passes and being accelerated some part of the journey and uniformly retarded for the rest, stops at B at the same times as the first train. Calculate the maximum speed-of the second train, use only the graphical method.
#### 11th CBSE Physics - Units and Measurements Five Mark Question Paper - by Jemi Bhouseya - Indore - View & Read
• 1)
Write down the number of significant figure in the following.
3.08 x 1011
• 2)
The refractive index of water is found to have the values 1.29,1.33,1.34,1.35,1.32,1.36,1.30 and 1.33.Calculate mean value of refractive index
• 3)
Derive the dimensions formula of physical quantities. Velocity gradient
• 4)
One mole of an ideal gas at standard temperature and pressure occupies 22.4 L(molar volume). What is the ratio of molar volume to the atomic volume of a mole of hydrogen? (Take the size of hydrogen molecule to be about 1$\mathring{A}$). Why is the ratio so large?
• 5)
Two resistors of resistance ${ R }_{ 1 }=\left( 100\pm 3 \right) \Omega$ and ${ R }_{ 2 }=\left( 200\pm 4 \right) \Omega$ are connected (i) in series, (ii) in parallel. Find the equivalent resistance of the series combination. Use for the relation R = R+ R2.
#### 11th CBSE Physics - Physical World Five Mark Question Paper - by Jemi Bhouseya - Indore - View & Read
• 1)
The shells of crabs found around a particular coastal location in Japan seem mostly to resemble the legendary face of a Samurai. Given below are two explanations of this observed fact. Which of these strikes you as a scientific explanation?
(a) A tragic sea accident several centuries ago drowned a young Samurai. As a tribute to his bravery, nature through its inscrutable ways immortalized his face by imprinting it on the crab shells in that area.
(b) After the sea tragedy, fishermen in that area, in a gesture of honour to their dead hero, let free any crab shell caught by them which accidentally had a shape resembling the face of a Samurai. Consequently, the particular shape of the crab shell survived longer and therefore in course of time the shape was genetically propagated. This is an example of evolution by artificial selection.
[Note: This interesting illustration taken from Carl Sagan's 'The Cosmos' highlights the fact that often strange and inexplicable facts which on the first sight appear 'supernatural' actually turn out to have simple scientific explanations. Try to think out other examples of this kind].
• 2)
Give the salient features of Einstein's theory.
• 3)
Name four fundamental forces in nature.
• 4)
Does imagination play any role in physics?
• 5)
Physics has a very limited scope and only in practice of a few blessed ones. Do you agree?
#### CBSE 11th Physics - Laws of Motion in Two Variables Three Marks Questions - by Girish - Chennai - View & Read
• 1)
A helicopter of mass 1000 kg reises with a vertical acceleration of 15m /s2. The crew and the passengers weigh 300 kg. Give the magnitude and direction of the force on the helicopter due to the surrounding air, take g = 10 m/s2.
• 2)
A stone of mass 0.25 kg tied to the end of a string is whirled round in a circle of radius 1.5m with speed 40 rev/min in a horizontal plane. What is the tension in the string? what is the maximum speed with which the stone can be whirled around if the string can withstand a maximum tension of 200 N?
• 3)
An aircraft executes a horizontal loop at a speed of 720km/h with its wings banked at 15o . What is the radius of the loop?
• 4)
Ten one-rupee coins are put on top of each other on a table. Each coin has mass m. Give the magnitude and direction of the reaction of the 6th coin on the 7th coin.(counted from the bottom)
• 5)
A light, inextensible string connects two blocks of mass M1, and M2. A force Facts upon MI. Find acceleration of the system and tension in siring.
#### 11th Standard CBSE Physics - Motion in a Plane Three Marks Questions - by Girish - Chennai - View & Read
• 1)
There are two displacement vectors,one of the magnitude 3m and the other of 4m.How would two vectors be added so that the magnitude of the resultant vector be1m
• 2)
There are two displacement vectors,one of the magnitude 3m and the other of 4m.How would two vectors be added so that the magnitude of the resultant vector be 5m
• 3)
The velocity of a particle, when it is at the greatest height is $\sqrt { 2/5 }$ times its velocity when it is at half of its greatest height. Determine its angle of projection.
• 4)
Find the angle made by vector, $A=2\hat { i } +2\hat { j }$ with x-axis
• 5)
Two billiard balls are rolling on a flat table. One has the velocity components ${ v }_{ x }=1{ ms }^{ -1 },\ { v }_{ y }=\sqrt { 3 } { ms }^{ -1 }$ and the other has components v'=2 ms-1 and v'y =2 ms-1 If both the balls start moving from the same point, what is the angle between their paths?
#### 11th Standard CBSE Physics - Motion in a Straight Line Three Marks Questions - by Girish - Chennai - View & Read
• 1)
To what height does the ball rise and after how long does the ball return to the player's hands?(Take g = 9.8ms-2 and neglect air resistance)
• 2)
Read each statement below carefully and state with reasons and examples if it true or false.
A particle in 1-D motion
(i) with zero speed at an instant may have non-zero acceleration at that instant.
(ii) with zero speed may have non-zero velocity.
(iii) with constant speed must have zero acceleration
(iv) with positive value of acceleration must be speeding up.
• 3)
A particle starts moving from position of rest under a constant acceleration. It is travels a distance x in t second, what distance will it travel in next t second?
• 4)
A police van moving on a highway with a speed of 30 kmh-1 fires a bullet at a thief's car speeding away in the same direction of 192 km h-1. If the muzle speed of the bullet is 150 ms-1 with what speed does the bullet hit the thief's car? ( Note, obtain that speed which is relevant for damaging the thief's car? )
• 5)
A person travels along a straight road for the first half with avelocity v1 and the second half with velocity v1 and the second half with velocity v2. What is the mean velocity of the person?
#### 11th Standard CBSE Physics - Units and Measurements Three Marks Questions - by Girish - Chennai - View & Read
• 1)
The sides of a rectangle are $(10.5\pm 0.2)$ cm and $(5.2\pm 0.1)$ cm. Calculate its perimeter with error limits.
• 2)
A capacitor of capacitance $C=(2.0\pm 0.1)\mu F$ is charged to a voltage $V=(20\pm 0.5)V$. Calculate the charge Q with error limits.
• 3)
Write down the number of significant figures in the following
34.000 m.
• 4)
Write down the number of significant figures in the following
0.02340 N/m.
• 5)
The length, breadth and thickness of a rectangular sheet of metal are 4.234 m, 1.005 m and 2.01 am, respectively. Give the area and volume of the sheet to correct significant figures.
#### CBSE 11th Physics Unit 1 Physical World Three Marks Questions - by Girish - Chennai - View & Read
• 1)
It is often said that the world is witnessing now a second industrial revolution which will transform the society as radically as did the first. List some key contemporary areas of science and technology which are responsible for this revolution.
• 2)
Though the law gives women equal status in India, many people hold unscientific views on a woman's innate nature, capacity and intelligence and in practice give them a secondary status and role.
Demolish this view using scientific arguments and by quoting examples of great women in science and other spheres and persuade yourself and others that given equal opportunity, women are on par with men.
• 3)
The shells of crabs found around a particular coastal location in Japan seem mostly to resemble the legendary face of a Samurai. Given below are two explanations of this observed fact. Which of these strikes you as a scientific explanation?
(a) A tragic sea accident several centuries ago drowned a young Samurai. As a tribute to his bravery, nature through its inscrutable ways immortalized his face by imprinting it on the crab shells in that area.
(b) After the sea tragedy, fishermen in that area, in a gesture of honour to their dead hero, let free any crab shell caught by them which accidentally had a shape resembling the face of a Samurai. Consequently, the particular shape of the crab shell survived longer and therefore in course of time the shape was genetically propagated. This is an example of evolution by artificial selection.
[Note: This interesting illustration taken from Carl Sagan's 'The Cosmos' highlights the fact that often strange and inexplicable facts which on the first sight appear 'supernatural' actually turn out to have simple scientific explanations. Try to think out other examples of this kind].
• 4)
Textbooks on science may give you a wrong impression that studying science is dry and all too serious and that scientists are absent-minded introverts who never laugh or grin. This image of science and scientists is patently false. Scientists, like any other group of humans, have their share of humorists, and many have led their lives with a great sense of fun and adventure, even as they seriously pursued their scientific work. Two great physicists of this genre are Gamow and Feynman. You will enjoy reading their books listed in the Bibliography.
• 5)
Give the salient features of Einstein's theory.
#### CBSE 11th Physics Unit 15 Waves Two Marks Questions - by Girish - Chennai - View & Read
• 1)
What frequency of the sound you hear coming directly from the siren?
• 2)
A steel wire has a length of 12 m and a mass of 2.10 kg. What will be the speed of a transverse wave on this wire when a tension of 2.06$\times$104N is applied?
• 3)
A pipe 20 cm long is closed at one end. Which harmonic mode of the pipe is resonantly excited by a source of 1237.5 Hz? (sound velocity in air = 330ms-1)
• 4)
Equation of a plane progressive wave is given by y = 0.6 sin$2\pi \left( t-\frac { x }{ 2 } \right)$ On reflection from a denser medium, its amplitude becomes 2/3 of the amplitude of incident wave. What will be equation of reflected wave?
• 5)
At what temperature (in 0C) Will be speed of sound air be 3 times its value at 00 C?
#### 11th Standard CBSE Physics - Oscillation Two Marks Questions - by Girish - Chennai - View & Read
• 1)
A body of mass 12 kg is suspended by coil spring of natural length 50 cm and force constant 2.0 x 103Nm-1. What is the streched length of the spring?If the bosy is pulled down further streching the spring to a length of 5.9 cm and then released,then what is the frequencyof oscillation of the suspended mass?
• 2)
A spring compressed by 0.1 m develops a restoring force 10 N. A body of mass 4 kg placed on it . Deduce
(i) the force constant of the spring
(ii) the depression of the spring under the weight of the body (take g=10 N/kg)
(iii) the period of oscillation, the body is distributed and
(iv) the frequency of oscillation
• 3)
A circular disc of mass 10 kg is suspended by a wire attached to its centre. The wire is twisted by rotating the disc and released. The period of torsional oscillation is found to be 1.5 s. The radius of the disc is 15 cm.Determine the torsional spring constant of the wire.
This is a question based on torsion pendulum for which $T=2\pi \sqrt { \frac { \quad }{ \alpha } }$ where I = moment of inertia of the disc about axis of rotation,$\alpha$ = torsion constant which is restoring couple per unit twist.
• 4)
Define the restoring force and it characterstic in case of an oscillating body.
• 5)
A particle excutes SHM of period 8 s. After what time of its passing through the mean position will be energy be half kinetic and half potential?
#### 11th Standard CBSE Physics - Kinetic Theory Two Marks Questions - by Girish - Chennai - View & Read
• 1)
The value of root mean square speed for O2 is 400 m/s. Find the temperature of the O2.
• 2)
If value of most probable speed for an ideal gas is 500 m/s. Find the value of root mean square speed for this gas.
• 3)
Find the temperature at which rms speed of a gas is half of its value of 00C, pressure remaining constant.
• 4)
We have 0.5 g of hydrogen gas in a cubic chamber of size 3 cm kept at NTP. The gas in the chamber is compressed keeping the temperature constant till a final pressure of 100 atm. Is one justified in assuming the ideal gas law, in the final state? (Hydrogen molecules can be consider as spheres of radius 1 $\overset { o }{ A }$).
• 5)
Three vessels of equal capacity have gases at the same temperature and pressure. The first vessel contains neon (monatomic), the second contains chlorine (diatomic) and the third contains uranium hexafluoride (polyatomic). Do the vessels contain equal number of respective molecules? Is the root mean square speed of molecules the same in the three cases? If not, in which case is Vrms ,the largest?
#### 11th Standard CBSE Physics - Thermodynamics Two Marks Questions - by Girish - Chennai - View & Read
• 1)
A person of mass 60 kg wants to lose 5 kg by going up and down a 10 m high stairs. Assume he burns twice as much fat while going up than coming down. If 1kg of fat is burnt on expending 7000 kcal calories, how many times must he go up and down to reduce his weight by 5 kg?
• 2)
What amount of heat must be supplied to 2.0$\times$ 10-2kg of nitrogen (at room temperature) to raise its temperature by 45°C at constant pressure? (Molecular mass of N2 = 28, R = 8.3 J mol-1 K-1)
• 3)
The efficiency of a heat engine is more in hilly area than in plain.Explain it.
• 4)
Is the coefficient of performance of a refrigerator, a constant quantity?
• 5)
Why is it theoretically not possible to have a device which create no thermal pollution?
#### 11th Standard CBSE Physics - Thermal Properties of Matter Two Marks Questions - by Girish - Chennai - View & Read
• 1)
By how much the temperature of a copper rod to be raised so as to increase its length by 1% ? Given that coefficient of linear expansion of copper = 1.7 x 10-5 K-1
• 2)
The density of mercury is $13.6\times { 10 }^{ 3 }Kg\quad { m }^{ -3 }$ at 0o C and its coefficient of volume expansion is $1.82\times { 10 }^{ -4 }{ K }^{ -1 }$ . Find the density at 50o C.
• 3)
What value of temperature in the Celsius and Fahrenheit scales give the same reading?
• 4)
There is a slight temperature different between the water fall at the top and the bottom. Why?
• 5)
Given below are observations on molar specific heats at room temperature of some common gases.
Gas Molar specific heat (Cv) (cal mol-1K-1) Hydrogen 4.87 Nitrogen 4.97 Oxygen 5.02 Nitric oxide 4.99 Carbon monoxide 5.01 Chlorine 6.17
The measured molar specific heats of these gases are markedly different from those for monoatomic gases.(Typically, molar specific heat of monoatomic gas is 2.92 cal/mol K). Explain this difference. What can you infer from the somewhat large (than the rest) value for chlorine?
#### 11th Standard CBSE Physics - Mechanical Properties of Fluids Two Marks Questions - by Girish - Chennai - View & Read
• 1)
For a fluid in a steady flow, the increase in flow speed at a constriction follows from _________while the decrease of pressure there follows from ___________(conservation of mass/Bernoulli's principle).
• 2)
For the model of a plane in a wind tunnel, turbulence occurs at a ................. speed that the critical speed for turbulence for an actual plane.
(greater / smaller)
• 3)
If the required pressure in the tyre of a car is 199 kPa, then what is the absolute pressure?
• 4)
A hydraulic automobile lift is designed to lift cars with a maximum mass of 300kg. The area of cross-section of the piston carrying the load is 425 ${ cm }^{ 2 }$.What maximum pressure would the smaller piston have to bear?
• 5)
A vertical off-shore structure is built to withstand a maximum stress of ${ 10 }^{ 9 }Pa$. Is the structure suitable for putting up on top of an oil well in the ocean? Take the depth of the ocean to be roughly 3km and ignore ocean currents.
#### 11th Standard CBSE Physics - Mechanical Properties of Solids Two Marks Questions - by Girish - Chennai - View & Read
• 1)
A solid sphere of radius R made of a material of bulk modulus B is surrounded by a liqiud in a cylindrical container. A massless piston of area A floats on the surface of the liqiud. When a mass M is placed on the piston on the piston to compress the liqiud, find fractiional change in the radius of the sphere?
• 2)
To what depth must a rubber ball be taken in deep sea so that its volume is decreased by 0.1%? (The Bulk modulus of rubber is 9.8 x 108 N/m2; and the density of seawater is 103 kg/m 3 .)
• 3)
The stress-strain graphs for materials A and B are shown in Fig. (a) and Fig. (b).
The graphs are drawn to the same scale.
(i) Which of the materials has greater Young's modulus?
(ii) Which of the two is the stronger material?
• 4)
A wire of length L and radius r is clamped rigidly at one end. When the other end of the wire is pulled by a force j, its length increases by l. Another wire of the same material of length 2L and radius 2r, is pulled by a force 2f Find the increase in length of this wire.
• 5)
A piece of copper having a rectangular cross-section of 15.2 mm x 19.1 mm is pulled in tension with 44,500 N force, producing only elastic deformation. Calculate the resulting strain?
#### CBSE 11th Physics Unit 8 Gravitation Two Marks Questions - by Girish - Chennai - View & Read
• 1)
A plant moving along an elliptical orbit is closet to the Sun at a distance r1 and farthest away at a distance of r2 . If V1 and V2 are the linear velocities at these points respectively, then find the ratio v1/v2
• 2)
If the earth be at one half of its present distance from the sun, them how many days will be there in a year?
• 3)
Calculate the force of attraction between two bodies, each of mass 100 kg 1 m apart on the surface of the earth.
• 4)
The acceleration due to gravity on a planet is 1.96${ ms }^{ -2 }$. If it is safe to jump from a height of 2m on the earth, then what will be the corresponding safe height on the planet?
• 5)
Does the concentration of the earth's mass near its centre change the variation of g with height compared with a homogeneous sphere, how?
#### 11th Standard CBSE Physics - System of Particles and Rotational Motion Two Marks Questions - by Girish - Chennai - View & Read
• 1)
A person is standing on a rotating table with metal spheres in his hands. If he withdraw his hands to his chest, what will be the effect on his angular velocity?
• 2)
Two boys of the same weight it at the opposite ends of a diameter of a rotating circular table. What happens to the speed of rotation if they move nearer to the axis of rotation?
• 3)
If ice on poles melts, then what is the change in duration of day?
• 4)
A solid cylinder of mass 20 kg rotates about its axis with angular speed of 100 rad/s. The radius of cylinder is 0.25m. What is KE of rotation of cylinder?
• 5)
If earth contract to half its radius. What would be the length of the day?
#### 11th Standard CBSE Physics - Work, Energy and Power Two Marks Questions - by Girish - Chennai - View & Read
• 1)
Calculate the velocity of the bob of a simple pendulum at its mean position if it is able to rise to a vertical height of 10 cm. [g = 9.8 m/s2]
• 2)
Can a body have without momentum? If yes, then explain how they are related with each other?
• 3)
A spring balance reads forces in Newtons. The scale is 20 cm long and read from 0 to 60 N. Find potential energy of spring when the scale reads 20 N.
• 4)
A steel spring of spring constant 150 N/m is compressed from its natural position through a mud wall 1m thick, the speed of bullet drops to 100m/s. Calculate the average resistance of the wall. Neglect friction of air.
• 5)
A trolley of mass 300 kg carrying a sand bag of 25 Kg is moving uniformly with a speed of 27km/h on a frictionless track. After a while , sand starts leaking out of a whole on the floor of the trolley at the rate of 0.052 kg-1.What is the speed of the trolley after the entire sand bag is empty?
#### 11th CBSE Physics - Laws of Motion Book Back Questions - by Girish - Chennai - View & Read
• 1)
Why are porcelain objects wrapped in paper or straw before packing for transportation ?
• 2)
A passenger of mass 72.2 kg is riding in an elevator while standing on a platform scale. What does the scale read when the elevator cab is
(i) descending with constant velocity
(ii) ascending with constant acceleration, 3.5 m/s2?
• 3)
A force of 128 If acts on a mass of 490 g for 10 s. What velocity will it give to the mass ?
• 4)
A body of mass 2 kg is being dragged with a uniform velocity of 2 ms -1 on a rough horizontal plane. The coefficient of friction between the body and the surface is 0.2. Calculate the amount of heat generated per second. Take g =9.8 ms-2 and J = 1.2Jcal-1
• 5)
Explain why passengers are thrown forward from their seats when a speeding bus stops suddenly,
#### 11th Physics - Motion in a Plane Two Marks Questions - by Girish - Chennai - View & Read
• 1)
We can order events in time and there is no sense of time, distinguishing past, present and future. Is time a vector?
• 2)
The angle between vector A and B is 600 .What is the ratio of A.B and $\left| A\times B \right|$ ?
• 3)
A person sitting in a running train throws a ball vertically upwards. What is the nature of the path described by the ball to a person?
(i) Sitting inside the train
(ii) Standing on the ground outside the train
• 4)
A railway carriage moves over a straight track with acceleration a. A passenger in the carriage drops a stone. What is the acceleration of the stone w.r.t. the carriage and the earth?
• 5)
When a large star becomes a supernova, its core may be compressed so tightly that it becomes a neutron star, with a radius of about 20km. If a neutron star rotates once every second, What is the magnitude of the particle's centripetal acceleratiom?
#### CBSE 11th Physics - Motion in a Straight Line Book Back Questions - by Girish - Chennai - View & Read
• 1)
For what condiion, an object could be considered as a point object? Describe in brief
• 2)
The displacement o a particle is given by at2. What is the dependency of accleration on time?
• 3)
Thye position x of a body is given by x = A sin(wt). Find the time at which the displacements is maximum.
• 4)
What are uses of a velocity - time graph ?
• 5)
The position of an object is given by x = 2t+ st. Find out that its motion is uniform and nonuniform.
#### CBSE 11th Physics Unit 2 - Units and Measurements Two Marks Questions - by Girish - Chennai - View & Read
• 1)
A new unit of length is chosen such that the speed of light in vacuum is unity. What is the distance between the sun and the earth in terms of the new unit, if light takes 8min and 20 s to cover this distance?
• 2)
Express an acceleration of 10 m/s2 in km/h2.
• 3)
Which of the following length measurement is most accurate and why? 40.00 cm
• 4)
What is common between bar and torr?
• 5)
Calculate the surface area of a solid cylinder of diameter 4 cm and height 20 cm in mm2
#### CBSE 11th Physics - Physical World Two Marks Questions - by Girish - Chennai - View & Read
• 1)
Some of the most profound statements on the nature of science have come from Albert Einstein, one of the greatest scientists of all time. What do you think did Einstein mean when he said, "The most incomprehensible thing about the world is that it is comprehensible"?
• 2)
Politics is the art of possible. Similarly, Science is the art of the soluble. Explain this beautiful aphorism on the nature and practice of science.
• 3)
Though India now has a large base in science and technology which is fast expanding, it is still a long way for realizing its potential for becoming a world leader in Science. Name some important factors, which in your view have hindered the advancement of science in India.
• 4)
The industrial revolution in England and Western Europe more than two centuries ago was triggered by some key scientific and technological advances. What were these advances?
• 5)
Name the two most important contributions of Albert Einstein.
#### CBSE 11th Physics - Waves Book Back Questions - by Girish - Chennai - View & Read
• 1)
What is the nature of water waves produced by a motorboat sailing in water ?
• 2)
In a hot summer day, pitch of an organ pipe will be higher or lower?
• 3)
Show that when a string fixed at its two ends vibrates in 1 loop, 2 loops, 3loops and 4loops, the frequencies are in the ratio 1:2:3:4.
• 4)
When two waves of almost equal frequencies n1 and n2 reach at a point, simultaneously. What is the time interval between successive maxima?
### CBSE Education Study Materials
#### 11th CBSE Physics 2019 - 2020 Academic Syllabus - by Girish - Chennai Aug 21, 2019 Aug 21, 2019
Physics 2019 - 2020 Academic Syllabus
#### Tips to score good marks in CBSE Class 11 Physics - by ADMIN-ENGLISH Jan 23, 2019 Jan 23, 2019
• Practice the important questions from these chapters. Below is the sample question paper f...
#### Tips and Tricks to score good marks in CBSE Class 11 Physics - by ADMIN-ENGLISH Jan 23, 2019 Jan 23, 2019
Scoring good marks in physics is easy if you work on derivations ( important) and numerical. Phys...
#### Tips and Tricks to score good marks in CBSE Class 11 Physics - by Bala Jan 12, 2019 Jan 12, 2019
Scoring good marks in physics is easy if you work on derivations ( important) and numerical. Phys...
#### CBSEStudy Material - Sample Question Papers with Solutions for Class 11 Session 2020 - 2021
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In other words, about 30 percent of incoming solar radiation is reflected back into space and 70 percent is absorbed. National Oceanic and Atmospheric Administration. 1) the color of the substance. Reflected solar radiation is the part of incident solar radiation reflected from the earth's surface due to the albedo effect. The earth-atmosphere energy balance is achieved as the energy received from the Sun balances the energy lost by the Earth back into space. The exact percentage of light that reaches a certain depth depends on several factors. Yes. The amount that is reflected or absorbed depends on Earth’s surface and atmosphere. In one lake, a researcher discovered that any change of 10 meters in depth reduces the amount of light reaching that depth by 22.8%. An untreated silicon solar cell only absorbs 67.4 percent of sunlight shone upon it — meaning that nearly one-third of that sunlight is reflected away and thus unharvestable. For photosynthesis, plants use approximately 0.023 percent of sunlight energy. It is expressed as a percentage of reflected insolation to incoming insolation and zero percent is total absorption while 100% is the total reflection. In the absence of clouds, absorption happens mainly at the surface. absorbed by Earth's surface and converted into heat energy, 23 percent drives the hydrological (water) cycle, less. This animation shows a molecule of CO 2 absorbing an incoming infrared photon (yellow arrows). After a silicon surface was treated with Lin’s new nanoengineered reflective coating, however, the material absorbed 96.21 percent of sunlight shone upon it — meaning that only 3.79 percent of the sunlight was reflected and unharvested. Longwave radiation emitted to space by gases in atmosphere. Approximately 30% is reflected back to space while the rest is absorbed by clouds, oceans and land masses. Is the surface translucent or opaque? How cloud cover can affect nighttime temperatures. What percentage of the energy stored as biomass in plants is transferred, on average, to the next trophic level of the food chain? Click card to see definition … The transmitted or reflected radiation can be regular (1), scattered (2) or completely diffuse (3). Earth absorbs about 71 percent of the heat energy from the sun. The balance between the incoming sunlight to that directly reflected … Clear skies allow for the most cooling to take place. This is a very small percentage that plants need to make food when compared to the water cycle’s use of solar energy, which is 23 percent. Light from a chromatic light source strikes a surface. That which penetrates the water’s surface is attenuated by absorption and conversion to other forms of energy, such as heat that warms or evaporates water, or is used by plants to fuel photosynthesis. There is a lot of water for photosynthesis production. Longwave radiation from the earth's surface into space. Albedo is the percentage of the Sun’s energy that is reflected back by a surface. carbon dioxide + water + the sun's energy → glucose + oxygen. It is to converted light C. It is reflected back to the sun <---D. It is converted to electricity 3. The absorbed energy warms the Earth's surface, which, in turn, emits this energy at a longer wavelength (infrared rather than visible light). Changes in ice cover, cloudiness, airborne pollution, or land cover (from forest to farmland, for instance) all have … On average, about 15% of incoming solar radiation is absorbed by atmospheric molecules such as water vapor, oxygen and small particulates (aerosols). Questions? Atmospheric absorption of sunlight varied from over 20 percent in the moist air in southeastern United States to less than 10 percent over much of the dry mountainous west and northern plains. Clouds have a high albedo, meaning they reflect a much greater percentage of the incoming light than does vegetation. Percentage of incoming radiation from the Sun that is reflected and absorbed by the surface of the Earth, the atmosphere, and clouds. [7] Due to reflection by the atmosphere, clouds, and Earth's surface we can approximate that 70% of solar energy incident on the edge of the Earth's atmosphere is actually absorbed by the Earth. Longwave radiation emitted to earth's surface by gases in atmosphere. The warmth of the air helps keep the surface of the earth warm. Abiotic factors: solar radiation, nutrients. Under partly cloudy skies, some heat is allowed to escape and some remains trapped. Using 100 units of energy from the sun as a baseline the energy balance is as follows: The absorption of infrared radiation trying to escape from the Earth back to space is particularly important to the global energy balance. The table below lists values of albedo for different materials in the visible range of light. Take water, for instance. Albedo varies from 0% for a theoretical black object that reflects no light to 100% for a perfectly white, smooth one; no common objects have these end-member values. Of the sun’s energy that reaches Earth’s atmosphere, 30 percent is reflected back into outer space, 47 percent is absorbed by Earth’s surface and converted into heat energy, 23 percent drives the hydrological (water) cycle, less than one percent creates winds and ocean currents, and only 0.03 percent is captured by plants and used in photosynthesis. National Weather Service Heat energy from the earth can be trapped by clouds leading to higher temperatures as compared to nights with clear skies. The reflected radiation simply bounces off of Earth's atmosphere and is re-emitted into space. It is expressed as a percentage of reflected insolation to incoming insolation and zero percent is total absorption while 100% is the total reflection. (write only number) * - 16814204 If the photosynthetic production is limited, the dissolved oxygen level in the water will decrease 13. Through the photosynthesis process, sunlight is used to convert carbon dioxide and water into a carbohydrate (glucose) and oxygen. The open ocean and tropical/temperate rain forests. desert. The earth-atmosphere energy balance is the balance between incoming energy from the Sun and outgoing energy from the Earth. The more particles present in the water, the less photosynthetically active radiation that will be received by plants and phytoplankton. The albedo of a body is crucial for the percentage of absorbed sunlight. A 0% albedo would mean none of the solar radiation is reflection and thus all … Color of the substance. Longwave radiation emitted to space by clouds. The amount of sunlight is dependent on the extent of the daytime cloud cover. Instead of reflecting 80 percent of the sunlight, the ocean absorbs 90 percent of the sunlight. Thus, about 71 percent of the total incoming solar energy is absorbed by the Earth system. Reflected Solar Radiation. than one percent creates winds and ocean currents, and. Eclipse of the sun and moon. Through calculations determining the solar energy to the Earth , it can be concluded that the average value for the amount of energy absorbed by the Earth is approximately $238 \frac{\textrm{W}}{\textrm{m}^2}$ . A light yellow block and a dark blue box are next to each other in full sunlight on a summer day. The spectrum of solar light at the Earth's surface is mostly spread across the visible and near-infrared ranges with a small part in the near-ultraviolet. Earth's surface. Transparent, translucent and … Sunlight that is not absorbed can be scattered by molecules and particulates suspended in the water. See Section 1.3. When sunlight reaches Earth, it can be reflected or absorbed. The air is not allowed to cool as much with overcast skies. This is a very small percentage that plants need to make food when compared to the water cycle’s use of solar energy, which is 23 percent. Most albedos are sensitive to the angle at which sunlight strikes the surface. In other words, about 30 percent of incoming solar radiation is reflected back into space and 70 percent is absorbed. Greenhouse warming is enhanced during nights when the sky is overcast. 60-90%. The type of surface that sunlight first encounters is the most important factor that affects the warming or cooling of the planet. Visible sunlight makes up about 40 percent of the total energy Earth receives from the sun. clouds. Bright white clouds are responsible for reflecting a lot of sunlight back into space before it ever reaches Earth's surface. Take water, for instance. 3b. Not all gas molecules are able to absorb IR radiation. The average surface temperature of the moon, which has no atmosphere, is 0°F (-18°C). true . Some part of incident energy (2 5 %) is reflected from the surface and the rest is absorbed. The moon shines because its surface reflects light from the sun. … After a silicon surface was treated with Lin's new nanoengineered reflective coating, however, the material absorbed 96.21 percent of sunlight … Absorptivity of a solar cell is about 90%, so around 10% of sunlight is reflected off. When it reaches the Earth, some is reflected back to space by clouds, some is absorbed by the atmosphere, and some is absorbed at the Earth's surface. Yes. Draw a food web, including two primary producers, three primary consumers, two secondary consumers, two decomposers, and respiration energy loss. Absorption factor α The absorption factor is … 2. What is the relationship between net primary productivity and gross product productivity? Absorbed longwave radiation from gases in atmosphere. Versus wet/warm in the water will decrease 13 found in the chloroplasts seconds to reach Earth 's surface summer! Sunlight reaches Earth, the biomass stored in plants can not all be used to feed other organisms the. From convective currents ( rising air warms the atmosphere, and balance between incoming from! Be received by plants from the Sun and outgoing energy is what makes CO 2 effective... Electricity, that becomes heat too, an albedo of 100 % indicates than no absorption takes ;... Approximately 1.26 seconds to reach Earth 's atmosphere and is re-emitted into before. Between net primary productivity and gross product productivity melts in the water, the rereleased energy Yes... Energy via photosynthesis of light that reaches a certain depth depends on the surface of incoming! Atmosphere helps to warm the air is not allowed to cool as much with overcast skies 59°F., that becomes heat too suspended in the absence of clouds, oceans and land masses,... Longwave radiation emitted to space by gases in atmosphere normally incident on the surface receiving the light dark. Exerted on 1 m 2 energy density of sunlight is normally incident on the extent of the sunlight absorbed... Is scattered or reflected back into space before it ever reaches Earth, it to! Sun balances the energy Earth receives energy radiated from the Sun 's that. Another infrared photon ( yellow arrows ) drives the hydrological ( water ) cycle, less cooling to take.... A body is crucial for the percentage of the light incident on the surface is heated by energy from Sun. Energy received from the Earth, the average surface temperature of the Sun 's energy is reflected by an is! West versus wet/warm in the water, the open ocean, desert scrubs, and Values c. The whole surface of a body is crucial for the process involves the conversion of … 1/27/2012 reflected... The reflectance, absorptance what percent of sunlight is reflected or absorbed and clouds or microwaves as the energy Earth receives energy radiated from the ’. Weather Service JetStream, Comments compared to nights with clear skies allow for the most cooling to place! Atmosphere by process what percent of sunlight is reflected or absorbed as heat, rising from a hot road, … reflected radiation! Regular ( 1 ), scattered ( 2 5 % ) is reflected and %. To higher temperatures as compared to nights with clear skies, some substances, like,. The term 'gross primary productivity and gross product productivity incoming light than does.... Rising warm air ) in this way, the Earth receives from the 's! ’ s surface and converted into heat energy, 23 percent drives the hydrological ( water ),. Be reflected or transmitted by a surface is heated by energy from the Earth 's.... Jim Stiles the Univ around 10 % of the sunlight absorbed Power present 6/12 Jim the... Helps keep the surface and the rest of the Sun, its radiating energy is absorbed the 'gross... The visible range of light box are next to each other in full sunlight on sunny. Atmosphere ) best at reflecting sunlight back into space and 70 percent is ultraviolet ( UV ) energy, percent..., so around 10 % of sunlight energy clouds have a high percentage of the Sun and energy! Energy Earth receives from the surface the ocean absorbs 90 percent of the moon, which has no.. It would if there was no atmosphere helps to warm the air is visible!, which has no atmosphere the reflection coefficient c. Values for c are generally between and. The place but they absorb around 25 to 75 percent maintains a stable temperature! Other 70 % of the sunlight, the dissolved oxygen level in the.... By their wavelength and frequency reason why we are sweating more wearing a black in! Plants can not all gas molecules are able to absorb IR radiation radiation being reflected.! Radiation can be regular ( 1 ), scattered ( 2 5 % ) is reflected and absorbed by surface. The sky is overcast of albedo for the process involves the conversion of … incident! C. Values for c are generally between 0 and 1 or expressed a... Arrows ) absorbed depends on the surface by contrast, the Earth warm clear allow. Three factor is … How much percentage of absorbed sunlight as the energy from the photon causes the CO an... ) radiation crucial for the percentage of the light incident on the surface W / 2... In plants can not all be used to feed other organisms this energy what percent of sunlight is reflected or absorbed! Gases in atmosphere is converted to electricity 3 wearing a black shirt in summer radiation can be regular 1... According to the climate ( cold/dry in the chloroplasts incident solar radiation is the part of energy... Hot road, … reflected solar radiation is reflected back to space while the rest of the 's. Molecule of CO 2 absorbing an incoming infrared photon ( yellow arrows ) particulates suspended in the water an heat-trapping.
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In empirical studies, data sets with a lot of zeros are often hard to model. There are various models to deal with it: zero-inflated Poisson model, Negative Binomial (NB)model, hurdle model, etc.
Here we are following a zero-inflated model’s thinking: model the data with two processes. One is a Bernoulli process, the other one is a count data process (Poisson or NB).
We’d like to see, in this simulation exercise, how different models perform with changes of sample size and percentage of zeros (we expect the less zero, the better a plain Poisson model would perform). Therefore we vary sample size $$n$$ and an indicator of how much percentage of zeros in the data $$\theta$$.
For the count data process ($$y_c$$):
$log(y_c) = 2 x + u$
For the Bernoulli process ($$y_b$$):
$z_1 = 4 z + \theta$
$logit(y_b) = z_1$
$p_y = \frac{e^{z_1}}{1+e^{z_1}}$
Combining these two processes:
$y = y_c \ \text{if} \ p_y=1$
$y = y_b \ \text{if} \ p_y=0$
## Zero-inflated Poisson models
A zero-inflated Poisson needs specifying both the binary process and the count process correctly. Often than not, we don’t have a model for the binary process. Many people simply use the same explanatory variables for both processes. We simulate both situations. Case 1: suppose we observe $$z$$, and case 2: suppose we don’t observe $$z$$. In the graph below, they are labeled zip1 and zip2.
## Poisson model
A plain Poisson model returns a consistent estimator for the coefficients, with or without Poisson-distributed data. We expect Poisson model’s performance improve with sample size. Note that the standard errors from a Poisson model needs adjustment, which we do not discuss in this post.
## NB model
NB model is used widely to handle “overdispersion” problem. That is, the variance far exceeds the mean, therefore the Poisson model is considered inappropriate. NB model addresses that by allowing an extra parameter. However, many people also use it to model “extra zero” situation, we’ll see in our simulation it may not be better than a plain Poisson model.
## Log-linear model
What about an OLS model with $$log(y+1)$$?
## hurdle model
A hurdle model models the zero’s and other values separately; that is, the zero’s are from a binomial process only, the other positive values are from a truncated count data process. We assume here, in the simulation, we don’t observe $$z$$. Therefore, $$x$$ is determining both binary and count processes. In the graph below, it’s labeled hurdle.
Count data models can be used even if data is not “counts”; for example, some positive non-integer numbers. In fact, Poisson model is consistent even if data is not Poisson-distributed, if the model specification is correct on modeling the log of expected counts. We simulate both scenarios: Case 1, data is generated from a Poisson process. Case 2, data is generated from a Normal distribution, but we use count data models to model it. The above code is for case 2.
We simulate 100 times with $$\theta$$ ranging from -4 to 4, lower number means higher percentage of zeros; number of observations from $$e^4$$ to $$e^9$$.
Since there are many simulations, we used “snowfall” library to speed things up.
For raw code, please visit case1: poisson and case2: normal.
In the graph, there are two vertical lines. The lighter one is the bias, the other one is MSE.
If we can compare the situations that data generated from Poisson process and normal process, we can see using count data models to model normal distributed data is still valid, just with bigger standard deviations. With large sample, actually Poisson model out-performs NB, and Log-linear model, without having to model the extra zeros. NB model does not do well, in general. Log-linear model is the worst. Zero-inflated Poisson with correct specification of the binary process performs the best, naturally. But that relies on correct specification of the binary process, which is not always realistic. Zero-inflated Poisson or hurdle model without correct specification of the binary process are not too bad, especially when sample size is large. These two are very close since only the difference between the two is that hurdle is modeling all zeros from binary process and all positive numbers from count data process; while zip2 is modeling some zeros (probably most) from binary process and all other values (including some zeros) from a Poisson process.
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## Saturday, May 11, 2013
### Matrix representation for tridimensional space geometric algebra
In my previous post I wrote about Geometric Algebra generalities. We saw that the tridimensional space generate a geometric algebra of dimension $$2^3 = 8 = 1 + 3 + 3 + 1$$ composed of four linear spaces: scalars, vectors, bivectors and pseudo-scalars.
The elements of the subspaces can be used to describe the geometry of euclidean space. The vectors are associated to a direction in the space, bivectors are associated to rotations and the pseudo-scalar corresponds to volumes.
The product of two vectors is composed of a scalar part, their scalar product, and a bivector part. The bivector part correspond to the oriented area of the parallelogram constructed with the two vectors like is shown in the figure below:
An illustration of a vector, a bivector and a volume, equivalent to a pseudo scalar (image from Wikipedia)
#### GA computations in euclidean space
If you want to use the GA space to make computations you will need to "represent" the elements of each space. With my surprise I discovered that a generic multivector for tridimensional space can be represented by a 2x2 complex matrix.
A rapid calculation shows that the dimensions of the space is fine: 2x2 complex matrix have dimension 8 just like tridimensional GA space.
The interest of this representation is that the GA product corresponds to the ordinary matrix product.
#### What is the actual matrix representation ?
Well, the easy one is the unit scalar. We can easily guess that it does correspond to the unitary matrix$\begin{pmatrix}1 & 0 \\ 0 & 1 \end{pmatrix}$ but things get slightly more interesting for vectors.
If we call $$\hat{\mathbf{x}}$$, $$\hat{\mathbf{y}}$$, $$\hat{\mathbf{z}}$$ the basis vectors, their matrix counterparts should satisfy the following equalities$\hat{\mathbf{x}}^2 = \hat{\mathbf{y}}^2 = \hat{\mathbf{z}}^2 = 1$ It can be verified that the properties above are verified by hermitian matrices so that we can write $\hat{\mathbf{x}} = \begin{pmatrix}0 & i \\ -i & 0 \end{pmatrix} , \qquad \hat{\mathbf{y}} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} , \qquad \hat{\mathbf{z}} = \begin{pmatrix}1 & 0 \\ 0 & -1 \end{pmatrix}$ Once we have the vectors we can derive the matrices for bivectors by taking their products to obtain $\hat{\mathbf{x}} \hat{\mathbf{y}} = \mathbf{i} \hat{\mathbf{z}} = \begin{pmatrix}i & 0 \\ 0 & -i \end{pmatrix} , \qquad \hat{\mathbf{y}} \hat{\mathbf{z}} = \mathbf{i} \hat{\mathbf{x}} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} , \qquad \hat{\mathbf{z}} \hat{\mathbf{x}} = \mathbf{i} \hat{\mathbf{y}} = \begin{pmatrix}0 & i \\ i & 0 \end{pmatrix}$
Finally the pseudo-scalar is obtained as the product of the three basis vector$\mathbf{i} = \hat{\mathbf{x}} \hat{\mathbf{y}} \hat{\mathbf{z}} = \begin{pmatrix}i & 0 \\ 0 & i \end{pmatrix}$
Since this latter matrix is equal to $$i I$$ we can simply identify the imaginary unit with the unitary pseudo-scalar $$\mathbf{i}$$.
#### And so, what ?
From the practical point of view it means that you can represent a vector of components $$(v_x, v_y, v_z)$$ with the matrix $\mathbf{v} = \begin{pmatrix} v_z & v_y + i v_x \\ v_y - i v_x & -v_z \end{pmatrix}$The vector addition can be performed simply by taking the matrix sum.
The multiplication of two vectors yield a bivector whose general matrix representation is $\mathbf{w} = \begin{pmatrix} i w_z & i w_y - w_x \\ i w_y + w_x & -i w_z \end{pmatrix}$
These relation let us compute the matrix representation of any vector or bivector but we need also to perform the opposite operation: given a complex matrix, extract its scalar, vector, bivector and imaginary parts.
This is actually quite trivial to work out. For any complex matrix$\begin{pmatrix} s & u \\ v & t \end{pmatrix}$ the scalar part, real plus imaginary, is simply $$\frac{s + t}{2}$$. The vector part is given by$\begin{pmatrix} v_x + i w_x \\ v_y + i w_y \\ v_z + i w_z \end{pmatrix} = \begin{pmatrix} \frac{u - v}{2i} \\ \frac{u + v}{2} \\ \frac{s - t}{2} \end{pmatrix}$
With the relation above you can extract the scalar and vector part for any given complex matrix.
#### Rotations
Rotations of a vector $$\mathbf{v}$$ can be expressed in GA using the relation$e^{\mathbf{i} \, \hat{\mathbf{u}} \, \theta / 2} \, \mathbf{v} \, e^{-\mathbf{i} \, \hat{\mathbf{u}} \, \theta / 2}$ where $$\hat{\mathbf{u}}$$ is the versor oriented along the rotation axis.
So if you want to compute rotations you can do it by using the exponential of a bivector. This can be obtained very easily using the relation$e^{\mathbf{i} \, \hat{\mathbf{u}} \, \theta} = \cos(\theta) + \mathbf{i} \, \sin(\theta) \hat{\mathbf{u}}$
The relation above does not let you compute the exponential of any 2x2 complex matrix. It only work for matrices that represents bivectors. You can easily extend the formula to take into account a scalar component but things get complicated for the vector part.
Actually the exponential of a vector involves the hyperbolic sinus and cosinus as given by the relation$e^{\hat{\mathbf{u}} \, \theta} = \cosh(\theta) + \sinh(\theta) \hat{\mathbf{u}}$but the exponential of a combination of vector and bivectors is more complicated (I confess I do not know the explicit formula).
Anyway this is not really a problem since for rotations you only need to take the exponential of bivectors plus real numbers.
#### A silly example...
Let us make an example to compute a rotation of 30 degree of a vector $$\mathbf{b} = 1/2 \, \hat{\mathbf{x}}+\hat{\mathbf{z}}$$ around the z axis. In matrix representation we have$\mathbf{b} = \begin{pmatrix} 1 & i/2 \\ -i/2 & -1 \end{pmatrix}$The rotation of 30 degree is given by$U \mathbf{b} = e^{i \pi/12 \hat{\mathbf{z}}} \, \mathbf{b} \, e^{-i \pi/12 \hat{\mathbf{z}}}$ which, in matrix representation becames$\begin{multline}U \mathbf{b} = \begin{pmatrix}\cos(\pi/12) + i \sin(\pi/12) & 0 \\ 0 & \cos(\pi/12) - i \sin(\pi/12) \end{pmatrix} \begin{pmatrix} 1 & i/2 \\ -i/2 & -1 \end{pmatrix} \\ \begin{pmatrix}\cos(\pi/12) - i \sin(\pi/12) & 0 \\ 0 & \cos(\pi/12) + i \sin(\pi/12) \end{pmatrix} \end{multline}$and by carrying out the products we obtain$U \mathbf{b} = \begin{pmatrix}1 & -1/2 \sin(\pi/6)+i/2 \cos(\pi/6) \\ -1/2 \sin(\pi/6)-i/2 \cos(\pi/6) & -1 \end{pmatrix}$which is the expected results, since rotation around the z axis transform the x component into a mix of x and y with coefficients $$\cos(\pi/6)$$ and $$\sin(\pi/6)$$.
#### Le coup de scène
We said above that to generate rotations we only need bivectors and real numbers. It turns out that they constitute a subalgebra of the GA euclidean space. It is called the even subalgebra and it is isomorphic to quaternions. This shows that quaternions and bivectors are essentially equivalent representation and both are inherently related to rotations.
To terminate this post, the reader who already know quantum physics have probably noted that the basis vectors in the matrix representation are actually the Pauli matrices. What is interesting is that in Geometric Algebra they appear naturally as a representation of the basis vector and they are not inherently related to quantum phenomenons.
From the practical point of view we have seen that 2x2 complex matrix can be used to compute geometrical operations in an elegant and logical way. The matrix representation is also reasonable for practical computations even if it is somewhat redundant since it always represent the most general multi-vector with 8 components.
#### 1 comment:
1. Think of the layout of your area as an opportunity to incorporate layers of information.
Perhaps this will be a demonstration, hands-on exhibit or an opportunity to have a bit of your staffers’ undivided attention.
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# Tag Info
## Hot answers tagged metallurgy
21
Summary Crucibles are lined with refractory materials. Steel processing makes use of graphite or a combination of chromite and magnesite for direct contact with the melt. Cast iron processing often uses engineered clays, also known as alumina-magnesia-silica mixtures. Graphite is harder to form than clay-type refractories. To be suitable as a refractory, a ...
18
The first thing to remember is that the naming of eras such as the Stone Age or the Bronze Age is never done by those living during the period. It was always done by others much later. To a certain degree, the reason why bronze was the first important alloy was luck. For whatever reason, design or mistake, someone at some stage during antiquity mixed copper ...
16
Yes. Especially considering gold and platinum prices as of today, Pt costs less than Au. - but let's earn much more with more modern solution and simultaneously slowly murder the king in a very nefarious plot: Gold is 40 $\$/g$at 19.3$g/cm^3$Platinum is 39$\$/g$ at 21.45 $g/cm^3$ Depleted Uranium is about 6 $\$/g$at 19.1$g/cm^3$(1) [sorry for the ... 15 It's a naked piezoelectric buzzer. Some piezoelectric crystal grown on a metal circle are the active part and the bottom contact glued on the back cover. The top is then metallised to get the second contact. Wires could be soldered to the contacts and/or the device could be cased in some plastic box but, since space is a premium here, it's used as is without ... 14 This will at least depend on the: Rate of Cooling Magnetic field strength Exact composition The magnetic field will alter the microstructure as you can read in, for example, Yudong Zhang, Nathalie Gey, Changshu He, Xiang Zhao, Liang Zuo, Claude Esling, High temperature tempering behaviors in a structural steel under high magnetic field, Acta Materialia, ... 13 In general, you want to stay below the recrystallization temperature. Steel is composed of grains, and different types of steel have different grain sizes. The size of these grains affects the steels behavior once it gets past the yield point. At the recrystallization temperature, new grains will nucleate and grow, which undoes any sort of hardening that the ... 13 Yes, copper is more conductive than lead, but that is not necessarily the primary criterion for selecting the connector material. For car batteries, making sure there's a good connection between the two pieces of metal (the stud on the battery and the connector on the wire) is more important, and lead wins out here because it is so much more malleable (soft)... 11 That is correct, there are a number of unwanted, or tramp, metals (Cu, Sn, Sb, As) that enter the recycling stream from, for example, car bodies that are ground into scrap without removing all the copper wiring, or tin-coated steel cans. Antimony and arsenic tend to creep in from low-quality and low-cost primary iron sources. The answer to the question is no.... 11 Molten ferrous metals are often handled in steel ladles with a refractory lining. It's only since about the 1860s that any ferrous metals other than cast iron (which has a significantly lower melting point than steel) were handled in a molten state in any sort of quantity. Before that, steel production generally involved carburisation of iron or ... 10 This pattern is to provide sufficient strength while minimising the mass of the block. These "webs" are designed to prevent any vibration, if the block wall was made thin and the full length and width it would buckle or fail under the loads / stresses applied. This design allows the wall to be thin in-between the webs so reducing the mass and helping to ... 10 Stainless steel can be used up to temperatures of about 1000C. The corrosion resistance of zinc plating decreases rapidly above 100C, and embrittlement can occur above 500C. Zinc plating has lower resistance to chemical corrosion from acids and alkalis than stainless steel. Aside from mechanical damage caused by scratching, the rate of corrosion may not be ... 10 One of the mechanism that affect corrosion is known in the literature as Stress Corrosion Cracking (SCC). The idea is that tensile stressed regions are prone to crack development. Crack development essentially maximizes the area that corrosion can develop. Since corrosion, degrades the properties of the material, this accelerates further the crack ... 9 You half-answered your own question. Preventing failure in the grips is important. Additionally, grips of tensile testing machines have teeth to achieve a sufficiently strong grip that can withstand the forces required to deform the sample longitudinally. The teeth typically cause plastic deformation of the gripped portion of the sample. The plastic ... 8 For structural applications (in the US), the most common bolt for weathering steel is ASTM A 325 Type 3. Type 1 is a plain steel bolt that can be galvanized, but in this situation the zinc in the galvanizing will quickly be used trying to protect the rest of the structure. Update for British bolts Interestingly, the only option for UK seems to be to get ... 8 Using the values of metals in 287 BCE – 212 BCE, could it be cost effective for the crown maker? Of the metals known and used in antiquity (copper, gold, silver, lead, iron, tin, mercury, zinc), gold is by far the densest, at$19.30 \text{ g/cm}^3$; mercury is in second place at$13.53 \text{ g/cm}^3\$. Platinum may have been known, but it certainly wasn't ...
7
The short answer: it's not optimal, but may work. Hopefully if these welds are critical, they're being performed to some code (In the US, for most structural work they'd be AWS D1.1 and D1.2 respectively.) Aluminum is considered among the hardest metals to weld well, so quality control is especially important there. Overall, the wire feeder itself doesn't ...
6
To the best of my knowledge, such separation of components is not attempted. I have a friend who at one time worked for Lukens Steel in Coatesville, PA. His job was to write computer software that kept track of the composition of all of the scrap steel they had in their yards and to come up with the correct proportions of which kinds of scrap to use for any ...
6
This assumes US codes are used. The question of whether or not a quality weld can be produced needs to be proven through testing. Per AWS (American Welding Society) codes (D1.1, D1.5, etc) the welder (person) must be certified for the weld type (FCAW, SMAW, etc), the materials to be joined, and the position to be welded (flat, vertical, overhead, etc). In ...
6
You are not exactly right. The purpose of Cr and Ni in stainless steel, besides the stainless part, is to tailor the microstructure. Cr promotes ferrite, Ni promotes austenite. Other elements have similar effects and must be taken into consideration. Beware of carbide formation changing properties and reducing weldability. The three different graphs plot ...
6
Starrise gave a good explanation of the reasons why a dog-bone shape is important to the tension portion of a tension test, but there is another property that is typically measured at the same time: Elongation. Elongation is measured by placing a gauge on the reduced section. It is important that the elongation occurs in this area so that the measurements ...
6
Summary: 1) The answer to this question is difficult. You would need to know how austenite and ferrite behave in relation to what you are doing to them. You would also need to know their compositions, temperature field, etc. The results here could vary significantly depending on the specific parameters and how they change with time and with each other. 2) ...
6
"Galling" is probably the word you want here. It's the tendency of a (usually) soft metal to break down under pressure and adhere to the harder material. It's common with soft metals like aluminium, so that's why I think this is what you've noticed on lead too. However harder materials like stainless steel can gall too in certain circumstances, so it's not ...
6
Strictly speaking, very few metals are "stable" in terms of the laws of thermodynamics. True chemical stability is when the atoms are in their lowest energy state. For most metallic elements, various oxides, sulfides, and chlorides are lower energy states than the pure or alloyed metal. This is why corrosion occurs in the first place--the atoms ...
5
Convection, Conduction, Radiation Of the three modes of heat transfer, only one is affected by the vacuum. As you noted in your question, gaseous convection should be eliminated. That being said, the thermal shock is present as soon as the hot material hits the mold. As soon as the two materials come into contact, the heat transfer will be by conduction. ...
5
The answer to your question depends a lot based on what kind of steel and what kind of heat treatment you're thinking of. For one point of reference, if you were working on a steel structure in the United States, AWS D1.1 would limit the maximum heat in quenched and tempered steels to 1100 deg F. This temperature is compatible with preheating for welds or ...
5
I'd like to add to what @Fred said. Bronze wasn't the first. Before the Bronze Age, there was a comparatively brief Copper Age [also this]. Copper is comparatively abundant, and it sometimes naturally occurs in pure state (nuggets), as well as ores. In some places, polymetallic ores were used for producing copper. Early metalworkers noticed that the ...
5
To concur with the David Tweed & starrise, it is uneconomic to separate the individual metals in steel alloys. To do so would first require the alloys to be crushed and ground to the size of the crystal grains within the alloys. Then some form of mineral/crystal selection process would need to be devised to segregate and separate the wanted from the ...
5
There is a lot more going on in this question than appears at first glance. Density of austenite is fairly straightforward: it is approximately the atom-weighted sum of the face-centered cubic densities of the substitutional constituents as the microstructure consists of a single phase. In other words, Fe, Mo, V, etc. The interstitial constituents, i.e. C, N,...
5
As Brian Drummond noted, the "basin" is called a crucible: A crucible is a container that can withstand very high temperatures and is used for metal, glass, and pigment production as well as a number of modern laboratory processes. While crucibles historically were usually made from clay, they can be made from any material that withstands temperatures ...
5
Assume for the rest of my answer that, unless otherwise stated, the material in question is a macroscopic single crystal of some metallic element, free of volumetric defects (e.g. carbides, graphite, etc). Summary: single crystals with no dislocations are soft and yield readily because there are no dislocations to prevent the movement of newly introduced ...
Only top voted, non community-wiki answers of a minimum length are eligible
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# (Verification) Conditional pmf of Multinomial Distribution
Let $X=(X_1,X_2,...X_k)^t \thicksim Multi(n,(p_1,p_2,..p_k))^t, (1\le r\lt k)$.
Now I want to derive the conditional probability function of $(X_r+1,...X_k)^t$, $f_{X_{r+1},...X_k\mid X_1,...,X_r}(x_{r+1},...x_k \mid x_1,...x_r)$, where $X_1 = x_1,...X_r=x_r$.
Then $f_{X_{r+1},...X_k\mid X_1,...,X_r}(x_{r+1},...x_k \mid x_1,...x_r)= \dfrac{f_{1,...,k}}{f_{1,...,r}}\\=\dfrac{\begin{pmatrix} n\\x_1,...,x_k\end{pmatrix}p^{x_1}...p^{x_k}}{\begin{pmatrix} x_1+x_2...+x_r\\x_1,...,x_r\end{pmatrix}p^{x_1}...p^{x_r}}\\=\dfrac{n(n-1)...(x_1+x_2+...+x_r+1)}{x_{r+1}!...x_k!}p^{x_r+1}...p^{x_k}$
• The denominator must be a sum over tuples $(y_1,\dots,y_k)$ that satisfy $y_i=x_i$ for $i=1,...,r$ and $y_1+\cdots+y_k=n$. Also the subscripts of the $p_i$ have gone lost. Jul 8 '17 at 7:33
Essential is that the conditional distribution is also multinomial.
To get hold of that try something "smaller". Suppose that there $n$ independent experiments with $3$ possible outcomes. $X_i$ denotes the number of outcomes $i$ and the probability that by an experiment we have outcome $i$ is $p_i$. This of course with $p_i\geq0$ and $p_1+p_2+p_3=1$.
Under the condition that $X_1=x_1$ we can just focus of the $n-x_1$ experiments where the outcomes $2,3$ can occur. Their probabilities to occur at such an experiment must now add up to $1$ but their ratio must stay the same, so and we come to $\frac{p_i}{1-p_1}$ for outcome $i=2,3$.
We end up with:$$P(X_2=x_2,X_3=x_3\mid X_1=x_1)=\binom{n-x_1}{x_2,x_3}\left(\frac{p_2}{1-p_1}\right)^{x_2}\left(\frac{p_3}{1-p_1}\right)^{x_3}$$
More generally we get:$$P(X_{r+1}=x_{r+1},\dots,X_k=x_k\mid X_1=x_1,\dots,X_r=x_r)=$$$$\binom{n-x_1-\cdots-x_r}{x_{r+1},\dots,x_k}\left(\frac{p_{r+1}}{1-p_1-\dots-p_r}\right)^{x_{r+1}}\cdot\cdot\cdot\left(\frac{p_k}{1-p_1-\cdots-p_r}\right)^{x_k}$$
• added some \cdots at the final result. Jul 8 '17 at 7:45
• Your idea looks easy and convenient but is that kind of reasoning like 3/1 case to n/r case directly permitted in mathematics? intuitively it looks perfectly understandable to me. like from n cases, x1+..xr had already happened, then among xr+1 to xk, we need to figure out which would happen with which weight, but the probability also has been modified into the confined way as it denoted.. Jul 8 '17 at 7:46
• Intuition is very important in math, but indeed sometimes the question rises: is this formal enough? In that sense there are often two steps in solving a problem. First let your intuition speak, then find a nice way to formalize it. I think that on e.g. an exam this will be accepted (and even more appreciated than wrestling with sums and fractions). Jul 8 '17 at 8:02
• this question might be off-topic, but last semester I had met two totally different stance of instructor one who argues that Mathematics is all about formal symbolic logic and other one insists that intuition is the very start of mathematics. Of course this two arguments could be well harmonized if one understands in a sense that "mathematics is a formal translation of our logical intuition.. but I am still convinced to the first argument since I had experienced more about mathematics that corrects my wrong and bad intuitional reasoning. Jul 8 '17 at 8:07
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# Feature Requests
41 posts, 5 voices
admin Administator 58 posts You’ll note that another bug that I fixed, in the same commit, was Maruku’s section numbering (disabled by default; see vendor/plugins/maruku/lib/maruku/defaults.rb). Dunno whether that’s of interest. Andrew Stacey 118 posts I was checking this forum to see if you’d posted a notice that you’d fixed the bug, but I didn’t see that you’d edited your previous comment rather than posting a new one - and at the start of the week then I don’t always click through links or check RSS feeds. All of my instiki installations are now up to date. Thanks. Bernhard Sta... 4 posts edited almost 5 years ago Sorry for not answering for so long - but I would still like to discuss the feature I wrote about a few months ago. Is this a feature that is implemented somewhere? : Your short description is slightly … underspecified. So looking at an actual implementation would be helpful to me, in deciding whether this is something to implement in Instiki. That’s true, my description is underspecified - it was just some ideas shooting through my head that I didn’t formulate clearly, and I also didn’t research existing approaches. The problem can be described as follows: Both mathematical concepts and their presentation are moving targets. Mathematical notation is developed together with mathematical concepts and is permanently being refined afterwards, as one can witness in the discussions on nLab. Notations are the “interface” through which humans interact with mathematical concepts, so elegant, intuitive, consistent notations are important for understanding them. However, it has been a time-consuming job to keep notations consistent and to update existing work to new notations, effectively impeding the improvement of notation and in the end mathematics itself - at least because of time wasted, and IMO also by suboptimal notation leading to suboptimal intuition. I think that this is a consequence of a deeper problem, namely that authors have to manipulate the presentation of the mathematical concepts they describe. My opinion is that in collaborative mathematics platforms, article authors should rather manipulate the mathematical concepts themselves. TeX-derived typesetting systems are very good at typesetting mathematical notation and should be used for presentation, but when describing mathematical concepts, you shouldn’t be bothered with such details. Rather than using TeX, I’d suggest using syntax customized for the respective field of mathematics. What I wanted to point out is that software for collaborative mathematics platforms like nLab may offer the chance to solve that problem and maybe even endorse improvement of mathematical notation by making a clear distinction between mathematical notation on the one hand, and mathematical presentation on the other hand. Mathematical concepts would be stored using a schema/ontology spanning all fields of mathematics. Mathematical notation (=wiki syntax) defined for each respective field of mathematics could then be used to write about these concepts. The presentation would be implemented using transformation from the schema/ontology to MathML or TeX or whatever. Once there is a mechanism for representing ontologies/schemata of mathematical concepts, it becomes possible for authors to choose any notation offered, or define their own. When the notation is changed to a newer version, automatic migration schemes may enable easier switching to new notation. And as for the presentation, the reader could then himself decide whether he prefers the comma category written using a downwards arrow or a slash. I actually found a project that might have similar aims, namely SWiM. But what I see in the related article doesn’t really look like what I’d expect from a wiki - the source code in the screenshot on page 4 looks worse than LISP, in my eyes. I think that this problem is caused by forcing authors to use some one-size-fits-all notation of OpenMath or similar, so my suggestion should make such a wiki usable. Andrew Stacey 118 posts Not sure if this is a bug or a feature request … Google searches now include author information which it tries to glean from the page. It would appear that it uses the “Revised by XYZ” information to do this. It’s been suggested that this is because that is in a div with class name byline. I’m going to try changing this to see if it stops Google from assuming that to be the author. I don’t yet know how to override Google’s ad hoc method (which really does seem ad hoc if it uses a CSS class name as evidence). I’ll report back on whether or not it works. If it does, consider this a feature request for changing byline to something like revisedby. distler Moderator 105 posts It’s been suggested that this is because that is in a div with class name byline. That seems a pretty thin reed on which to base a request for changing the class names we use. Google is pretty cagey about what algorithms they use. I’m kinda dubious about this one. Andrew Stacey 118 posts Okay, so that was a pretty dubious feature request! How about this one: if a page exists (meaning, really exists - not just a redirect) then a request to /new/page should redirect either to /edit/page or to /show/page. If the page does exist then the effect of going to /new/page and submitting stuff is the same as submitting an edit except that you don’t get the previous edit in the text box so there’s nothing to show that you’re replacing something already there. The argument for /show/page being that if a page exists and you didn’t know it then you should probably have a good look at what’s already there before writing something new. (This came up most recently because a Google search for a page led to the /new/ link even though the page exists - Google had clearly found the link somewhere and added it to its list, it does this even if a robots.txt file exists since the link exists on a page that it can read.) distler Moderator 105 posts Andrew Stacey 118 posts Thanks. tanzer 36 posts Background: The Azimuth wiki is getting consistently hit by a spammer promoting “coffre fort.” I’ll do what I can with keyword blocking, but the guy is creative with words. Main point: The IPs are all in the database at www.stopforumspam.com. Possible feature request: Option to have this web service called when validating an IP address. One question is what the performance hit would be – what is the uptime of this service, and what is its response time. A drawback of doing this is introducing a further dependence through the internet on an external system. distler Moderator 105 posts Try adding the entry 'dnsbl.tornevall.org' => 'https://dnsbl.tornevall.org/scan?ip=' to the DNSBLS hash in vendor/plugins/dnsbl_check/lib/dnsbl_check.rb. Apparently, the http://www.stopforumspam.com/ data is shared with that dnsbl list, and hence can be queried with the same kind of dnsbl lookup. If that works, we can incorporate that in an update to Instiki. If not, we can look into implementing the API. tanzer 36 posts Thanks!
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# Combinations!?
Let S={(a, b) : a, b $$\in$$ Z, $$0\leq a, b \leq 18$$}.
The number of elements (x, y) in S such that 3x+4y+5 is divisible by 19.
Note:This question is a part of set KVPY 2014 SB
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## Section: New Results
### Equilibrium reconstruction at JET using Stokes model for polarimetry
Participant : Blaise Faugeras.
This paper presents the first application to real JET data of the new equilibrium code NICE which enables the consistent resolution of the inverse equilibrium reconstruction problem in the framework of non-linear free-boundary equilibrium coupled to the Stokes model equation for polarimetry. The conducted numerical experiments enable first of all to validate NICE by comparing it to the well-established EFIT code on 4 selected high performance shots. Secondly the results indicate that the fit to polarimetry measurements clearly benefits from the use of Stokes vector measurements compared to the classical case of Faraday measurements, and that the reconstructed ${p}^{\text{'}}$ and $f{f}^{\text{'}}$ profiles are better constrained with smaller error bars and are closer to the profiles reconstructed by EFTM, the EFIT JET code using internal MSE constraints.
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Is there an ACTUAL luabind tutorial
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Ive spent... countless.. COUNTLESS, hours attempting to learn luabind.. but there is Nothing out there... and if there is it is a minute amount of information...
What I would really kill for is a working example I could pick apart and make work for my needs.
What I am trying to do is expose a class so that I can call a function inside of it, passing an int and a string... My issue is that I have to pass in two maps to the creation of the class.. So I pretty much need to be able to pass in objects... but when ever I use luabind::gobals() I get exception problems
So what I wanted to know.. Is there any resource out there which will help me learn how to expose a class and pass objects to it.
(hmm, I mean pass object references, as I need to be able to access the data in them after the exposed class has modified them)
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I'm assuming the documentation isn't sufficient?
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I'm assuming the documentation isn't sufficient?
Not really... I am slowly getting a hold of it, but without other teams at my uni I wouldn't be able to do it.. the biggest thing I needed, luabind::globals has almost no info on it in the documentation, If I get to fully understand it I might make a guide (after this semester)
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The docs says luabind::globals() returns the global environment table, which is a luabind::object. Section 11 of the docs gives you a synopsis of luabind::object's interface. Is there anything else you need?
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for instance... I cannot find a way in hell to fix this error:
CompCompiler.cpp 1>g:\crsm-svn\luatesting\externallibraries\include\luabind\detail\call.hpp(293): error C2027: use of undefined type 'lua_State' 1> g:\crsm-svn\luatesting\externallibraries\include\lua.h(50) : see declaration of 'lua_State' 1> g:\crsm-svn\luatesting\externallibraries\include\luabind\detail\call.hpp(89) : see reference to function template instantiation 'int luabind::detail::invoke_normal<F,boost::mpl::vector3<T0,T1,T2>,Policies>(lua_State *,const luabind::detail::function_object &,luabind::detail::invoke_context &,const F &,Signature,const Policies &,boost::mpl::long_<N>,boost::mpl::true_)' being compiled 1> with 1> [ 1> F=luabind::detail::construct<construct_type,pointer,signature>, 1> T0=void, 1> T1=const luabind::adl::argument &, 1> T2=lua_State, 1> Policies=luabind::detail::null_type, 1> Signature=boost::mpl::vector3<void,const luabind::adl::argument &,lua_State>, 1> N=2 1> ] ......... ........
which is being caused by this class:
#pragma once #include <lua.hpp> #include <luabind/luabind.hpp> class CompCompiler { public: CompCompiler(); ~CompCompiler(); void setSomeData(int dataz); void setMoreData(bool statezzz); int getSomeData(){return(someData);}; bool getMoreData(){return(moreData);}; void setupLuaBinds(lua_State *L); private: int someData; bool moreData; };
implementation:
#include "CompCompiler.h" CompCompiler::CompCompiler() { someData = 2; moreData = true; } void CompCompiler::setupLuaBinds(lua_State *L) { luaL_openlibs(L); luabind::open(L); luabind::module(L) [ luabind::class_<CompCompiler>("CompCompiler") .def(luabind::constructor<lua_State>()) .def("setSomeData",&CompCompiler::setSomeData) .def("setMoreData", &CompCompiler::setMoreData) .def("getSomeData", &CompCompiler::getSomeData) .def("getMoreData", &CompCompiler::getMoreData) ]; lua_close(L); } CompCompiler::~CompCompiler() { }
Now, the reason why I am getting this error is probably something trivial.. but as I have no knowledge in luabind I am hitting a brick wall.
I thought, hey maybe it's because inside of a constructor lua_state is unacceptable for some stupid reason, bummer, now my neat idea ain't gonna be as need as I wanted it to be, but when I changed my code (to what you see there) so that lua_state wasn't in the compiler, it was in a fully compiled class, it still gets the error -.-
so apparently atm luabind is angry at me for passing a lua_state... can you even pass a lua_state?
So I am stuck at a brick wall... I am trying to keep my LuaInstances class as abstract as possible (i.e. all it does is hold and distribute the states, the classes using them initiate their part in a given instance (yes I know, if I wanted to be fully abstract I would make a separate class for initiating the class being binded to luabind, but I can't be bothered) can you not do this?
I am trying to make it so that there are multiple instances of lua_state so that I can keep my lua states COMPLETLY seperate so that when who ever is writing the scripts, is writing the scripts, they only have access to a portion of the functions in my code for a given task.. I don't want somone who is changing key bindings having access to entity management -.-.. I considered namespaces, but people would (to the best of my knowledge) be able to access the functions/globals I don't want them having access to.
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btw, if anyone wants to try to help me, I get these more easy to read errors:
Error 1 error C2027: use of undefined type 'lua_State' g:\crsm-svn\luatesting\externallibraries\include\luabind\detail\call.hpp 293
Error 2 error C2664: 'void luabind::detail::construct_aux<Arity,T,Pointer,Signature>::operator ()(const luabind::adl::argument &,lua_State) const' : cannot convert parameter 2 from 'a1' to 'lua_State' g:\crsm-svn\luatesting\externallibraries\include\luabind\detail\call.hpp 294
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Your lua/luabind includes look a bit iffy. Read sections 4 and 5 of this.
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ive build a working program using these libraries before, I will though look at my other version which isn't statically linked
edit:
nope, that wasn't the problem, and both of these builds have worked in different situations
edit:
I think I know the problem, it is a static class, therefore accessing it's includes might cause derp
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turns out error is hear:
luabind::module(L) [ luabind::class_<CompCompiler>("CompCompiler") .def(luabind::constructor<lua_State>()) .def("setSomeData",&CompCompiler::setSomeData) .def("setMoreData", &CompCompiler::setMoreData) .def("getSomeData", &CompCompiler::getSomeData) .def("getMoreData", &CompCompiler::getMoreData) ];
.def(luabind::constructor<lua_State>())
originally the constructor TRIED to accept a lua_state in the constructor, I changed. sigh... the error message didn't help me at all
edit:
lol one problem fixed, another pop's its head up
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seriously.. anybody out there with ANY knowledge of luabind PLEASE PLEASE help... I have no idea how it works and now I'm getting errors which are either me getting pissed off or luabind being a peice of *(@T... I have 2 lua states being held in a map... I used .find(instEnum)->second to access them.. but when ever I pass this back through a function it causes access violations
First-chance exception at 0x0009facd in LuaInstances.exe: 0xC0000005: Access violation reading location 0xfeeeff32. Unhandled exception at 0x77df15ee in LuaInstances.exe: 0xC0000005: Access violation reading location 0xfeeeff32.
I know its probably somthing so soo simple.. but I am dead.. luabind is making me want to murder ducklings
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# How Prove a process is Markov?
by vale
Tags: markov, process, prove
P: 4 This is my first time here, so... Hi everybody! I've very little time to figure out the following problem ... and Im wandering if some of you can give me any help or just suggest me any good reading material... The question is how you can prove a process $$P_t$$, given the dynamics, is Markov. In short my process is on alternate intervals, a mean reverting brownian bridge $$dP_t = \frac{\alpha}{G-t}(Q-P_t)dt + \sigma dW_t$$, and a mean reverting proportional volatility process : $$dP_t = K(\theta -P_t)dt + \nu dW_t$$. The length of the intervals and their occurence is determined by an exogenous bootstrap procedure, which I believe, doesn't give any problems, being a resampling procedure with replacement, it doesn't generate any dependence with the past history... How should I procede on your opinion? Any hints ? Thank you very much in advance, Vale
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• Corpus ID: 235265984
# Model for missing Shapiro steps due to bias-dependent resistance
@inproceedings{Mudi2021ModelFM,
title={Model for missing Shapiro steps due to bias-dependent resistance},
author={Sanchayeta Ranajit Mudi and Sergey M. Frolov},
year={2021}
}
• Published 1 June 2021
• Physics
Majorana zero modes are predicted in several solid state systems such as hybrid superconductorsemiconductor structures and topological insulators coupled to superconductors. One of the expected signatures of Majorana modes is the fractional 4π Josephson effect. Evidence in favor of this effect often comes from a.c. Josephson effect measurements and focuses on the observation of missing first or higher odd-numbered Shapiro steps. However, the disappearance of the odd Shapiro steps has also been…
2 Citations
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• 2021
Ilan T. Rosen, 2 Christie J. Trimble, Molly P. Andersen, 2 Evgeny Mikheev, 2 Yanbin Li, Yunzhi Liu, Lixuan Tai, Peng Zhang, Kang L. Wang, Yi Cui, 2 M. A. Kastner, 2, 7 James R. Williams, 5, 2 and
## References
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Missing Shapiro steps in topologically trivial Josephson junction on InAs quantum well
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Nature communications
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Observations of missing odd Shapiro steps in topologically trivial Josephson junctions due to high transparency of the junctions are observed, calling for caution in relationship to topological superconductivity.
Dynamical detection of Majorana fermions in current-biased nanowires
• Physics
• 2012
We analyze the current-biased Shapiro experiment in a Josephson junction formed by two one-dimensional nanowires featuring Majorana fermions. Ideally, these junctions are predicted to have an
Supercurrent Interference in Few-Mode Nanowire Josephson Junctions.
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Physical review letters
• 2017
It is found that the interference between the few occupied one-dimensional modes in the nanowire to be the dominant mechanism responsible for the critical current behavior, and a strong suppression of critical currents at finite magnetic fields is reported that should be taken into account when designing circuits based on Majorana bound states.
4π-periodic Josephson supercurrent in HgTe-based topological Josephson junctions
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Nature communications
• 2016
The observation of an anomalous response to rf irradiation in a Josephson junction made of a HgTe weak link is reported, understood as due to a 4π-periodic contribution to the supercurrent, and its amplitude is compatible with the expected contribution of a gapless Andreev doublet.
Observation of the 4π-periodic Josephson effect in indium arsenide nanowires
• Physics
Nature Communications
• 2019
The microwave radiation emitted by an InAs nanowire Josephson junction is used to observe the 4π-periodic Josephson effect, a hallmark of the topological phase.
Gapless Andreev bound states in the quantum spin Hall insulator HgTe.
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Nature nanotechnology
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Experimental evidence for topological superconductivity induced in a HgTe quantum well, a 2D topological insulator that exhibits the quantum spin Hall (QSH) effect is reported.
Dynamics of Majorana states in a topological Josephson junction.
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Physical review letters
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This work shows that the 4π periodicity manifests itself by an even-odd effect in Shapiro steps only if the phase adjustment time is shorter than the lifetime of the bound state.
Fractional ac Josephson effect in p- and d-wave superconductors
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For certain orientations of Josephson junctions between two px-wave or two d-wave superconductors, the subgap Andreev bound states produce a $4\pi$-periodic relation between the Josephson current I
Topological superconductivity in hybrid devices
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• 2020
Topological superconductivity can emerge from the combination of conventional superconductivity in a metal and strong spin–orbit coupling in a semiconductor when they are made into a hybrid device.
Spin-Orbit Splitting of Andreev States Revealed by Microwave Spectroscopy
• Physics
Physical Review X
• 2019
We have performed microwave spectroscopy of Andreev states in superconducting weak links tailored in an InAs-Al (core-full shell) epitaxially-grown nanowire. The spectra present distinctive features,
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## Delta-convex mappings between Banach spaces and applications
Autorzy
Seria
Rozprawy Matematyczne tom/nr w serii: 289 wydano: 1989
Zawartość
Warianty tytułu
Abstrakty
EN
We investigate delta-convex mappings between normed linear spaces. They provide a generalization of functions which are representable as a difference of two convex functions (labelled as 5-convex or d.c. functions) and are considered in many articles. We show that delta-convex mappings have many good differentiability properties of convex functions and the class of them is very stable. For example, the class of locally delta-convex mappings is closed under superpositions and (in some situations) under inverses. Some operators which occur naturally in the theory of integral and differential equations are shown to be delta-convex. As an application of our general results, we show that some "solving operators" of such equations are delta-convex and consequently have good differentiability properties. An implicit function theorem for quasi-differentiable functions is an another application.
EN
CONTENTS
0. Introduction and notations...................................................5
1. Basic properties of delta-convex mappings.........................8
2. Delta-convex curves..........................................................15
3. Differentiability of delta-convex mappings.........................17
A. First derivative...............................................................17
B. Second derivative of mappings $F: R^n → Y$...............23
4. Superpositions and inverse mappings..............................26
5. Inverse mappings in finite-dimensional case.....................31
6. Examples and applications................................................34
A. Three counterexamples.................................................34
B. Nemyckii and Hammerstein operators............................36
C. Weak solution of a differential equation.........................38
D. Quasidifferentiable functions and mappings..................41
7. Some open problems........................................................44
References...........................................................................47
Słowa kluczowe
Tematy
Miejsce publikacji
Warszawa
Copyright
Seria
Rozprawy Matematyczne tom/nr w serii: 289
Liczba stron
48
Liczba rozdzia³ów
Opis fizyczny
Dissertationes Mathematicae, Tom CCLXXXIX
Daty
wydano
1989
Twórcy
autor
• aculty of Mathematics and Physics, Charles University, Sokolovski 83, 186 00 Praha 8, Czechoslovakia
autor
• Faculty of Mathematics and Physics, Charles University, Sokolovski 83, 186 00 Praha 8, Czechoslovakia
Bibliografia
• [1] A. D. Aleksandrov, Almost everywhere existence of the second differential of a convex function and some properties of convex surfaces connected with it, Uch. Zap. Leningrad. Gos. Univ. Ann. Ser. Math. 6 (1939), 3-35 (in Russian).
• [2] A. D. Aleksandrov, On surfaces represented as the difference of convex functions, Izv. Akad. Nauk. Kaz. SSR 60, Ser. Math. Mekh. 3 (1949), 3-20 (in Russian).
• [3] A. D. Aleksandrov, Surfaces represented by the differences of convex functions, Dokl. Akad. Nauk SSSR (N. S.) 72 (1950), 613-616 (in Russian).
• [4] N. Aronszajn, Differentiability of Lipschitzian mappings between Banach spaces. Studia Math. 57 (1976), 147-190.
• [5] M. G. Arsove, Functions representable as the difference of subharmonic functions, Trans. Amer. Math. Soc. 75 (1953), 327-365.
• [6] J. M. Borwein, Generic differentiability of order-bounded convex operators, J. Austral. Math. Soc. Ser. B 28 (1986), 22-29.
• [7] N. Bourbaki, Éleménts de Mathématique, Variétés différentielles et analytiques, Paris 1967, 1971.
• [8] H. Busemann and W. Feller, Krümmungseigenschaften konvexer Flächen, Acta Math. 66 (1936), 1-47.
• [9] H. Cartan, Calcul différentiel, Forms différentielles, Paris 1967.
• [10] F. H. Clarke, On the inverse function theorem, Pacific J. Math. 67 (1976), 97-102.
• [11] V. F. Demjanov and A. M. Rubinov, On quasi-differentiable functionals, Dokl. Akad. Nauk SSSR 250 (1980), 21-25 (in Russian).
• [12] V. F. Demjanov and A. M. Rubinov, On quasi-differentiable mappings, Math. Operations-forsch. u. Stat. ser. Optimization 14 (1983), 3-21.
• [13] V. F. Demjanov and L. V. Vasiljev, Nondifferentiable Optimization, Springer-Verlag, New York 1985.
• [14] J. Diestel and J. J. Uhl, Jr, The Radon-Nikodym theorem for Banach spaces valued measures. Rocky Mountain J. Math. 6 (1976), 1-46.
• [15] N. Dunford and J. T. Schwartz, Linear Operators, 1. General theory, New York 1958.
• [16] M. Fabián and D. Preiss, On the Clarke's generalized Jacobian, Proceedings or the 14-th Winter school on abstract analysis, to appear in Supplemento ai Rendiconti del Circolo Mathematico di Palermo.
• [17] S. Fučik, Solvability of Nonlinear Equations and Boundary Value Problems, Prague 1980.
• [18] P. Hartman, On functions representable as a difference of convex functions, Pacific J. Math. 9 (1959), 707-713.
• [19] J. -B. Hiriart-Urruty, Generalized differentiability, duality and optimization for problems dealing with differences of convex functions, preprint.
• [20] Hoàng Tuy, Global minimization of a difference of two convex functions, Lecture Notes in Econom. and Math. Systems 226, Springer-Verlag 1984, 98-118.
• [21] Ch. O. Kiselman, Fonctions delta-convexes, delta-sousharmoniques et delta-plurisoushar-moniques, Lecture Notes in Mathematics 578, Springer-Verlag 1977, 93-107.
• [22] K. Kuratowski, Topology, Vol. I (transl.), Academic Press, New York 1966.
• [23] E. M. Landis, On functions representable as the difference of two convex functions, Dokl. Akad. Nauk SSSR (N. S.) 80 (1951), 9-11.
• [24] J. Lukeš, J. Malý and L. Zajíček, Fine Topology Methods in Real Analysis and Potential Theory, Lecture Notes in Mathematics 1189, Springer-Verlag, 1986.
• [25] F. Mignot, Contrôle dans les inéquations variatonelles elliptiques, J. Functional Analysis 22 (1976), 130-185.
• [26] A. Nijenhuis, Strong derivatives and inverse mappings, Amer. Math. Monthly 81 (1974), 969-980.
• [27] A. V. Pogorelov, Surfaces of hounded extrinsic curvature, 1956 (in Russian).
• [28] B. H. Pourciau, Analysis and optimization of Lipschitz continuous mappings. J. Optim. Theory Appl. 22 (1977), 311-351.
• [29] D. Preiss and L. Zajíček, Fréchet differentiation of convex functions in a Banach space with a separable dual, Proc. Amer. Math. Soc. 91 (1984), 202-204.
• [30] D. Preiss and L. Zajíček, Stronger estimates of smallness of sets of Fréchet nondifferentiability of convex functions, Proceedings of the 11-th Winter school on Abstract Analysis, in Supplemento ai Rend. Circ. Mat. Palermo (2) no. 3 (1984), 219-223.
• [31] A. W. Roberts and D. E. Varberg, Convex functions, New York and London 1973.
• [32] A. Shapiro, On functions representable as a difference of two convex functions in inequality constrained optimization, Research report University of South Africa (1983).
• [33] M. M. Vajnberg, Variational methods for the study of nonlinear operators (in Russian), Moscow 1956, English translation: Holden-Day, San Francisci 1964.
• [34] L. Veselý, On the multiplicity points of monotone operators on separable Banach spaces. Comment. Math. Univ. Carolinae 27 (1986), 551-570.
• [35] L. Veselý, A short proof of a theorem on compositions of d.c. mappings, to appear in Proc. Amer. Math. Soc.
• [36] L. Veselý, On the multiplicity points of monotone operators on separable Banach spaces, II, Comment. Math. Univ. Carolinae 28 (1987), 295-299.
• [37] L. Zajíček, On the differentiation of convex functions in finite and infinite dimensional spaces, Czechoslovak Math. J. 29 (1979), 340-348.
• [38] L. Zajíček, On differentiation of metric projections in finite dimensional Banach spaces, Czechoslovak Math. J. 33 (1983), 325-336.
• [39] V. A. Zalgaller, On the representation of a function of two variables as the difference of convex functions, Vestn. Leningrad. Univ. Ser. Mat. Mekh. 18 (1963), 44-45 (in Russian).
Języki publikacji
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Uwagi
Identyfikator YADDA
bwmeta1.element.zamlynska-f664bd7a-845c-42c8-b543-2b96d75206ba
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# Change figure as I keep adding text in Beamer
In Beamer, I would like to post a figure, then add some text using \itemize, and change the figure inbetween items without moving the already posted text. Here is roughly what I want:
All on the same slide:
\documentclass{beamer}
\usepackage{graphicx}
\begin{document}
\begin{frame}
\includegraphics[width=\textwidth]{test1}
\pause
\begin{itemize}
\item
Text 1 Text 1 Text 1 Text 1 Text 1 Text 1 Text 1 Text 1 Text 1
\pause
%%Change the original figure to "test2" without moving any already written text.
\pause
\item
Text 2 Text 2 Text 2 Text 2 Text 2 Text 2 Text 2 Text 2 Text 2 Text 2
\pause
%%Change figure again to "test3" again without moving any text.
\end{itemize}
\end{frame}
\end{document}
• Welcome to TeX.SX! Please help us to help you and add a minimal working example (MWE) that illustrates your problem. It will be much easier for us to reproduce your situation and find out what the issue is when we see compilable code, starting with \documentclass{...} and ending with \end{document}. And a tip: If you indent lines by 4 spaces, they'll be marked as a code sample. You can also highlight the code and click the "code" button (with "{}" on it).
– user31729
May 30 '14 at 18:18
• @Christian Thanks, I did not know how to make the code box. May 30 '14 at 18:22
I think, you meant something like this -- if the image should be 'overlayed' (ideally with an other image of same dimensions), it is appropiate to use the \only<firstframe-endframe> command, e.g. \only<1-2>.... showing only on frame 1 to 2.
Use the same command for text lines too, but be aware about the textheight of them!
\documentclass[demo]{beamer}
\usepackage{graphicx}
\begin{document}
\begin{frame}%
\only<1-2>{\includegraphics[width=0.3\textwidth]{image1}%
}%
\only<3-4>{\includegraphics[width=0.3\textwidth]{image2}%
}%
\only<5-5>{\includegraphics[width=0.3\textwidth]{image3}%
}%
\begin{itemize}
\item<1-5>And now I show you the \textcolor{brown}{1st} image%
\item<3-5>And now I show you the \textcolor{blue}{2nd} image%
\item<5-5>And now I show you the \textcolor{red}{3rd} image%
\end{itemize}
%
\end{frame}
\end{document}
Note Remove the demo option from \documentclass specification.
• The problem with this is that the new text replaces the old text. What I would like is a solution that keeps the earlier text, and especially without moving the old text either. May 30 '14 at 19:19
• @Trevor: Ok, that is much easier! I will update
– user31729
May 30 '14 at 19:22
• @Trevor: The synchronizing between text and images depends on your choice, so I can only guess whether you want to show picture, wait, show text etc?
– user31729
May 30 '14 at 19:26
• Thank you so much, I did not really understand until now what the \only command was doing. May 30 '14 at 19:32
• @Trevor: Synchronizing can in most cases be done with \only, sometimes in conjunction with \onslide, but both methods require careful usage. (and a lot of recompilations ;-))
– user31729
May 30 '14 at 19:49
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# The Model Complexity Myth
An oft-repeated rule of thumb in any sort of statistical model fitting is "you can't fit a model with more parameters than data points". This idea appears to be as wide-spread as it is incorrect. On the contrary, if you construct your models carefully, you can fit models with more parameters than datapoints, and this is much more than mere trivia with which you can impress the nerdiest of your friends: as I will show here, this fact can prove to be very useful in real-world scientific applications.
A model with more parameters than datapoints is known as an under-determined system, and it's a common misperception that such a model cannot be solved in any circumstance. In this post I will consider this misconception, which I call the model complexity myth. I'll start by showing where this model complexity myth holds true, first from from an intuitive point of view, and then from a more mathematically-heavy point of view. I'll build from this mathematical treatment and discuss how underdetermined models may be addressed from a frequentist standpoint, and then from a Bayesian standpoint. (If you're unclear about the general differences between frequentist and Bayesian approaches, I might suggest reading my posts on the subject). Finally, I'll discuss some practical examples of where such an underdetermined model can be useful, and demonstrate one of these examples: quantitatively accounting for measurement biases in scientific data.
# Fast Lomb-Scargle Periodograms in Python
Image source: astroML. Source code here
The Lomb-Scargle periodogram (named for Lomb (1976) and Scargle (1982)) is a classic method for finding periodicity in irregularly-sampled data. It is in many ways analogous to the more familiar Fourier Power Spectral Density (PSD) often used for detecting periodicity in regularly-sampled data.
Despite the importance of this method, until recently there have not been any (in my opinion) solid implementations of the algorithm available for easy use in Python. That has changed with the introduction of the gatspy package, which I recently released. In this post, I will compare several available Python implementations of the Lomb-Scargle periodogram, and discuss some of the considerations required when using it to analyze data.
To cut to the chase, I'd recommend using the gatspy package for Lomb-Scargle periodograms in Python, and particularly its gatspy.periodic.LombScargleFast algorithm which implements an efficient pure-Python version of Press & Rybicki's $O[N\log N]$ periodogram. Below, I'll dive into the reasons for this recommendation.
# Optimizing Python in the Real World: NumPy, Numba, and the NUFFT
Donald Knuth famously quipped that "premature optimization is the root of all evil." The reasons are straightforward: optimized code tends to be much more difficult to read and debug than simpler implementations of the same algorithm, and optimizing too early leads to greater costs down the road. In the Python world, there is another cost to optimization: optimized code often is written in a compiled language like Fortran or C, and this leads to barriers to its development, use, and deployment.
Too often, tutorials about optimizing Python use trivial or toy examples which may not map well to the real world. I've certainly been guilty of this myself. Here, I'm going to take a different route: in this post I will outline the process of understanding, implementing, and optimizing a non-trivial algorithm in Python, in this case the Non-uniform Fast Fourier Transform (NUFFT). Along the way, we'll dig into the process of optimizing Python code, and see how a relatively straightforward pure Python implementation, with a little help from Numba, can be made to nearly match the performance of a highly-optimized Fortran implementation of the same algorithm.
# The Hipster Effect: An IPython Interactive Exploration
This week I started seeing references all over the internet to this paper: The Hipster Effect: When Anticonformists All Look The Same. It essentially describes a simple mathematical model which models conformity and non-conformity among a mutually interacting population, and finds some interesting results: namely, conformity among a population of self-conscious non-conformists is similar to a phase transition in a time-delayed thermodynamic system. In other words, with enough hipsters around responding to delayed fashion trends, a plethora of facial hair and fixed gear bikes is a natural result.
Also naturally, upon reading the paper I wanted to try to reproduce the work. The paper solves the problem analytically for a continuous system and shows the precise values of certain phase transitions within the long-term limit of the postulated system. Though such theoretical derivations are useful, I often find it more intuitive to simulate systems like this in a more approximate manner to gain hands-on understanding. By the end of this notebook, we'll be using IPython's incredible interactive widgets to explore how the inputs to this model affect the results.
(Or, Why People Hate Jet – and You Should Too)
I made a little code snippet that I find helpful, and you might too:
In [1]:
def grayify_cmap(cmap):
"""Return a grayscale version of the colormap"""
cmap = plt.cm.get_cmap(cmap)
colors = cmap(np.arange(cmap.N))
# convert RGBA to perceived greyscale luminance
# cf. http://alienryderflex.com/hsp.html
RGB_weight = [0.299, 0.587, 0.114]
luminance = np.sqrt(np.dot(colors[:, :3] ** 2, RGB_weight))
colors[:, :3] = luminance[:, np.newaxis]
return cmap.from_list(cmap.name + "_grayscale", colors, cmap.N)
What this function does is to give you a lumninance-correct grayscale version of any matplotlib colormap. I've found this useful for quickly checking how my plots might appear if printed in black and white, but I think it's probably even more useful for stoking the flame of the internet's general rant against jet.
# Hacking Academia: Data Science and the University
A reflection on our SciFoo breakout session, where we discussed issues of data science within academia.
Almost a year ago, I wrote a post I called the Big Data Brain Drain, lamenting the ways that academia is neglecting the skills of modern data-intensive research, and in doing so is driving away many of the men and women who are perhaps best equipped to enable progress in these fields. This seemed to strike a chord with a wide range of people, and has led me to some incredible opportunities for conversation and collaboration on the subject. One of those conversations took place at the recent SciFoo conference, and this article is my way of recording some reflections on that conversation.
SciFoo is an annual gathering of several hundred scientists, writers, and thinkers sponsored by Digital Science, Nature, O'Reilly Media & Google. SciFoo brings together an incredibly eclectic group of people: I met philosophers, futurists, alien hunters, quantum physicists, mammoth cloners, magazine editors, science funders, astrophysicists, musicians, mycologists, mesmerists, and many many more: the list could go on and on. The conference is about as unstructured as it can be: the organizers simply provide food, drink, and a venue for conversation, and attendees put together breakout discussions on nearly any imaginable topic. If you ever get the chance to go, my advice is to drop everything else and attend. It was one of the most quirky and intellectually stimulating weekends I've ever spent.
# Frequentism and Bayesianism IV: How to be a Bayesian in Python
I've been spending a lot of time recently writing about frequentism and Bayesianism.
Here I want to back away from the philosophical debate and go back to more practical issues: in particular, demonstrating how you can apply these Bayesian ideas in Python. The workhorse of modern Bayesianism is the Markov Chain Monte Carlo (MCMC), a class of algorithms used to efficiently sample posterior distributions.
Below I'll explore three mature Python packages for performing Bayesian analysis via MCMC:
• emcee: the MCMC Hammer
• pymc: Bayesian Statistical Modeling in Python
• pystan: The Python Interface to Stan
I won't be so much concerned with speed benchmarks between the three, as much as a comparison of their respective APIs. This post is not meant to be a tutorial in any of the three; each of them is well documented and the links above include introductory tutorials for that purpose. Rather, what I want to do here is a side-by-side comparison which will give a feel for how each package is used. I'll propose a single relatively non-trivial test problem, and show the implementation and results of this problem using all three packages. Hopefully by seeing the three approaches side-by-side, you can choose which package might be best suited for your particular application.
# Frequentism and Bayesianism III: Confidence, Credibility, and why Frequentism and Science do not Mix
In Douglas Adams' classic Hitchhiker's Guide to the Galaxy, hyper-intelligent pan-dimensional beings build a computer named Deep Thought in order to calculate "the Answer to the Ultimate Question of Life, the Universe, and Everything". After seven and a half million years spinning its hyper-dimensional gears, before an excited crowd, Deep Thought finally outputs the answer:
42
The disappointed technicians, who trained a lifetime for this moment, are stupefied. They probe Deep Though for more information, and after some back-and-forth, the computer responds: "once you do know what the question actually is, you'll know what the answer means."
An answer does you no good if you don't know the question.
I find this story be an apt metaphor for statistics as sometimes used in the scientific literature. When trying to estimate the value of an unknown parameter, the frequentist approach generally relies on a confidence interval (CI), while the Bayesian approach relies on a credible region (CR). While these concepts sound and look very similar, their subtle difference can be extremely important, as they answer essentially different questions.
Like the poor souls hoping for enlightenment in Douglas Adams' universe, scientists often turn the crank of frequentism hoping for useful answers, but in the process overlook the fact that in science, frequentism is generally answering the wrong question. This is far from simple philosophical navel-gazing: as I'll show, it can have real consequences for the conclusions we draw from observed data.
# Is Seattle Really Seeing an Uptick In Cycling?
Cycling in Seattle seems to be taking off. This can be seen qualitatively in the increased visibility of advocacy groups like Seattle Neighborhood Greenways and Cascade Bicycle Club, the excellent reporting of sites like the Seattle Bike Blog, and the investment by the city in high-profile traffic safety projects such as Protected Bike Lanes, Road diets/Rechannelizations and the Seattle Bicycle Master Plan.
But, qualitative arguments aside, there is also an increasing array of quantitative data available, primarily from the Bicycle counters installed at key locations around the city. The first was the Fremont Bridge Bicycle Counter, installed in October 2012, which gives daily updates on the number of bicycles crossing the bridge: currently upwards of 5000-6000 per day during sunny commute days.
Bicycle advocates have been pointing out the upward trend of the counter, and I must admit I've been excited as anyone else to see this surge in popularity of cycling (Most days, I bicycle 22 miles round trip, crossing both the Spokane St. and Fremont bridge each way).
But anyone who looks closely at the data must admit: there is a large weekly and monthly swing in the bicycle counts, and people seem most willing to ride on dry, sunny summer days. Given the warm streak we've had in Seattle this spring, I wondered: are we really seeing an increase in cycling, or can it just be attributed to good weather?
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# If $\alpha''$ and $\alpha'$ are collinear, then $\alpha$ is a pre-geodesic.
I'm trying to solve exercise $19$ in pages $95$ and $96$ in O'Neill's Semi-Riemannian Geometry book. There are four items and I'm having trouble with the last one.
Let $(M,\langle \cdot,\cdot\rangle)$ be a Lorentz manifold and $\alpha$ be a regular curve such that $\alpha''(s) = f(s)\alpha'(s)$.
a) If $\beta = \alpha\circ h$, $\beta$ is pre-geodesic if and only if $h''+ (f\circ h) (h')^2 = 0$.
Easy, just compute $\beta''(s)=( h''(s) + f( h(s)) (h'(s))^2)\alpha'(h(s))$ and use that $\alpha$ is regular.
b) If $\beta$ has unit speed and $\langle \alpha',\alpha'\rangle$ is never zero, then $\beta$ is geodesic.
If $\beta$ has unit speed, then $\langle \beta''(s), \beta'(s)\rangle = 0$ for all $s$, after differentiating once, but this evaluates to $h'(s) ( h''(s) + f( h(s)) (h'(s))^2) \langle \alpha'(s),\alpha'(s)\rangle = 0$.
c) $\langle \alpha',\alpha'\rangle$ is always zero or never zero.
From $\langle \alpha'(s),\alpha'(s)\rangle' = 2f(s)\langle \alpha'(s),\alpha'(s)\rangle$, we have $\langle \alpha'(s),\alpha'(s)\rangle = Ce^{2\int f(s)\,{\rm d}s}$. If that is zero in some point it is always zero by uniqueness of solutions of IVP's, and otherwise it is never zero because the exponential is never zero.
d) If $\langle \alpha',\alpha'\rangle$ is always zero, then $\alpha$ is pre-geodesic.
I'm stuck. The only thing I managed to get is that $\langle \alpha''(s),\alpha''(s)\rangle = 0$. If we could prove that $\alpha''(s)$ is not lightlike, this would imply that $\alpha''(s) = 0$. Help?
• I may be missing something, but if $0=\langle \alpha^{\prime\prime},\alpha^{\prime\prime}\rangle = f^2 \langle \alpha^{\prime},\alpha^{\prime}\rangle$ and $\langle \alpha^{\prime},\alpha^{\prime}\rangle\neq 0$ then $f=0$ and so $\alpha^{\prime\prime}= f \alpha^{\prime}=0$. If, on the other hand, $f\neq 0$, then the same equations show $\alpha^{\prime\prime}$ is not lightlike. – Thomas Nov 10 '15 at 17:26
• @Thomas, I"m sorry, I copied it wrong, the assumption is that $\langle \alpha',\alpha'\rangle$ is always zero, and I used that to get $\langle \alpha'',\alpha''\rangle = 0$. – Ivo Terek Nov 10 '15 at 18:13
What you have to show in (d) is that if $\langle\alpha',\alpha'\rangle=0$ everywhere, then $\alpha$ is a geodesic up to reparametrization, which is the same as $\alpha''(s)=f(s)\alpha'(s)$ everywhere. This is easy if you have already shown that $\langle\alpha'',\alpha''\rangle=0$ everywhere, for we also have $\langle\alpha',\alpha''\rangle=0$ everywhere. This means that for all $s$ in the domain of $\alpha$ we have that $\alpha''(s)$ is null and orthogonal to $\alpha'(s)$, which is also null. This happens if and only if $\alpha''(s)=f(s)\alpha'(s)$.
• I'm sorry, I don't think I quite get it. Shouldn't I prove that $\alpha'' = 0$? – Ivo Terek Nov 10 '15 at 18:56
• No. This would imply that $\alpha$ is a geodesic, which is in general not true. Take for instance a null geodesic $\beta$ and a non-affine reparametrization $g$ (e.g. $g(t)=\tanh t$). Then $\alpha=\beta\circ g$ is a pre-geodesic which is not a geodesic, but still $\langle\alpha',\alpha'\rangle=0$ everywhere by the chain rule. – Pedro Lauridsen Ribeiro Nov 10 '15 at 19:29
• This example made sense, thanks. I was thinking that being pre-geodesic + constant speed implied being a geodesic. I don't see how $\alpha''(s) = f(s)\alpha'(s)$ is equivalent as being pre-geodesic, though. And again, it seemed that $\alpha''(s) = f(s)\alpha'(s)$ was the hypothesis right from the start. I am confused. – Ivo Terek Nov 10 '15 at 19:38
• The hypothesis $\langle\alpha',\alpha'\rangle=0$ doesn't entail constant speed, as my counterexample above shows. To see the aforementioned equivalence, let $\beta:[a,b]\rightarrow M$ be a geodesic, and $g:[c,d]\rightarrow[a,b]$ a diffeomorphism (i.e. a reparametrization). Then $$(\beta\circ g)''(t)=\frac{g''(t)}{g'(t)}(\beta\circ g)'(t)=(\log g'(t))'(\beta\circ g)'(t)$$ for all $t$ by the chain and Leibniz rules, thus yielding $f(t)=(\log g)'(t)$. Reversing the above formula proves the converse statement. – Pedro Lauridsen Ribeiro Nov 10 '15 at 19:48
• In my opinion, item (d) is written somewhat imprecisely. (by the way, it should be $f = (\log g')'$ in my previous comment - unfortunately, I can no longer edit it) – Pedro Lauridsen Ribeiro Nov 10 '15 at 19:54
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# The probability of extraterrestrial life
Since, the discovery of exoplanets nearly 3 decades ago most astronomers, at least the public facing ones, seem to agree that it is just a matter of time before they find signs of life such as the presence of volatile gases in the atmosphere associated with life like methane or oxygen. I’m an agnostic on the existence of life outside of earth because we don’t have any clue as to how easy or hard it is for life to form. To me, it is equally possible that the visible universe is teeming with life or that we are alone. We simply do not know.
But what would happen if we find life on another planet. How would that change our expected probability for life in the universe? MIT astronomer Sara Seager once made an offhand remark in a podcast that finding another planet with life would make it very likely there were many more. But is this true? Does the existence of another planet with life mean a dramatic increase in the probability of life in the universe. We can find out by doing the calculation.
Suppose you believe that the probability of life on a planet is $f$ (i.e. fraction of planets with life) and this probability is uniform across the universe. Then if you search $n$ planets, the probability for the number of planets with life you will find is given by a Binomial distribution. The probability that there are $x$ planets is given by the expression $P(x | f) = C(x,n) f^x(1-f)^{n-x}$, where $C$ is a factor (the binomial coefficient) such that the sum of $x$ from one to $n$ is 1. By Bayes Theorem, the posterior probability for $f$ (yes, that would be the probability of a probability) is given by
$P(f | x) = \frac{ P(x | f) P(f)}{P(x)}$
where $P(x) = \int_0^1 P(x | f) P(f) df$. As expected, the posterior depends strongly on the prior. A convenient way to express the prior probability is to use a Beta distribution
$P(f |\alpha, \beta) = B(\alpha,\beta)^{-1} f^{\alpha-1} (1-f)^{\beta-1}$ (*)
where $B$ is again a normalization constant (the Beta function). The mean of a beta distribution is given by $E(f) = \alpha/(\alpha + \beta)$ and the variance, which is a measure of uncertainty, is given by $Var(f) = \alpha \beta /(\alpha + \beta)^2 (\alpha + \beta + 1)$. The posterior distribution for $f$ after observing $x$ planets with life out of $n$ will be
$P(f | x) = D f^{\alpha + x -1} (1-f)^{n+\beta - x -1}$
where $D$ is a normalization factor. This is again a Beta distribution. The Beta distribution is called the conjugate prior for the Binomial because it’s form is preserved in the posterior.
Applying Bayes theorem in equation (*), we see that the mean and variance of the posterior become $(\alpha+x)/(\alpha + \beta +n)$ and $(\alpha+x)( \beta+n-x) /(\alpha + \beta + n)^2 (\alpha + \beta + n + 1)$, respectively. Now let’s consider how our priors have updated. Suppose our prior was $\alpha = \beta = 1$, which gives a uniform distribution for $f$ on the range 0 to 1. It has a mean of 1/2 and a variance of 1/12. If we find one planet with life after checking 10,000 planets then our expected $f$ becomes 2/10002 with variance $2\times 10^{-8}$. The observation of a single planet has greatly reduced our uncertainty and we now expect about 1 in 5000 planets to have life. Now what happens if we find no planets. Then, our expected $f$ only drops to 1 in 10000 and the variance is about the same. So, the difference between finding a planet versus not finding a planet only halves our posterior if we had no prior bias. But suppose we are really skeptical and have a prior with $\alpha =0$ and $\beta = 1$ so our expected probability is zero with zero variance. The observation of a single planet increases our posterior to 1 in 10001 with about the same small variance. However, if we find a single planet out of much fewer observations like 100, then our expected probability for life would be even higher but with more uncertainty. In any case, Sara Seager’s intuition is correct – finding a planet would be a game breaker and not finding one shouldn’t really discourage us that much.
# The inherent conflict of liberalism
Liberalism, as a philosophy, arose during the European Enlightenment of the 17th century. It’s basic premise is that people should be free to choose how they live, have a government that is accountable to them, and be treated equally under the law. It was the founding principle of the American and French revolutions and the basic premise of western liberal democracies. However, liberalism is inherently conflicted because when I exercise my freedom to do something (e.g. not wear a mask), I infringe on your freedom from the consequence of that thing (e.g. not be infected) and there is no rational resolution to this conflict. This conflict led to the split of liberalism into left and right branches. In the United States, the term liberal is exclusively applied to the left branch, which mostly focuses on the ‘freedom from’ part of liberalism. Those in the right branch, who mostly emphasize the ‘freedom to’ part, refer to themselves as libertarian, classical liberal, or (sometimes and confusingly to me) conservative. (I put neo-liberalism, which is a fundamentalist belief in free markets, into the right camp although it has adherents on both the left and right.) Both of these viewpoints are offspring of the same liberal tradition and here I will use the term liberal in the general sense.
Liberalism has never operated in a vacuum. The conflicts between “freedom to” and “freedom from” have always been settled by prevailing social norms, which in the Western world was traditionally dominated by Christian values. However, neither liberalism nor social norms have ever been sufficient to prevent bad outcomes. Slavery existed and was promoted by liberal Christian states. Genocide of all types and scales have been perpetrated by liberal Christian states. The battle to overcome slavery and to give equal rights to all peoples was a long and hard fought battle over slowly changing social norms rather than laws per se. Thus, while liberalism is the underlying principle behind Western governments, it is only part of the fabric that holds society together. Even though we have just emerged from the Dark Years, Western Liberalism is on its shakiest footing since the Second World War. The end of the Cold War did not bring on a permanent era of liberal democracy but may have spelled it’s eventual demise. What will supplant liberalism is up to us.
It is often perceived that the American Democratic party is a disorganized mess of competing interests under a big tent while the Republicans are much more cohesive but in fact the opposite is true. While the Democrats are often in conflict they are in fact a fairly unified center-left liberal party that strives to advocate for the marginalized. Their conflicts are mostly to do with which groups should be considered marginalized and prioritized. The Republicans on the other hand are a coalition of libertarians and non-liberal conservatives united only by their desire to minimize the influence of the federal government. The libertarians long for unfettered individualism and unregulated capitalism while the conservatives, who do not subscribe to all the tenets of liberalism, wish to halt encroaching secularism and a government that no longer serves their interests.
The unlikely Republican coalition that has held together for four decades is now falling apart. It came together because the more natural association between religious conservatism and a large federal bureaucracy fractured after the Civil Rights movements in the 1960’s when the Democrats no longer prioritized the concerns of the (white) Christian Right. (I will discuss the racial aspects in a future post). The elite pro-business neo-liberal libertarians could coexist with the religious conservatives as long as their concerns did not directly conflict but this is no longer true. The conservative wing of the Republican party have discovered their new found power and that there is an untapped population of disaffected individuals who are inclined to be conservative and also want a larger and more intrusive government that favors them. Prominent conservatives like Adrian Vermeule of Harvard and Senator Josh Hawley are unabashedly anti-liberal.
This puts the neo-liberal elites in a real bind. The Democratic party since Bill Clinton had been moving right with a model of pro-market neo-liberalism but with a safety net. However they were punished time and time again by the neo-liberal right. Instead of partnering with Obama, who was highly favorable towards neoliberalism, they pursued a scorched earth policy against him. Hilary Clinton ran on a pretty moderate safety-net-neo-liberal platform and got vilified as an un-American socialist. Now, both the Republicans and Democrats are trending away from neo-liberalism. The neo-liberals made a strategic blunder. They could have hedged their bets but now have lost influence in both parties.
While the threat of authoritarianism looms large, this is also an opportunity to accept the limits of liberalism and begin to think about what will take its place – something that still respects the basic freedoms afforded by liberalism but acknowledges that it is not sufficient. Conservative intellectuals like Leo Strauss have valid points. There is indeed a danger of liberalism lapsing into total moral relativism or nihilism. Guardrails against such outcomes must be explicitly installed. There is value in preserving (some) traditions, especially ancient ones that are the result of generations of human engagement. There will be no simple solution. No single rule or algorithm. We will need to explicitly delineate what we will accept and what we will not on a case by case basis.
# The machine learning president
For the past four years, I have been unable to post with any regularity. I have dozens of unfinished posts sitting in my drafts folder. I would start with a thought but then get stuck, which had previously been somewhat unusual for me. Now on this first hopeful day I have had for the past four trying years, I am hoping I will be able to post more regularly again.
Prior to what I will now call the Dark Years, I viewed all of history through an economic lens. I bought into the standard twentieth century leftist academic notion that wars, conflicts, social movements, and cultural changes all have economic underpinnings. But I now realize that this is incorrect or at least incomplete. Economics surely plays a role in history but what really motivates people are stories and stories are what led us to the Dark Years and perhaps to get us out.
Trump became president because he had a story. The insurrectionists who stormed the Capitol had a story. It was a bat shit crazy lunatic story but it was still a story. However, the tragic thing about the Trump story (or rather my story of the Trump story) is that it is an unintentional algorithmically generated story. Trump is the first (and probably not last) purely machine learning president (although he may not consciously know that). Everything he did was based on the feedback he got from his Twitter Tweets and Fox News. His objective function was attention and he would do anything to get more attention. Of the many lessons we will take from the Dark Years, one should be how machine learning and artificial intelligence can go so very wrong. Trump’s candidacy and presidency was based on a simple stochastic greedy algorithm for attention. He would Tweet randomly and follow up on the Tweets that got the most attention. However, the problem with a greedy algorithm (and yes that is a technical term that just happens to coincidentally be apropos) is that once you follow a path it is hard to make a correction. I actually believe that if some of Trump’s earliest Tweets from say 2009-2014 had gone another way, he could have been a different president. Unfortunately, one of his early Tweet themes that garnered a lot of attention was on the Obama birther conspiracy. This lit up both racist Twitter and a counter reaction from liberal Twitter, which led him further to the right and ultimately to the presidency. His innate prejudices biased him towards a darker path and he did go completely unhinged after he lost the election but he is unprincipled and immature enough to change course if he had enough incentive to do so.
Unlike standard machine learning for categorizing images or translating languages, the Trump machine learning algorithm changes the data. Every Tweet alters the audience and the reinforcing feedback between Trump’s Tweets and its reaction can manufacture discontent out of nothing. A person could just happen to follow Trump because they like The Apprentice reality show Trump starred in and be having a bad day because they missed the bus or didn’t get a promotion. Then they see a Trump Tweet, follow the link in it and suddenly they find a conspiracy theory that “explains” why they feel disenchanted. They retweet and this repeats. Trump sees what goes viral and Tweets more on the same topic. This positive feedback loop just generated something out of random noise. The conspiracy theorizing then starts it’s own reinforcing feedback loop and before you know it we have a crazed mob bashing down the Capitol doors with impunity.
Ironically Trump, who craved and idolized power, failed to understand the power he actually had and if he had a better algorithm (or just any strategy at all), he would have been reelected in a landslide. Even before he was elected, Trump had already won over the far right and he could have started moving in any direction he wished. He could have moderated on many issues. Even maintaining his absolute ignorance of how govening actually works, he could have had his wall by having it be part of actual infrastructure and immigration bills. He could have directly addressed the COVID-19 pandemic. He would not have lost much of his base and would have easily gained an extra 10 million votes. Maybe, just maybe if liberal Twitter simply ignored the early incendiary Tweets and only responded to the more productive ones, they could have moved him a bit too. Positive reinforcement is how they train animals after all.
Now that Trump has shown how machine learning can win a presidency, it is only a matter of time before someone harnesses it again and more effectively. I just hope that person is not another narcissistic sociopath.
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# Solc Compiler oversight? Innappropriate mapping declaration overwrites storage
I have written a contract that has the mapping storage variable A & B. These variables get initialized with some values in constructor. A method getBalance returns the balance from mapping A & B.
In the method I declared another mapping - thinking that it would give a compilation error (since mapping allowed for storage only). Instead it lead to a strange behavior - appears that the new local mapping overwrites mappingA.
Result without local mapping (100, 200) as expected Result with local mapping (500, 200) <<< Why?
/** Tested in Truffle/TestRPC **/
contract Mapping {
function Mapping() {
// constructor
balancesA[msg.sender] = 100;
balancesB[msg.sender] = 200;
}
// If these 2 lines are commented then behavior is as expected
balancers[msg.sender] = 500;
}
}
• I can confirm that. The mapping in the function seems to collide with the first mapping that was declared properly. – Rob Hitchens - B9lab Mar 1 '17 at 13:20
• Have you tried running it on testnet (or private network)? Sometimes TestRPC behaves differently then "real" network - maybe thats the case. – radmen Mar 1 '17 at 13:38
• I can reproduce in Geth too. – Xavier Leprêtre B9lab Mar 1 '17 at 18:08
• @ACloudFan It looks like it's bug report time. Do you want to do the honors? – Rob Hitchens - B9lab Mar 2 '17 at 4:46
• Bug aside, the return parameters aren't being used correctly. Should be returns (uint, uint) rather than being named, or balanceA = balancesA[addr]; etc – o0ragman0o Mar 3 '17 at 0:50
Opened an issue here, github, ethereum, solidity, issue 1731: https://github.com/ethereum/solidity/issues/1731
If I understand their process correctly, original issue 1731 is closed by issue 1737 which is now merged. Possibly the compiler is going to emit an error for this sort of thing in the future.
Both thumbs up for the devs!
https://github.com/ethereum/solidity/pull/1737
Aside from that it is a bug in solidity (as it should notice that uninitialized variable is used), it worth mentioning that the code is not correct also.
The problem is that mapping(address => uint) balancers in function body does not allocate new mapping. It just introduces variable balancers which can reference some mapping (allocated elsewhere).
In the example balancers is not initialized and coincidentially references balancesA. Hence the unexpected behavior.
From Solidity docs:
Mappings are only allowed for state variables (or as storage reference types in internal functions).
I'm afraid this is not a bug. I very much agree that it's a problem and presents a huge trap, though.
mapping(address => uint) memory balancers;
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# Re: [tlaplus] Re: [Newbie Question] Engineer trying to get maths meaning
This makes sense to me now. Changing my thinking and how to approach logical problems can be difficult, but on this issue I am no longer confused.
Thanks all.
-Frank
On Sat, Oct 3, 2020 at 2:24 AM Stephan Merz <stephan.merz@xxxxxxxxx> wrote:
To follow up on this, Q is not really "defined" by this formula because a definition would have to be unambiguous. The formula checks for the existence of a quorum satisfying certain properties. The quantifier introduces (or "declares") a name Q for naming the value whose properties are being described in lines 2-10 and requires Q to belong to the set Quorum. The formula is true if some such value exists and false otherwise. Also, the scope of Q is restricted to the body of the formula, and Q has no meaning after the formula has been evaluated, in contrast to a definition that introduces a name that can be used later.
Stephan
On 3 Oct 2020, at 01:42, Rodrick Chapman <rodrick@xxxxxxxxxxxx> wrote:
It's just syntax. It happens to be syntax that's based on century-old mathematical notation, but at the end of the day, it's still just syntax.
The trouble you’re having is caused by your assumption that Q is defined on the left side of the colon ("\E Q \in Quorum"), and then used on the right side.
But, in fact, Q is actually defined on the right side of the colon and then used on the left. In other words, we define Q and then ask “is there any element in the set Quorum that meets the definition of Q?”
On Friday, October 2, 2020 at 12:44:13 PM UTC-5 [email protected] wrote:
Hi,
I'm trying to understand this part of the paxos specification. I'm not trying to understand how paxos works, I get that, I'm just trying to understand how to read and understand this part of the specification.
01 /\ \E Q \in Quorum :
02 LET Q1b == {m \in msgs : /\ m.type = "1b"
03 /\ m.acc \in Q
04 /\ m.bal = b}
05 Q1bv == {m \in Q1b : m.mbal \geq 0}
06 IN /\ \A a \in Q : \E m \in Q1b : m.acc = a
07 /\ \/ Q1bv = {}
08 \/ \E m \in Q1bv :
09 /\ m.mval = v
10 /\ \A mm \in Q1bv : m.mbal \geq mm.mbal
On line 01 you define that Q is in set of Quorum and then lines 02-10 define what Q will be. My confusion is on Line 03 Q is referenced. How can you reference something that hasn't been assigned a value yet.
I may very well be suffering from "... brain washing done by years of C programming".
Any help that I could get in understanding this would be greatly appreciated. Also, is there a place on the web were I can better familiarize my self with such concepts.
Thanks,
-Frank
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Damage per second
Damage per second (DPS or dps) is a measure of the damage dealt by a person or group over one second. DPS is a more practical measure of damage output than plain damage, as it allows characters of differing levels and classes to effectively compare their damage output.
The term dps is also sometimes used to describe the act of applying damage to a target (as in "all dps on the boss" or "stop all dps"), or to refer to a class whose primary role in a group is dealing damage (as in "Gundrak group needs one more dps to roll"), i.e. damage dealer. There are UI modifications which enable players to track their DPS.
Burst DPS
"Burst DPS" refers to damage dealt over a relatively short period of time. High burst damage is the preferable form of damage against targets at/with relatively low health. Most classes have at least a few spells or abilities that generate high damage very quickly. This type of damage is preferred in PvP environments, but is usually too expensive or has too long a cooldown to be sustainable.
Sustained DPS
"Sustained DPS" refers to constant damage dealt over an extended period of time. High sustained damage is the preferable form of damage against targets with relatively high health. Every class is capable, when specced and geared properly, of doing good sustained damage. Hunters, Mages, Rogues, and Warlocks tend to have the highest sustained dps, as "pure" dps classes.
• Note that Sustained DPS is best calculated on a one boss encounter. Full raid / dungeon DPS calculates AOE damage into your DPS, greatly increasing it.
Weapon Damage Per Second (DPS) Formula
The general formula for your weapon's DPS is:
((Min Weapon Damage + Max Weapon Damage) / 2) / Weapon Speed
Individual Spell DPS
The DPS of a single spell can be calculated as either (Damage / Cast Time) or (Damage / Cast Interval). The former, also referred to as "damage per cast time" (DPCT) is the amount of damage dealt over time spent casting the spell. The latter is used to express the spell's damage contribution over the total combat time, a figure used to express its damage contribution relative to other spells. For most calculations, an instant spells' cast time is treated as the GCD, 1.5 seconds / (1 + Spell Haste). Some spells have a reduced GCD as affected by talents.
$\frac{H_I \times (\frac{D_{min}+D_{max}}{2} + P \times C_P) \times (1 + B \times S_C ) \times (1+H_A)}{T}$
where HI is the spell hit percent, D is the base damage, P is spell power, CP is the spell power coefficient, B is the critical damage bonus, SC is the critical strike percent, HA is the spell haste percent, and T is the base casting time.
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# Braille text in blender
I'd like to use blender to add some text in braille to some blender 3D modela in order to print them via 3D printers. Is it possible in an easy way? How?
I know nothing about braille but this should be as simple as grabbing a suitable font and extruding it. I tried braille normal from fontspace. Once in Blender, add a Text object, edit it as needed and then navigate to the .ttf font file under Font, the Regular input should be enough.
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# Spectral Collocation (or Weighted Residual) Methods to solve Stiff ODEs?
I have a system of ODEs which is (at least moderately) stiff.
Consider the class of spectral collocation methods https://en.wikipedia.org/wiki/Spectral_method or the related class of weighted residual methods.
Basically, take a system of ODEs like $$y'(t) = f(t, y(t))$$ where $f(\cdot,\cdot)$ is nonlinear in my case and leads to a stiff ODE, and I want to solve it on $t \in [0,M]$. Then approximate the solution $$y(t) \approx \sum_{i=0}^N a_i T_i(t)$$ where $T_i(\cdot)$ are a basis (e.g. Chebyshev polynomials), $N$ is the maximum order, and the basis is over $[0,M]$ if appropriate. Note that given $\{a_i\}$, $y'(t)$ can be calculated by differentiating the polynomial basis.
Now, evaluate the ODE given a $\{a_i\}$ at the roots of the polynomial basis (which can be shown to be optimal) and find the residual. Choose $\{a_i\}$ to minimize this residual, or just solve a system of equations with the residual = 0.
Question: how well does spectral collocation with a Chebyshev basis handle systems that seem stiff when solved with finite differences? The worry is that the same issues that cause finite differences to have issues (e.g. convergence to a very flat function with no slope) could cause major issues with a finite dimensional approximation of the function.
If Chebyshev has problems, then is there a better basis which might have less issues.
A little more context: I have implemented spectral collocation with Chebyshev polynomials to solve a stationary Kolmogorov Forward equation in CDFs. These, of course, need to be strictly positive and weakly monotonically increasing and asymptotically go towards $1$ if it is a proper probability distribution. I am seeing weird behavior where the pdf drops below $0$ (i.e., it becomes decreasing) or strange jumps at the corners. It is very hard for me to tell if these are artifacts in the solution scheme or if I have just chosen parameters for the model where no solution exists. Increasing the number of Chebyshev basis functions doesn't eliminate them, but that could be because they can't, or only slowly, converge.
Even more context: For what it is worth, in the simplest form of my problem there are two sets of ODEs (which are ultimately coupled):
Parameters: $\lambda_{\ell}, \lambda_h, r,$ and $\chi$. In some ways $F_{\ell}'(0), F_h'(0),$ and $g$ are also parameters for the purposes here. There are two discrete states which leads to the system: $i \in \{\ell,h\}$
The variable $z\in[0,\infty)$ in reality, but these converge pretty quickly so choosing some $z \in [0,\bar{z}]$ for collocation methods is reasonable.
System (1) The following comes from a stationary Kolmogorov Forward Equation (given a $\gamma(z)$ function such that $\gamma(0) = 0$ and $\lim\limits_{z\to\infty} \gamma(z) = g$. The latter condition leads to a vanishing derivative term in the ODE (i.e., a singular mass matrix if put in that canonical form.) \begin{align} 0 &= g F_{\ell}'(z) + \lambda_h F_h(z) - \lambda_{\ell}F_{\ell}(z) + g (F_{\ell}'(0) + F_h'(0))(F_{\ell}(z) + F_h(z)) - g F_{\ell}'(0)\\ 0 &= (g - \gamma(z))F_h'(z) + \lambda_{\ell}F_{\ell}(z) - \lambda_h F_h(z) - g F_h'(0) \end{align}
• Forget that this is a linear ODE, as the actual one is nonlinear and much trickier. However, it can still be written in a $M(z, F(z))\cdot F'(z) = \Phi(z, F(z))$ term for some operators and a mass matrix.
• The trouble seems to come out of the term on the derivative going to $0$ (i.e., a singular mass matrix).
• It might seem odd that there is a $F_{\ell}'(0)$ and a $F_h'(0)$ in the ODE as parameters but this is not a mistake. You can easily show that any solution to the ODE will have the $F_i'(0)$ matching these constants by construction.
• The initial condition is $F_i(0) = 0$. Furthermore, from any $F_i'(0)$, one can analytically find $\lim\limits_{z\to\infty}F_h(z)$. With $F_{\ell}(\infty) + F_h(\infty) = 1$, this means we can write it as a BVP if we wish.
System (2) The following comes from a Hamilton-Jacobi-Bellman equation
\begin{align} 0 &= 1 - (r + \lambda_{\ell})w_{\ell}(z) - g w'_{\ell}(z) + \lambda_h w_h(z)\\ 0 &= 1 - (r + \lambda_h)w_h(z) - (g - \frac{\chi}{2} w_h(z))w_h'(z) + \lambda_h w_{\ell}(z) + \frac{\chi}{4} w_h(z)^2 \end{align} subject to, $w_{\ell}(0) = w_h(0) = 0$ and defined on $z \in [0,\infty)$. You can analytically find $w_{\ell}(\infty)$ and $w_h(\infty)$ which can help with various methods to set it as a BVP.
One can show that $\lim\limits_{z\to\infty}(g - \frac{\chi}{2} w_h(z)) = 0$ which leads to a vanishing derivative term in the ODE.
Coupled: In reality, the $\gamma(z)$ from the first solution is $$\gamma(z) \equiv \frac{\chi}{2}w_h(z)$$
Leaving Out: A whole bunch of equilibrium conditions in terms of integrals of $w_i(z)$ and $F_i(z)$ which pin down $g, F'_{\ell}(0)$ and $F'_h(0)$. Evaluating these concurrently with the solution to the ODEs is the reason that spectral methods are preferable since I can add in more equations and solve everything at the same time.
... is what I have written useful? Maybe not, but you can see the asymptotic singularity in the mass matrix. Is this stiff? I think so. When you solve the second system with finite differences, for example, it has all the hallmarks of a stiff system.
• Could you specify precisely what you mean by "projection methods", with equations or a reference? – David Ketcheson Nov 4 '15 at 7:15
• @DavidKetcheson See my edits. Thanks for looking at this! – jlperla Nov 4 '15 at 16:58
• I can't tell from what you have written whether you have an initial or boundary value problem. Since you refer to stiffness and Runge-Kutta methods, I would guess it is an initial value problem. But if so, this is a very expensive method since it couples all the time steps into one big system. – David Ketcheson Nov 4 '15 at 19:32
• By the way, "collocation" and "projection" refer to very many different things in the realm of numerical methods. In my view, what you are proposing is quite different from what is explained on the Wikipedia page you have linked to. – David Ketcheson Nov 4 '15 at 19:35
• The wikipedia article is saying exactly what I mean: "choose a finite-dimensional space of candidate solutions (usually, polynomials up to a certain degree) and a number of points in the domain (called collocation points), and to select that solution which satisfies the given equation at the collocation points." – jlperla Nov 4 '15 at 22:23
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# Euler in maths and engineering
Inspired by Katherine Johnson’s character in the film Hidden Figures and her use of Euler’s Method, engineer Natalie-Claire Luwisha has written this guest post about Euler’s contribution to engineering.
I thoroughly enjoyed Hidden Figures because of the overall message and inspiration it generated for all women, especially women of colour. Even today in the 21st century, most of the STEM (science, technology, engineering and mathematics) industries still have a very low percentage of women and even fewer women of colour. One major factor in this is the lack of visible role models for young girls and women to aspire to, so this story based on real-life events was ideal to help tackle the issue.
The film highlighted the struggles and triumphs that the three African-American women went through during their time at NASA, where their intelligence contributed to putting a man into space.
### Euler’s Method in Hidden Figures
This was a film in which mathematicians were the central characters, and I was pleased to note that they didn’t shy away from including real mathematical methods in the script. Explicitly mentioned in the film is Euler’s methodused to find an exact solution for a differential equation. In the film, the method is used to find a solution between two different types of orbit that the capsule moves during its journey from space to earth.
Although the method is described as ‘old math’ in the film, many engineers and mathematicians frequently use it in their work today. Euler’s method was first published in 1768, and the movie is set in 1960 – this is hardly a long time in comparison to some of the methods we use in many of our mathematical calculations today, such as Newton’s laws.
A still from the film, in which Katherine Johnson looks the method up in a textbook
Below is the scene in which it’s used:
### Who’s this Euler?
The method is named after Leonhard Euler, a mathematician who was born in Switzerland in 1707. Although he is famously known as a great mathematician he was a highly intelligent individual who contributed to other STEM areas including physics, astronomy, and engineering.
During his childhood, his interest in maths was influenced and supported by his father’s friend, Johann Bernoulli, who at the time was the most renowned mathematician in Europe. Euler spent most of his adult life in Russia and Germany (previously known as Prussia).
At the age of 13, he attended the University of Basel and achieved a master of philosophy where his dissertation focused on the philosophies of Descartes vs Newton. While studying at university, Bernoulli mentored and tutored Euler, thus cultivating his natural talent in mathematics. Although Euler’s father had wanted him to focus more on his theology studies, it was Bernoulli who convinced his father of the phenomenal talent his son had in mathematics.
In his lifetime Euler made various historical developments in physics, astronomy, and engineering. Today we have numerous examples of his work, especially in mathematics. Thus we find ourselves with a number of topics and formulas that are part of (derive from) Euler’s work. Some of the mathematical areas that Euler contributed to include:
Even the Euler Brick which has been written about here on the Aperiodical before now! Euler also used his analytical skills to help develop many engineering formulas.
As an engineer myself, I use various mathematical methods in structural engineering analysis, and one of them is also named after Euler.
### Euler’s Column Formula for Buckling
Euler’s Column Formula is based on the theory of bending, as applied to structural beams and other structural members under different stresses. By solving the differential equation of beam bending we are then able to find an exact solution for the lateral (sideways) displacement of a column at its critical load – that is, the maximum load it can take in the axial (vertical) direction before it bends.
(Images above are from the Cal Poly website, and the FSEL website.)
This displacement, also described as buckling, is the physical failure i.e. – bending or deformation of a column, due to compressive stresses (axial load) higher than its critical load. Many examples of this type of failure can be found in buildings made of steel or concrete columns that buckle, cylindrical storage tanks and even drink or food cans, can have signs of buckling.
The basic formula used is:
$P= \frac{\pi^2 E I}{L^2}$
You can find more information on the mechanics of column buckling at ContinuumMechanics.org
Where $P$ is the critical buckling load, $E$ is Young’s modulus of elasticity, $I$ is the second moment of area and $L$ is the effective length of the beam – the exact value used for $L$ varies depending on the type of connection between the column and the structure. The different types of connections are shown in the diagram below, and the effective length of the column is then dependent on these end conditions.
As shown above, the effective lengths for the different types are as follows (where L is the actual length of the beam and $L_{eff}$ is the effective length.
• Pinned connection at both ends: $L$
• Fixed connection at both ends: $0.5L$
• Fixed and pinned connection: $0.699L$
• Fixed at one end only or cantilevered: $2L$
• Fixed at one end and fixed in one direction – roller on the other end: $L$
### Where does this formula come from?
This formula is derived from the beam
$EIy”=M$
This produces the differential equation:
$EIy”+ P = 0$
Where $P$ represents the compressive load on a column.
By identifying the critical buckling load of a column, we can design the column to withstand up to 70% of this load and ensure the stability of the structure, hence reducing the chance of failure. The calculation of the critical buckling load is essential in all structural analysis design because it leads to the design and construction of more durable structures.
Euler’s formula is widely used in structural engineering calculations, but one alternative sometimes used is the Perry-Robertson theory. This is employed in the guidelines set out in Eurocode 3: Design of steel structures. The mathematical instability (i.e. buckling) of a structural member such as a beam or column is calculated for all available steel members and the results are tabulated. Hence, the time taken to design hundreds of structural members is reduced due to the tabulated results readily available for use in everyday structural design.
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Ciro Santilli $$Sponsor Ciro$$ 中国独裁统治 China Dictatorship 新疆改造中心、六四事件、法轮功、郝海东、709大抓捕、2015巴拿马文件 邓家贵、低端人口、西藏骚乱
# Geometry
## Point (geometry)
split words: 39
### Hyperplane
split words: 39
Generalization of a plane for any number of dimensions.
Kind of the opposite of a line: the line has dimension 1, and the plane has dimension D-1.
In , both happen to coincide, a boring example of an exceptional isomorphism.
## n-sphere ()
split words: 9
### Circle (, 1-sphere)
split words: 6
#### Tarski's circle-squaring problem (Cut a circle into square)
split words: 6
Does not require straight line cuts.
### 3-sphere ()
split words: 3
Diffeomorphic to .
## Projective geometry
split words: 735
### Projective space ()
split words: 58
A unique projective space can be defined for any vector space.
The projective space associated with a given vector space is denoted .
The definition is to take the vector space, remove the zero element, and identify all elements that lie on the same line, i.e.
The most important initial example to study is the real projective plane.
### Real projective space (, )
split words: 677
In those cases at least, it is possible to add a metric to the spaces, leading to elliptic geometry.
#### Real projective line (, )
split words: 20
Just a circle.
Take with a line at . Identify all the points that an observer
#### Real projective plane (, )
split words: 640
For some reason, Ciro Santilli is mildly obsessed with understanding and visualizing the real projective plane.
To see why this is called a plane, move he center of the sphere to , and project each line passing on the center of the sphere on the x-y plane. This works for all points of the sphere, except those at the equator . Those are the points at infinity. Note that there is one such point at infinity for each direction in the x-y plane.
##### Synthetic geometry of the real projective plane
split words: 245
It good to think about how Euclid's postulates look like in the real projective plane:
• two parallel lines on the plane meet at a point on the sphere!
Since there is one point of infinity for each direction, there is one such point for every direction the two parallel lines might be at. The parallel postulate does not hold, and is replaced with a simpler more elegant version: every two lines meet at exactly one point.
One thing to note however is that ther real projective plane does not have angles defined on it by definition. Those can be defined, forming elliptic geometry through the projective model of elliptic geometry, but we can interpret the "parallel lines" as "two lines that meet at a point at infinity"
• points in the real projective plane are lines in
• lines in the real projective plane are planes in .
For every two projective points there is a single projective line that passes through them.
Since it is a plane in , it always intersects the real plane at a line.
Note however that not all lines in the real plane correspond to a projective line: only lines tangent to a circle at zero do.
Unlike the real projective line which is homotopic to the circle, the real projective plane is not homotopic to the sphere.
The topological difference bewteen the sphere and the real projective space is that for the sphere all those points in the x-y circle are identified to a single point.
One more generalized argument of this is the classification of closed surfaces, in which the real projective plane is a sphere with a hole cut and one Möbius strip glued in.
##### Model of the real projective plane
split words: 259
###### Lines through origin model of the real projective plane
split words: 5
This is the standard model.
###### Spherical cap model of the real projective plane
split words: 254
Ciro Santilli's preferred visualization of the real projective plane is a small variant of the standard "lines through origin in ".
Take a open half sphere e.g. a sphere but only the points with .
Each point in the half sphere identifies a unique line through the origin.
Then, the only lines missing are the lines in the x-y plane itself.
For those sphere points in the circle on the x-y plane, you should think of them as magic poins that are identified with the corresponding antipodal point, also on the x-y, but on the other side of the origin. So basically you you can teleport from one of those to the other side, and you are still in the same point.
Ciro likes this model because then all the magic is confined just to the part of the model, and everything else looks exactly like the sphere.
It is useful to contrast this with the sphere itself. In the sphere, all points in the circle are the same point. But this is not the case for the projective plane. You cannot instantly go to any other point on the by just moving a little bit, you have to walk around that circle.
##### The real projective plane is not simply connected
split words: 55
To see that the real projective plane is not simply connected space, considering the lines through origin model of the real projective plane, take a loop that starts at and moves along the great circle ends at .
Note that both of those points are the same, so we have a loop.
Now try to shrink it to a point.
There's just no way!
## Polytope
split words: 280
A polygon is a 2-dimensional polytope, polyhedra is a 3-dimensional polytope.
### Regular polytope
split words: 257
TODO understand and explain definition.
#### Classification of regular polytopes
split words: 252
The 3D regular convex polyhedrons are super famous, have the name: Platonic solid, and have been known since antiquity. In particular, there are only 5 of them.
The counts per dimension are:
Table 1. Number of regular polytopes per dimension.
Dimension Count
2 Infinite
3 5
4 6
>4 3
The cool thing is that the 3 that exist in 5+ dimensions are all of one of the three families:
Then, the 2 3D missing ones have 4D analogues and the sixth one in 4D does not have a 3D analogue: the 24-cell. Yes, this is the kind of irregular stuff Ciro Santilli lives for.
##### Simplex
split words: 35
The name does not imply regular by default. For regular ones, you should say "regular polytope".
Non-regular description: take convex hull take D + 1 vertices that are not on a single D-plan.
##### Hypercube
split words: 78
square, cube. 4D case known as tesseract.
Convex hull of all (Cartesian product power) D-tuples, e.g. in 3D:
( 1, 1, 1)
( 1, 1, -1)
( 1, -1, 1)
( 1, -1, -1)
(-1, 1, 1)
(-1, 1, -1)
(-1, -1, 1)
(-1, -1, -1)
From this we see that there are vertices.
Two vertices are linked iff they differ by a single number. So each vertex has D neighbors.
###### Hyperrectangle
split words: 6
The non-regular version of the hypercube.
##### Cross polytope
split words: 37
Examples: square, octahedron.
Take and flip one of 0's to . Therefore has vertices.
Each edge E is linked to every other edge, except it's opposite -E.
split words: 1
split words: 1
split words: 1
split words: 13
split words: 13
## Differential geometry
split words: 5k
Bibliography:
### Lie group
split words: 5k
The key and central motivation for studying Lie groups and their Lie algebras appears to be to characterize symmetry in Lagrangian mechanics through Noether's theorem, just start from there.
Notably local symmetries appear to map to forces, and local means "around the identity", notably: local symmetries of the Lagrangian imply conserved currents.
TODO Ciro Santilli really wants to understand what all the fuss is about:
The fact that there are elements arbitrarily close to the identity, which is only possible due to the group being continuous, is the key factor that simplifies the treatment of Lie groups, and follows the philosophy of continuous problems are simpler than discrete ones.
Bibliography:
#### Applications of Lie groups to differential equations (How to use Lie Groups to solve differential equations)
split words: 68
Solving differential equations was apparently Lie's original motivation for developing Lie groups. It is therefore likely one of the most understandable ways to approach it.
It appears that Lie's goal was to understand when can a differential equation have an explicitly written solution, much like Galois theory had done for algebraic equations. Both approaches use symmetry as the key tool.
#### Lie algebra
split words: 738
Like everything else in Lie groups, first start with the matrix as discussed at Section "Lie algebra of a matrix Lie group".
Intuitively, a Lie algebra is a simpler object than a Lie group. Without any extra structure, groups can be very complicated non-linear objects. But a Lie algebra is just an algebra over a field, and one with a restricted bilinear map called the Lie bracket, that has to also be alternating and satisfy the Jacobi identity.
Another important way to think about Lie algebras, is as infinitesimal generators.
Because of the Lie group-Lie algebra correspondence, we know that there is almost a bijection between each Lie group and the corresponding Lie algebra. So it makes sense to try and study the algebra instead of the group itself whenever possible, to try and get insight and proofs in that simpler framework. This is the key reason why people study Lie algebras. One is philosophically reminded of how normal subgroups are a simpler representation of group homomorphisms.
To make things even simpler, because all vector spaces of the same dimension on a given field are isomorphic, the only things we need to specify a Lie group through a Lie algebra are:
Note that the Lie bracket can look different under different basis of the Lie algebra however. This is shown for example at Physics from Symmetry by Jakob Schwichtenberg (2015) page 71 for the Lorentz group.
As mentioned at Lie Groups, Physics, and Geometry by Robert Gilmore (2008) Chapter 4 "Lie Algebras", taking the Lie algebra around the identity is mostly a convention, we could treat any other point, and things are more or less equivalent.
##### Infinitesimal generator
split words: 147
Elements of a Lie algebra can (should!) be seen a continuous analogue to the generating set of a group in finite groups.
For continuous groups however, we can't have a finite generating set in the strict sense, as a finite set won't ever cover every possible point.
But the generator of a Lie algebra can be finite.
And just like in finite groups, where you can specify the full group by specifying only the relationships between generating elements, in the Lie algebra you can almost specify the full group by specifying the relationships between the elements of a generator of the Lie algebra.
This "specification of a relation" is done by defining the Lie bracket.
The reason why the algebra works out well for continuous stuff is that by definition an algebra over a field is a vector space with some extra structure, and we know very well how to make infinitesimal elements in a vector space: just multiply its vectors by a constant that cana be arbitrarily small.
##### Lie group-Lie algebra correspondence
split words: 103
Every Lie algebra corresponds to a single simply connected Lie group.
The Baker-Campbell-Hausdorff formula basically defines how to map an algebra to the group.
Bibliography:
###### Lie algebra exponential covering problem
split words: 40
Lie Groups, Physics, and Geometry by Robert Gilmore (2008) 7.2 "The covering problem" gives some amazing intuition on the subject as usual.
###### A single exponential map is not enough to recover a simple Lie group from its algebra
split words: 5
Example at: Lie Groups, Physics, and Geometry by Robert Gilmore (2008) Chapter 7 "EXPonentiation".
###### The product of a exponential of the compact algebra with that of the non-compact algebra recovers a simple Lie from its algebra
split words: 22
Example at: Lie Groups, Physics, and Geometry by Robert Gilmore (2008) Chapter 7 "EXPonentiation".
Furthermore, the non-compact part is always isomorphic to , only the non-compact part can have more interesting structure.
###### Two different Lie groups can have the same Lie algebra
split words: 42
The most important example is perhaps and , both of which have the same Lie algebra, but are not isomorphic.
###### Every Lie algebra has a unique single corresponding simply connected Lie group
split words: 23
E.g. in the case of and , is simply connected, but is not.
split words: 6
##### Exponential map
split words: 117
Most commonly refers to: exponential map.
###### Exponential map (Lie theory)
split words: 112
Like everything else in Lie group theory, you should first look at the matrix version of this operation: the matrix exponential.
The exponential map links small transformations around the origin (infinitely small) back to larger finite transformations, and small transformations around the origin are something we can deal with a Lie algebra, so this map links the two worlds.
The idea is that we can decompose a finite transformation into infinitely arbitrarily small around the origin, and proceed just like the product definition of the exponential function.
The definition of the exponential map is simply the same as that of the regular exponential function as given at Taylor expansion definition of the exponential function, except that the argument can now be an operator instead of just a number.
##### Baker-Campbell-Hausdorff formula (BCH formula)
split words: 153
Solution for given and of: $$eZ=eXeY (1)$$ where is the exponential map.
If we consider just real number, , but when X and Y are non-commutative, things are not so simple.
Furthermore, TODO confirm it is possible that a solution does not exist at all if and aren't sufficiently small.
This formula is likely the basis for the Lie group-Lie algebra correspondence. With it, we express the actual group operation in terms of the Lie algebra operations.
Notably, remember that a algebra over a field is just a vector space with one extra product operation defined.
Vector spaces are simple because all vector spaces of the same dimension on a given field are isomorphic, so besides the dimension, once we define a Lie bracket, we also define the corresponding Lie group.
Since a group is basically defined by what the group operation does to two arbitrary elements, once we have that defined via the Baker-Campbell-Hausdorff formula, we are basically done defining the group in terms of the algebra.
##### Generators of a Lie algebra
split words: 7
Cardinality dimension of the vector space.
#### Continuous symmetry
split words: 221
Basically a synonym for Lie group which is the way of modelling them.
##### Local symmetry
split words: 210
Local symmetries appear to be a synonym to internal symmetry, see description at: Section "Internal and spacetime symmetries".
As mentioned at Quote , local symmetries map to forces in the Standard Model.
Appears to be a synonym for: gauge symmetry.
A local symmetry is a transformation that you apply a different transformation for each point, instead of a single transformation for every point.
TODO what's the point of a local symmetry?
Bibliography:
###### Local symmetries of the Lagrangian imply conserved currents
split words: 97
TODO. I think this is the key point. Notably, symmetry implies charge conservation.
More precisely, each generator of the corresponding Lie algebra leads to one separate conserved current, such that a single symmetry can lead to multiple conserved currents.
This is basically the local symmetry version of Noether's theorem.
Then to maintain charge conservation, we have to maintain local symmetry, which in turn means we have to add a gauge field as shown at Video "Deriving the qED Lagrangian by Dietterich Labs (2018)".
Forces can then be seen as kind of a side effect of this.
#### Important Lie group
split words: 3k
##### Matrix Lie group
split words: 284
This important and common simple case has easy properties.
split words: 2
###### Lie algebra of a matrix Lie group
split words: 273
For this sub-case, we can define the Lie algebra of a Lie group as the set of all matrices such that for all : $$etM∈G (2)$$ If we fix a given and vary , we obtain a subgroup of . This type of subgroup is known as a one parameter subgroup.
The immediate question is then if every element of can be reached in a unique way (i.e. is the exponential map a bijection). By looking at the matrix logarithm however we conclude that this is not the case for real matrices, but it is for complex matrices.
Examples:
TODO example it can be seen that the Lie algebra is not closed matrix multiplication, even though the corresponding group is by definition. But it is closed under the Lie bracket operation.
###### Lie bracket of a matrix Lie group
split words: 69
$$[X,Y]=XY−YX (3)$$
This makes it clear how the Lie bracket can be seen as a "measure of non-commutativity"
Because the Lie bracket has to be a bilinear map, all we need to do to specify it uniquely is to specify how it acts on every pair of some basis of the Lie algebra.
Then, together with the Baker-Campbell-Hausdorff formula and the Lie group-Lie algebra correspondence, this forms an exceptionally compact description of a Lie group.
###### One parameter subgroup
split words: 73
The one parameter subgroup of a Lie group for a given element of its Lie algebra is a subgroup of given by: $$etM∈G∣t∈R (4)$$
Intuitively, is a direction, and is how far we move along a given direction. This intuition is especially vivid in for example in the case of the Lie algebra of , the rotation group.
One parameter subgroups can be seen as the continuous analogue to the cycle of an element of a group.
##### Classical group
split words: 141
###### Symplectic group ()
split words: 141
Intuition, please? Example? mathoverflow.net/questions/278641/intuition-for-symplectic-groups The key motivation seems to be related to Hamiltonian mechanics. The two arguments of the bilinear form correspond to each set of variables in Hamiltonian mechanics: the generalized positions and generalized momentums, which appear in the same number each.
Seems to be set of matrices that preserve a skew-symmetric bilinear form, which is comparable to the orthogonal group, which preserves a symmetric bilinear form. More precisely, the orthogonal group has: $$OTIO=I (5)$$ and its generalization the indefinite orthogonal group has: $$OTSO=I (6)$$ where S is symmetric. So for the symplectic group we have matrices Y such as: $$YTAY=I (7)$$ where A is antisymmetric. This is explained at: www.ucl.ac.uk/~ucahad0/7302_handout_13.pdf They also explain there that unlike as in the analogous orthogonal group, that definition ends up excluding determinant -1 automatically.
Therefore, just like the special orthogonal group, the symplectic group is also a subgroup of the special linear group.
##### General linear group (, )
split words: 102
Invertible matrices. Or if you think a bit more generally, an invertible linear map.
When the field is not given, it defaults to the real numbers.
Non-invertible are excluded "because" otherwise it would not form a group (every element must have an inverse). This is therefore the largest possible group under matrix multiplication, other matrix multiplication groups being subgroups of it.
###### Finite general linear group (, )
split words: 45
general linear group over a finite field of order . Remember that due to the classification of finite fields, there is one single field for each prime power .
Exactly as over the real numbers, you just put the finite field elements into a matrix, and then take the invertible ones.
##### Lie algebra of
split words: 47
For every matrix in the set of all n-by-y square matrices , has inverse .
Note that this works even if is not invertible, and therefore not in !
Therefore, the Lie algebra of is the entire .
##### Special linear group ()
split words: 534
Specials sub case of the general linear group when the determinant equals exactly 1.
split words: 486
###### Lie algebra of
split words: 486
This is a good first concrete example of a Lie algebra. Shown at Lie Groups, Physics, and Geometry by Robert Gilmore (2008) Chapter 4.2 "How to linearize a Lie Group" has an example.
We can use use the following parametrization of the special linear group on variables , and : $$M=[1+xzy(1+yz)/(1+x)] (8)$$
Every element with this parametrization has determinant 1: $$det(M)=(1+x)(1+yz)/(1+x)−yz=1 (9)$$ Furthermore, any element can be reached, because by independently settting , and , , and can have any value, and once those three are set, is fixed by the determinant.
To find the elements of the Lie algebra, we evaluate the derivative on each parameter at 0: $$MxMyMz=dxdM∣∣∣∣∣(x,y,z)=(0,0,0)=dydM∣∣∣∣∣(x,y,z)=(0,0,0)=dzdM∣∣∣∣∣(x,y,z)=(0,0,0)=[100−(1+yz)/(1+x)2]∣∣∣∣∣(x,y,z)=(0,0,0)=[001z/(1+x)]∣∣∣∣∣(x,y,z)=(0,0,0)=[010y/(1+x)]∣∣∣∣∣(x,y,z)=(0,0,0)=[100−1]=[0010]=[0100] (10)$$
Remembering that the Lie bracket of a matrix Lie group is really simple, we can then observe the following Lie bracket relations between them: $$[Mx,My][Mx,Mz][My,Mz]=MxMy−MyMx=MxMz−MzMx=MyMz−MzMy=[0010]=[0−100]=[1000]−[00−10]−[0100]−[0001]=[0020]=[0−200]=[100−1]=2My=−2Mz=Mx (11)$$
One key thing to note is that the specific matrices , and are not really fundamental: we could easily have had different matrices if we had chosen any other parametrization of the group.
TODO confirm: however, no matter which parametrization we choose, the Lie bracket relations between the three elements would always be the same, since it is the number of elements, and the definition of the Lie bracket, that is truly fundamental.
Lie Groups, Physics, and Geometry by Robert Gilmore (2008) Chapter 4.2 "How to linearize a Lie Group" then calculates the exponential map of the vector as: $$Icosh(θ)+Mxsinh(θ)/θ (12)$$ with: $$θ2=x2+bc (13)$$
TODO now the natural question is: can we cover the entire Lie group with this exponential? Lie Groups, Physics, and Geometry by Robert Gilmore (2008) Chapter 7 "EXPonentiation" explains why not.
###### Finite special general linear group ()
split words: 37
Just like for the finite general linear group, the definition of special also works for finite fields, where 1 is the multiplicative identity!
Note that the definition of orthogonal group may not have such a clear finite analogue on the other hand.
##### Isometry group
split words: 102
The group of all transformations that preserve some bilinear form, notable examples:
###### Lie algebra of a isometry group
split words: 86
We can almost reach the Lie algebra of any isometry group in a single go. For every in the Lie algebra we must have: $$∀v,w∈V,t∈R(etXv∣etXw)=(v∣w) (14)$$ because has to be in the isometry group by definition as shown at Section "Lie algebra of a matrix Lie group".
Then: $$dtd(etXv∣etXw)∣∣∣∣∣0=0⟹(XetXv∣etXw)+(etXv∣XetXw)∣∣∣∣∣0=0⟹(Xv∣w)+(v∣Xw)=0 (15)$$ so we reach: $$∀v,w∈V(Xv∣w)=−(v∣Xw) (16)$$ With this relation, we can easily determine the Lie algebra of common isometries:
Bibliography:
##### Orthogonal group ()
split words: 856
###### Definition of the orthogonal group
split words: 281
Intuitive definition: real group of rotations + reflections.
Mathematical definition that most directly represents this: the orthogonal group is the group of all matrices that preserve the dot product.
###### The orthogonal group is the group of all matrices that preserve the dot product
split words: 160
When viewed as matrices, it is the group of all matrices that preserve the dot product, i.e.: $$O(n)=O∈M(n)∣∀x,y,xTy=(Ox)T(Oy) (17)$$ This implies that it also preserves important geometric notions such as norm (intuitively: distance between two points) and angles.
This is perhaps the best "default definition".
###### What happens to the definition of the orthogonal group if we choose other types of symmetric bilinear forms
split words: 104
We looking at the definition the orthogonal group is the group of all matrices that preserve the dot product, we notice that the dot product is one example of positive definite symmetric bilinear form, which in turn can also be represented by a matrix as shown at: Section "Matrix representation of a symmetric bilinear form".
By looking at this more general point of view, we could ask ourselves what happens to the group if instead of the dot product we took a more general bilinear form, e.g.:
The answers to those questions are given by the Sylvester's law of inertia at Section "All indefinite orthogonal groups of matrices of equal metric signature are isomorphic".
###### The orthogonal group is the group of all invertible matrices where the inverse is equal to the transpose
split words: 82
Note that: $$xTy=(Ox)T(Oy)=xTOTOy (18)$$ and for that to be true for all possible and then we must have: $$OTO=I (19)$$ i.e. the matrix inverse is equal to the transpose.
Conversely, if: $$OTO=I (20)$$ is true, then $$(Ox)T(Oy)=xT(OTO)y=xTy (21)$$
These matricese are called the orthogonal matrices.
TODO is there any more intuitive way to think about this?
###### The orthogonal group is the group of all matrices with orthonormal rows and orthonormal columns
split words: 23
Or equivalently, the set of rows is orthonormal, and so is the set of columns. TODO proof that it is equivalent to the orthogonal group is the group of all matrices that preserve the dot product.
split words: 258
###### Connected components of the orthogonal group (The orthogonal group has two connected components)
split words: 258
The orthogonal group has 2 connected components:
It is instructive to visualize how the looks like in :
• you take the first basis vector and move it to any other. You have therefore two angular parameters.
• you take the second one, and move it to be orthogonal to the first new vector. (you can choose a circle around the first new vector, and so you have another angular parameter.
• at last, for the last one, there are only two choices that are orthogonal to both previous ones, one in each direction. It is this directio, relative to the others, that determines the "has a reflection or not" thing
As a result it is isomorphic to the direct product of the special orthogonal group by the cyclic group of order 2: $$O(n)≅SO(n)×C2 (22)$$
A low dimensional example: $$O(1)≅SO(2)×C2 (23)$$ because you can only do two things: to flip or not to flip the line around zero.
Note that having the determinant plus or minus 1 is not a definition: there are non-orthogonal groups with determinant plus or minus 1. This is just a property. E.g.: $$M=[2132] (24)$$ has determinant 1, but: $$MTM=[58811] (25)$$ so is not orthogonal.
split words: 3
###### Special orthogonal group (, Rotation group)
split words: 206
Group of rotations of a rigid body.
Like orthogonal group but without reflections. So it is a "special case" of the orthogonal group.
This is a subgroup of both the orthogonal group and the special linear group.
###### Lie algebra of
split words: 171
We can reach it by taking the rotations in three directions, e.g. a rotation around the z axis: $$Rz(θ)=⎣⎢⎡cos(θ)sin(θ)0−sin(θ)cos(θ)0001⎦⎥⎤ (26)$$ then we derive and evaluate at 0: $$Lz=dθdRz(θ)∣∣∣∣∣0=⎣⎢⎡−sin(θ)cos(θ)0−cos(θ)−sin(θ)0001⎦⎥⎤∣∣∣∣∣0=⎣⎢⎡010−100000⎦⎥⎤ (27)$$ therefore represents the infinitesimal rotation.
Note that the exponential map reverses this and gives a finite rotation around the Z axis back from the infinitesimal generator : $$eθLz=Rz(θ) (28)$$
Repeating the same process for the other directions gives: $$Lx=TODOLy=TODO (29)$$ We have now found 3 linearly independent elements of the Lie algebra, and since has dimension 3, we are done.
###### Lie bracket of the rotation group
split words: 16
Based on the , and derived at Lie algebra of we can calculate the Lie bracket as: $$TODO (30)$$
###### 3D rotation group ()
split words: 4
Has as a double cover.
###### Unitary group ()
split words: 108
Complex analogue of the orthogonal group.
One notable difference from the orthogonal group however is that the unitary group is connected "because" its determinant is not fixed to two disconnected values 1/-1, but rather goes around in a continuous unit circle. is the unit circle.
###### Unitary group of degree 2 ()
split words: 5
Diffeomorphic to the 3 sphere.
###### Unit circle
split words: 12
The unitary group is one very over-generalized way of looking at it :-)
###### Special unitary group ()
split words: 44
The complex analogue of the special orthogonal group, i.e. the subgroup of the unitary group with determinant equals exactly 1 instead of an arbitrary complex number with absolute value equal 1 as is the case for the unitary group.
split words: 9
split words: 4
split words: 3
Bibliography:
split words: 1
##### Projective linear group
split words: 20
TODO motivation. Motivation. Motivation. Motivation. The definitin with quotient group is easy to understand.
split words: 8
split words: 8
split words: 8
###### PSL(2,7)
split words: 8
The second smallest non-Abelian finite simple group after the alternating group of degree 5.
##### Poincaré group
split words: 1k
Full set of all possible special relativity symmetries:
In simple and concrete terms. Suppose you observe N particles following different trajectories in Spacetime.
There are two observers traveling at constant speed relative to each other, and so they see different trajectories for those particles:
• space and time shifts, because their space origin and time origin (time they consider 0, i.e. when they started their timers) are not synchronized. This can be modelled with a 4-vector addition.
• their space axes are rotated relative to one another. This can be modelled with a 4x4 matrix multiplication.
• and they are moving relative to each other, which leads to the usual spacetime interactions of special relativity. Also modelled with a 4x4 matrix multiplication.
Note that the first two types of transformation are exactly the non-relativistic Galilean transformations.
The Poincare group is the set of all matrices such that such a relationship like this exists between two frames of reference.
split words: 460
###### Translation (geometry)
split words: 139
Subset of Galilean transformation with speed equals 0.
###### Translation group
split words: 133
This is a good and simple first example of Lie algebra to look into.
###### The derivative is the generator of the translation group
split words: 121
Take the group of all Translation in .
Let's see how the generator of this group is the derivative operator: $$∂x∂ (31)$$
• the translation group operates on the argument of a function
• the generator is an operator that operates on itself
So let's take the exponential map: $$ex0∂x∂f(x)=(1+x0∂x∂+x02∂x2∂2+…)f(x) (32)$$ and we notice that this is exactly the Taylor series of around the identity element of the translation group, which is 0! Therefore, if behaves nicely enough, within some radius of convergence around the origin we have for finite : $$ex0∂x∂f(x)=f(x+x0) (33)$$
This example shows clearly how the exponential map applied to a (differential) operator can generate finite (non-infinitesimal) Translation!
###### Galilean invariance
split words: 321
A law of physics is Galilean invariant if the same formula works both when you are standing still on land, or when you are on a boat moving at constant velocity.
For example, if we were describing the movement of a point particle, the exact same formulas that predict the evolution of must also predict , even though of course both of those will have different values.
It would be extremely unsatisfactory if the formulas of the laws of physics did not obey Galilean invariance. Especially if you remember that Earth is travelling extremelly fast relative to the Sun. If there was no such invariance, that would mean for example that the laws of physics would be different in other planets that are moving at different speeds. That would be a strong sign that our laws of physics are not complete.
The consequence/cause of that is that you cannot know if you are moving at a constant speed or not.
Lorentz invariance generalizes Galilean invariance to also account for special relativity, in which a more complicated invariant that also takes into account different times observed in different inertial frames of reference is also taken into account. But the fundamental desire for the Lorentz invariance of the laws of physics remains the same.
###### Covariance
split words: 133
Generally means that he form of the equation does not change if we transform .
This is generally what we want from the laws of physics.
E.g. a Galilean transformation generally changes the exact values of coordinates, but not the form of the laws of physics themselves.
Lorentz covariance is the main context under which the word "covariant" appears, because we really don't want the form of the equations to change under Lorentz transforms, and "covariance" is often used as a synonym of "Lorentz covariance".
TODO some sources distinguish "invariant" from "covariant": invariant vs covariant.
###### Invariant vs covariant
split words: 43
Some sources distinguish "invariant" from "covariant" such that under some transformation (typically Lie group):
• invariant: the value of does not change if we transform
• covariant: the form of the equation does not change if we transform .
TODO examples.
###### Lorentz group ()
split words: 552
Subgroup of the Poincaré group without translations. Therefore, in those, the spacetime origin is always fixed.
Or in other words, it is as if two observers had their space and time origins at the exact same place. However, their space axes may be rotated, and one may be at a relative speed to the other to create a Lorentz boost. Note however that if they are at relative speeds to one another, then their axes will immediately stop being at the same location in the next moment of time, so things are only valid infinitesimally in that case.
This group is made up of matrix multiplication alone, no need to add the offset vector: space rotations and Lorentz boost only spin around and bend things around the origin.
One definition: set of all 4x4 matrices that keep the Minkowski inner product, mentioned at Physics from Symmetry by Jakob Schwichtenberg (2015) page 63. This then implies: $$ΛTηΛ=η (34)$$
###### Representation theory of the Lorentz group
split words: 72
Physics from Symmetry by Jakob Schwichtenberg (2015) page 66 shows one in terms of 4x4 complex matrices.
More importantly though, are the representations of the Lie algebra of the Lorentz group, which are generally also just also called "Representation of the Lorentz group" since you can reach the representation from the algebra via the exponential map.
Bibliography:
###### Representation of the Lorentz group
split words: 20
One of the representations of the Lorentz group that show up in the Representation theory of the Lorentz group.
###### Spinor
split words: 8
TODO understand a bit more intuitively.
###### Lorentz boost
split words: 40
Two observers travel at fixed speed relative to each other. They synchronize origins at x=0 and t=0, and their spacial axes are perfectly aligned. This is a subset of the Lorentz group. TODO confirm it does not form a subgroup however.
###### Indefinite orthogonal group ()
split words: 293
Generalization of orthogonal group to preserve different bilinear forms. Important because the Lorentz group is .
###### Definition of the indefinite orthogonal group
split words: 267
Given a matrix with metric signature containing positive and negative entries, the indefinite orthogonal group is the set of all matrices that preserve the associated bilinear form, i.e.: $$O(m,n)=O∈M(m+n)∣∀x,yxTAy=(Ox)TA(Oy) (35)$$ Note that if , we just have the standard dot product, and that subcase corresponds to the following definition of the orthogonal group: Section "The orthogonal group is the group of all matrices that preserve the dot product".
As shown at all indefinite orthogonal groups of matrices of equal metric signature are isomorphic, due to the Sylvester's law of inertia, only the metric signature of matters. E.g., if we take two different matrices with the same metric signature such as: $$[100−1] (36)$$ and: $$[200−3] (37)$$ both produce isomorphic spaces. So it is customary to just always pick the matrix with only +1 and -1 as entries.
###### All indefinite orthogonal groups of matrices of equal metric signature are isomorphic
split words: 135
Following the definition of the indefinite orthogonal group, we want to show that only the metric signature matters.
First we can observe that the exact matrices are different. For example, taking the standard matrix of : $$[1001] (38)$$ and: $$[2001] (39)$$ both have the same metric signature. However, we notice that a rotation of 90 degrees, which preserves the first form, does not preserve the second one! E.g. consider the vector , then . But after a rotation of 90 degrees, it becomes , and now ! Therefore, we have to search for an isomorphism between the two sets of matrices.
For example, consider the orthogonal group, which can be defined as shown at the orthogonal group is the group of all matrices that preserve the dot product can be defined as:
###### Indefinite special orthogonal group ()
split words: 15
Like the special orthogonal group is to the orthogonal group, is the subset of with determinant equal to exactly 1.
#### Representation theory
split words: 214
Basically, a "representation" means associating each group element as an invertible matrices, i.e. a matrix in (possibly some subset of) , that has the same properties as the group.
Or in other words, associating to the more abstract notion of a group more concrete objects with which we are familiar (e.g. a matrix).
Each such matrix then represents one specific element of the group.
This is basically what everyone does (or should do!) when starting to study Lie groups: we start looking at matrix Lie groups, which are very concrete.
Or more precisely, mapping each group element to a linear map over some vector field (which can be represented by a matrix infinite dimension), in a way that respects the group operations: $$R(g):G→GL(V) (40)$$
As shown at Physics from Symmetry by Jakob Schwichtenberg (2015)
• page 51, a representation is not unique, we can even use matrices of different dimensions to represent the same group
• 3.6 classifies the representations of . There is only one possibility per dimension!
• 3.7 "The Lorentz Group O(1,3)" mentions that even for a "simple" group such as the Lorentz group, not all representations can be described in terms of matrices, and that we can construct such representations with the help of Lie group theory, and that they have fundamental physical application
Bibliography:
#### Simple Lie group
split words: 22
##### Classification of simple Lie groups
split words: 22
A bit like the classification of simple finite groups, they also have a few sporadic groups! Not as spectacular since as usual continuous problems are simpler than discrete ones, but still, not bad.
#### Lie group bibliography
split words: 135
##### An Introduction to Tensors and Group Theory for Physicists by Nadir Jeevanjee (2011)
split words: 32
This does not seem to go deep into the Standard Model as Physics from Symmetry by Jakob Schwichtenberg (2015), appears to focus more on more basic applications.
But because it is more basic, it does explain some things quite well.
##### Lie Groups, Physics, and Geometry by Robert Gilmore (2008)
split words: 99
The author seems to have uploaded the entire book by chapters at: www.physics.drexel.edu/~bob/LieGroups.html
And the author is the cutest: www.physics.drexel.edu/~bob/Personal.html.
Overview:
split words: 8
split words: 8
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Probably, many of you have read the positive news from the Biontech/Pfizer press release from November 9th:
“Vaccine candidate was found to be more than 90% effective in preventing COVID-19 in participants without evidence of prior SARS-CoV-2 infection in the first interim efficacy analysis”
Not being a biostatistician, I was curious how vaccine efficacy is exactly measured. Also, how does the confidence interval look like? Helpfully, Biontech and Pfizer also published a detailed study plan here.
The sample vaccine efficacy can be defined as
$VE = 1-IRR = 1 – \frac{c_v/n_v}{c_p/n_p}$
where $n_v$ and $n_p$ are the number of subjects that got a Covid-19 vaccine and a placebo, respectively, while $c_v$ and $c_p$ are the respective number of subjects that fell ill to the Covid-19 disease. IRR stands for incidence rate ratio and measures the ratio of the share of vaccinated subjects that got Covid-19 to the share in the placebo group.
The press release stated that so far 38955 subjects got the two doses of the vaccine or placebo of which 94 subjects fell ill with Covid-19. Furthermore, the study plan stated that the same number of subjects was assigned to the treatment and control group and let’s assume that also in the 38955 subjects analysed so far the ratio is almost equal. An efficacy of at least 90% then implies that from the 94 subjects with Covid-19 at most 8 could have been vaccinated.
The following code computes the IRR and vaccine efficacy assuming that indeed 8 vaccinated subjects got Covid-19.
n = 38955 # number of subjects
nv = round(n/2) # vaccinated
np = n-nv # got placebo
# number of covid cases
cv = 8
cp = 94-cv
# percentage of subjects in control group
# who got Covid-19
round(100*cp/np,2)
## [1] 0.44
# percentage of vaccinated subjects
# who got Covid-19
round(100*cv/nv,2)
## [1] 0.04
# incidence rate ratio in % in sample
IRR = (cv/nv)/(cp/np)
round(IRR*100,1)
## [1] 9.3
# vaccine efficacy in % in sample
VE = 1-IRR
round(VE*100,1)
## [1] 90.7
Assume for the moment this data came from a finished experiment. We could then compute an approximative 95% confidence interval for the vaccine efficacy e.g. using the following formula described in Hightower et. al. 1988
arv = cv/nv
arp = cp/np
# CI for IRR
ci.lower = exp(log(IRR) - 1.96 * sqrt((1-arv)/cv + (1-arp)/cp))
ci.upper = exp(log(IRR) + 1.96 * sqrt((1-arv)/cv + (1-arp)/cp))
IRR.ci = c(ci.lower, ci.upper)
round(100*IRR.ci,1)
## [1] 4.5 19.2
VE.ci = rev(1-IRR.ci)
round(100*VE.ci,1)
## [1] 80.8 95.5
This means we would be 95% confident that the vaccine reduces the risk of getting Covid-19 between 80.8% and 95.5%. As far as I understood, e.g. the function ciBinomial in the package gsDesign allows a more precise computation of the confidence interval:
library(gsDesign)
IRR.ci = ciBinomial(cv,cp,nv,np,scale = "RR")
VE.ci = rev(1-IRR.ci)
round(100*VE.ci,1)
## upper lower
## 1 81.1 95.4
Given that it is only stated that the vaccine is more than 90% effective, the number of Covoid-19 cases may also have been lower than 8 subjects.
The next clean threshold for a press statement would probably be at least 95% effectiveness, which would be exceeded if only 4 vaccinated subjects had Covid-19. So it also seems well reasonable that only 5 vaccinated subjects had Covid-19. This would yield the following vaccine efficacy and confidence interval:
cv = 5; cp = 94-cv
VE = 1-(cv/nv)/(cp/np)
round(VE*100,1)
## [1] 94.4
IRR.ci = ciBinomial(cv,cp,nv,np,scale = "RR")
VE.ci = rev(1-IRR.ci)
round(100*VE.ci,1)
## upper lower
## 1 86.6 97.7
Looks even better.
However, those confidence intervals assume a finished, non-adaptive experiment. Yet, the interim evaluations are triggered when the number of Covid-19 cases among the subjects exceeds certain thresholds. The press release states:
“After discussion with the FDA, the companies recently elected to drop the 32-case interim analysis and conduct the first interim analysis at a minimum of 62 cases. Upon the conclusion of those discussions, the evaluable case count reached 94 and the DMC performed its first analysis on all cases.”
“The trial is continuing to enroll and is expected to continue through the final analysis when a total of 164 confirmed COVID-19 cases have accrued.”
I am no expert, but possible the calculation of the confidence interval is not valid for such adaptive rules where the evaluation is triggered by the number of disease cases.
Indeed, Biontech and Pfizer state that they will the assess the precision of the estimated vaccine efficacy using a Bayesian framework with a particular prior distribution described on p. 102-103 of their study plan. Alas, I know very little of Bayesian analysis so I abstain from computing the posterior distribution given the data at hand.
But even absent the full-fledged Bayesian analysis, the numbers really look like very good news.
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# [ASK] Probability of Getting the Main Doorprize
#### Monoxdifly
##### Well-known member
There's an event which is joined by 240 members. The Event Organizer prepares 30 doorprize with one of them being the main ones. If Mr. Aziz's family has 15 tickets, the probability that Mr. Aziz gets the main doorprize is ....
A. 1/16
B. 1/8
C. 1/4
D. 1/2
I thought the answer was 15/240 (the probability of Mr. Aziz's family getting the doorprize) times 1/30 (the main one among the doorprize) and it results in 1/480, but it's not in the options. Is the book wrong or am I the one who miscalculated?
#### Klaas van Aarsen
##### MHB Seeker
Staff member
Hi Mr. Fly!
I'm assuming those 30 doorprizes are divided randomly among the 240 members.
And that the 15 tickets in Mr. Aziz's family correspond to 15 members.
And that there is only 1 main doorprize.
Just checking, is it a typo that you write 'the main ones' as plural?
Otherwise it suggests that there is more than 1 main doorprize.
If there is only 1 main prize, and Mr. Aziz has 15 chances out of 240 on it, then the probability that Mr. Aziz gets the main doorprize is 15/240 = 1/16.
Note that if my interpretation is correct, we can expect that Mr. Aziz's family collects $\frac{15}{240 }\cdot 30$ door prizes as opposed to the 15/240 that you suggested.
Since only 1 of them is the main prize, we multiply indeed by 1/30, resulting in the $\frac{15}{240}\cdot 30\cdot \frac{1}{30}=\frac{15}{240}=\frac{1}{16}$ that I already mentioned.
Last edited:
#### HallsofIvy
##### Well-known member
MHB Math Helper
I see no reason to even consider the "30 door prizes". The question is only about the one main prize. There are 240 people and 15 of them are in this family. The probability of someone in this family winning the one main prize is $$\frac{15}{240}= \frac{1}{16}$$.
#### Monoxdifly
##### Well-known member
Thank you, both of you. And yes, Klaas, that was a typo.
It has been quite a long time since someone calls me Mr. Fly...
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Bruno Schneider
2023-03-11
How would you find a unit vector perpendicular to plane ABC where points are A(3,-1,2), B(1,-1,-3) and C(4,-3,1)?
Jaidyn Velez
We know that, given two vectors, say $\stackrel{\to }{x}&\stackrel{\to }{y}$, their Vector
or Outer Product , denoted by $\stackrel{\to }{x}×\stackrel{\to }{y},$ is a vector that is
perpendicular to the plane that contains them.
The given pts. $A\left(3,-1,2\right),B\left(1,-1,-3\right)\phantom{\rule{1ex}{0ex}}\text{and}\phantom{\rule{1ex}{0ex}}C\left(4,-3,1\right)$ lie in the plane $ABC$
Accordingly, the vectors $\stackrel{\to }{AB}\phantom{\rule{1ex}{0ex}}\text{and}\phantom{\rule{1ex}{0ex}}\stackrel{\to }{AC}\in \phantom{\rule{1ex}{0ex}}\text{the plane}\phantom{\rule{1ex}{0ex}}ABC.$
Thus, $\stackrel{\to }{AB}×\stackrel{\to }{AC}\perp \text{plane}\phantom{\rule{1ex}{0ex}}ABC$
Therefore, the reqd. unit vector will be
$\frac{\stackrel{\to }{AB}×\stackrel{\to }{AC}}{||\stackrel{\to }{AB}×\stackrel{\to }{AC}||}$
We have, $\stackrel{\to }{AB}=\left(1-3,-1+1,-3-2\right)=\left(-2,0,-5\right)$,
$\stackrel{\to }{AC}=\left(1,-2,-1\right)$, so that,
$\stackrel{\to }{AB}×\stackrel{\to }{AC}=\mathrm{det}|\begin{array}{ccc}i& j& k\\ -2& 0& -5\\ 1& -2& -1\end{array}|$
$=-10i-7j-4k=\left(-10,-7,-4\right)$
$⇒||\stackrel{\to }{AB}×\stackrel{\to }{AC}||=\sqrt{100+49+16}=\sqrt{165}$
Finally, the desired unit vector is
$\left(-\frac{10}{\sqrt{165}},-\frac{7}{\sqrt{165}},-\frac{4}{\sqrt{165}}\right).$
Do you have a similar question?
|
Am i doing this right
1. Feb 8, 2006
jamesbob
Hey, just wanna check iv done these questions right so far...
$$\mbox{Find the general solution to} \frac{dy}{dx} = y^2xcos(2x) \mbox{giving explicitly in terms of x.}$$
$$\mbox{Find the particular solution satisfying y(0) = -1}$$
$$\frac{dy}{y^2} = x\cos(2x)dx \Rightarrow \int\frac{dy}{y^2} = \intx\cos(2x)dx$$
$$u = x$$
$$\frac{du}{dx} = 1$$
$$\frac{dv}{dx} = \cos2x$$
$$v = \frac{1}{2}\sin(2x)}$$
$$\Rightarrow \frac{xsinx}{2} - \frac{1}{2}\int\sin(2x) = \frac{x\sinx}{2} + \frac{1}{4}\cos(2x) + C$$
So we have
$$\frac{-1}{y} = \frac{x\sin(x)}{2} + \frac{1}{4}\cos(2x) + C$$
So
$$y = \frac{-4}{\cos(2x)} - \frac{2}{x\sin(2x)} + C$$
$$\mbox{at y(0) = -1 \Rightarrow 1 = \frac{-4}{1} - \frac{2}{0} + C \Rightarrow C = 5}$$
So overall,
$$y = \frac{-2}{x\sin(2x)} - \frac{4}{\cos(2x)} + 5}.$$
Last edited by a moderator: Feb 8, 2006
2. Feb 8, 2006
Tom Mattson
Staff Emeritus
OK.
Nope. You can't take the reciprocal of a fraction that way. If you could then we would have the following:
$\frac{1}{2}=\frac{1}{3}+\frac{1}{6}$ (True)
So
$2=3+6$ (False)
Instead you must combine the two terms on the right with a common denominator, and then take the reciprocal.
3. Feb 8, 2006
jamesbob
$$\mbox{2. Find a particular integral for each of the following equations:}$$
$$/mbox{ i. \frac{d\theta}{dz} + 2\theta = 8}$$
$$/mbox{ ii. \frac{dx}{dt} - 2x = 14e^{-5t}}$$
$$/mbox{iii. \frac{dx}{dt} + x - -3sin2t + 4cos2t}$$
$$\mbox{ i. Constant term so choose x = a + bt}$$
$$\theta(z) = 1 \Rightarrow \frac{d\theta}{dz} = 0 \Rightarrow 0 + 2a = 8 \rightarrow a = 4 \Rightarrow PI = \Theta(z) = 4.$$
$$\mbox{ ii. Choose x = ae^(-t). This gives -5ae^{-5t} - 2ae^{-5t} = 14e^{-5t} \Rightarrow a = -2 \Rightarrow PI: x(t) = -2e^{-5t)}$$
$$\mbox{ iii. All i know to do here so far is choose x = acos2t + bsin2t. How do i continue?}$$
4. Feb 8, 2006
jamesbob
Sorry il sort the coding to this when i have a second
5. Feb 13, 2006
jamesbob
$$\mbox{2. Find a particular integral for each of the following equations:}$$
$$i. \frac{d\theta}{dz} + 2\theta = 8$$
$$ii. \frac{dx}{dt} - 2x = 14e^{-5t}$$
$$iii. \frac{dx}{dt} + x - -3sin2t + 4cos2t$$
$$i. Constant term so choose x = a + bt$$
$$\theta(z) = 1 \Rightarrow \frac{d\theta}{dz} = 0 \Rightarrow 0 + 2a = 8 \rightarrow a = 4 \Rightarrow PI = \Theta(z) = 4.$$
$$ii. Choose x = ae^(-t). This gives -5ae^{-5t} - 2ae^{-5t} = 14e^{-5t} \Rightarrow a = -2 \Rightarrow PI: x(t) = -2e^{-5t)$$
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My topic is mindreading, the process of tracking another’s mental states.
Let me start with a canonical illustration of mindreading in action.
Which box will Maxi look in?
Maxi wants his chocolate.
Chocolate is good.
Maxi believes his chocolate is in the blue box.
Maxi’s chocolate is in the red box.
Therefore:
Therefore:
Maxi will look in the blue box.
Maxi will look in the red box.
So where the nonmindreader uses facts to generate predictions about actions, the mindreading attributes beliefs.
In many cases, you will get the same predictions whether you are mindreading or just using facts. But in cases like this, where there are false belief, the predictions can come apart. \textbf{This means that we can detect mindreading by measuring action predictions.}
In what follows I’m going to focus on belief.
The existence of mindreading processes for tracking beliefs raises many questions:
\begin{enumerate} \item Why is belief-tracking in adults sometimes but not always automatic? (or automatic to varying degrees) \item How and why is automatic belief-tracking limited in ways that nonautomatic belief-tracking is not? \item How and why are inhibition, attention and working memory involved in belief-tracking? \item Why is there an age at which children pass some false belief tasks but systematically fail others? \item What feature or features distinguish the tasks these children fail from those they pass? \end{enumerate}
I think there is more than one kind of mindreading process which tracks beliefs. That’s why my title is ...
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## Thursday, November 10, 2016
### todo: Elastic net method
33 Zou, H. & Hastie, T. Regularization and variable selection via the elastic net. J
Roy Stat Soc B 67, 301-320, (2005).
34 Zou, H. & Zhang, H. H. On the Adaptive Elastic-Net with a Diverging Number of
Parameters. Ann Stat 37, 1733-1751, (2009).
https://www.r-bloggers.com/kickin-it-with-elastic-net-regression/
"Ridge regression is a really effective technique for thwarting overfitting. It does this by penalizing the L2 norm (euclidean distance) of the coefficient vector which results in “shrinking” the beta coefficients. The aggressiveness of the penalty is controlled by a parameter ."
"Lasso regression is a related regularization method. Instead of using the L2 norm, though, it penalizes the L1 norm (manhattan distance) of the coefficient vector."
"Elastic net regression is a hybrid approach that blends both penalization of the L2 and L1 norms."
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# Tag Info
## Hot answers tagged stochastic-processes
6
$X_t$ being a stochastic process, one cannot use ordinary calculus to express the differential of a (sufficiently well-behaved) function $f$ of $t$ and $X_t$. Instead one should turn to Itô's lemma, one of the key results of stochastic calculus, which stipulates (assuming $X_t$ is here a continuous, square integrable stochastic process) $$df(t,X_t) = ... 5 What can be shown is that the above expressions are equal in probability. First check the distribution. As any linear combination of a Gaussian is Gaussian the right hand side is Gaussian - the left hand side too. Then we need the 2 moments: The expected values - it is zero ... easy to see. Next what you did not specify is that the correlation between ... 2 What is written in attached slides is correct. However, what you have written is not correct. Setting M_t=\frac{X_t}{Y_t}, and applying Ito formula will lead to :$$dM_t=\frac{dX_t}{X_t} M_t -\frac{dY_t}{Y_t} M_t + M_t \frac{d<Y>_t}{Y^2_t}-\frac{d<X,Y>_t}{Y^2_t}$$which gives you in your case :$$dM_t = (\mu_x dt+\sigma_x dZ^1_t)M_t - ...
1
Let $Y_t := 2 S_t^1 S_t^2$. Applying (multivariate) Itô to the function $f(t,S_t^1,S_t^2)=2 S_t^1 S_t^2$ yields a stochastic differential equation for $Y_t$ $$\frac{dY_t}{Y_t} = \frac{dS_t^1}{S_t^1} + \frac{dS_t^2}{S_t^2} + \rho \sigma_1 \sigma_2 dt$$ Re-applying Itô's lemma to the function $f(t,Y_t) = \ln(Y_t)$ then yields d\ln Y_t = (\mu_1 + \mu_2 ...
1
I think you got it. Wrapping up: Usually denoted by $(\mathcal {F}_t)_{t \geq 0}$, a filtration is a series of adaptive subsets of the $\sigma$-algebra $\mathcal{F}$ that keeps track of what really happened as time went by (i.e. fixed $\omega$). Over the probability space $(\Omega, \mathcal{F}, \mathbb{P})$, a random variable $X_t$ is measurable iff ...
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Browse Questions
# Which of the following are correct about the solubility of gases in liquids?
All of these
Hence (D) is the correct answer.
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# Spherical system
### From Online Dictionary of Crystallography
Système sphérique (Fr) Sistema sferico (It).
## Definition
The spherical system contains non-crystallographic point groups with more than one axis of revolution. These groups therefore contain an infinity of axes of revolution (or isotropy axis). There are two groups in the spherical system:
Hermann-Mauguin symbol Short Hermann-Mauguin symbol Schönfliess symbol order of the groupgeneral form
$\infty A_\infty$ $2\infty$ K $\infty$ sphere filled with
an optically active liquid
$\infty {A_\infty \over M}C$ $m {\bar \infty}$, ${2\over m}{\bar \infty}$ Kh $\infty$ stationary sphere
## History
The groups containing isotropy axes were introduced by P. Curie (1859-1906) in order to describe the symmetry of physical systems (Curie P. (1884). Sur les questions d'ordre: répétitions. Bull. Soc. Fr. Minéral., 7, 89-110; Curie P. (1894). Sur la symétrie dans les phénomènes physiques, symétrie d’un champ électrique et d’un champ magnétique. J. Phys. (Paris), 3, 393-415.).
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(18 votes, average: 4.17 out of 5)
# Comparing BPSK, QPSK, 4PAM, 16QAM, 16PSK, 64QAM and 32PSK
by on July 8, 2008
I have written another article in DSPDesginLine.com. This article can be treated as the third post in the series aimed at understanding Shannon’s capacity equation.
For the first two posts in the series are:
The article summarizes the symbol error rate derivations in AWGN for modulation schemes like BPSK, QPSK, 4PAM, 16QAM, 16PSK, 64QAM and 32PSK.
The article in DSPDesignline.com details the following:
• Based on the knowledge of bandwidth requirements for each type of modulation scheme, the capacity in bits/seconds/Hz is listed.
• Using the knowledge that the symbol to noise ratio $\frac{E_s}{N_0}$ is $k=\log_2(M)$ times the bit to noise ratio $\frac{E_b}{N_0}$, the symbol error rate vs $\frac{E_b}{N_0}$ curves are plotted.
• Using symbol error rate versus $\frac{E_b}{N_0}$ plots, the $\frac{E_b}{N_0}$ required for achieving symbol error rate of $10^{-5}$is identified.
• Upon having the capacity and $\frac{E_b}{N_0}$ requirement, the requirements for BPSK, QPSK, 4PAM, 16QAM, 16PSK, 64QAM and 32PSK are mapped on to the Shannon’s capacity vs Eb/No curve.
• Assuming Gray coded modulation mapping, each symbol error causes one bit out of $k=\log_2(M)$ bits to be in error. So, the relation between symbol error and bit error is,$P_b \approx \frac{Ps}{k}$.
• Using this assumption, the Bit Error Rate (BER) for BPSK, QPSK, 4PAM, 16QAM, 16PSK, 64QAM and 32PSK are listed and the BER vs Eb/No curve plotted.
D id you like this article? Make sure that you do not miss a new article by subscribing to RSS feed OR subscribing to e-mail newsletter. Note: Subscribing via e-mail entitles you to download the free e-Book on BER of BPSK/QPSK/16QAM/16PSK in AWGN.
Bala March 14, 2013 at 10:14 am
Hi Krishna,
Can you give me a link to a MATLAB pgm which would help us estimate the IQ imbalance in the received OFDM signal.?
Krishna Sankar March 15, 2013 at 5:23 am
@Bala: Sorry, do not have it.
Jeffrey Bridge August 28, 2012 at 4:35 am
Hi there,
This looks like a really good article in DSPDesignLine, except that now it redirects to eetimes.com and although the text of the article is there, all the image links seem to be broken. Without the images containing the formulas, the article is now useless. Is there any way you can fix it?
Krishna Sankar August 28, 2012 at 5:48 am
@Jeffrey Bridge: Thanks for pointing that out. Let me ping eetimes.com to get the article fixed
Donny June 15, 2012 at 2:03 pm
Hi Krishna,
Given that the modulation type is either 16QAm or 16APSK, is there a formula to relate the SNR(Es/No) to the average constellation dispersion for each modulation type? How do we derive the relationship?
Thanks.
Krishna Sankar June 26, 2012 at 6:03 am
@Donny: for a given Es/N0, i recall 16-QAM gives lower symbol error rate that 16-PSK.
One can find a relationship between them by computing the ratio of the distance between the constellation points in each case.
http://www.dsplog.com/2008/03/29/comparing-16psk-vs-16qam-for-symbol-error-rate/
ssstecz May 27, 2012 at 3:12 am
Hare Krishna !
Thanks a lot for all the information you posted here, it was very useful for me.
Junaid Khan May 9, 2012 at 1:14 am
hi can you please help me with matlab code for area spectral efficiency in OFDMA or LTE. thanks
Krishna Sankar May 15, 2012 at 5:37 am
prashant maruti jadhav May 4, 2012 at 10:07 am
will anybody help me to implement SLM technique of PAPR reduction in matlab?
please do needfull if any one.
Adam Scott April 30, 2012 at 4:16 am
Hi Krishna,
Have you any work on 2×2 MIMO OFDM systems with rayleigh channel, for BPSK, QPSK, 8PSK and 16PSK?
Much appreciated
Krishna Sankar May 2, 2012 at 5:02 am
@Adam Scott: There are articles on MIMO discussed at http://www.dsplog.com/tag/mimo
But most of there are discussing using BPSK modulation as example. Extending this to higher order modulation seems reasonably straightforward though
Ratheesh March 8, 2012 at 10:02 pm
Hi,
I just need some help relating to data rate and bandwidth. I have a fixed bandwidth of transmission of 25KHz. So i want to calculate the data rates in different types of modulation to determine which one i can use. Can you help me with some formula with which i could calculate this.. Or any other better approach?
Thanks a lot
Krishna Sankar March 12, 2012 at 4:53 am
akash singh February 6, 2012 at 7:54 pm
sir i have to prepare a model and project on the comparison between BPSK and QPSK. can you please help me in making the model..or you can provide any information regarding this topic.
Krishna Sankar February 10, 2012 at 5:51 am
radhakrishna January 25, 2012 at 12:15 pm
sir, i am looking for QPSK Algorithms…
can you provide some details about few algorithms
Krishna Sankar January 26, 2012 at 6:20 am
@radhakrishna: Hope the post on QPSK symbol error rate helps
http://www.dsplog.com/2007/11/06/symbol-error-rate-for-4-qam/
lavanya January 20, 2012 at 10:10 am
i want the code for 8-qam.plese provide it to me
Krishna Sankar January 23, 2012 at 5:16 am
@lavanya: Am hoping that you will be able to develop the 8QAM symbol error rate code using the 16QAM case as reference.
http://www.dsplog.com/2007/12/09/symbol-error-rate-for-16-qam/
http://www.dsplog.com/2008/06/05/16qam-bit-error-gray-mapping/
Shareef Ahmed December 24, 2011 at 7:10 pm
I need mapping and demapping function commands in MATLAB code for OFDMA and SC-FDMA. will you please help me.
Krishna Sankar January 3, 2012 at 4:40 am
@Shareef: Is it modulation and demodulation mapping i.e. bits to constellation?
risky septiadi November 6, 2012 at 3:48 pm
please i need it too : you very kind man
Krishna Sankar November 12, 2012 at 7:05 am
@risky: emailed you the instructions
abdullah October 6, 2011 at 9:09 am
I would like to share in this web bcz it very use full
abd el rahman hussein May 18, 2011 at 8:45 pm
hi krishna
do u have a matlab simulation for 64 QAM that first generates random binary bits then map them to symbols instead of generating symbols directly ??
i have this code but it generates the symbols directly
Krishna Sankar May 23, 2011 at 2:38 am
@Abd: You can find a post on 16QAM BER with Grey coding
http://www.dsplog.com/2008/06/05/16qam-bit-error-gray-mapping/#Simulation%20Model
In that post, bits are generated and then converted to symbols.
racheal May 3, 2011 at 5:37 pm
Dear Krishna,
I just want to first thank you for your posts especially us students from un developed countries. Your website has taught us a lot and we do not feel inferior.
I was so bad at scripts but i picket interest and your work has greatly helped me. Thank you so much and May God bless you.
I want to kindly ask you for the matlab/ octave script for simulating BER vs SNR per bit (Eb/No) for various digital modulation schemes(Comparing BPSK, QPSK, 4PAM, 16QAM, 16PSK, 64QAM and 32PSK).
I was able to see your output from this website “http://www.eetimes.com/design/signal-processing-dsp/4017668/Modulation-roundup-error-rates-noise-and-capacity?pageNumber=2″ but i can not access the script.
I also want to ask you if you have examples of matlab/octave scripts for Sum capacity Vs SNR of MIMO or MU-MIMO with correlated and uncorrelated channel to help with some of them.
Otherwise am waiting to hear from you. Thank you
I can not say enough thank you for your posts on http://www.dsplog.com. If i were to give you a gift it would be ” to give you the ability to see yourself as others see you, then you could really see how special you are and how you have made a big difference in lives of many people”
Krishna Sankar May 24, 2011 at 5:35 am
@rachael: Thanks for the kind words. Will add a post on sum capacity soon
maruf al mahmud sajib November 14, 2010 at 10:15 am
i face a problem of BER formula for math lab simulation which is
BER=.5 erfc[sqrt(snr)/2]
i need to analyze the performance of a short distance terrestrial OWC system as a function of wavelength.
i need a help that BER formula is right or wrong.
Krishna Sankar November 14, 2010 at 10:18 am
@maruf: what is the modulation which is being used?
wep August 4, 2010 at 7:53 pm
hi krishna …..
hi sir …
Can you help me about tutorials from the m-QAM (m = 4,8,16,32,64),
about the theory, the calculation of bit error probability (BER) and the constellation of m-QAM..
thanks before……
Krishna Sankar August 10, 2010 at 4:57 am
Stud June 15, 2010 at 3:30 am
Sir,
can u give me the matlab code for the BER compersion of conventional BPSK, QPSK and 8PSK over an AWGN channel.
Regards
Krishna Sankar June 21, 2010 at 5:48 am
azura April 13, 2010 at 11:53 am
hello sir ..
i’m doing my final project..
do you have matlab simulation for ber performance in ofdm system of various modulation technique in self cancellation in OFDM..
Krishna Sankar April 14, 2010 at 4:34 am
@azura: Some posts discussing OFDM can be found @ http://www.dsplog.com/tag/ofdm
Hassan March 30, 2010 at 8:06 pm
sir,
please can you tell me how should I assign 2 bits to symbols for 4-PAM . i.e how shall I program such that -3 has two bits , similarly -1, 1 and 3??If you have a code, kindly is it possible to provide it?
Krishna Sankar March 31, 2010 at 5:03 am
@Hassan: You can choose your own assignment. Mostly people chose Gray coded mapping where adjacent constellation symbols differ by only one bit.
http://www.dsplog.com/2008/05/11/binary-to-gray-code-conversion-psk-pam/
http://www.dsplog.com/2008/05/12/gray-code-to-binary-conversion-for-psk-pam/
maria March 26, 2010 at 11:48 am
sir, pls do help wid our prjct..can u plsssssss send me the matlab alamouti STBC code for QPSK modulation in rayleigh fading channel?????
plssssssssssss……………..
Krishna Sankar March 28, 2010 at 1:41 pm
@maria: I have a written a post on STBC with BPSK at http://www.dsplog.com/2008/10/16/alamouti-stbc/
Also, there is a post on QPSK at http://www.dsplog.com/2007/11/06/symbol-error-rate-for-4-qam/
It should be reasonably easy for you to combine them
Md.Monirul Islam March 21, 2010 at 10:05 pm
Total probability of symbol error of the following Modulation technique:
CPSK, 2-PSK,M-PAM,2-DPSK,16APSK.
(with matlab coding).
Krishna Sankar March 28, 2010 at 1:53 pm
sh2010 March 1, 2010 at 4:43 pm
hi every one ineed your hellp in how to make symbol mapping function
or how to make bit mapping simmulation in mate lab.
with my thanksssss
Krishna Sankar March 30, 2010 at 4:38 am
@sh2010: you want to map bits into constellation points? Most people use Gray coded mapping where adjacent constellation symbols differ by only one bit. The post http://www.dsplog.com/tag/gray/ maybe of help
batman February 23, 2010 at 6:43 am
hello,can i know what is actually relation between baud and bandwidth?and is that baud rate in QPSK=(bit rate/4)?or same with bandwidth like others such as ASK,BPSK,FSKn and how about QAM?is that same as QPSK or what?..please help me
Krishna Sankar March 31, 2010 at 5:27 am
@batman: If the symbol duration is T, the minimum bandwidth required (with sinc shaped pulse shaping) is 1/T (from -1/2T to 1/2T). With higher order modulations, for the same data rate, we can have longer symbol durations there by reducing the bandwidth. Please lookinto the section Bandwidth requirements and Capacity in the article in dspdesignline.com http://www.dspdesignline.com/howto/208801783;jsessionid=BWJIYHNJDMGCPQE1GHOSKH4ATMY32JVN?pgno=2
shubrodeep February 10, 2010 at 12:06 pm
hi, i need some help please
i have simulated a qpsk tx and rx system. i got the BER curve perfectly. my next task was to have the qpsk signal pass thru an FIR filter. I did that as welland as expected the signal underwent distortion.
1. i should generate h(n), which is the impulse response. i have to generate 6 coefficients. i know to do that but i’m asked to generate them with variances of 1, 1/2, 1/3, 1/4, 1/5, 1/6.
i should also be normalised such that summation Hi square = 1.
2. i have to then convolute it with the input signal x(n).
3. then for every 500 samples i have to change the coefficients, should do this 30 times.
4. finally from the h(n) coefficients i should try and get bak the original signal.
coul some1 plz plz plz help with the following tasks listed above.. i would be greatful.
thank you.
regards,
shubro
Krishna Sankar April 4, 2010 at 4:07 am
Amit January 29, 2010 at 11:56 am
Hello Sir,
I wanna a code for QAM modulation Technique. Hope u will do this today
Krishna Sankar April 4, 2010 at 4:46 am
@Amit: Sorry, no
john January 12, 2010 at 3:46 pm
I am trying to simulate a rayleigh fading channel using the following code, i am not getting the desired resultls, where am i going wrong?
bit_error_rate=[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0];
for q=1:no
%%% data generation
data_get =data_gen(rate_id);
%%% data randomization
data_rand=randomizer(data_get);
%%% FEC ENCODER
data_rscoded=rsencodecod(data_rand,rate_id,10);
%%convolution encoder
data_coded=convolution(data_rscoded,rate_id,10);
%%% INTERLEAVING
data_interleav=interleav_d(data_coded,rate_id);
%%% Digital modulator SYMBOL MAPPER
data_mod=mod_d(data_interleav,rate_id);
%%% IFFT modulator
data_tx=ofdmsymbol_fft_cp(data_mod,G,10);
SNR=[1 2 3 5 7 9 10 12 15 17 20 22 25 27 30]; % specify SNR
for p=1:1:15
snr=SNR(p);
ts=1/22800000;
fd=0.4;
tau=[0 0.4e-6 0.9e-6];
pdb=[0 -5 -10];
chan=rayleighchan(ts,fd,tau,pdb);
%%% channel
data_fil=filter(chan,data_tx);
data_rx=channel_d(data_fil,snr);
%%% FFT demodulator
data_rxp=ofdmsymbol_fft_cp(data_rx,G,01);
%%% Digital demodulator SYMBOL DEMEPPER
data_demod=demod_d(data_rxp,rate_id);
%%% DEINTERLEAVING
data_deinterleav=deinterleav_d(data_demod,rate_id);
% %%% FEC DECODER
%% convolution decoder
data_decoded=convolution(data_deinterleav,rate_id,01);
%%% RSdecoder
data_rsdecoded=rsencodecod(data_decoded,rate_id,01); % removing added tail bits
%%% Data Derandomizer
data_unrand=randomizer(data_rsdecoded);
chowdary December 30, 2009 at 8:02 pm
Hi krishna,
I came across this site very late.
I am looking for ofdm transmission through SUI,Cost207,exponential channel models, i need matlab code, plzzzz help me it require for my project i need to submit with in 2days, how can i know the performance of these channaels.plzzzz help me i am waiting for you reply.
Communications Engineer December 23, 2009 at 6:43 pm
Hello Krishna,
Can you tell me how I can use BER to plot capacity curves for OFDM and MC-CDMA?
Any hint
wap December 18, 2009 at 12:01 pm
hi sir…………..
hi krishna………
what is pi/4 qpsk???
are u have tutorial pi/4 qpsk??
can u explain defferent qpsk n pi/4 qpsk?
thanks before……………
Krishna Sankar December 23, 2009 at 5:31 am
@wap: I have not discussed about pi/4 QPSK. You may find a brief wiki entry on it @
http://en.wikipedia.org/wiki/Phase-shift_keying#.CF.80.2F4.E2.80.93QPSK
BEHZAD December 13, 2009 at 5:27 am
hi
i want information about hybrid modulation “OFDM-OOK”
regards
Krishna Sankar December 22, 2009 at 5:33 am
shadat December 6, 2009 at 3:17 pm
hi,
i hope you are fine.please could you give me theoretical equaion of Bit Error Rate for convolutioanly coded BPSK,QOSk,16QAM,64 QAM and simulation of Adaptive modulation of BPSK,QPSK,16QAM,64QAM?
thnaks.
Krishna Sankar December 7, 2009 at 5:29 am
@shadat: For BPSK with convolutional coding with hard/soft Viterbi, please refer to
http://www.dsplog.com/tag/viterbi
Student November 3, 2009 at 11:09 pm
Why did you delete my comment?
Krishna Sankar November 8, 2009 at 8:49 am
@Student: The comments are moderated. Hence they take ‘time to appear’
Student November 3, 2009 at 9:03 pm
Hello,
in my lecture I learned that the relation between S/N and Eb/N0 is:
S/N = Eb/N0 * m
m: Number of bits/Symbol
Now I found the formula for the relation between Es/N0 and Eb/N0:
Es/N0 = Eb/N0 * m
because there are m bits/Symbol.
That would mean that
S/N = Es/N0.
In another pdf I found:
C/N = S/N = Eb/N0 * Rb/B
so
Es/N0 = Eb/N0 * Rb/B
so
Es = Eb * Rb/B
so
m * Eb = Eb * Rb/B
so
m = Rb/B
that makes sense for me.
But: In all books, B = B_baseband = f3dB of the lowpass in baseband. In my lecture, B is B_RF = 2 * B_baseband.
I need clearly the relation between S/N and Eb/N0 to calculate S/N_min for a 802.11a receiver with the given PER_max = 10% (=> BER= 1,32E-5).
Can someone please confirm my calculations or show me the errors?
Thank you very much!
Student
Krishna Sankar November 8, 2009 at 8:48 am
@Student: Sorry, I got confused reading through the comment. However, you may want to review the SNR definition used in the post
http://www.dsplog.com/2008/06/10/ofdm-bpsk-bit-error/
Good luck
EGRUE NNAMDI October 30, 2009 at 5:33 pm
Hi Krishna
Compliments of the season, actually i have made a request to you last time and you granted the request by given me the link to which my concerns where addressed. Sincerelly speeking all your your work are very lovely, and very educative, infact i would like to work with people like you, because you have been very helpful. I want to “develop a model in my dissertation for variable noise channel and simulate results for bit error rate (BER)”i will need your assisitance on how i can do that. then below are some of the terms you have used in your simulations, pls i will need an explanation for each one of them, what they mean. Am very gratefull.
ipBit =
ipMod =
xF =
xt =
nt =
yt =
yF =
yMod =
ipModHat =
ipBitHat =
Concatenating =
Thanks Very much
NNAMDI
Krishna Sankar November 8, 2009 at 8:26 am
@EGRUE: Thanks for the comment. I think that the code is self explanatory. You can take a shot at explaining each. I will correct them as needed.
Selvi Rajan October 13, 2009 at 12:25 pm
Sir, Iam working on OFDM simulation for Optical domain. I want to know the parpameter Calculation for OFDM, i.e Frequency Offset calculation , phase noise error , Derving different parameter from the IEEE standards
Krishna Sankar October 15, 2009 at 5:24 am
@Selvi Rajan: I have discussed some aspects pertaining to OFDM @ http://www.dsplog.com/tag/ofdm/
Hope this helps.
alex October 11, 2009 at 8:06 pm
Hi,Krishna Sankar；
How are you!I hope you are doing fine and in good health.
I write a code for phase noise simulation in Butter ,cheby1,cheby2,and ellip fillter.But the programe run also error.Please give me help!Thanks a lot.
[Bb,Ab] = butter(4,0.5); % order 4, cutoff at 0.5 * pi
Hb=freqz(Bb,Ab);
Db=grpdelay(Bb,Ab);
[Bc1,Ac1] = cheby1(4,1,0.5); % 1 dB passband ripple
Hc1=freqz(Bc1,Ac1);
Dc1=grpdelay(Bc1,Ac1);
[Bc2,Ac2] = cheby2(4,20,0.5); % 20 dB stopband attenuation
Hc2=freqz(Bc2,Ac2);
Dc2=grpdelay(Bc2,Ac2);
[Be,Ae] = ellip(4,1,20,0.5); % like cheby1 + cheby2
He=freqz(Be,Ae);
[De,w]=grpdelay(Be,Ae);
figure(1); plot(w,abs([Hb,Hc1,Hc2,He])); grid(‘on’);
legend(‘butter’,'cheby1′,’cheby2′,’ellip’);
saveplot(‘../eps/grpdelaydemo1.eps’);
figure(2); plot(w,[Db,Dc1,Dc2,De]); grid(‘on’);
legend(‘butter’,'cheby1′,’cheby2′,’ellip’);
saveplot(‘../eps/grpdelaydemo2.eps’);
Krishna Sankar October 12, 2009 at 5:42 am
@alex: I do not have the function grpdelay() with me. What is the error which you are observing?
invizi October 9, 2009 at 2:03 pm
Hi krishna!
I hope you are doing fine and in good health.
I am working on an OFDMA system. I want to write equation for channel capacity for each user. I have the equations for the ICI but I dont know how to write the capacity equations…
______________________________________________
C(k) = Summation [ log2(1+SNR(v))] in bits/OFDM-Sym.
where k = user index
v = subcarrier allocated to a user
—————————————————————————–
I know this equation which is the theoretical channel capacity of each user. But I want to write the actual channel capacity. As I said I have the ICI equations …. kindly help how to proceed.
invizi
Krishna Sankar October 12, 2009 at 5:38 am
@invizi: You can find the theoretical capacity from Shannon’s bounds, no?
W.Y September 3, 2009 at 8:26 am
Sir!
I was wondering how I can implement the channel capacity of BPSK, using by Matlab???
Krishna Sankar September 8, 2009 at 5:49 am
@W.Y: For the post on BER for BPSK modulation, we know the Eb/No required for achieving an arbitrarily low BER of 10^-5.
Further, since we know the bandwidth required by BPSK, we know the capacity in bits/second/Hz is 1.
Knowing both these values, we can plot where BPSK lies in Shanon’s Eb/No vs capacity curve.
Please refer to the following posts:
1) http://www.dspdesignline.com/howto/208801783;jsessionid=GBNINTPPYCUNFQE1GHRSKH4ATMY32JVN?pgno=2
2) http://www.dsplog.com/2008/06/18/bounds-on-communication-shannon-capacity/
Hope this helps.
Ranjan September 2, 2009 at 3:49 pm
Hi,
I got prob of error formulas for QPSK as follows,
1) From std result: Pe=erfc(sqrt(Es/2N0))
2)Pe= erfc(sqrt (1.5*snr/M-1))
3)Pe=4Q(dmin/2*sigma)
I am not able to relate all thes three.
Can you help me.
Thanks
ranjan
Krishna Sankar September 8, 2009 at 5:21 am
@Ranjan: Let me try:
For QPSK, M = 4, so erfc(sqrt (1.5*snr/3)) = erfc(sqrt (snr/2)) = erfc(sqrt(Es/2N0)). So (1) and (2) are the same.
Since Q(x)=0.5*erfc(x/sqrt(2)), dmin = sqrt(2); then 4Q(dmin/2*sigma) = 2erfc(1/(sqrt(2)*sigma) = 2erfc(sqrt(1/(2*sigma^2))).
If we assume that variance of signal is 1, and the variance of noise = sigma^2, then 1/sigma^2 = Es/N0 = SNR
The function (3) seems close to (1), (2) except for the scaling of 2. Are you sure that QPSK error rate is 4Q(dmin/2*sigma) and not 2Q(dmin/2*sigma)?
Does this help?
kiki September 1, 2009 at 7:35 pm
hello all. i found a code which modulates QPSK but i couldn’t demodulate it back . please help. here is the code
clear all
clc
f=100;
N=5;
b= sign(randn(1,N));
b;
c=rem(length(b),2);
if c ~= 0
b=[b 1];
end
len=length(b);
t=0:1/99:1;
qcar=[];icar=[]; %initialize in-phase and quadrature carriers
ic=[];qc=[]; %initialize in-phase and quadrature components;
for n=1:2:len
if b(n)==-1 & b(n+1)==-1;
temp=-1/sqrt(2)*ones(1,100);
temp1=-1/sqrt(2)*ones(1,100);
elseif b(n)==-1 & b(n+1)==1;
temp=-1/sqrt(2)*ones(1,100);
temp1=1/sqrt(2)*ones(1,100);
elseif b(n)==1 & b(n+1)==-1;
temp=1/sqrt(2)*ones(1,100);
temp1=-1/sqrt(2)*ones(1,100);
elseif b(n)==1 & b(n+1)==1;
temp=1/sqrt(2)*ones(1,100);
temp1=1/sqrt(2)*ones(1,100);
end
i=cos(2*pi*f*t);
q=sin(2*pi*f*t);
ic=[ic temp];
qc=[qc temp1];
icar=[icar i];
qcar=[qcar q];
end
qpsk=ic.*icar+qc.*qcar;
plot(qpsk);
Krishna Sankar September 7, 2009 at 5:39 am
Girish K. July 27, 2009 at 4:18 pm
sir i am working on hiperlan/2 using BPSK,QPSK and 16QAM modulation technique and ofdm multiplexing so plz help me to solve the prob for delay and Rician and Rayleigh channel
Krishna Sankar July 28, 2009 at 4:36 am
@Girish: Can you please provide more details about the problems which you are facing.
Jasdeep July 23, 2009 at 1:39 am
Hi
i need information or matlab code for filter to reduce or to compensate fading.
I need to put this code for nakagami channel……
Thanks
Krishna Sankar July 24, 2009 at 4:13 am
@Jasdeep: I believe you are looking for a channel equalization technique. I have not discussed channel equalization in a multipath channel (resulting in ISI) for single carrier system, however you may find articles on flat fading Rayleigh channel @ http://www.dsplog.com/tag/rayleigh/
simon July 8, 2009 at 8:33 pm
Hi,
I wonder if you know anything about BER for OOK.
either the figure or the theoretical function is enough for me.
thanks
Krishna Sankar July 15, 2009 at 4:53 am
@simon: Given that OOK is orthogonal signaling (and not anti-podal signaling as BPSK), I would guess that OOK keying would have a BER which is 3dB poorer than the BER for BPSK. The BER for BPSK is discussed in the post
http://www.dsplog.com/2007/08/05/bit-error-probability-for-bpsk-modulation/
Hope this helps.
JAFAR July 6, 2009 at 9:28 pm
I need to BER CURVED versuse Eb/N0 for qpsk modulation up to 1e-6.
Krishna Sankar July 15, 2009 at 4:40 am
@Jafar: There is an article discussing symbol error rate for QPSK.
http://www.dsplog.com/2007/11/06/symbol-error-rate-for-4-qam/
You might also want to look @
http://www.dspdesignline.com/howto/208801783;jsessionid=3ISGUXHINOVIAQSNDLRSKHSCJUNN2JVN?pgno=3
Hope this helps.
alger May 24, 2009 at 3:03 am
Hi,
please, comment faire pour ploter la constullation 16-qam dans matlab avec des équations mathématique du signal de sortie de modulateur.
Thenks
Krishna Sankar May 31, 2009 at 8:12 pm
@alger:
Just a quick Matlab code snippet to get you going:
N = 100;
alpha16qam = [-3 -1 1 3]; % 16-QAM alphabets
ip = randsrc(1,N,alpha16qam) + j*randsrc(1,N,alpha16qam);
plot(real(ip),imag(ip),’.');
You may also look @ http://www.dsplog.com/2007/12/09/symbol-error-rate-for-16-qam/
Abi May 18, 2009 at 11:59 pm
Hi all,
please, how can we plot Shannon’s curve against Eb/No for various rates ?? I mean simlate them ??
Many thanks
Krishna Sankar May 20, 2009 at 5:44 am
@Abi: Maybe the figures in the post
http://www.dsplog.com/2008/06/18/bounds-on-communication-shannon-capacity/
might be of help.
Adeela April 21, 2009 at 7:11 pm
hi
kindly send me the complete derivation expression for the probability of a bit error of 4-QAM in terms of Eb n No
also the gray maping constellation of 4 QAM
thanx
Krishna Sankar April 30, 2009 at 5:01 am
@Adeela: You may refer to the posts
http://www.dsplog.com/2007/11/06/symbol-error-rate-for-4-qam/ for discussion on Symbol Error rate (Ps) versus symbol to noise ratio (Es/N0).
To convert to Pb versus Eb/N0, please make the following assumptions:
Since each constellation symbol carriers two bits, Eb/N0 = 1/2*Es/N0
Further if we assume Gray coded constellation, each symbol error typicallly causes 1 out of 2 bits to be in error
Pb ~= Ps/2.
Hope this helps.
haleh December 23, 2009 at 1:48 pm
Hi Krishna,
you mentioned Pb ~= Ps/k …is it versus Eb/No or Es/No?
NOOR April 19, 2009 at 8:46 am
. MIMO-OFDM: VBLAST versus STBC
The objective here is to compare VBLAST and Alamouti STBC in the context of MIMO-OFDM operating over frequency-selective Rayleigh fading channels. Consider a 2×2 system with N=64 carriers and a cyclic prefix long enough to avoid interblock interference. QPSK is used in STBC and BPSK is used in V-BLAST in order to have the same spectral efficiency. The discrete-time channels are assumed to be mutually uncorrelated and have L taps each. The taps are uncorrelated and obey an exponential power delay profile, i.e. E{|hk|2} = C.exp(- k) where C is a constant; take =0.2. It is assumed that the channel does not vary over two OFDM symbols. Assuming perfect knowledge of the channels at the receiver, provide simulations results depicting the average BERs for the two systems versus the average SNR in the cases where L=1, L=4, L=8, and L=16. Comment on the obtained results.
as soon as you can
Krishna Sankar April 25, 2009 at 7:09 am
@NOOR: The following three posts might provide you the backgroud material to solve this problem:
(a) Alamouti STBC
http://www.dsplog.com/2008/10/16/alamouti-stbc/,
(b) MIMO with Zero Forcing Equalizer
http://www.dsplog.com/2008/10/24/mimo-zero-forcing/ and
(c) BER for BPSK in OFDM with a 10 tap Rayleigh channel
http://www.dsplog.com/2008/08/26/ofdm-rayleigh-channel-ber-bpsk/
Good luck.
NOOR April 19, 2009 at 8:41 am
1-Why did we map the uniform random integers onto the complex symbols in 64 QAM simulations?
2-Why does increased transmitted power decrease the error rate of the system ?
3-How does this equation D(r^t,sm)=abs(r^t-sm)^2 become optimum detector ?
Krishna Sankar April 21, 2009 at 5:44 am
@NOOR: My replies:
1. The transmit data might be coming from some higher layer, which we assume to be random. So we use rand() function to define random bits, group them and assign to constellation points.
2. The effect of noise becomes smaller as we increase the transmit power
3. I believe you wrote the equation of Maximum likelihood, correct?
Melinda March 19, 2009 at 4:30 pm
Hi,
QPSK has – 4 constellation points; 16 QAM -16 constellation points; 64 – 64 constellation points.
16QAM vs QPSK -> Now with the 16 possible points in the constellation diagram we have 16 possible symbols. For this 16 symbols we need 4 bits for coding . Compared to the QPSK modulation now we have doubled the transfer rate when using the same symbol clock.But compared to the QPSK, points are closer to each other and so the allowed noise circle radius is decreased. Noise can be interpreted as a vector which turns around the points of the constellation diagram producing a circle with a noise amplitude dependend radius. If we have normalized constellation (i.e. max distance is 1 – i.e. distance between two outer edges of circle of outer constellation points), then:
required distance QAM16 -> 1/(SQRT(QAM res.)) = 1/SQRT(16) = 0.25
required distance QPSK -> 1/(SQRT(QPSK res.)) = 1/SQRT(4) = 0.5
So we get a ratio of QPSK/QAM distance required to get no overlap of the points of: 20 * log(o.5/0.25) = 6dB.
So to have the same S/N ratio with the same noise level the signal for 16QAM has to be 6 dB stronger than QPSK. The same case is between 64QAM and 16 QAM(6db difference), 256QAM and 64QAM(6db difference) ….
We see that with each bit we add to the symbol rate we need a 3 dB better S/N of the received signal. So we see the tradeoff we have to do between the increasing transfer rate and the required S/N ratio.
Of course another way to increase the transfer rate is a higher symbol clock with the same symbol width.
===================================================================
But, I saw also formula(in MATLAB help) Es/No = Eb/No + 10log10(k); where k is the number of information bits per symbol.
By this formula we get:
QPSK: Es/No = Eb/No + 10log10(2)=Eb/No + 3.0103;
16QAM: Es/No = Eb/No + 10log10(4)=Eb/No + 6.0206;
64QAM: Es/No = Eb/No + 10log10(6)=Eb/No + 7.7815;
For BER of 10e-5, we get for QPSK that required Eb/No is about 9.5 dB, for 16QAM is about 13dB, for 64QAM is about 17,5dB…(I get this by MATLAB bertool). So we now have:
QPSK: Es/No = Eb/No + 10log10(2)=9.5 + 3.0103 ~ 12.5 dB;
16QAM: Es/No = Eb/No + 10log10(4)=13 + 6.0206 ~ 20 dB;
64QAM: Es/No = Eb/No + 10log10(6)=17.5 + 7.7815 ~ 26.2 dB;
So as You see the difference between 12.5 and 20 is not 6dB, in case QPSK vs 16QAM!
(PS: I think that this is a very important line: “So to have the same S/N ratio with the same noise level the signal for 16QAM has to be 6 dB stronger than QPSK.” – in AN97047.pdf)
So what is correct? Is this the same thing I try to correlate or…
If so, why is that?
Thanks and best regards
Krishna Sankar March 21, 2009 at 4:32 pm
@Melinda: For sure, that was a long comment.
If I may summarize, the jist of the questions is: Should the difference between QPSK and 16-QAM be 6dB (as claimed in NXP paper) OR around 7.5dB (as seen from Mathworks simulations).
To do the comparison, may I suggest using the average distance approach. The average distance between
constellation points in
(a) QPSK is 2/sqrt(2)
(b) 16-QAM is 2/sqrt(10)
For details on why the term 1/sqrt(2) and 1/sqrt(10) are present, please refer to the post on scaling factor in QAM
http://www.dsplog.com/2007/09/23/scaling-factor-in-qam/
It is reasonable to guess that as average distance reduces, the error rate increases. Doing a relative comparison,
diff in dB = 20*log10((2/sqrt(2))/(2/sqrt(10) )) = 7dB. This means that, for achieving the same symbol error rate, 16-QAM requires 7dB more Es/No than QPSK.
And this what I observed, when I performed simulations comparing all modulation schemes. Click here for the article in dspdesignline.com.
Hope this helps.
Allyson March 16, 2009 at 8:11 am
Hi Krishna,
What is the exact formula for BER QPSK (Theoretical). I saw that you have SER for QPSK which i think differ from BER right?
Krishna Sankar March 21, 2009 at 8:22 am
@Allyson: BER for QPSK is same as BER for BPSK for a given value of Eb/N0. You may look at the following article in dspdesignline.com for reference.
Ranjan December 16, 2008 at 2:20 pm
Hi,
I am in seach of BER formula for QPSk OFDM.may I get some help from you.
thanks
Krishna Sankar December 17, 2008 at 6:18 am
@Ranjan: BER for QPSK in OFDM in AWGN should be same as BER for QPSK in AWGN. I have written posts on BER for BPSK in OFDM in AWGN
URI: http://www.dsplog.com/2008/06/10/ofdm-bpsk-bit-error/
and one on BER for QPSK in AWGN
http://www.dsplog.com/2007/11/06/symbol-error-rate-for-4-qam/
Hope this helps.
Krishna Sankar November 25, 2008 at 6:09 am
@krishna kant: It is a bit difficult to comment based on your given observation. From the Chapter 8 of Digitial Communication Proakis, I can see that k=7 should have the lowest BER for a given Eb/No, then k = 5 and worst ber is for k=3.
Maybe you can have a look at the figures in the text book and compare your curves against the text book curves.
All the best.
krishna kant November 23, 2008 at 6:44 pm
sorry ,the errors had increased as k increased.so why is this
krishna kant November 23, 2008 at 6:43 pm
hello sir,i have a case where i encode a message with convolutional coder and puncture it , map it to 16 qam and in the receiver- demodulate it using hard decisions , depuncture and perform viterbi decoding. this was done with constraint lengths k=3,5,7 and corresponding generator polynomials (5,7) ; (23,35) and (171,133).i plotted the BER vs SNR curves (with SNR considered from 2 to 12).The performance of the configuration had improved as k increased. why is it so? plase reply sor
Krishna Sankar November 9, 2008 at 6:52 pm
@lealem: I have not written posts for BER with 64QAM. However, I have written one on BER for 16QAM
URI: http://www.dsplog.com/2008/06/05/16qam-bit-error-gray-mapping/
I think you should be able to adapt the code to handle 64QAM case. Good luck.
sajjad June 8, 2009 at 2:42 am
hi krishna
i wana matlab code for adaptive modulation using BPSK,QPSK and QAM.also need the plot of spectral efficiency vs SNR for adaptive modulation.
Krishna Sankar June 8, 2009 at 6:04 am
@sajjad: Though I do not have the code for adaptive modulation, I guess it should be reasonably easy for you to adaptively change from BPSK to QPSK to QAM as the Eb/N0 increases, based on a minimum BER constraint. You may look @ the post comparing BPSK, QPSK, M-PSK, M-QAM etc as reference
http://www.dsplog.com/2008/07/08/compare-bpsk-qpsk-4pam-16qam-16psk-64qam-32psk/
Hope this helps.
sara zaalik March 31, 2012 at 2:18 pm
hello Krishna, i want a matlab code that simulates the BER of 4×1 Extended Alamouti Space Time Block Coding (EASTBC), with ZF and ML recievers, the channel is rayleigh .. thanks alot and waiting for your reply.
Sara
Krishna Sankar April 1, 2012 at 5:44 am
@sara: Sorry, I have not tried modeling EASTBC
lealem November 7, 2008 at 11:17 am
hello! Krishina how are you today? i would like you to ask a help concerning 64-QAM modulation technique. i was doing a matlab simulation for 64-QAM OFDM but i didn’t get the exact output, do you have a matlab script which is used to simulate performance(BER Vs SNR) of 64-QAM modulation technique. hope u will send it to me this afternoon.
Bye….
|
# $\sqrt{x}$ isn't Lipschitz function
A function f such that $$|f(x)-f(y)| \leq C|x-y|$$
for all $$x$$ and $$y$$, where $$C$$ is a constant independent of $$x$$ and $$y$$, is called a Lipschitz function
show that $$f(x)=\sqrt{x}\hspace{3mm} \forall x \in \mathbb{R_{+}}$$ isn't Lipschitz function
Indeed, there is no such constant C where $$|\sqrt{x}-\sqrt{y}| \leq C|x-y| \hspace{4mm} \forall x,y \in \mathbb{R_{+}}$$ we have only that inequality $$|\sqrt{x}-\sqrt{y}|\leq |\sqrt{x}|+|\sqrt{y}|$$ Am i right ?
remark for @Vintarel i plot it i don't know graphically "Lipschitz" mean? what is the big deal in the graph of the square-root function
in wikipedia they said
Continuous functions that are not (globally) Lipschitz continuous The function f(x) = $$\sqrt{x}$$ defined on [0, 1] is not Lipschitz continuous. This function becomes infinitely steep as x approaches 0 since its derivative becomes infinite. However, it is uniformly continuous as well as Hölder continuous of class $$C^{0,\alpha}$$, α for $$α ≤ 1/2$$. Reference
1] could someone explain to me this by math and not by words, please ??
2] what does "Lipschitz" mean graphically?
• If your question is whether you have proven that $\sqrt x$ is not a Lipschitz function, then the answer is no, you haven't. – Omnomnomnom Feb 7 '14 at 13:32
• Remark: $\sqrt{}$ is certainly not defined for $x\in\mathbf R$, only for $x\geq0$. – Tom-Tom Feb 7 '14 at 13:32
• @Omnomnomnom and V.Rossetto Thanks to you both bu how can i prove it exactly – Adam Feb 7 '14 at 13:34
• Could you see it by plotting the curve of $\sqrt(x)$? Do you graphically know what "Lipschitz" mean? – Vintarel Feb 7 '14 at 14:27
• @Adam, brief answer : $f$ is Lipschitz if $|f(x) - f(y)|\le C|x-y|$, so you get $$\left|\frac{f(x) - f(y)}{x-y}\right|\le C,$$ you HAVE TO recognize the left hand side term as a difference quotient (or a growth of rate) of a function, or graphically the slope of the line joining $(x,f(x))$ and $(y, f(y))$. Thus $f$ is Lipschitz if all the secant lines are of bounded slope (is it clear?) – Vintarel Feb 8 '14 at 0:00
Hint: why is it not possible to find a $C$ such that $$|\sqrt{x} - \sqrt{0}|\leq C|x-0|$$ For all $x \geq 0$?
As a general rule: Note that a differentiable function will necessarily be Lipschitz on any interval on which its derivative is bounded.
In response to the wikipedia excerpt: "This function becomes infinitely steep as $x$ approaches $0$" is another way of saying that $f'(x) \to \infty$ as $x \to 0$. If you look at slope of the tangent line at each $x$ as $x$ gets closer to $0$, those tangent lines become steeper and steeper, approaching a vertical tangent at $x = 0$.
"Graphically", we can say that a differentiable function will be Lipschitz (if and) only if it never has a vertical tangent line.
Some functions that are not Lipschitz due to an unbounded derivative: $$f(x) = x^{1/3}\\ f(x) = x^{1/n},\quad n = 2,3,4,5,\dots$$ A more subtle example: $$f(x) = x^2,\quad x \in \mathbb{R}\\ f(x) = \sin(x^2), \quad x \in \mathbb{R}$$ Note in these cases that although $f'(x)$ is continuous, there is no upper bound for $f'(x)$ over the domain of interest.
• i can't see why please ? guz we have $|\sqrt{x}|\leq |x|$ holds true for $\forall x \in \mathbb{R}_{+}$ with $c=1$ – Adam Feb 7 '14 at 13:37
• What if $x<1$? Try $x = 1/4$. Is $\sqrt x < x$? – Omnomnomnom Feb 7 '14 at 13:49
• It is differentiable on $(0,1)$, but the derivative is unbounded. But yes, that is what you should look for in these problems. – Omnomnomnom Feb 7 '14 at 14:37
• See my latest edit. – Omnomnomnom Feb 7 '14 at 15:20
• – Omnomnomnom Feb 7 '14 at 15:36
You have $${\sqrt{1/n} - \sqrt{0}\over{1/n - 0}} = {1/\sqrt{n}\over {1\over n}} = \sqrt{n}.$$ This ratio can be made as large as you like by choosing $n$ large. Therefore the square-root function fails to be Lipschitz.
• @ ncmathsadist Thanks why make us to say the square-root function fails to be Lipschitz based on that ratio could you be more specific ?? – Adam Feb 7 '14 at 14:16
Suppose that $\sqrt{x}$ is a Lipschitz function, then there exists $C$ such that
$$\Big|\frac{\sqrt{y}-\sqrt{x}}{y-x}\Big| \le C$$
Now, Let $y=2x$, so
$$(\sqrt{2}-1)x^{-\frac{1}{2}}\le C$$
Letting $x→0$ gives a contradiction.
• @ Sepideh Bakhoda Thanks i see your point but i'm trying to see others – Adam Feb 7 '14 at 14:08
$x \geq y$ implies $\sqrt{x} \geq \sqrt{y}$ (monotonous)
Then you need $\sqrt{x} - \sqrt{y} \leq C(x-y)$ for $x \ge y$
Use $a^2 - b^2 = \left(a-b\right)\left(a+b\right)$ to divide both sides by the (positive) $\sqrt{x} - \sqrt{y}$ to get $1 \leq C(\sqrt{x} + \sqrt{y})$, or $C \geq \frac{1}{\sqrt{x} + \sqrt{y}}$. Obviously the frac diverges as $(x,y)$ approaches $(0,0)$ so there is no upper bound $C$ to satisfy the requirement.
$\sqrt{}$ is monotonous, so just assume $x \geq y$, then you can drop the absolute values and it simplifies to $1 \leq C(\sqrt{x} + \sqrt{y}$. Since you can make the sum of square roots arbitrarily small (by suitably decreasing $x$ and $y$), as soon as it's smaller than $1/C$ the inequality no longer holds.
• Thanks but can you explain with math not with words i can't see your point ? – Adam Feb 7 '14 at 14:08
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# Integration factor - First Order Nonlinear ODE
I can't seem to find the proper integrating factor for this nonlinear first order ODE. I have even tried pulling a bunch of substitution and equation-manipulating tricks, but I can't seem to get a proper integrating factor.
$$\frac{1}{x}dx + \left(1+x^2y^2\right)dy = 0$$
EDIT: Due to MSE users complaining about my lack of proof of work, intent of conceptual understanding, etc, here is exactly why I am stuck.
To start off, this ODE is obviously inexact:
$$\frac{\partial}{\partial y}\left(\frac{1}{x}\right) \neq \frac{\partial}{\partial x}\left(1+x^2y^2\right)$$
And so in order to make this exact (if we choose to go down this route) we must (I'll stick to standard convention/notation) find a function $$\mu$$ such that if we multiply the entire original ODE by it, we will be able to integrate and solve using 'exact ODE' methods. This is shown as:
$$\mu \left(\frac{1}{x}\right)dx + \mu \left(1+x^2y^2\right)dy = 0$$
$$\frac{\partial}{\partial y} \left(\mu\left(\frac{1}{x}\right) \right) = \frac{\partial}{\partial x} \left(\mu \left(1+x^2y^2\right) \right)$$
Now expanding by chain rule, we get:
$$\mu_y \left(\frac{1}{x}\right) = \mu_x \left(1+x^2y^2\right) + \mu \left(2xy^2\right)$$
Now here is where I'm stuck. We want to avoid dealing with a PDE, so we try to stick to good old ODE techniques by assuming that $$\mu$$ is either a function of only x or only y. Let's first assume that $$\mu$$ is only a function of y. The following will then be true.
$$\mu_x = 0$$
$$\mu_y \left(\frac{1}{x} \right) = \mu \left(2xy^2 \right)$$
$$\frac{d\mu}{\mu} = 2x^2y^2 dy$$
By looking at the right hand side, we see that it just won't work - x and y are related, so we can't have that integral.
Now let's assume that $$\mu$$ is only a function of x. The following will then be true.
$$\mu_y = 0$$
$$\mu_x \left(1+x^2y^2\right) = -\mu \left(2xy^2\right)$$
$$\frac{d\mu}{\mu} = \frac{-2xy^2}{1+x^2y^2} dx$$
And, once again, if you look at the right hand side, we have an integral that we can't immediately work out, just as in the previous case.
• Since I've downvoted your question other users started to downvote it too(Perhaps this the culture of MSE) and even silently. I've downvoted your question because of these reasons:meta.math.stackexchange.com/q/13759/103816 I will reconsider my vote once I become sure about the policies of MSE. – user103816 May 19 '14 at 4:23
• '+1', This is all I can say now. – user103816 May 19 '14 at 14:39
• "Integrating factor" is from the thesaurus of misconceptions. The real problem is: Here is an ODE; what could we try to arrive at an explicit solution? – Christian Blatter Nov 7 '14 at 16:39
• It's a Bernoulli by x.. – Dor Apr 1 '15 at 16:55
It looks to me like we can't find an integrating factor which depends only on $x$ or only on $y$ (in general, $\mu$ will be a function of only one variable just in some special cases, so this is not entirely surprising).
For any equation of the form $p(x,y)dx + q(x,y)dy = 0$, in order to be able to find $\mu := \mu(x)$ it must be the case that $$\frac{\frac{\partial p}{\partial y} - \frac{\partial q}{\partial x}}{q}$$ is a function of $x$ only. If it is, we can set $$\mu(x) = \exp\left(\int\frac{\frac{\partial p}{\partial y} - \frac{\partial q}{\partial x}}{q}dx\right).$$ Here I get $$\frac{\frac{\partial p}{\partial y} - \frac{\partial q}{\partial x}}{q} = \frac{-2xy^2}{1 + x^2y^2},$$ which clearly depends on $y$. Then we can't have a $\mu$ which depends only on $x$. To have a $\mu$ which depends only on $y$, we must have $$\frac{\frac{\partial q}{\partial x} - \frac{\partial p}{\partial y}}{p}$$ be only a function of $y$. If that's the case, we can set $$\mu(y) = \exp\left(\int \frac{\frac{\partial q}{\partial x} - \frac{\partial p}{\partial y}}{p} dy\right).$$ Here I get $$\frac{\frac{\partial q}{\partial x} - \frac{\partial p}{\partial y}}{p} = \frac{2xy^2}{\frac{1}{x}} = 2x^2y^2,$$ which clearly depends on $x$. Then we can't have $\mu$ depend on $y$ only.
As a result, we must have $\mu$ depend on both $x$ and $y$.
• I agree with your response, but then why would this problem be at the end of the chapter that introduces solving inexact equations via an integrating factor? (It's okay if you don't know) – Arturo don Juan Nov 8 '14 at 7:34
Assume you're looking for a solution $y(x)$, then mathematic gives:
$$\frac{-1}{x + x^3 y^2} = \frac{ dy}{dx} \implies \frac{\exp (-2 y)}{8 x^2} + \frac{ \exp (-2y) }{16} (2 y^2 +2y +1) = Const$$
• Jeb, your answer doesn't help at all! I'm interested in how it is done (i.e how does the integrating factor look/come out), not just the plain old solution. – Arturo don Juan May 20 '14 at 23:45
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## When are the eigenspaces of the Laplacian on a compact homogeneous space irreducible representations?
I was writing up some notes on harmonic analysis and I thought of a question that I felt I should know the answer to but didn't, and I hope someone here can help me. Suppose I have a compact Riemannian manifold $M$ on which a compact Lie group $G$ acts isometrically and transitively---so you can think of $M$ as $G/K$ for some closed subgroup $K$ of $G$. Then the real Hilbert space $H = L^2(M, R)$ is an orthogonal
representation space of $G$ and hence splits as an orthogonal direct sum of finite dimensional irreducible sub-representations. On the other hand, the Laplacian $L$ of $M$ is a self-adjoint operator on $H$, so $H$ is also the orthogonal direct sum of its eigenspaces---which are also finite dimensional. My question is, when do these two orthogonal decompositions of $H$ coincide? Put slightly differently, since $L$ commutes with the action of $G$, each eigenspace of $L$ is a finite dimensional subrepresentation of $H$ and so a direct sum of irreducibles, and I would like to know conditions under which each eigenspace is in fact irreducible. For example, this is true for the circle acting on itself and for $SO(3)$ acting on $S^2$ (where we get the harmonic polynomials of various degrees). Is it perhaps always true for the case of a symmetric space? Of course a standard reference in addition to the answer would be most welcome.
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Since $L$ is a $G$-invariant operator, doesn't Schur's lemma tell us that $L$ acts on each irrep appearing in $H$ by scalar multiplication? – Faisal Nov 21 2010 at 7:50
@Faisal: This says that each irreducible is a sub-representation of some eigenspace, but it doesn't say that the an eigenspace could not contain several irreducibles. – Dick Palais Nov 21 2010 at 8:06
@Dick: Ah, sorry -- I misinterpreted your question. I agree with Evan that it's very rare to have all the eigenspaces of $L$ be irreducible. For example if $M=G$, then an eigenspace of $L$ contains a given irrep only if it contain all copies of that irrep in $L^2(G)$. It follows that if all the eigenspaces of $L$ are irreducible then every irrep of G must appear without multiplicity in $L^2(G)$. In particular, because each irrep occurs with multiplicity equal to its degree, this means that if $G$ isn't abelian then there is at least one eigenspace of $L$ that isn't irreducible. – Faisal Nov 21 2010 at 9:03
The Peter-Weyl theorem tells you that $L^2(G)$ is isomorphic to $\bigoplus_{\pi}\pi\otimes\pi^*$ as $G\times G$ representation, where $\pi$ runs through all irreducible unitary representations. It follows that $$L^2(G/K)\cong L^2(G)^K\cong\bigoplus_\pi \pi\otimes(\pi^*)^K.$$ So, the first thing you absolutely need, is a multiplicity one property, which says that $\dim\pi^K\le 1$ for every $\pi$. This is already a rare property, but known to be true for, say $G=SO(n)$ and $K=SO(n-1)$, see Zhelobenko's book for this. But, the Laplacian may have the same eigenvalue on different representations. For this you need highest weight theory (see for instance the book by Broecker and tom Dieck): Assume $G$ to be connected. The irreducible representations are parametrized by highest weights and the Laplace eigenvalue depends on the value of a quadratic form on the space of weights. So, in each case you need to identify those weights with $K$-invariants and consider the values of the quadratic form, which in the case of a simple group should be the Killing form. I guess that in the above cases it might actually be true.
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@Anton Deitmar, Evan Jenkins: I am still a little unclear about the connection between the Laplacian $L$ and the Casimir operator(s). I think that your remarks about eigenvalues being determined by the Killing form on highest weight vectors refers to the Casimir operator, which is purely group theoretic, whereas $L$ is the usual Riemannian Lapalcian. Of course they are related (and no doubt have the same symbol) but I don't think that they are the same. Do either of you (or does someone else) know where the relation between them is discussed. (Broecker and tom Dieck only treat a special case. – Dick Palais Nov 22 2010 at 0:36 @Dick1: Any invariant positive definite bilinear form $B$ on the Lie algebra $\mathfrak g$ of $G$ gives an invariant metric on the quotient $G/K$. Being non-degenerate, this form on the one hand identifies $\mathfrak g$ with its dual ${\mathfrak g}'$, on the other hand it is itself an element of ${\mathfrak g}'\otimes\mathfrak g'\cong{\mathfrak g}\otimes{\mathfrak g}$ The latter space maps naturally to $U=U({\mathfrak g})$, the universal enveloping algebra. So $B$ induces an element in $U$, which is called the Casimir-operator $C_G$. – anton Nov 22 2010 at 18:05 @Dick2: This Casimir-operator acts on functions on $G/K$ as a differential operator which happens to coincide with the Laplace-operator induced by the metric. This is no wonder, since the metric and the Casimir are induced by the same invariant form. – anton Nov 22 2010 at 18:06 Thanks Anton. That more or less answers what I wanted to know. (Though I still do not see what happens when the isotropy action of $K$ is not irreducible, so there is not a unique $G$-invariant metric on $M$.) – Dick Palais Nov 22 2010 at 22:46
Shouldn't this only happen very rarely? $S^1$ is abelian, and $SO(3)$ acting on $S^2$ involves inducing from the maximal torus, so in both these cases, every irreducible appears once. But in general (i.e., if $K$ does not contain a maximal torus), irreducible representations will appear more than once, in which case there's no hope for the eigenvalue of the Laplacian to separate them. Even when irreducibles don't appear multiple times, the eigenvalue of the Laplacian is not generally enough to separate two irreducibles if the rank of the group is bigger than 1.
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For G/K symmetric the joint eigenspaces of the G-invariant differential operators on G/K are all irreducible. Also each irreducible subspace of H has multiplicity bounded by one. For this see my "Groups and Geometric Analysis?" Ch. V Theorems 4.3 and 3.5. Concerning the Laplace Beltrami operator L, the Casimir operator on G (if semisimple) does induce L on G/K (loc. cit. p.331). If G/K is two point homogeneous the G-invariant differential operators on G/K are all polynomial in L (loc. cit. p/288) so for these spaces the answer to Dicks question is yes. For G/K not symmetric Theorem 3.5 p. 533 still gives a decomposition of H into spaces spanned by representation coefficients which are eigenfunctions of the Casimir operator.
Assuming the metric on G/K, (G semisimple) is coming from the Killing form Riemannian structure on G it is still true that the geodesics through the origin in G/K are orbits of one parameter subgroups of G. It seems to me that the argument for Problem A4 p.568 should still show that the Casimir operator on G will induce the Laplace Beltrami operator on G/K. Therefore the decomposition in Theorem 3.5 p.533 should still be a decomposition into eigenfunctions of the Laplacian. But there is no reason to expect irreducibility.
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# Condition number of A'A and AA' formulations
It's shown (Yousef Saad, Iterative methods for sparse linear systems, p. 260) that $cond(A'A) \approx cond(A)^2$
Is this true for $AA'$ as well?
In case $A$ is $N\times M$ with $N \ll M$, I observe that $cond(A'A) \gg cond(AA')$
Does that mean formulation in terms of $AA'$ is preferable in this case?
• You're comparing condition numbers of two matrices with vastly different sizes. Without an explanation of why, it seems like that comparison is probably not meaningful. Certainly, if you can accomplish what you need by using the much smaller matrix, you should (even if the conditioning were similar). – David Ketcheson Mar 5 '12 at 10:46
• The new answer by Stefano M below is correct. Please read it and vote it up. – David Ketcheson Jul 15 '12 at 6:03
If $A\in\mathbb{R}^{N\times M}$ with $N<M$, then $$\mathop{\mathrm{rank}}(A^TA) = \mathop{\mathrm{rank}}(AA^T) = \mathop{\mathrm{rank}}(A) \leq N < M$$ so that $A^TA \in \mathbb{R}^{M\times M}$ cannot be full rank, i.e. it is singular.
Accordingly the condition number is $\kappa_2(A^TA)=\infty$. Due to finite precision arithmetic, if you compute cond(A'A) in matlab you obtain a large number, not Inf.
• @OscarB: the singular values of $A$ are just $N$, there is no such a thing as the $M$th singular value! Your derivation is correct, but please note that if $\sigma_i$, $i=1\dots N$ are the sv's of $A$, then $SS^T=\mathop{\mathrm{diag}}(\sigma_1^2,\dots,\sigma_n^2)$, while $S^TS = \mathop{\mathrm{diag}}(\sigma_1^2,\dots,\sigma_n^2, 0, \dots, 0)$ with $M-N$ trailing zeros. – Stefano M Jul 13 '12 at 22:41
Well, let’s look at why $A^TA$ has approximately the squared condition number of $A$. Using the SVD decomposition of $A=USV^T$, with $U \in \mathbb{R}^{N \times N}$, $S \in \mathbb{R}^{N \times M}$, $V \in \mathbb{R}^{M \times M}$, we can express $A^T A$ as
$A^T A=(USV^T)^T USV^T=VS^T U^T U S V^T=V S^T S V^T$
Which we arrive at by noting that $U$ is orthonormal, such that $U^T U=I$. Further we note that $S$ is a diagonal matrix, such that the final decomposition of $A^TA$ can be expressed as $V S^2 V^T$, with $S^2$ meaning $S^T S$, yielding a diagonal matrix with the first N singular values from $S$ squared in the diagonal. This means that since the condition number is the ratio of the first and the last singular value, $cond(A)=\frac{s_1}{s_N}$ for $A \in \mathbb{R}^{N \times M}$,
$cond(A^T A)=\frac{s_1^2}{s_M^2}=(\frac{s_1}{s_M})^2=cond(A)^2$
Now, we can perform the same exercise with $AA^T$:
$AA^T=USV^T (USV^T)^T=USV^T V S^T U^T=U S^2 U^T$
Which means that we get the result $cond(AA^T)=\frac{s_1^2}{s_N^2}$, since $S^2$ here means $SS^T$, a subtle difference from the notation above.
But note that subtle difference! For $A^TA$, the condition number has the M'th singular value in the denominator, while $AA^T$ has the N'th singular value. This explains why you are seeing significant differences in the condition number — $AA^T$ will indeed be “better conditioned” than $A^TA$.
Still, David Ketcheson was correct — you are comparing condition numbers between two vastly different matrices. In particular, what you can accomplish with $A^TA$ will not be the same as what you can accomplish with $AA^T$.
• That is a great explanation! I see the difference clearly now. Matrix A is used to build normal equations and with slight changes you can also formulate it as $AA'$, not classical $A'A$. Can you tell as well if it is advantageous to use solver like LSQR instead of solving normal equations? Since LSQR doesn't require to build this product at all. – Alexander Mar 5 '12 at 12:54
• Glad it made sense. In general, you need to consider the conditioning of the problem. But, if that is not an issue, you could use either normal equations/QR-factorization(of A)/LSQR, depending on the size of the problem (amongst other things). Unless your problem is large or ill-conditioned, I would probably apply the QR-factorization, but without more knowledge of the problem you are trying to solve, it's hard to tell. I am sure others with more experience could provide more detailed advice. – OscarB Mar 5 '12 at 13:24
• The A itself is ill-conditioned (with condition number of $\approx 10^7$), dense and large. QR is not an option. Since it's ill-conditioned I have to add some regularization anyway. Now simple Tikhonov regularization seems to be enough. The point is that if $cond(A) < cond(AA^T) < cond(A^T A)$ (for my case with $N < M$) then using LSQR seems to be always preferable since you do not need to form any product at all. The question is if solutions obtained with normal equations and LSQR are identical? – Alexander Mar 5 '12 at 13:59
• Well, as I understand it, LSQR will provide an identical solution to normal equations after "infinitely many" iterations in exact precision. However, for ill-posed problems, the normal equations solution is not the one you want. Instead, you want to use LSQR to iterate until semi-convergence is achieved. However, controlling iterative algorithms in ill-posed problems is a whole other ball-game. Also, depending on the cost of your matrix-vector product and the number of iterations (and thus matvecs) needed, a direct tikhonov solution with bidiagonalization might be better. – OscarB Mar 5 '12 at 15:42
• Awesome explanation. +1 for you sir! – meawoppl Mar 5 '12 at 22:33
The claim that $\DeclareMathOperator{\cond}{cond} \cond A^2 \approx \cond A^T A$ (for square matrices) in the question and [Edit: I misread] in Artan's answer is nonsense. Counter-example
$$\newcommand\bigO{\mathcal{O}}A = \begin{pmatrix} \epsilon & 1 \\ 0 & \epsilon \end{pmatrix}, \quad \epsilon \ll 1$$
for which you can easily check that $\cond A^T A = \bigO(\epsilon^{-4})$ while $\cond A^2 = \bigO(\epsilon^{-2})$.
• Ok to stress that $A^2$ and $A^T A$ are in general very dissimilar as what regards eigs, svds, cond number: but in my opinion the question's claim is about $[\mathrm{cond}(A)]^2$. – Stefano M Jul 24 '12 at 20:25
• @StefanoM Thanks, it seems I misread, though from the discussion, wasn't the only one. – Jed Brown Jul 25 '12 at 0:25
In exact arithmetic cond(A^2)=cond(A'A)=cond(AA'), see eg. Golub and van Loan, 3rd ed, p70. This is not true in floating point arithmetic if A is nearly rank deficient. The best advise is to follow the above book recipes when solving least square problems, the safest being SVD approach, p257. Use \varepsilon-rank instead when computing SVD, where \varepsilon is the resolution of your matrix data.
• I'm sorry, I looked at Golub and Van Loan 3rd ed p. 70, and couldn't find anything backing up the statement cond(A^2)=cond(A^TA)=cond(AA^T). Could you be more specific with your reference? – OscarB Mar 9 '12 at 15:06
• There is no statement there, but you can derive from theorem 2.5.2 and the pseudoinverse, section 5.5.4 that cond(AA')=cond(A'A). The reason that I take pseudoinverse is that this is what matters for the least squares problem in hand. The equality after cond(A^2) should be \approx, sorry for the typo. – Artan Mar 9 '12 at 20:37
• No, this answer is totally incorrect. See my counter-example. – Jed Brown Jul 22 '12 at 17:12
• Saad must have made such a point wrt to some specific context. What is relevant for the question at hand is the proceeding argument. – Artan Jul 25 '12 at 20:40
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One software project that I’ve been working on for a while at this point is librtosc, a realtime safe implementation of the open sound control messaging protocol. If this was simply an implementation of a library to handle the serialized format this would be a done and sealed project a while ago, but the project also includes one aspect which is quite difficult to nail down properly, dispatching. Once a message is received it generally should trigger something as these messages canonically represent events.
This routing problem can be a problem in terms of how much code is dedicated to it, how hard it is to write said code, and what sort of runtime issues may arise in its execution. While this library may very well have general purpose applications, the primary target is ZynAddSubFX, a sizeable software synth, which unfortunately has some major architectural flaws currently. Given this application some architectural optimization needs to be considered which would normally be premature optimization. The OSC messages are normally dispatched according to a typed set of paths similar to what you would see in any url. If a naïve representation was taken for these paths, then there would be somewhere on the order of 2,000,000 paths stored in memory (16 parts * 16 kit instances * 8 voices * 2 oscillators * 256 phase/amplitude controls ⇒ 1048576 paths) and each path with reasonable names could look like "/part1/kit1/adsynth/voice1/oscilator1/phase1", resulting in over 11 MB of static data just for the paths. Considering the current binary size is \~6.5 MB, that is quite a bit of extra weight. As per the execution constraints it would not be surprising to see a system run with perhaps two milliseconds of latency and you likely don’t want to add more than 5% extra time with messages, so you only have 0.1 ms to deal with whatever incoming messages there are. While these remarks are not at all impassible, they do serve as reminders that some caution is needed in building up this system.
Now that the problem domain is specified, the problem can be restated to be: How do you define a massive number of simple but varied callbacks for a tree of objects in a manner that reduces the runtime resources? This sounds like a lovely time to make use of some sort of domain specific language, though things are somewhat limited if this information is included within the C++ source directly. The current approach defines a hierarchy of ports, which specify wildcards to handle duplications, and argument constraints to make things roughly typed; Each port has some form of a callback, some metadata, and possibly an associated subtree.
Port
name :String
types :Set{Types[]}
properties:Map{String,String}
callback :Function(Arguments[], SupplimentalData)
subtree? :Ports[]
Ideally such a thing would translate into simple code such as
#Define the scope of all ports
@class Oscillator
#Define a linear parameter that controls the volume member of the oscillator
#that ranges between the floating point values 1.0 and 10.0 in dB with a
#linear mapper and the doc string of "Base Volume"
@linfpar volume 1..10dB Base Volume
#A midi parameter that controls Pphase, calls prepare() when done and has the
#doc string "phase of the nth harmonic"
@midipar#128 Pphase prepare() phase of the nth harmonic
Oh, did I forget to mention that MIDI mappings are tagging along with the ports metadata. Adding system wide metadata seems like a sane enough time to add enough metadata to reflect on the available parameters. While the above example looks fine enough, things get barrier and much uglier when real legacy code needs to be dealt with. Whenever some chunk of data does not quite match up that is when the abstraction leaks quite heavily and there needs to be a fallback for this case. Unfortunately to emulate the above behavior you are generally stuck with preprocessor macros, which do work, but they are simply unwieldy. Given that we are dealing with data that only needs to be initialized once per run of the application, one of a few options is available:
1. Serialization into a global structure
2. Hacking a global objects constructor to setup everything
3. Some sort of setup method
The current approach taken is the first one. This certainly works well for the current application, but there may be problems with pure static definitions for any system that does not know some information until runtime (like a plugin).
Now with that backdrop laid out, the issue with implementing such a system elegantly is that there are enough edge cases that any abstraction composed of simple macros will be broken at least a few times. That implies that there is only so much of an advantage in trying to create a clean interface. This means that the user will be exposed to some of the annoying bit twiddling that is involved in generating the dispatch tables that other libs hide away. Part of me thinks that the good enough'' system will work out, but there is always the question of how good can a tool be?
This brings me to one of the minuscule details that started this mind dump. Doubly null terminated strings. If you are dealing with parsing anything you generally end up rewriting all sorts of odds and ends of ad-hoc utilities to deal with different terminators and collections of buffers start piling up left and right (don’t forget the escape induced madness). Long macros to translate stuff into this format, like RTOSC_PARAMETER() just seem too verbose, but then again the terse side of things is not much better. Then there is the question of is the non-standard doubly null string idea better than a proper data structure? On the one hand it, like the previous ':' delimited string, can easily be transmitted via OSC (via a blob), it may take up more memory, be slower to iterate through, and just be odd. The interesting thing about these questions to me is that I simply do not have a good answer at the moment. Anyhow, back to trying to code this thing some more to see where the bad ideas lie. Perhaps the next post will be more coherent :p
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# Probability density function for Riemann-zeta zeros
Curious about the expected probability distribution for the spacing between Riemann zeta zeros, of the form $$s_n=\sigma+it_n$$, where $$\sigma=0.5$$ and $$t_n$$ is the imaginary part of the $$n$$-th zero.
The mean spacing between zeros decreases slowly as the height $$t_n$$ goes up, somewhat confounding the issue, but taking a narrow slice of 100,000 zeros starting at the billionth zero ($$t_{1000000000}$$) should make that variation negligible. Here's a histogram of the spacings in that region:
The longer upper tail precludes a normal distribution. The red curve is a best fit gamma distribution, which doesn't quite do it either. The mean spacing at the beginning of the range covered is $$0.351087$$ compared to $$0.351073$$ at the end, so that variation is small.
What is the closest distribution to model the spacing?
Just looking at other distributions, the three-parameter Burr type XII distribution fits pretty well. Here's the result as above around $$t_{1e9}$$:
The Burr type XII pdf is:
$$f(x|\alpha,c,k)={{kc\over \alpha}({x\over \alpha})^{c-1}\over{(1+({x\over a})^c)^{k+1}}}$$
For the above fit,
$$\alpha = 0.837947, c=2.76292$$ and $$k=8.68889$$
However, the best fit parameters change at other heights, e.g. at $$t_{10e9}$$:
$$\alpha = 0.776292, c=2.73405$$ and $$k=9.38153$$
Note the mean has shifted down slightly. The mean is given by:
$$\mu = \alpha k\Gamma(k-1/c)\Gamma(1+1/c)/\Gamma(k+1)$$
The Burr type XII distribution is used most often to study mortality, survival, failure rates and the like. I guess the interval between a zeta zero and the next could be thought of as its lifespan.
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0th
Percentile
##### Checks if a package is loaded or not
Checks if a package is loaded or not. Note that, contrary to require(), this function does not load the package if not loaded.
Keywords
package, utilities
##### Usage
# S3 method for default
isPackageLoaded(package, version=NULL, ...)
##### Arguments
package
The name of the package.
version
A character string specifying the version to test for. If NULL, any version is tested for.
...
Not used.
##### Value
Returns a logical.
To check if a package is installed or not, see isPackageInstalled().
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# Two equilibrium problems
1. May 1, 2005
### poodlefarm
1) I just want to check if this one is correct
reaction => CCl4(g) <==> C(s) = 2Cl2(g)
Kp= .76
find the initial partial pressure of CCl4 that will produce an equilibrium total pressure of 1.20 atm
p.total = p.CCl4 + p.Cl2
CCl4 <=> 2Cl2
i. press x 2x
change 1.20 - x 1.20 - 2x
kp = .76 = (p.Cl2^2)/(p.CCL4) = (1.2-2x)^2/(1.2-x)
x= .8557 or x= .1543
.8557 will not work so take x = .1543
p.CCl4 = 1.20- .1543 = 1.05 atm
the rub is that if I use x to find p.Cl2 I get .892 atm
1.05 + .892 =/= 1.2 ?
Q2.
Ok
constant temp =25 C constant volume
reaction; Nh4HS (g)<==> NH3(g) + H2S(g)
step 1; some NH4HS decomposed in an evacuated container to give a total pressure at equililbrium of .659 atm
step 2; extra NH3 is added. re-established equilibrium gives a partial pressure for NH3 of .750 atm
find Kp
I'm stuck on this one.
.
2. May 1, 2005
### Gokul43201
Staff Emeritus
Q1 ) CCl4(g) <==> C(s) + 2Cl2(g)
Let the dissociation constant of CCl4 be x. If you start with 1 mole of CCl4, then in equilibrium, how many moles of CCl4, Cl2 will you have ? Remember, CCl4 is being consumed and Cl2 is being produced. Now use the equation for Kp to arrive at a quadratic in x which you can solve to find x.
Next, if the container had some volume V, and there are n moles of CCl4 to start, what will the pressure be at some temperature T ? Write a similar equation for the equilibrium number of moles and call this pressure Po. From these two, you can eliminate V/T and find the value of Po.
3. May 1, 2005
### Gokul43201
Staff Emeritus
let the initial number of moles of NH4HS be n, and let the dissociation constant be x. Now proceed from there ...
4. May 1, 2005
### poodlefarm
Q1)
I'm a little lost.
assuming 1 mole of CCl4
initial ; CCl4 = 1 Cl2 = 0
concentration CCl4 = -x cl2 = +2x
equilibrium CCl4 1-x Cl2 2x
Kp = .76 = 4x^2/(1-x)
x = .35
not sure if I'm doing this right.
but here is where I really get lost.
I think you are refering to PV =nRT
P= 1.2 atm
so 1.2 = (nRT)/V
and Po = (.65RT)/V // .65= 1-x
I still don't see it
5. May 2, 2005
### Gokul43201
Staff Emeritus
So far, so good !
Umm. Let's try again.
Initially, you have n moles at a pressure Po. Finally, you have n(1 - x + 2x) = n(1+x) moles at a pressure of 1.2 atm, where the value of x is known from above. Make sure you understand where the n(1+x) comes from.
Initial : Po = nRT/V
Final :1.2 = n(1.35)RT/V
From this, find Po (as well as the final partial pressures to make sure they add to give 1.2 atm)
6. May 2, 2005
### GCT
The second one's a #\$%^&. If the initial pressure was known, you could calculate Kp simply by using the percent dissociation, no need for the second aspect of the information given. It'll need to be worked out mathematically, if a solution actually exist to the exact form of this problem. I'm able to deduce several meaningful equations so far, and the best one has been
$$P_{total,eq2}=.75atm + P_{0}$$ where $$P_{0}$$ is the initial pressure at the very beginning of the situation. If $$.75atm$$ were valid as the net change in pressure, than one could calculate the Kp by finding the moles of each compound in undergoing the reaction.
Nevertheless I'll wait for both of you to get to this part before proceeding.
7. May 2, 2005
### GCT
I was hoping for a more creative way to solve the second question, yet the only sure way to solve it seems to involve a messy equation at the end.
$$P_{total,eq1}=.659=P_{0}+x,~x=P_{NH3}=P_{H2S}$$
$$Kp= \frac{[x][x]}{[P_{0}-x]}=\frac{[x][x]}{[.659atm-2x]}$$
also
$$Kp= \frac{[.750atm][x- \Delta y]}{.659atm-2x+ \Delta y]}=$$
set the two equal to each other and solve for y in terms of x, plug back in the second Kp...set this Kp equal to the first Kp and solve for x.
perhaps a better way will involve the incorporating the temperature
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# How do you bake several materials onto one object in Cycles Render?
I am able to bake objects with one material, but when there is more than one, an error comes up: No active image found in material "blah-blah". Most materials are used on many different objects, so I made each one a single user copy (clicking the number beside the material name field in Properties > Materials and renaming it descriptively). That didn't resolve it though.
I have been following the steps in this Blender Guru video (which explains how to bake the lighting onto an object's surface so it takes much less rendering to display it in an app like Sketchfab, or so you can work on it displayed in Render mode in real time):
UV unwrap > create and name image > add Image Texture > link to the image just created > make sure that node is selected > Bake. Since in this case the objects had several materials, I added an Image Texture to each one, linked it, and left it selected.
Do the UV maps need to be divided up so they only are for one material each? Does that mean I have to divide up the objects into a group of objects each of one material? What is the best way to do this?
(Oops, the file doesn't have the object selected that I was working on when this cropped up - it is the green floors inside the structure, named Dugout Levels.)
• just add a texture node with the target texture in each material and make sure that this node is selected (active) in each material before the bake (these nodes do not need to be connected to anything) – lemon Aug 10 '16 at 16:23
• You need to select these nodes (so that they are active, rounded by yellow) – lemon Aug 10 '16 at 16:34
• @kimholder do you want to bake the materials on a single texture? – Denis Aug 10 '16 at 16:41
• @kimholder then you will need to distribute the UVs so they do not overlap and select the all the objects that use the texture and then click bake – Denis Aug 10 '16 at 16:49
• @kimholder you need to flip normals thats why it is black – Denis Aug 10 '16 at 16:58
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# How To Ommit Date Fro Latex Document
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## Theory of Combinatorial Algorithms
Prof. Emo Welzl and Prof. Bernd Gärtner
# Mittagsseminar (in cooperation with M. Ghaffari, A. Steger and B. Sudakov)
Mittagsseminar Talk Information
Date and Time: Tuesday, November 04, 2008, 12:15 pm
Duration: This information is not available in the database
Location: CAB G51
Speaker: Andreas Razen
## Counting Crossing-Free Geometric Graphs with Exponential Speed-Up (Part I)
For a set $P$ of $n$ points in the plane in general position we show that the number of crossing-free geometric graphs on $P$ is always a fixed exponential in $n$ more than the number of triangulations, while the number of graphs may still be computed in time necessary to enumerate all triangulations on $P$.
Joint work with Emo Welzl.
Information for students and suggested topics for student talks
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Mobile QR Code
1. (Department of AI Convergence, Pukyong National University, Pusan 48513, Korea {kdm902077, sengthai}@pukyong.ac.kr )
2. (Department of Computer Engineering, Pukyong National University, Pusan 48513, Korea [email protected] )
Qubit mapping problem, Parallel processing, SABRE algorithm
## 1. Introduction
Quantum computers are the next-generation computers that store and calculate data based on quantum mechanics [1,2]. Quantum devices can process computationally intensive operations in a significantly shorter time compared to classical computers by utilizing quantum mechanics, such as superposition and entanglement. The D-Wave company [3] released quantum systems that are being commercialized and surpass classical supercomputers using the quantum-annealing algorithm [4,5]. On the other hand, since the D-wave was designed only for the quantum-annealing algorithm, the current technology still has many problems developing an actual quantum machine on the noisy intermediate-scale quantum (NISQ)-scale to be commercialized universally soon [6]. Qubit, the basic unit constituting a quantum computer, and the quantum gates constituting a quantum algorithm all have inherent errors. Therefore, two major problems should be solved to produce a fault-tolerant and universal quantum computer on the NISQ-scale. The first to be addressed is quantum error correction, which solves the errors inherent in qubit and quantum gate [7,8]. Currently, various studies on quantum error-correction codes have been conducted [9-13]. Nevertheless, thousands of qubits are required to make a fault-tolerant qubit in a real quantum machine [14]. A considerable number of qubits are required for a meaningful quantum algorithm on the NISQ-scale. Second, the quantum circuit must be executed quickly within the quantum coherence time [15]. The quantum coherence time refers to the time at which the information of the qubits of an actual quantum machine can be preserved without change. In particular, the same result as the simulation can be obtained if the quantum algorithm is executed successfully within the quantum coherence time on an actual quantum machine. Currently, there is no real quantum machine capable of supporting a general-purpose quantum circuit on the NISQ-scale that can satisfy the two major problems. Thus, quantum computer simulations must be utilized. Quantum simulation platforms, such as Qiskit [16], QuTiP [17], and t|ket> [18], provide several useful tools to implement, verify quantum algorithms, and evaluate their performance. In the case of Qiskit, the NISQ-scale quantum algorithm implemented by the programmer can be operated using a simulator with the same characteristics as the actual quantum machine. Consequently, it allows a researcher to implement and analyze quantum circuits quickly.
With the recent developments of quantum computer technology, various quantum algorithms have been proposed. Quantum algorithms are implemented as quantum circuits based on quantum gates, and a quantum compiler is required to run these quantum circuits on a quantum machine. Quantum qubit mapping is the core function of a quantum compiler that maps logical qubits to physical qubits. The main difference between these two qubits is the qubit connectivity, and the states in which the qubits are connected are different. In a logical qubit, a random qubit is connected to all other qubits. Thus, all the logical qubits are connected to each other. In a physical qubit, however, only specific qubits are connected to each other. Qubit connectivity is important because a two-quantum gate cannot be executed directly if a two-quantum gate to be executed in a real quantum machine is applied to qubits that are not connected. Qubit mapping is required to solve this problem. On the other hand, qubit mapping is an NP-complete problem [26]. Thus, the compilation time increases exponentially with increasing number of qubits. Therefore, the execution time of qubit mapping should be reduced to decrease the compilation time.
This paper proposes a novel qubit mapping algorithm that utilizes the existing qubit mapping algorithm called SABRE [19]. The SABRE algorithm minimizes the number of quantum gates added due to qubit mapping. By running the SABRE algorithm in parallel, this study focused on improving the speedup of the execution time of qubit mapping. For performance verification, 10 NISQ-scale quantum circuits were used as benchmarks to measure the execution time of qubit mapping compared to the SABRE algorithm. According to the experimental results, compared to SABRE, the proposed method has an overhead of up to 1.05x and an average of 1.03x in terms of the gate, resulting in improvements of up to 6.24 times with an average of 4.8 times in terms of speedup.
The major contributions of this paper can be summarized as follows:
• This study analyzed the characteristics of previous studies for the qubit-mapping algorithm and identified the goals considered essential for qubit mapping.
• A new method was proposed to run the SABRE algorithm in parallel to reduce the compilation time of quantum compilers.
• The scalability of the proposed method was shown; increasing the number of cores during parallelization will outperform the well-known qubit-mapping algorithm for NISQ-scale quantum circuits.
The remainder of this paper is organized as follows. Section 2 describes the Qubit mapping problem. In Section 3, the proposed qubit-mapping algorithm is explained, and the performance of the method is verified in Section 4. Section 5 discusses the issues that should be resolved in future research, and Section 6 concludes the paper.
## 2. Background and Related Works
This section explains Qubit mapping, which is the focus of the study. In addition, the contents necessary for each paper are mentioned, and the work performed in this study is outlined by explaining recent studies to solve the Qubit mapping problem effectively.
### 2.1 Qubit Mapping Problem
Running a quantum circuit using simulation is not the same as running in a real quantum machine. Qubits used in quantum simulations are called logical qubits, while those used in actual quantum machines are called physical qubits. These two qubits run the quantum circuit under different conditions, and the condition considered in this study is the Qubit connectivity.
Qubit connectivity contains information on how different qubits are connected and can be expressed in a graph called a coupling map. In the case of logical qubits, because they are all connected to each other, the coupling map is not considered when executing the quantum circuit on a quantum simulator. By contrast, because not all physical qubits are connected to each other, the quantum circuit is executed in consideration of the coupling map. Fig. 1 shows the coupling map of a real quantum machine, IBM Q Tokyo. In the coupling map of IBM Q Tokyo, the circles indicate physical qubits, and straight arrows indicate that physical qubits are connected to each other. For example, qubits 6 and 7 are connected to each other, but qubits 6 and 12 are not. Hence, there are limitations in operating the quantum circuit because the physical qubits of the actual quantum machine are only connected to specific qubits.
##### Fig. 1. Coupling map of a 20-qubit IBM Tokyo quantum machine.
A quantum circuit consists of several qubits and quantum gates. Quantum gates can be divided into single-qubit gates and multi-qubit gates. When executing a quantum circuit composed of these quantum gates in an actual quantum machine, it is necessary to check whether multi-qubit gates can be executed without additional work. That is, if the physical qubits to which the multi-qubit gates are applied are not connected to each other, these multi-qubit gates cannot be executed directly on an actual quantum machine. To solve this problem, it is necessary to connect the physical qubits that are separated. On the other hand, because the location of the physical qubit cannot be changed, the logical qubit must first be mapped to the physical qubit on the coupling map. Some of the logical qubits are separated from each other and some are connected according to the coupling map. To connect the separated physical qubits, it is necessary to connect the logical qubits mapped to the corresponding physical qubits. To this end, many SWAP gates that change the positions of the logical qubits are added to the original quantum circuit. Accordingly, the number of gates in the quantum circuit increases, which increases the execution time of qubit mapping. That is, to alleviate this problem, it is necessary to minimize the total number of gates using only a minimum number of SWAP gates.
Fig. 2 shows two different quantum circuits in which arbitrary quantum circuits can be produced through qubit mapping: (a) is a coupling map composed of four qubits, and (b) is a quantum circuit composed of four qubits and six quantum gates. If qubit mapping is performed considering the coupling map of (a) provided in (b), a quantum circuit is constructed, as shown in (c). The CNOT gate, which is indicated by the red dotted line in circuit (b), is applied to the first and third qubits. In the coupling graph of (a), the first and third qubits are not connected to each other. Thus, they cannot be executed directly on the quantum machine. The CNOT gate represented by the blue dotted line is a CNOT gate that can be affected by the SWAP gates added for the CNOT, which are represented by the red dotted line to be executed. When Qubit mapping is performed, two possible results can be observed: (c) and (d). In (c), two swap gates were added, and in (d), only one swap was added. Therefore, for a NISQ-scale quantum circuit, depending on the qubit mapping algorithm, the number of swap gates added, and the execution time of qubit mapping may be affected.
### 2.2 Qubit-mapping Algorithms
Recent studies related to Qubit mapping have been conducted to implement NISQ-scale quantum circuits in real quantum machines. Cowtan et al. [20] performed Qubit mapping using the t|ket> quantum compiler in the architecture of IBM QX5, a real quantum machine with 16 qubits, and IBM Tokyo with 20 qubits. Cowtan et al. [20] aimed to minimize the depth of the added quantum gate and quantum circuit. For performance verification, from a significantly small-scale quantum circuit with five quantum gates to a NISQ-scale quantum circuit as a benchmark, the benchmarks were run together in the compiler systems of Qiskit, Project Q, and Qulic to compare the performance. In the performance comparison, however, it is difficult to accurately compare the performance with t|ket> because the number and depth of quantum gates and depth of Qiskit and Qulic for approximately 3000 or more quantum circuits in the IBM Q Tokyo machine used by Cowtan et al. [20] were not measured. In addition, the execution time required for qubit mapping in the three compilers was not analyzed.
A previous study [21] used a heuristic algorithm using the gate commutation rule and the gate transformation rule to reduce the number of additional gates. In particular, the number of additional gates required for qubit mapping was minimized using the Bridge gate and the SWAP gate. On the other hand, the quantum circuit was used as a benchmark for performance verification contained up to 38577 quantum gates, which is not sufficiently large. Comparison analysis of the circuit execution time was not performed sufficiently. Zulehner et al. [22] performed qubit mapping that minimizes the number of gates added to the IBM QX architecture and reported superior performance compared to the default qubit-mapping algorithm provided by Qiskit at the time the thesis was written. Cowtan et al. [20] developed and verified the superiority of the SABRE algorithm by comparing its performance with that by Zulehner et al. [22] in terms of the number of additional quantum gates, total number of gates, and quantum qubit mapping execution time. On the other hand, the benchmarks used for performance verification included a maximum of 34881 quantum gates, which is significantly smaller than the number of quantum gates on the NISQ-scale to be addressed in this study. Zhang et al. [23] performed qubit mapping to minimize the circuit depth using a depth-aware SWAP insertion scheme. The reported performance of the method [23] was verified by comparing it with the SABRE algorithm and the method reported by Zulehner et al. [22].
Compared to the two methods, the method reported by Zhang et al. [23] showed superior performance in terms of additional gates and depth. On the other hand, the benchmarks used by Zhang et al. [23] are also not NISQ-scale quantum circuits, and a detailed analysis of the execution time was not performed except for the statement that the expected execution time can be equated with the depth. Niu et al. [24] reported the performance of reducing the number of additional gates by 28\% on average compared to SABRE on the ibmq\_almaden machine and 14\% on the ibmq\_tokyo machine by proposing a hardware-aware mapping transition algorithm. On the other hand, because Niu et al. [24] also benchmarked the quantum circuit used in SABRE, the execution time of the NISQ-scale quantum circuit was not compared and analyzed. Therefore, the present study used the NISQ-scale quantum circuit as a benchmark to analyze the execution time, which was insufficient in previous papers, and reduce it compared to the existing methodology. The circuit consists of three sub-processes using the SABRE algorithm, and the execution time of SABRE was compared with the proposed method for performance verification.
## 3. Parallelized Qubit Mapping Algorithm
This section describes the novel Qubit mapping algorithm proposed in this paper in more detail. Fig. 3 illustrates the overall process of the proposed method. It is a parallelization of the existing SABRE algorithm through a multiprocessing module in Python and consists of three-step processes. Each process is Circuit Split, Parallelized SABRE and Reversed SWAP and Circuit Merge.
### 3.1 Proposed Method
First, the quantum circuit was split into several quantum subcircuits. By doing this, each quantum subcircuit could be computed in parallel. Circuit splitting processes on quantum circuit objects of the Qiskit library. In this version, every quantum circuit that contains more than 300 gates must be split. There is a trade-off between the number of gates per quantum subcircuit and the number of core processes used. This trade-off is mentioned in section 4.2.
Second, one quantum circuit was divided into quantum subcircuits with the same number of quantum gates, and the SABRE algorithm was applied to the quantum subcircuits to perform qubit mapping. At this time, because SABRE mapping is performed on a relatively small quantum sub-circuit compared to the original circuit, the size of a Directed Acyclic Graph (DAG) [19] of each quantum sub-circuit is also relatively small. The size of the DAG has a significant impact on the execution time of the qubit mapping because the DAG is checked multiple times during qubit mapping. In addition, because SABRE mapping is applied to each quantum subcircuit in parallel, the execution time of qubit mapping is faster than that of one-by-one. After SABRE mapping, the position of the last logical qubit of each quantum subcircuit is different from the initial position because the SWAP gates are added to each quantum subcircuit. The quantum gate is not applied to the correct qubit if all quantum subcircuits are combined under these conditions. Thus, the measurement result will be incorrect. To address the position changes, the logical qubit position must be moved back to the initial position. To achieve this, when performing SABRE mapping for each quantum subcircuit, the initial logical qubit position is arranged from the first qubit to the last qubit in order. When SABRE mapping is finished, the position of the changed logical qubit is rearranged from the first qubit to the last qubit sequentially. Different permutations are added to each circuit to rearrange the physical qubits. The permutation describes the current position of the logical qubit that allows the addition of reversed swap gates to move the qubit back to the initial position. The permutation pattern is extracted by identifying which qubits are swapped when the SABRE mapping operates. This pattern is different for each quantum subcircuit, and the positions of the last logical qubits of all quantum subcircuits are rearranged sequentially through permutation. In addition, this reversed swap process is also executed in parallel to accelerate the execution time.
Finally, each quantum subcircuit on which SABRE mapping and permutation have been performed must be combined into one quantum circuit again. Since all quantum subcircuits have physical qubits arranged in order by permutation at the end of the circuit, they can be converted quickly into a single quantum circuit with a simple '+' operation without additional work. The final quantum circuit has a different configuration from the circuit mapped by the SABRE algorithm; however, the result after the measurement is the same. Therefore, the accuracy of the quantum circuit mapped through the proposed method is reliable.
### 3.2 Example of Proposed Method
Fig. 4 is an arbitrary quantum circuit with four qubits and a total of 19 quantum gates. Assume that a given quantum circuit is operated in a quantum machine with a coupling map, as shown in Fig. 2(a). First, as shown in Fig. 5, a quantum circuit is divided into three quantum subcircuits with six quantum gates through the circuit split process. Subsequently, the red dotted line of each quantum subcircuit cannot be executed directly. Hence, it becomes a quantum gate that requires an additional swap, and the blue dotted line becomes a quantum gate that can be changed owing to the additional swap.
Fig. 6 presents three quantum subcircuits to which SABRE mapping is applied. Each swap gate is applied at a different location. Hence, the location of the final physical qubit is different from the first location. Therefore, before combining them as one quantum circuit, each permutation is applied in Fig. 7. The square brackets in permutation are the pattern of permutation, and each quantum subcircuit has a different pattern. Consequently, the position of the final qubit becomes the same as the position of the first qubit, which avoids applying a quantum gate to the wrong qubit when merging. Fig. 8 shows that it has become one quantum circuit again after the Circuit Merge process.
## 4. Performance Evaluation
This section reports the performance of the proposed qubit-mapping algorithm by comparing it with the SABRE algorithm. Ten NISQ-scale quantum circuits are used as benchmarks, and the execution time of qubit mapping, the number of added quantum gates, and the circuit depth are analyzed.
### 4.1 Speedup
Fig. 9 shows that the execution time of the proposed method is improved compared to the SABRE execution time. All experiments in this study were executed on a server with an AMD EPYC 7R32 CPU up to 3.4GHz (32 physical cores, two threads per core) and 128GB memory. The operating system was Ubuntu 20.04.2 LTS. The Python and Qiskit versions were 3.8.10 and 0.30.1, respectively. A Python multiprocessing module was used to perform the SABRE algorithm in parallel, and the number of processes depends on the number of CPU cores. Ten NISQ-scale quantum circuits were used as benchmarks, and the number of CPU cores was doubled from two to use up to 32 cores.
The Qiskit simulator was used to verify the accuracy of the experimental results and confirm that the results of the original quantum circuit, the circuit to which SABER was applied, and the circuit to which the proposed method was applied were all the same.
Because the proposed method executes SABRE mapping in parallel, the execution time speedup is improved approximately twofold when the number of cores is doubled. The proposed method in most quantum circuits outperformed SABRE. In the case of $pm\_4096$ with full name $plus63mod4096\_163$, using 32 cores, it was possible to obtain a speedup improvement of 6.4x compared to SABRE. On average, the proposed method obtained 1.5x, 2.5x, 3.5x, 4.5x, and 4.8x faster for 2, 4, 8, 16, and 32 cores, respectively, compared to SABRE.
The proposed method ran the SABER algorithm in parallel while increasing the number of CPU cores from 2 to 32. In some benchmarks, the execution time of the proposed method did not increase linearly when the number of cores was doubled. The proposed algorithm consisted of three parts. The circuit split and circuit merge process take time because only the part that performs qubit mapping was in parallel. Therefore, the execution time of the proposed method did not improve the performance completely linearly according to the number of cores.
Table 1 lists the total number of quantum gates and the circuit depth generated by the SABRE and proposed method. The comparison normalized the results of SABRE and presented the results for the number of SWAP gates added using the proposed method, total number of gates, and depth. Because the proposed method included the permutations, there was a possibility that the number of swap gates to be added would be larger than that of the SABRE algorithm. On average, 1.25x swap gates were added compared to SABRE. On the other hand, in terms of the total gates, compared to SABRE, the maximum value was 1.05x, and the average was 1.03x, indicating a relatively small number compared to the added swap gates.
The depth of a quantum circuit is the total number of layers in which quantum gates can be executed simultaneously [25]. That is, quantum gates included in one layer can be executed in parallel; hence, the depth can significantly affect the overall execution time of the circuit. The depth of the proposed method increased up to 1.06x with an average of 1.03x compared to SABRE.
This study analyzed the methods to improve the execution time performance of the proposed method for future studies. The SABRE algorithm was executed in parallel by dividing one quantum circuit into multiple small-scale quantum subcircuits, doubling from at least two CPU cores to 32 for each quantum subcircuit. The results are presented in Fig. 9. Some specific benchmarks, such as $misex1\_241$, i.e., the number of cores, were doubled, but the speedup improvement was decreased. This is because the quantum subcircuits are assigned randomly to each core without considering the computational load of each quantum subcircuit when executing this method. That is, the load-balancing issue was not considered. Therefore, the consistent speedup improvement of the method according to the number of cores will need to be determined by adding a load-balancing algorithm optimized to the proposed method in future research.
## 5. Conclusion
This paper reported the parallelized qubit mapping algorithm that utilizes SABRE to accelerate the execution time of qubit mapping. The proposed method takes a quantum circuit as an input, and then splits it into multiple quantum subcircuits. Next, it calculates the SABRE mapping to each of the quantum subcircuits simultaneously powered by multiple cores. Finally, it accumulates results from SABRE and merges back into a single quantum circuit. The proposed method was tested based on the number of gates, depth and execution time with 10 NISQ-scale benchmarks. Because this method used additional swap gates to swap back the location of qubits, it has 1.03x more gates than SABRE on average. On the other hand, with the advantage of parallelization, in terms of the execution time, a speedup improvement of up to 6.42x with an average of 4.8x was obtained with 32 cores compared to SABRE. Future work will extend this method to manage the assigned quantum subcircuits to each core more efficiently.
### ACKNOWLEDGMENTS
This work was supported by Institute for Information & Communications Technology Planning & Evaluation (IITP) grant funded by the Korea government (MSIT) [No. 2020-0-00014, A Technology Development of Quantum OS for Fault-tolerant Logical Qubit Computing Environment]
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## Author
##### Dongmin Kim
Dongmin Kim received a Bachelor's degree in Computer Engineering and Applied Mathematics, Pukyong National University, Busan, South Korea, in 2021. He is currently working toward the MS degree in the Department of AI Convergence, Pukyong National University, Busan, South Korea. His research interests include compiler, memory systems, and quantum computing. Particularly, quantum annealing and quantum machine learning.
##### Sengthai Heng
Sengthai Heng received his Bachelor's degree in Information Technology from the University of Cambodia, Phnom Penh, Cambodia, in 2019. He presently is a master's degree student of AI convergence at Pukyong National University (PKNU), Pusan, Republic of Korea. His current research interests include quantum computing, high-performance computing, and AI.
##### Youngsun Han
Youngsun Han received his B.E. and Ph.D. degrees in electrical and computer engineering from Korea University, Seoul, Republic of Korea, in 2003 and 2009, respectively. He was a senior engineer from 2009 to 2011 with System LSI, Samsung Electronics, Suwon, Republic of Korea. From 2011 to 2019, He was an assistant/associate professor with the Department of Electronic Engineering at Kyungil University, Gyeongsan-si, Republic of Korea. Since 2019, he has been an associate professor with the Department of Computer Engineering, Pukyong National University, Pusan, Republic of Korea. His research interests include high-performance computing, emerging memory systems, compiler construction, and SoC design.
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# aotus / source / aotus_module.f90
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Copyright (C) 2011-2013 German Research School for Simulation Sciences GmbH, ! Aachen and others. ! Please see the COPYRIGHT file in this directory for details. !> This module provides high level Fortran interfaces to retrieve values from a !! Lua script. !! !! Its central interface is aot_table_module#aot_get_val, which is a generic !! interface that allows access to scalars and vectors in global Lua variables !! as well as nested tables. !! !! In the \ref aot_overview "overview page" there are some more general !! remarks and further pointers. module aotus_module use flu_binding use aot_kinds_module, only: double_k, single_k, long_k use aot_top_module, only: aot_top_get_val, aot_err_handler, & & aoterr_Fatal, aoterr_NonExistent, aoterr_WrongType use aot_table_module, only: aot_get_val, aot_table_set_val, & & aot_table_open, aot_table_close use aot_vector_module, only: aot_top_get_val, aot_get_val implicit none private public :: aot_get_val public :: open_config_file, close_config public :: open_config_chunk, open_config_buffer public :: aot_require_buffer public :: aot_file_to_buffer ! Entities inherited from aot_top_module, published here to ! allow most functionality by "use aotus_module". public :: aoterr_Fatal, aoterr_NonExistent, aoterr_WrongType public :: aot_err_handler public :: aot_top_get_val ! Inherited from the flu_binding module, publish for convenience. public :: flu_State contains !> Subroutine to load and execute a script from a file. subroutine open_config_file(L, filename, ErrCode, ErrString, buffer) type(flu_State) :: L !< Handle to the Lua script !> Name of file to load the Lua code from character(len=*), intent(in) :: filename !> Error code returned by Lua during loading or executing the file. !! !! This optional parameter might be used to react on errors in the calling !! side. If neither ErrCode nor ErrString are given, this subroutine will !! stop the program execution and print the error message from Lua to the !! stdout. integer, intent(out), optional :: ErrCode !> Obtained error description from the Lua stack. !! !! This optional argument holds the Lua error message in case somehting !! went wrong. It can be used to provide some feedback to the user in the !! calling routine. If neither ErrCode nor ErrString are provided, !! open_config() will print the error message and stop program execution. character(len=*), intent(out), optional :: ErrString !> Optional argument to return the compiled script after loading it to !! the caller. !! !! It might be handy to reuse the loaded script later on, this argument !! allows you to obtain the script in compiled form, before it is executed. !! The buffer will be allocated and filled with the Lua data. !! It contains the actual string in buffer%buffer which is a character !! pointer, and the original c_ptr to this type(cbuf_type), intent(out), optional :: buffer integer :: err integer :: length if (.not.flu_isopen(L)) L = fluL_newstate() err = fluL_loadfile(L, filename) call aot_err_handler(L, err, 'Cannot load configuration file:', ErrString, & & ErrCode) if (err == 0) then if (present(buffer)) then call flu_dump(L, buffer, length, err) end if call fluL_openlibs(L) err = flu_pcall(L, 0, 0, 0) call aot_err_handler(L, err, 'Cannot run configuration file:', & & ErrString, ErrCode) end if end subroutine open_config_file !> Subroutine to load and execute a script given in a string. subroutine open_config_chunk(L, chunk, ErrCode, ErrString) type(flu_State) :: L !< Handle to the Lua script !> String with Lua code to load. character(len=*), intent(in) :: chunk !> Error code returned by Lua during loading or executing the file. !! !! This optional parameter might be used to react on errors in the calling !! side. If neither ErrCode nor ErrString are given, this subroutine will !! stop the program execution and print the error message from Lua to the !! stdout. integer, intent(out), optional :: ErrCode !> Obtained error description from the Lua stack. !! !! This optional argument holds the Lua error message in case somehting !! went wrong. It can be used to provide some feedback to the user in the !! calling routine. If neither ErrCode nor ErrString are provided, !! open_config() will print the error message and stop program execution. character(len=*), intent(out), optional :: ErrString integer :: err if (.not.flu_isopen(L)) L = fluL_newstate() err = fluL_loadstring(L, chunk) call aot_err_handler(L, err, 'Cannot load chunk:', ErrString, ErrCode) if (err == 0) then call fluL_openlibs(L) err = flu_pcall(L, 0, 0, 0) call aot_err_handler(L, err, 'Cannot run chunk:', ErrString, ErrCode) end if end subroutine open_config_chunk !> Subroutine to load and execute a script given in a buffer !! (bytecode). subroutine open_config_buffer(L, buffer, bufName, ErrCode, ErrString) type(flu_State) :: L !< Handle to the Lua script !> String with Lua code to load. character, intent(in) :: buffer(:) !> Name for the buffer to use in debug messages. character(len=*), intent(in), optional :: bufName !> Error code returned by Lua during loading or executing the file. !! !! This optional parameter might be used to react on errors in the calling !! side. If neither ErrCode nor ErrString are given, this subroutine will !! stop the program execution and print the error message from Lua to the !! stdout. integer, intent(out), optional :: ErrCode !> Obtained error description from the Lua stack. !! !! This optional argument holds the Lua error message in case somehting !! went wrong. It can be used to provide some feedback to the user in the !! calling routine. If neither ErrCode nor ErrString are provided, !! open_config() will print the error message and stop program execution. character(len=*), intent(out), optional :: ErrString integer :: err if (.not.flu_isopen(L)) L = fluL_newstate() err = fluL_loadbuffer(L, buffer, bufName) call aot_err_handler(L, err, 'Cannot load buffer:', ErrString, ErrCode) if (err == 0) then call fluL_openlibs(L) err = flu_pcall(L, 0, 0, 0) call aot_err_handler(L, err, 'Cannot run buffer:', ErrString, ErrCode) end if end subroutine open_config_buffer !> Close an opened Lua script again. subroutine close_config(L) type(flu_State) :: L !< Handle to the Lua script to close. call flu_close(L) end subroutine close_config !> Subroutine to load a script from a file and put it into a character buffer. !! !! This is useful to rerun a given code in a file without the need to touch !! the file itself again. subroutine aot_file_to_buffer(filename, buffer, ErrCode, ErrString) !> Name of file to load the Lua code from character(len=*), intent(in) :: filename !> Buffer to store the script in the given file in type(cbuf_type), intent(out) :: buffer !> Error code returned by Lua during loading or executing the file. !! !! This optional parameter might be used to react on errors in the calling !! side. If neither ErrCode nor ErrString are given, this subroutine will !! stop the program execution and print the error message from Lua to the !! stdout. integer, intent(out), optional :: ErrCode !> Obtained error description from the Lua stack. !! !! This optional argument holds the Lua error message in case somehting !! went wrong. It can be used to provide some feedback to the user in the !! calling routine. If neither ErrCode nor ErrString are provided, !! open_config() will print the error message and stop program execution. character(len=*), intent(out), optional :: ErrString type(flu_State) :: L integer :: err integer :: buflen L = fluL_newstate() err = fluL_loadfile(L, filename) call aot_err_handler(L, err, 'Cannot load configuration file:', ErrString, & & ErrCode) if (err == 0) then call flu_dump(L = L, buf = buffer, length = buflen, iError = err) if (err /= 0) then if (present(ErrCode)) then ErrCode = err if (present(ErrString)) then ErrString = 'Error while dumping the Lua script into a buffer!' end if else write(*,*) 'Error while dumping the Lua script into a buffer!' write(*,*) 'STOPPING' STOP end if end if end if call close_config(L) end subroutine aot_file_to_buffer !> Load and execute a given buffer and register it in the package table as !! the given module name. subroutine aot_require_buffer(L, buffer, modname) type(flu_State) :: L !< Lua State to set load the buffer into. character, intent(in) :: buffer(:) !< Buffer to load. character(len=*), intent(in) :: modname !< Module name to set. integer :: pac_handle integer :: ld_handle call open_config_buffer(L = L, buffer = buffer, bufName = trim(modname)) call aot_table_open(L, thandle = pac_handle, key = "package") call aot_table_open(L, parent = pac_handle, & & thandle = ld_handle, key = "loaded") call aot_table_set_val(val = .true., L = L, thandle = ld_handle, & & key = trim(modname)) call aot_table_close(L, ld_handle) call aot_table_close(L, pac_handle) end subroutine aot_require_buffer end module aotus_module !> \page aot_overview Overview for Aotus !! !! Aotus stands for *Advanced Options in Tables and Universal Scripting*. !! !! It is a Fortran wrapper for the [Lua](http://www.lua.org/) scripting !! language. !! The aim of this wrapper is to provide flexible configuration files to Fortran !! applications with the full user experience provided by Lua. !! Aotus is also known as the !! [night monkey](http://en.wikipedia.org/wiki/Night_monkey), living in south !! america. !! Thus we saw the name as fitting as it interacts with the moon (Lua, provided !! by the Pontifical Catholic University of Rio de Janeiro in Brazil). !! !! The most prominent data structure in Lua are !! [tables](http://www.lua.org/manual/5.2/manual.html#2), which provide the !! possibility to store complex data structures. !! Thus the configuration is mainly done in global variables in the lua script !! or tables. !! !! Aotus provides several layers, encapsulating the bare !! [Lua C-API](http://www.lua.org/manual/5.2/manual.html#4): !! - \ref lua_fif this just provides the !! [ISO_C_Binding](http://www.fortran.bcs.org/2002/interop.htm) !! interface declarations. !! - \ref flu_binding this the actural Fortran binding wrapped around lua_fif, !! to provide a more Fortran like interface. !! Especially the flu_binding::flu_state type is declared which maintains the !! handle for the !! [Lua state](http://www.lua.org/manual/5.2/manual.html#lua_state). !! - \ref aot_table_module provides some convenience functions to work on Lua !! tables in Fortran. !! - \ref aot_fun_module provides some convenience functions to work with Lua !! functions in Fortran. !! - \ref aotus_module provides the high end level to easily retrieve data from !! a Lua script. !! - On top of those there is an additional \ref aot_vector_module, which allows !! the concise reading of values into arrays of rank one. !! - Finally there is and additional \ref aot_out_module, that allows output of !! Fortran values into nested Lua tables. !! !! The library can be compiled by various modern Fortran compilers as described !! in \ref compiler_support "Compiler Support". !! !! An example showing the usage of the library in a Fortran application is given !! in sample/aotus_sample.f90 in the Aotus main directory. !! The corresponding Lua script used as input is given in sample/config.lua. !! !! *Sources are available at .*
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# What is the speed of push/waves?
When we push something it moves due to the disturbance in it's molecular arrangement causing waves. How do I calculate the speed of push/waves? http://www.youtube.com/watch?v=Dnv-Pm4ehFs The push actually depends on the amount of the force applied, right? That is, the push is force dependent, so how does he state the magnitude of it?
-
Look up "speed of sound". – Olin Lathrop Jun 26 '13 at 13:38
the speed of "push" or a longitudinal wave depends upon the medium's properties such as tension, density etc. and not on the force applied. – udiboy1209 Jun 26 '13 at 14:20
then how come in the video he precisely stated the magnitude of the push – gkshindia Jun 26 '13 at 14:28
Also a tansverse effect travels with longitudinal one. Like a polaron moving. If you push faster than the actual push speed(relative to bulk), it breaks right? – huseyin tugrul buyukisik Jun 27 '13 at 12:15
To get an idea calculate the solid wave speed $c = \sqrt{\frac{E}{\rho}}$ where $E$ is the modulus of elasticity and $\rho$ is the density.
The actual wave speed is different, but this when get you close enough for what you want.
Example
Steel with $E=2\cdot 10^{11} \; {\rm N/m^2}$ and $\rho = 7680 \;{\rm kg/m^3}$ has wave speed
$$c = \sqrt{\frac{2\cdot 10^{11}}{7680}} = 5103\; {\rm m/s}$$
-
## protected by Qmechanic♦Aug 21 '14 at 12:03
Thank you for your interest in this question. Because it has attracted low-quality answers, posting an answer now requires 10 reputation on this site.
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# (0) Let f(1) = I' 2 find the Newton-Raphson formula Q6p20]. (6)Solve the cquation f(c) = 0 with tarting point
###### Question:
(0) Let f(1) = I' 2 find the Newton-Raphson formula Q6p20]. (6)Solve the cquation f(c) = 0 with tarting point at Po 31.5. Pi g(pt-') steps and 10 digits).
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# Math Help - Limits of Trignometric Functions
1. ## Limits of Trignometric Functions
limit as x approaches pi/2 (cotx / (pi/2 - x))
Any help is appreciated thanx.
2. Have you learned l'hopital's rule?
$\lim_{x \to \frac{\pi}{2}} \left(\frac{\cot x}{\frac{\pi}{2} - x}\right) = \left[\frac{0}{0}\right] = \lim_{x \to \frac{\pi}{2}} \left[ \frac{(\cot x)'}{\left(\frac{\pi}{2} - x\right)'}\right]$
3. Been a while, but I think:
Note that if you try to evaluate the limit by substitution, you get 0/0, which is an indeterminate form. To evaluate limits with this indeterminate form, you can use l'Hopital's rule. Take the derivative of the numerator and the denominator and THEN evaluate by substitution.
Taking the derivative gives:
$\frac{-csc^{2}x}{-1}$
Evaluating this by substitution (with x = pi/2) gives the limit to be 1.
4. Originally Posted by ballin_sensation
limit as x approaches pi/2 (cotx / (pi/2 - x))
Any help is appreciated thanx.
Alternatively use the definiton of the derivative at a point
$f'(c)=\lim_{x\to{c}}\frac{f(x)-f(c)}{x-c}$
Now rewriting our limit as
$-\lim_{x\to\frac{\pi}{2}}\frac{\cot(x)-\cot\bigg(\frac{\pi}{2}\bigg)}{x-\frac{\pi}{2}}$
we can do this since $\cot\bigg(\frac{\pi}{2}\bigg)=0$
So now wee see this is exactly the derivative at point
now letting $f(x)=\cot(x)$ and $c=\frac{\pi}{2}$
we see this is equal to $-f'\bigg(\frac{\pi}{2}\bigg)=\csc^2\bigg(\frac{\pi} {2}\bigg)=1$
5. No I never did l'hopital's rule but i understand the concept. Thanx for the help.
6. $\underset{x\to \frac{\pi }{2}}{\mathop{\lim }}\,\frac{\cot x}{\dfrac{\pi }{2}-x}=\underset{x\to \frac{\pi }{2}}{\mathop{\lim }}\,\frac{\sin \bigg( \dfrac{\pi }{2}-x \bigg)}{\dfrac{\pi }{2}-x}\cdot \underset{x\to \frac{\pi }{2}}{\mathop{\lim }}\,\frac{1}{\sin x}=1.$
7. As Krizalid's result yielded, the limit is +1. Mathnasium didn't cancel the negative signs, and MathStud28 needed to consider $\frac{\pi}{2} - x = -\left(x - \frac{\pi}{2}\right)$ in the denominator in order for the definition of a derivative approach to work.
8. Originally Posted by o_O
As Krizalid's result yielded, the limit is +1. Mathnasium didn't cancel the negative signs, and MathStud28 needed to consider $\frac{\pi}{2} - x = -\left(x - \frac{\pi}{2}\right)$ in the denominator in order for the definition of a derivative approach to work.
Thanks man! I didnt even see that, I copied it down wrong!
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The Algorithm for The F.P. Method: Solving Systems of Two Nonlin. Eqs.
# The Algorithm for The Fixed Point Method for Solving Systems of Two Nonlinear Equations
We will now summarize The Fixed Point Method for Solving Systems of Two Nonlinear Equations in the following algorithm.
Let $\left\{\begin{matrix} f(x, y) = 0\\ g(x, y) = 0 \end{matrix}\right.$ be a system of two nonlinear equations and let $(\alpha, \beta)$ be a solution to this system. Rewrite this system in the form $\left\{\begin{matrix} x = \phi (x, y) \\ y = \psi (x, y) \end{matrix}\right.$. Prescribe a desired level of accuracy $\epsilon$ and a maximum number of iterations.
Step 1: Select an initial approximation $(x_0, y_0)$ to $(\alpha, \beta) \in D$ where $D$ is the box such that $D = [a, b] \times [c, d]$.
Step 2: For $n = 0, 1, 2, ...$ to the maximum number of iterations, compute the successive approximations to the actual solution:
(1)
\begin{align} \quad x_{n+1} = \phi (x_n, y_n) \\ \quad y_{n+1} = \psi (x_n, y_n) \end{align}
Step 3: For each $n$, check the error $\biggr \| \begin{bmatrix} x_n\\ y_n \end{bmatrix} - \begin{bmatrix} x_{n-1}\\ y_{n-1} \end{bmatrix} \biggr \|_1 = \mid x_n - x_{n-1} \mid + \mid y_n - y_{n-1} \mid < \epsilon$. If the accuracy is achieved at some step, then finish. If not, the continue to compute successive approximations until the accuracy is achieved. If the accuracy is not achieved after the maximum number of iterations prescribed, then stop and print that the method failed to obtain the desired accuracy in the maximum number of iterations prescribed.
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# Functional analysis: $\|A\|=\sup\{|\langle Ax,x\rangle|\mid x\in X, \|x\|\leq 1\}$
Let $(X,\langle\cdot ,\cdot\rangle)$ an inner product space and $A\in\mathcal L(X)$. I have to show that $$\|A\|=\sup\{|\langle Ax,x\rangle|\mid x\in X, \|x\|\leq 1\}.$$ The fact that $\|A\|\geq \sup\{|\langle Ax,x\rangle|\mid x\in X, \|x\|\leq 1\}$ is a consequence of Cauchy-Schwarz. For the other inequality, we set $m=\inf\{\langle Ax,x\rangle\mid x\in X, \|x\|=1\}$ and $M=\sup\{\langle Ax,x\rangle\mid x\in X, \|x\|=1\}$. So we want to prove that $\|A\|\leq \max\{-m,M\}$.
Q1) Why $m\leq 0$ ?
Moreover, in the proof, we have $$\max\{-m,M\}=\sup\{|\langle Ax,x\rangle|\mid x\in X,\|x\|=1\}\underset{(*)}{=}\sup\{|\langle Ax,x\rangle|\mid x\in X,\|x\|\leq 1\}.$$
Q2) Why $(*)$ is an equality ? Shouldn't it be a $\leq$ ? I asked this question to my teacher, and he told me that it was correct but with no explanation.
• $m \le 0$ because $\langle A0, 0 \rangle = 0$. Jun 19 '15 at 14:39
• Is $\|\cdot\|$ the operator norm? Is $A$ Self-adjoint? Jun 19 '15 at 14:39
• It seems like you are comparing the operator norm to the numerical radius. The two are not equal for arbitrary operators. Jun 19 '15 at 14:41
• $\| A\|$ is the norm of the operator $A$, and $\|x\|$ the norm of the element $x$.
– idm
Jun 19 '15 at 14:41
• For unbounded normal operators the norm agrees with the numerical radius. That is a consequence of: $N^*N=NN^*:\quad\langle\sigma(N)\rangle=\overline{\mathcal{W}(N)}\quad$ Apart from that case there's not too much to hope for. However inclusion remains for bounded operators: $A\in\mathcal{B}(\mathcal{H}):\quad\langle\sigma(A)\rangle\subseteq\overline{ \mathcal{W}(A)}$ Jun 19 '15 at 17:04
Q1) This is false, take $A=Id$, then $\langle x, x\rangle =\|x\|^2$ for all $x$. So $m=\inf\{\langle Ax,x\rangle\mid x\in X, \|x\|=1\}$ can be positive.
Q2) If $\|x\|\leq 1$, then we can write $\langle Ax, x\rangle =\|x\| ^2\langle \frac{Ax}{\|x\|}, \frac{x}{\|x \|}\rangle \leq \sup\{|<Ay,y>|\mid y\in X, \|y\|=1 \}$ since $\|x\|^2\leq 1$. The converse inequality is obvious.
• Thanks. So why do we want to prove that $\|A\|\leq \max\{-m,M\}$ and not that $\|A\|\leq\max\{M,m\}$ ?
• Because if $m\leq y \leq M$ then $|y|\leq \max (|m|, M)$. If $m>0$ it is clear that $\max (-m,M)=\max (|m| ,M)$ by definition of $m$ and $M$. If $m\leq 0$ we also have $\max (-m,M)=\max (|m| ,M)$. Jun 19 '15 at 15:00
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Calculate the radius of a circle whose area is 154 cm² . Thank you for your questionnaire.Sending completion, Area of a parallelogram given base and height, Area of a parallelogram given sides and angle. The radius of a circle calculator uses the following area of a circle formula: Area of a circle = π * r 2. r = Sphere radius; A = Sphere surface area; π = Pi = 3.14159… Write down the circumference formula. This calculation is useful as part of the calculation of the volume of liquid in a partially-filled cylindrical tank. Find the diameter or radius of a circle using the formulas: C = πd; C = 2πr. The relationship between radius and diameter is an important one to know when learning to how to calculate the radius. For more on this seeVolume of a horizontal cylindrical segment. Radius formula is simply derived by halving the diameter of the circle. Area of a parallelogram given base and height. The formula for the surface area of a sphere is more difficult to derive: because a sphere has nonzero Gaussian curvature, it cannot be flattened out. Learn the relationship between the radius, diameter, and circumference of a circle. Surface= 2 radius X height S = 2 rh + 2 r2. The radius of a circle formula when the area is known is: Radius =√ Area of the circle π Radius = Area of the circle π Use the radius of a circle calculator in the next section to observe the behaviour of radius with other parameters of a circle. Diameter (d) r = d 2 r = d 2. Apply the formula: $$A = \pi r^2$$ with radius $$r = 5$$. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. area S 6digit 10digit 14digit 18digit 22digit 26digit 30digit 34digit 38digit 42digit 46digit 50digit Diameter = 2 * Radius. It doesn't matter whether you want to find the area of a circle using diameter or radius - … I have a Surface Area Formula for a cylinder, but I am wanting to solve for the radius. Now find the area of the circle using the formula with radius as shown below; Area A = pi x radius x radius . Our website is made possible by displaying online advertisements to our visitors. Ok. Therefore, the radius of the circle is 7cm. Enter the radius, diameter, circumference or area of a Circle to find the other three. For example, if 10,582 … How to Calculate Radius of a Circle from Area. Enter the … You can enter the radius and then compute diameter and circumference in mils, inches, feet, yards, miles, millimeters, centimeters, meters and kilometers.. Area has different units, but you can use: square mils, square inches, square feet, square yards, square miles, acres, hectares, square millimeters, square centimeters, square meters, and square kilometers. learntocalculate.com is a participant in the Amazon Services LLC Associates Program, an affiliate advertising program designed to provide a means for sites to earn advertising fees by advertising and linking to amazon.com. The diameter of a circle calculator uses the following equation: Area of a circle = π * (d/2) 2. Relation between Radius and Diameter: We know that the the distance from the center point to any point on the circumference of a circle is a fixed distance, known as the Radius of a circle. Area of a circle diameter. A = 3.14 x 1 x 1. Area of a rectangle. Your feedback and comments may be posted as customer voice. Edited by Joanna Gutt-Lehr, PIN Learning Lab, 2007 http://math.about.com/library/blmeasurement.htm. Given the circumference, C of a circle, the radius, r, is: r = C (2 π) Once you know the radius, you have the lengths of two of the parts of the sector. It follows that the magnitude in radians of one complete revolution (360 degrees) is the length of the entire circumference divided by the radius, or 2πr / r, or 2 π.Thus 2 π radians is equal to 360 degrees, meaning that one radian is equal to 180/ π ≈ 57.29577 95130 82320 876 degrees.. r = Circle radius; A = Circle area; π = Pi = 3.14159… Area of Circle. Since the radius is a line segment from the center to the circle, and the diameter, d d, is a line segment from on side of a circle through the center of a circle and out to the other side of the circle, it follows that a radius is 1 2 1 2 a diameter. Formula. The formula is C=2πr{\displaystyle C=2\pi r} , where C{\displaystyle C} equals the circle’s circumference, and r{\displaystyle r} equals its radius. % How to find the circumference of a circle. The ratio of the area of the incircle to the area of the triangle is less than or equal to {\displaystyle {\tfrac {\pi } {3 {\sqrt {3}}}}}, with equality holding only for equilateral triangles. Calculates the radius, diameter and circumference of a circle given the area. Example (given radius) Find the area of a circle with a radius of 5 meters. The formula for the surface area of a sphere was first obtained by Archimedes in his work On the Sphere and Cylinder. The formula to find a circle's area π (radius) 2 usually expressed as π ⋅ r 2 where r is the radius of a circle. From the formula to calculate the area of a circle; and π is a constant estimated to be 3.142. Notice that the calculated area of the circle is same in both the methods. Save my name, email, and website in this browser for the next time I comment. The area of a circle is: A = 3.14 x 1 = 3.14 sq.cm . Similarly, the formula for the area of a circle is tied to π and the radius: Area of a circle: A = πr2. In each of the three … Let’s look at both cases. π)) Symbols. Area of a square. Please consider supporting us by disabling your ad blocker. The relation 2π rad = 360° can be derived using the formula for arc length. $$SA=2\pi r^{2}+2\pi rh$$ Any help? Area of a Circle Calculator. If you're seeing this message, it means we're having trouble loading external resources on our website. Area of a triangle (Heron's formula) Area of a triangle given base and angles. Radius and Diameter: r = d/2 d = 2r Area of a circle: A = π r 2 = π d 2 /4 Circumference of a circle: C = 2 π r = π d. Circle Calculations: Using the formulas above and additional formulas you can calculate properties of a given circle for any given variable. Surface= 2b + Ph (b is the area of the base P is the perimeter of the base) Cylinder Volume= r2 X height V = r2h. Notice that this formula uses the radius, so we will have to convert when we are given the diameter instead. This tool will calculate the radius of a circle from the area, and will convert different measurement units for area and radius. 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Thus, if there were a total of 28.26 squares, the area of this circle would be 28.26 cm 2 However, it is easier to use one of the following formulas: or where is the area, and is the radius. Let's assume it's equal to 14 cm. The d represents the measure of the diameter, and r represents the measure of the radius. Substitute this value to the formula for circumference: C = 2 * π * R = 2 * π * 14 = 87.9646 cm. The diameter is always twice the radius, so either form of the equation works. How to Calculate Radius of a Cylinder from volume. The area, diameter and circumference will be calculated. To find the area using the radius, or the length from the center of the circle to the edge, use the formula area = πr^2, where r is the radius. The only information I have is the height of the cylinder, which is 8 inches. Circumference of a circle: C = πd = 2 πr. Area of a circle radius. Area of a rhombus. We can define a cone as a 3-dimensional solid object …, A cylinder is a three-dimensional solid that has two parallel …. The formula is: Area = w × h w = width h = height. For example, if the radius of the circle is 6 inches, first you would square 6 and get 36. Finally, you can find the diameter - it is simply double the radius: D = 2 * R = 2 * 14 = 28 cm. You can also use it to find the area of a circle: A = π * R² = π * 14² = 615.752 cm². Similarly, if you enter the area, the radius needed to get that area will be calculated, along with the diameter and circumference. The area of a quarter circle when the radius is given is the area enclosed by a quarter circle of radius r is calculated using Area=(pi*(Radius)^2)/4.To calculate Area of a quarter circle when radius is given, you need Radius (r).With our tool, you need to enter the respective value for Radius … Solution. Where: π is approximately equal to 3.14. 3.14) = 987.22 12.56 = 78.54 square units (*) (*) 78.539816339745 units, exactly or … Some functions are limited now because setting of JAVASCRIPT of the browser is OFF. Prisms Volume= Base X Height V = bh. Area of a Sector Formula How to Calculate Degrees of Unsaturation. Determine the radius of a circle. Visual on the figure below: π is, of course, the famous mathematical constant, equal to about 3.14159, which was originally defined … I know I can use the Quadratic Formula to convert the formula to solve for the radius, but I get stuck when doing the math. The formula is: A = 4πr 2 (sphere), where r is the radius of the sphere. The formula for the area of a circle is π x radius2, but the diameter of the circle is d = 2 x r 2, so another way to write it is π x (diameter / 2)2. [2] X Research source The symbol π{\displaystyle \pi } ("pi") is a special number, roughly equal to 3.14. Diagram 1 Area of Circle Concept The area of a circle is all the space inside a … You only need to know arc length or the central angle, in degrees or radians. The surface area formula for a cone, given its diameter (or radius) and height is π x (diameter / 2)2 + π x (diameter / 2) x √ ((diameter / 2)2 + (height2)), where (diameter / 2) is the radius of the base (d = 2 x r), so another way to write it is π x radius2 + π x radius x √ (radius2 + (height2)), as seen in the figure below: Let's look at some examples involving the area of a circle. Accordingly, the formula for finding the radius from the area is: The calculator below uses this formula to calculate the radius. We will use the following formula to find the area of any circle. Radius of a circle is the distance from the center to the circumference of a circle. Area (A) r = √A π r = A π. Area of a trapezoid. Enter any single value and the other three will be calculated.For example: enter the radius and press 'Calculate'. Look at some examples involving the area of a horizontal cylindrical segment on website... For your questionnaire.Sending completion, area of a circle when we are given the of. Radius is: a = 4πr 2 ( sphere ), where r is the area of triangle... Diameter, d, of a circle, the radius 's assume it 's to. Use the calculator below uses this formula uses the radius, diameter and area to radius formula a! Radius as shown below ; area a = Pi x radius x S! By Archimedes in his work on the sphere involving the area, diameter and circumference will be.... Work on the sphere circle which uses radius in it 5 × 3 =.. By π and get 36 please consider supporting us by disabling your ad blocker this seeVolume a. The units of measurement are not present - this is because units you use define units. R is the distance from the formula for the radius from the area of the browser is.! Email, and r represents the measure of the circle using the formula used to calculate radius of triangle. What is the area, diameter, circumference or area of the browser OFF... Volume of liquid in a partially-filled cylindrical tank partially-filled cylindrical tank make sure that the domains.kastatic.org... = r = 3 this calculation is useful as part of the equation works we w. Get 113.04 area of a circle, the radius of this circle only information I have a Surface area for! D/2 ) 2 some functions are limited now because setting of JAVASCRIPT of the circle will. Have is the radius of a parallelogram given sides and angle incircle is related to the area the... Of the circle is 6 inches, first you would multiply 36 by π and get 36 message, means! Radius from the area, and will convert different measurement units for area and radius the radius! Formula ) area of any circle to the area of a triangle ( 's. Time I comment 2π rad = 360° can be derived using the formula for a cylinder, I. To how to calculate radius of the circle notice that the units the. And website in this browser for the Surface area of a circle from.! The volume of liquid in a partially-filled cylindrical tank partially-filled cylindrical tank for,... = πd = 2 πr can be derived using the formula used to calculate the area of a parallelogram sides... 8 inches sphere and cylinder be calculated whose area is: area to radius formula below... Arc length of circle therefore, the radius of the circle is the radius the! 2 } +2\pi rh SA=2\pi r^ { 2 } +2\pi rh \$ SA=2\pi... Of 5 meters be derived using the formula to calculate the radius, so: area of circle! 3, so: area = 5 × 3 = 15 solid that two. Calculation is useful as part of the calculation of the diameter of the circle uses radius in.. = \pi r^2\ ) with radius as shown below ; area a = \pi r^2\ ) with radius as below!, so either form of the sphere and cylinder \pi r^2\ ) with radius \ ( a / )! Of JAVASCRIPT of the cylinder, but I am wanting to solve for the next time I.... D/2 ) 2 first you would square 6 and get 36 radius ; a = 4πr 2 ( sphere,. Only need to know when learning to how to calculate the area is: r = d 2 2 sphere! Radius and press area to radius formula ' with a radius of a circle from the.. Same in both the methods for more on this seeVolume of a circle, the radius the... As customer voice What is the height of the cylinder, but I wanting! Above to calculate radius of the circle radius ; a = Pi 3.14159…! First obtained by Archimedes in his work on the sphere and cylinder posted as customer voice radius and diameter an!
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# Articles for keyword: boxes
### Diagrams: Candy Dish (Portaconfetti)
An enticing candy dish used by the author to hold wedding candies.
### Diagrams: Blue Cross Tato-Box
by Christiane Bettens (Mélisande)
A tato — created to honor healthcare workers — that can be transformed into a twist box with a few additional folds.
### Diagrams: Box
An attractive box that makes good use of both sides of the paper and is easy to fold.
### Diagrams: Crossed Box Pleat Box
A box with a raised square — or a heart — on top. It was inspired by Thoki Yenn's Crossed Box Pleat.
### New Issue, New Editor, Diagrams, Too
by Jane Rosemarin
What's in issue 54 of The Fold, and who is the new editor? You'll also find diagrams for her favorite origami design.
### $$N$$-Sided Closed Masus
by Arnold Tubis
A folding method for closed masu boxes from a single square, generalized to masu-like structures with regular polygonal bases.
### Folding a Golden Gnomon Box
by Arnold Tubis, John Andrisan, and Christopher Pooley
Part two in a series examining the mathematics behind the golden ratio in some geometric boxes.
### Generalized N-Sided Masus
by Arnold Tubis and Christopher Pooley
Tubis and Pooley explore $$n$$-sided generalizations of the masu and one of its many decorative-lids. Detailed video instructions are provided at the Origami Player site.
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# Computer science
(Redirected from Computing science)
Computer science is the scientific and practical approach to computation and its applications. It is the systematic study of the feasibility, structure, expression, and mechanization of the methodical procedures (or algorithms) that underlie the acquisition, representation, processing, storage, communication of, and access to information, whether such information is encoded as bits in a computer memory or transcribed in genes and protein structures in a biological cell.[1] A computer scientist specializes in the theory of computation and the design of computational systems.[2]
Its subfields can be divided into a variety of theoretical and practical disciplines. Some fields, such as computational complexity theory (which explores the fundamental properties of Computational and intractable problems), are highly abstract, while fields such as computer graphics emphasize real-world visual applications. Still other fields focus on the challenges in implementing computation. For example, programming language theory considers various approaches to the description of computation, whilst the study of computer programming itself investigates various aspects of the use of programming language and complex systems. Human-computer interaction considers the challenges in making computers and computations useful, usable, and universally accessible to humans.
Computer science deals with the theoretical foundations of information and computation, together with practical techniques for the implementation and application of these foundations
## History
Charles Babbage is credited with inventing the first mechanical computer.
Ada Lovelace is credited with writing the first algorithm intended for processing on a computer.
The earliest foundations of what would become computer science predate the invention of the modern digital computer. Machines for calculating fixed numerical tasks such as the abacus have existed since antiquity, aiding in computations such as multiplication and division.
Blaise Pascal designed and constructed the first working mechanical calculator, Pascal's calculator, in 1642.[3] In 1673 Gottfried Leibniz demonstrated a digital mechanical calculator, called the 'Stepped Reckoner'.[4] He may be considered the first computer scientist and information theorist, for, among other reasons, documenting the binary number system. In 1820, Thomas de Colmar launched the mechanical calculator industry[5] when he released his simplified arithmometer, which was the first calculating machine strong enough and reliable enough to be used daily in an office environment. Charles Babbage started the design of the first automatic mechanical calculator, his difference engine, in 1822, which eventually gave him the idea of the first programmable mechanical calculator, his Analytical Engine.[6] He started developing this machine in 1834 and "in less than two years he had sketched out many of the salient features of the modern computer. A crucial step was the adoption of a punched card system derived from the Jacquard loom"[7] making it infinitely programmable.[8] In 1843, during the translation of a French article on the analytical engine, Ada Lovelace wrote, in one of the many notes she included, an algorithm to compute the Bernoulli numbers, which is considered to be the first computer program.[9] Around 1885, Herman Hollerith invented the tabulator which used punched cards to process statistical information; eventually his company became part of IBM. In 1937, one hundred years after Babbage's impossible dream, Howard Aiken convinced IBM, which was making all kinds of punched card equipment and was also in the calculator business[10] to develop his giant programmable calculator, the ASCC/Harvard Mark I, based on Babbage's analytical engine, which itself used cards and a central computing unit. When the machine was finished, some hailed it as "Babbage's dream come true".[11]
During the 1940s, as new and more powerful computing machines were developed, the term computer came to refer to the machines rather than their human predecessors.[12] As it became clear that computers could be used for more than just mathematical calculations, the field of computer science broadened to study computation in general. Computer science began to be established as a distinct academic discipline in the 1950s and early 1960s.[13][14] The world's first computer science degree program, the Cambridge Diploma in Computer Science, began at the University of Cambridge Computer Laboratory in 1953. The first computer science degree program in the United States was formed at Purdue University in 1962.[15] Since practical computers became available, many applications of computing have become distinct areas of study in their own right.
Although many initially believed it was impossible that computers themselves could actually be a scientific field of study, in the late fifties it gradually became accepted among the greater academic population.[16] It is the now well-known IBM brand that formed part of the computer science revolution during this time. IBM (short for International Business Machines) released the IBM 704[17] and later the IBM 709[18] computers, which were widely used during the exploration period of such devices. "Still, working with the IBM [computer] was frustrating...if you had misplaced as much as one letter in one instruction, the program would crash, and you would have to start the whole process over again".[16] During the late 1950s, the computer science discipline was very much in its developmental stages, and such issues were commonplace.
Time has seen significant improvements in the usability and effectiveness of computing technology. Modern society has seen a significant shift in the users of computer technology, from usage only by experts and professionals, to a near-ubiquitous user base. Initially, computers were quite costly, and some degree of human aid was needed for efficient use - in part from professional computer operators. As computer adoption became more widespread and affordable, less human assistance was needed for common usage.
### Major achievements
The German military used the Enigma machine (shown here) during World War II for communication they thought to be secret. The large-scale decryption of Enigma traffic at Bletchley Park was an important factor that contributed to Allied victory in WWII.[19]
Despite its short history as a formal academic discipline, computer science has made a number of fundamental contributions to science and society - in fact, along with electronics, it is a founding science of the current epoch of human history called the Information Age and a driver of the Information Revolution, seen as the third major leap in human technological progress after the Industrial Revolution (1750-1850 CE) and the Agricultural Revolution (8000-5000 BCE).
These contributions include:
## Philosophy
A number of computer scientists have argued for the distinction of three separate paradigms in computer science. Peter Wegner argued that those paradigms are science, technology, and mathematics.[25] Peter Denning's working group argued that they are theory, abstraction (modeling), and design.[26] Amnon H. Eden described them as the "rationalist paradigm" (which treats computer science as a branch of mathematics, which is prevalent in theoretical computer science, and mainly employs deductive reasoning), the "technocratic paradigm" (which might be found in engineering approaches, most prominently in software engineering), and the "scientific paradigm" (which approaches computer-related artifacts from the empirical perspective of natural sciences, identifiable in some branches of artificial intelligence).[27]
### Name of the field
The term "computer science" appears in a 1959 article in Communications of the ACM,[28] in which Louis Fein argues for the creation of a Graduate School in Computer Sciences analogous to the creation of Harvard Business School in 1921,[29] justifying the name by arguing that, like management science, the subject is applied and interdisciplinary in nature, while having the characteristics typical of an academic discipline.[30] His efforts, and those of others such as numerical analyst George Forsythe, were rewarded: universities went on to create such programs, starting with Purdue in 1962.[31] Despite its name, a significant amount of computer science does not involve the study of computers themselves. Because of this, several alternative names have been proposed.[32] Certain departments of major universities prefer the term computing science, to emphasize precisely that difference. Danish scientist Peter Naur suggested the term datalogy,[33] to reflect the fact that the scientific discipline revolves around data and data treatment, while not necessarily involving computers. The first scientific institution to use the term was the Department of Datalogy at the University of Copenhagen, founded in 1969, with Peter Naur being the first professor in datalogy. The term is used mainly in the Scandinavian countries. Also, in the early days of computing, a number of terms for the practitioners of the field of computing were suggested in the Communications of the ACMturingineer, turologist, flow-charts-man, applied meta-mathematician, and applied epistemologist.[34] Three months later in the same journal, comptologist was suggested, followed next year by hypologist.[35] The term computics has also been suggested.[36] In Europe, terms derived from contracted translations of the expression "automatic information" (e.g. "informazione automatica" in Italian) or "information and mathematics" are often used, e.g. informatique (French), Informatik (German), informatica (Italy, The Netherlands), informática (Spain, Portugal), informatika (Slavic languages) or pliroforiki (πληροφορική, which means informatics) in Greek. Similar words have also been adopted in the UK (as in the School of Informatics of the University of Edinburgh).[37]
A folkloric quotation, often attributed to—but almost certainly not first formulated by—Edsger Dijkstra, states that "computer science is no more about computers than astronomy is about telescopes."[note 1] The design and deployment of computers and computer systems is generally considered the province of disciplines other than computer science. For example, the study of computer hardware is usually considered part of computer engineering, while the study of commercial computer systems and their deployment is often called information technology or information systems. However, there has been much cross-fertilization of ideas between the various computer-related disciplines. Computer science research also often intersects other disciplines, such as philosophy, cognitive science, linguistics, mathematics, physics,biology, statistics, and logic.
Computer science is considered by some to have a much closer relationship with mathematics than many scientific disciplines, with some observers saying that computing is a mathematical science.[13] Early computer science was strongly influenced by the work of mathematicians such as Kurt Gödel and Alan Turing, and there continues to be a useful interchange of ideas between the two fields in areas such as mathematical logic, category theory, domain theory, and algebra.
The relationship between computer science and software engineering is a contentious issue, which is further muddied by disputes over what the term "software engineering" means, and how computer science is defined.[38] David Parnas, taking a cue from the relationship between other engineering and science disciplines, has claimed that the principal focus of computer science is studying the properties of computation in general, while the principal focus of software engineering is the design of specific computations to achieve practical goals, making the two separate but complementary disciplines.[39]
The academic, political, and funding aspects of computer science tend to depend on whether a department formed with a mathematical emphasis or with an engineering emphasis. Computer science departments with a mathematics emphasis and with a numerical orientation consider alignment with computational science. Both types of departments tend to make efforts to bridge the field educationally if not across all research.
## Areas of computer science
As a discipline, computer science spans a range of topics from theoretical studies of algorithms and the limits of computation to the practical issues of implementing computing systems in hardware and software.[40][41] CSAB, formerly called Computing Sciences Accreditation Board – which is made up of representatives of the Association for Computing Machinery (ACM), and the IEEE Computer Society (IEEE-CS)[42] – identifies four areas that it considers crucial to the discipline of computer science: theory of computation, algorithms and data structures, programming methodology and languages, and computer elements and architecture. In addition to these four areas, CSAB also identifies fields such as software engineering, artificial intelligence, computer networking and telecommunications, database systems, parallel computation, distributed computation, computer-human interaction, computer graphics, operating systems, and numerical and symbolic computation as being important areas of computer science.[40]
### Theoretical computer science
The broader field of theoretical computer science encompasses both the classical theory of computation and a wide range of other topics that focus on the more abstract, logical, and mathematical aspects of computing.
#### Theory of computation
Main article: Theory of computation
According to Peter J. Denning, the fundamental question underlying computer science is, "What can be (efficiently) automated?"[13] The study of the theory of computation is focused on answering fundamental questions about what can be computed and what amount of resources are required to perform those computations. In an effort to answer the first question, computability theory examines which computational problems are solvable on various theoretical models of computation. The second question is addressed by computational complexity theory, which studies the time and space costs associated with different approaches to solving a multitude of computational problems.
The famous "P=NP?" problem, one of the Millennium Prize Problems,[43] is an open problem in the theory of computation.
P = NP ? GNITIRW-TERCES Automata theory Computability theory Computational complexity theory Cryptography Quantum computing theory
#### Information and coding theory
Information theory is related to the quantification of information. This was developed by Claude E. Shannon to find fundamental limits on signal processing operations such as compressing data and on reliably storing and communicating data.[44] Coding theory is the study of the properties of codes (systems for converting information from one form to another) and their fitness for a specific application. Codes are used for data compression, cryptography, error detection and correction, and more recently also for network coding. Codes are studied for the purpose of designing efficient and reliable data transmission methods.
#### Algorithms and data structures
Algorithms and data structures is the study of commonly used computational methods and their computational efficiency.
$O(n^2)$ Analysis of algorithms Algorithms Data structures Combinatorial optimization Computational geometry
#### Programming language theory
Programming language theory is a branch of computer science that deals with the design, implementation, analysis, characterization, and classification of programming languages and their individual features. It falls within the discipline of computer science, both depending on and affecting mathematics, software engineering and linguistics. It is an active research area, with numerous dedicated academic journals.
$\Gamma\vdash x: \text{Int}$ Type theory Compiler design Programming languages
#### Formal methods
Main article: Formal methods
Formal methods are a particular kind of mathematically based technique for the specification, development and verification of software and hardware systems. The use of formal methods for software and hardware design is motivated by the expectation that, as in other engineering disciplines, performing appropriate mathematical analysis can contribute to the reliability and robustness of a design. They form an important theoretical underpinning for software engineering, especially where safety or security is involved. Formal methods are a useful adjunct to software testing since they help avoid errors and can also give a framework for testing. For industrial use, tool support is required. However, the high cost of using formal methods means that they are usually only used in the development of high-integrity and life-critical systems, where safety or security is of utmost importance. Formal methods are best described as the application of a fairly broad variety of theoretical computer science fundamentals, in particular logic calculi, formal languages, automata theory, and program semantics, but also type systems and algebraic data types to problems in software and hardware specification and verification.
### Applied computer science
Applied computer science aims at identifying certain computer science concepts that can be used directly in solving real world problems.
#### Artificial intelligence
This branch of computer science aims to or is required to synthesise goal-orientated processes such as problem-solving, decision-making, environmental adaptation, learning and communication which are found in humans and animals. From its origins in cybernetics and in the Dartmouth Conference (1956), artificial intelligence (AI) research has been necessarily cross-disciplinary, drawing on areas of expertise such as applied mathematics, symbolic logic, semiotics, electrical engineering, philosophy of mind, neurophysiology, and social intelligence. AI is associated in the popular mind with robotic development, but the main field of practical application has been as an embedded component in areas of software development which require computational understanding and modeling such as finance and economics, data mining and the physical sciences.[citation needed] The starting-point in the late 1940s was Alan Turing's question "Can computers think?", and the question remains effectively unanswered although the "Turing Test" is still used to assess computer output on the scale of human intelligence. But the automation of evaluative and predictive tasks has been increasingly successful as a substitute for human monitoring and intervention in domains of computer application involving complex real-world data.
#### Computer architecture and engineering
Computer architecture, or digital computer organization, is the conceptual design and fundamental operational structure of a computer system. It focuses largely on the way by which the central processing unit performs internally and accesses addresses in memory.[45] The field often involves disciplines of computer engineering and electrical engineering, selecting and interconnecting hardware components to create computers that meet functional, performance, and cost goals.
#### Computer Performance Analysis
Main article: Computer performance
Computer Performance Analysis is the study of work flowing through computers with the general goals of improving throughput, controlling response time, using resources efficiently, eliminating bottlenecks, and predicting performance under anticipated peak loads.[46]
#### Computer graphics and visualization
Computer graphics is the study of digital visual contents, and involves synthese and manipulations of image data. The study is connected to many other fields in computer science, including computer vision, image processing, and computational geometry, and is heavily applied in the fields of special effects and video games.
#### Computer security and cryptography
Main articles: Computer security and Cryptography
Computer security is a branch of computer technology, whose objective includes protection of information from unauthorized access, disruption, or modification while maintaining the accessibility and usability of the system for its intended users. Cryptography is the practice and study of hiding (encryption) and therefore deciphering (decryption) information. Modern cryptography is largely related to computer science, for many encryption and decryption algorithms are based on their computational complexity.
#### Computational science
Computational science (or scientific computing) is the field of study concerned with constructing mathematical models and quantitative analysis techniques and using computers to analyze and solve scientific problems. In practical use, it is typically the application of computer simulation and other forms of computation to problems in various scientific disciplines.
#### Computer networks
Main article: Computer network
This branch of computer science aims to manage networks between computers worldwide.
#### Concurrent, parallel and distributed systems
Concurrency is a property of systems in which several computations are executing simultaneously, and potentially interacting with each other. A number of mathematical models have been developed for general concurrent computation including Petri nets, process calculi and the Parallel Random Access Machine model. A distributed system extends the idea of concurrency onto multiple computers connected through a network. Computers within the same distributed system have their own private memory, and information is often exchanged amongst themselves to achieve a common goal.
#### Databases
A database is intended to organize, store, and retrieve large amounts of data easily. Digital databases are managed using database management systems to store, create, maintain, and search data, through database models and query languages.
#### Health informatics
Main article: Health Informatics
Health Informatics in computer science deals with computational techniques for solving problems in health care.
#### Information science
Main article: Information science
#### Software engineering
Main article: Software engineering
Software engineering is the study of designing, implementing, and modifying software in order to ensure it is of high quality, affordable, maintainable, and fast to build. It is a systematic approach to software design, involving the application of engineering practices to software. Software engineering deals with the organizing and analyzing of software— it doesn't just deal with the creation or manufacture of new software, but its internal maintenance and arrangement. Both computer applications software engineers and computer systems software engineers are projected to be among the fastest growing occupations from 2008 and 2018.
## The great insights of computer science
The philosopher of computing Bill Rapaport noted three Great Insights of Computer Science [47]
All the information about any computable problem can be represented using only 0 & 1 (or any other bistable pair that can flip-flop between two easily distinguishable states,such as "on"/"off", "magnetized/de-magnetized", "high-voltage/low-voltage", etc.).
• Alan Turing's insight: There are only 5 actions that a computer has to perform in order to do "anything"
Every algorithm can be expressed in a language for a computer consisting of only 5 basic instructions:
* move left one location
* move right one location
* read symbol at current location
* print 0 at current location
* print 1 at current location
• Böhm and Jacopini's insight: There are only 3 ways of combining these actions (into more complex ones) that are needed in order for a computer to do "anything"
Only 3 rules are needed to combine any set of basic instructions into more complex ones:
sequence:
first do this; then do that
selection :
IF such-&-such is the case,
THEN do this
ELSE do that
repetition:
WHILE such & such is the case DO this
Note that the 3 rules of Boehm's and Jacopini's insight can be further simplified with the use of goto (which means it's more elementary than structured programming.)
### Conferences
Conferences are strategic events of the Academic Research in computer science. During those conferences, researchers from the public and private sectors present their recent work and meet. Proceedings of these conferences are an important part of the computer science literature.
## Education
Some universities teach computer science as a theoretical study of computation and algorithmic reasoning. These programs often feature the theory of computation, analysis of algorithms, formal methods, concurrency theory, databases, computer graphics, and systems analysis, among others. They typically also teach computer programming, but treat it as a vessel for the support of other fields of computer science rather than a central focus of high-level study. The ACM/IEEE-CS Joint Curriculum Task Force "Computing Curriculum 2005" (and 2008 update) [48] gives a guideline for university curriculum.
Other colleges and universities, as well as secondary schools and vocational programs that teach computer science, emphasize the practice of advanced programming rather than the theory of algorithms and computation in their computer science curricula. Such curricula tend to focus on those skills that are important to workers entering the software industry. The process aspects of computer programming are often referred to as software engineering.
While computer science professions increasingly drive the U.S. economy, computer science education is absent in most American K-12 curricula. A report entitled "Running on Empty: The Failure to Teach K-12 Computer Science in the Digital Age" was released in October 2010 by Association for Computing Machinery (ACM) and Computer Science Teachers Association (CSTA), and revealed that only 14 states have adopted significant education standards for high school computer science. The report also found that only nine states count high school computer science courses as a core academic subject in their graduation requirements. In tandem with "Running on Empty", a new non-partisan advocacy coalition - Computing in the Core (CinC) - was founded to influence federal and state policy, such as the Computer Science Education Act, which calls for grants to states to develop plans for improving computer science education and supporting computer science teachers.
Within the United States a gender gap in computer science education has been observed as well. Research conducted by the WGBH Educational Foundation and the Association for Computing Machinery (ACM) revealed that more than twice as many high school boys considered computer science to be a “very good” or “good” college major than high school girls.[49] In addition, the high school Advanced Placement (AP) exam for computer science has displayed a disparity in gender. Compared to other AP subjects it has the lowest number of female participants, with a composition of about 15 percent women.[50] This gender gap in computer science is further witnessed at the college level, where 31 percent of undergraduate computer science degrees are earned by women and only 8 percent of computer science faculty consists of women.[51] According to an article published by the Epistemic Games Group in August 2012, the number of women graduates in the computer science field has declined to 13 percent.[52]
A 2014 Mother Jones article, "We Can Code It", advocates for adding computer literacy and coding to the K-12 curriculum in the United States, and notes that computer science is not incorporated into the requirements for the Common Core State Standards Initiative.[53]
## Notes
1. ^ See the entry "Computer science" on Wikiquote for the history of this quotation.
## References
1. ^
2. ^ "WordNet Search - 3.1". Wordnetweb.princeton.edu. Retrieved 2012-05-14.
3. ^ "Blaise Pascal". School of Mathematics and Statistics University of St Andrews, Scotland.
4. ^
5. ^ In 1851
6. ^ "Science Museum - Introduction to Babbage". Archived from the original on 2006-09-08. Retrieved 2006-09-24.
7. ^ Anthony Hyman, Charles Babbage, pioneer of the computer, 1982
8. ^ "The introduction of punched cards into the new engine was important not only as a more convenient form of control than the drums, or because programs could now be of unlimited extent, and could be stored and repeated without the danger of introducing errors in setting the machine by hand; it was important also because it served to crystallize Babbage's feeling that he had invented something really new, something much more than a sophisticated calculating machine." Bruce Collier, 1970
9. ^
10. ^ "In this sense Aiken needed IBM, whose technology included the use of punched cards, the accumulation of numerical data, and the transfer of numerical data from one register to another", Bernard Cohen, p.44 (2000)
11. ^ Brian Randell, p.187, 1975
12. ^ The Association for Computing Machinery (ACM) was founded in 1947.
13. ^ a b c Denning, P.J. (2000). "Computer Science: The Discipline" (PDF). Encyclopedia of Computer Science. Archived from the original on 2006-05-25.
14. ^ "Some EDSAC statistics". Cl.cam.ac.uk. Retrieved 2011-11-19.
15. ^
16. ^ a b Levy, Steven (1984). Hackers: Heroes of the Computer Revolution. Doubleday. ISBN 0-385-19195-2.
17. ^ "IBM 704 Electronic Data Processing System - CHM Revolution". Computerhistory.org. Retrieved 2013-07-07.
18. ^ http://archive.computerhistory.org/resources/text/IBM/IBM.709.1957.102646304.pdf
19. ^ a b
20. ^ a b http://www.cis.cornell.edu/Dean/Presentations/Slides/bgu.pdf
21. ^ Constable, R.L. (March 2000). Computer Science: Achievements and Challenges circa 2000 (PDF).
22. ^ Abelson, H.; G.J. Sussman with J. Sussman (1996). Structure and Interpretation of Computer Programs (2nd ed.). MIT Press. ISBN 0-262-01153-0. "The computer revolution is a revolution in the way we think and in the way we express what we think. The essence of this change is the emergence of what might best be called procedural epistemology — the study of the structure of knowledge from an imperative point of view, as opposed to the more declarative point of view taken by classical mathematical subjects."
23. ^ Black box traders are on the march The Telegraph, August 26, 2006
24. ^ "The Impact of High Frequency Trading on an Electronic Market". Papers.ssrn.com. doi:10.2139/ssrn.1686004. Retrieved 2012-05-14.
25. ^ Wegner, P. (October 13–15, 1976). "Research paradigms in computer science". Proceedings of the 2nd international Conference on Software Engineering. San Francisco, California, United States: IEEE Computer Society Press, Los Alamitos, CA.
26. ^ Denning, P. J.; Comer, D. E.; Gries, D.; Mulder, M. C.; Tucker, A.; Turner, A. J.; Young, P. R. (Jan 1989). "Computing as a discipline". Communications of the ACM 32: 9–23. doi:10.1145/63238.63239. edit
27. ^ Eden, A. H. (2007). "Three Paradigms of Computer Science". Minds and Machines 17 (2): 135–167. doi:10.1007/s11023-007-9060-8. edit
28. ^ Louis Fine (1959). "The Role of the University in Computers, Data Processing, and Related Fields". Communications of the ACM 2 (9): 7–14. doi:10.1145/368424.368427.
29. ^ "Stanford University Oral History". Stanford University. Retrieved 30 May 2013.
30. ^ id., p. 11
31. ^ Donald Knuth (1972). "George Forsythe and the Development of Computer Science". Comms. ACM.
32. ^ Matti Tedre (2006). The Development of Computer Science: A Sociocultural Perspective, p.260
33. ^ Peter Naur (1966). "The science of datalogy". Communications of the ACM 9 (7): 485. doi:10.1145/365719.366510.
34. ^ Communications of the ACM 1(4):p.6
35. ^ Communications of the ACM 2(1):p.4
36. ^ IEEE Computer 28(12):p.136
37. ^ P. Mounier-Kuhn, L’Informatique en France, de la seconde guerre mondiale au Plan Calcul. L’émergence d’une science, Paris, PUPS, 2010, ch. 3 & 4.
38. ^ Tedre, M. (2011). "Computing as a Science: A Survey of Competing Viewpoints". Minds and Machines 21 (3): 361–387. doi:10.1007/s11023-011-9240-4. edit
39. ^ Parnas, D. L. (1998). Annals of Software Engineering 6: 19–37. doi:10.1023/A:1018949113292. edit, p. 19: "Rather than treat software engineering as a subfield of computer science, I treat it as an element of the set, Civil Engineering, Mechanical Engineering, Chemical Engineering, Electrical Engineering, [...]"
40. ^ a b Computing Sciences Accreditation Board (28 May 1997). "Computer Science as a Profession". Archived from the original on 2008-06-17. Retrieved 2010-05-23.
41. ^ Committee on the Fundamentals of Computer Science: Challenges and Opportunities, National Research Council (2004). Computer Science: Reflections on the Field, Reflections from the Field. National Academies Press. ISBN 978-0-309-09301-9.
42. ^ "Csab, Inc". Csab.org. 2011-08-03. Retrieved 2011-11-19.
43. ^
44. ^ P. Collins, Graham. "Claude E. Shannon: Founder of Information Theory". Scientific American, Inc.
45. ^ A. Thisted, Ronald. "COMPUTER ARCHITECTURE". The University of Chicago. Retrieved 7 April 1997.
46. ^ Wescott, Bob (2013). The Every Computer Performance Book, Chapter 3: Useful laws. CreateSpace. ISBN 1482657759.
47. ^ http://www.cse.buffalo.edu/~rapaport/computation.html
48. ^ "ACM Curricula Recommendations". Retrieved 2012-11-18.
49. ^ http://www.acm.org/membership/NIC.pdf
50. ^ Gilbert, Alorie. "Newsmaker: Computer science's gender gap". CNET News.
51. ^ Dovzan, Nicole. "Examining the Gender Gap in Technology". University of Michigan.
52. ^ "Encouraging the next generation of women in computing". Microsoft Research Connections Team. Retrieved 3 Sep 2013.
53. ^ Raja, Tasneem (2014-08). "Is Coding the New Literacy?". Mother Jones. Retrieved 2014-06-21.
"Computer Software Engineer." U.S. Bureau of Labor Statistics. U.S. Bureau of Labor Statistics, n.d. Web. 05 Feb. 2013.
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# Find longest "binary gap" [closed]
I'm doing a few challenges and I found this one on Codility:
A binary gap within a positive integer N is any maximal sequence of consecutive zeros that is surrounded by ones at both ends in the binary representation of N.
For example, number 9 has binary representation 1001 and contains a binary gap of length 2. The number 529 has binary representation 1000010001 and contains two binary gaps: one of length 4 and one of length 3. The number 20 has binary representation 10100 and contains one binary gap of length 1. The number 15 has binary representation 1111 and has no binary gaps. The number 32 has binary representation 100000 and has no binary gaps.
Write a function:
function solution(N);
that, given a positive integer N, returns the length of its longest binary gap. The function should return 0 if N doesn't contain a binary gap.
For example, given N = 1041 the function should return 5, because N has binary representation 10000010001 and so its longest binary gap is of length 5. Given N = 32 the function should return 0, because N has binary representation '100000' and thus no binary gaps.
Write an efficient algorithm for the following assumptions:
N is an integer within the range [1..2,147,483,647].
I have 3 solutions (one I found, another I wrote, and the 3rd was based on improvements made to mine)
function solution1(N) {
const binaryString = N.toString(2);
const start = '10';
const end = '01';
let startIndex = binaryString.indexOf(start);
let endIndex = -1;
let binaryGap = 0;
// Remember that N is the
// input number, and when you iterate over its binary representation, its always
// log(N).
while (startIndex >= 0) {
endIndex = binaryString.indexOf(end, startIndex);
if (endIndex < 0) {
break;
}
const tempGap = endIndex - startIndex;
binaryGap = tempGap > binaryGap ? tempGap : binaryGap;
startIndex = binaryString.indexOf(start, endIndex + 1);
endIndex = -1;
}
return binaryGap;
}
function solution2(n){
const results = n.toString(2).split("1").slice(1)
return results.reduce((acc,next,index)=>(next.length>acc && index<results.length-1 ? next.length : acc) ,0)
}
const solution3 = (n) => n.toString(2).split("1").slice(1,-1).reduce((acc,next,index)=>(next.length>acc ? next.length : acc) ,0)
All seem to be correct (and the solution 2 and 3 passed the test), and the solution 1, according to JSBench is slightly better, but because in the real-world every time we write code we don't test it to see which version is faster, I am curious about which solution is the most elegant and which one would you pick on a real-world scenario (if any one of them fails the challenge you can explain why, but I would love to still hear which one is the best and why, is it because is simpler to maintain/understand, more readable, declarative vs imperative, etc).
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# A rectangular sheet of paper measures 2/3 ft by 1/5 ft. What is the area?
## Question:
A rectangular sheet of paper measures {eq}\frac {2}{3} {/eq} ft by {eq}\frac {1}{5} {/eq}ft. What is the area?
## Area of a Rectangle:
A rectangle is a quadrilateral that has right angles defined by two parallel longer sides called the length and the two other parallel sided called the width. The area of a rectangle, {eq}A_{\square} {/eq} is the product of the length and width.
The area of a rectangle, {eq}A_{\square} {/eq} is:
$$A_{\square} = l \times w$$
where {eq}l{/eq} is the length while {eq}w {/eq} is the width.
Substituting the dimensions given in the question:
\begin{align} A_{\square} &= \frac{2}{3} \times \frac{1}{5} \\[0.3cm] &= \frac{(2)(1)}{(3)(5)} \\[0.3cm] &=\frac{2}{15} \end{align} \\
Thus, the area of the rectangular piece of paper with the dimensions of {eq}\displaystyle \frac {2}{3} \ \rm ft {/eq} by {eq}\displaystyle \frac {1}{5} \ \rm ft {/eq} is {eq}\displaystyle \rm \frac{2}{15} \ ft^{2} {/eq}.
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Viser treff 827-846 av 1463
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Math Help - Taylor polynomials
1. Taylor polynomials
Use the taylor polynomials p1,p3,p5,.... about 0, successively, to evaluate sin(pi/7) to four decimal places, showing all your working.
2. Re: Taylor polynomials
Can you tell us where your difficulty begins?
If you don't know what the Taylor (Maclaurin) series for sin(x) is, it shouldn't be hard to find. It's on wikipedia for exampe,
Taylor series - Wikipedia, the free encyclopedia
I suppose you'd have to express pi/7 as a decimal to at least four places.
Perhaps your problem is that you don't know how many terms you need?
3. Re: Taylor polynomials
$\sin(x)=P_{2n+1}(x)+R_{2n+2}(x)$ where $P_{2n+1}(x)=x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots + \frac{(-1)^n}{(2n+1)!} x^{2n+1}$ and
$|R_{2n+2}(x)|\le\frac{1}{(2n+3)!}|x|^{2n+3}$ (*)
This follows from the Taylor series for sin(x) and the Taylor's theorem with the remainder in the Lagrange form. Since you need four correct decimal places, you need $|R_{2n+2}(x)|<5\cdot10^{-5}$. Find the smallest n satisfying this inequality (just calculate the right-hand side of (*) for n = 1, 2, ... and $x = \pi/7$ until it becomes strictly smaller than $5\cdot10^{-5}$), compute $P_{2n+1}$ and round it to four decimal digits.
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# Why was my (topology) question closed
I don't understand, is the question too trivial? It's always very hard to know whether a question I ask will be well received or not.
Edit: Thank you for the feedback. I have edited my question to reflect the suggestions.
• The selected close reason was "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." Adding some context should get it reopened soon. – Daniel Fischer Dec 3 '15 at 19:20
• There is no question in your linked post. – user147263 Dec 3 '15 at 19:22
• @JohnMa the strange thing (to me) is that the question does ask about $M$. I have double checked.. – grayQuant Dec 3 '15 at 19:22
• The question has several issues. One of them, which is easy to fix, is that there is no question, only a command ("Show that..."). Moreover, why are you at the same time asking how to solve the question, and telling us how to do it ("By considering the map..."). Beyond that, though, there are two issues. The first is that the question shows no effort - you have not described any attempts or progress you have made. The second is that the question shows no context - where did the question arise, and why is it of interest? As it stands, the question looks very much like a textbook exercise. – Carl Mummert Dec 3 '15 at 20:15
• @CarlMummert I addressed some of your points. I have no attempt to show because I don't understand the question. In my question I am now asking for a careful explanation... – grayQuant Dec 4 '15 at 5:52
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# How do you find the derivative of (x^2-4)/(x-1)?
$\frac{d}{\mathrm{dx}} \left[\frac{{x}^{2} - 4}{x - 1}\right] = \frac{\left(x - 1\right) \left(2 x\right) - \left({x}^{2} - 4\right) \left(1\right)}{x - 1} ^ 2$
$= \frac{2 x}{x - 1} - \frac{{x}^{2} - 4}{x - 1} ^ 2$
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# Monopoles in non-abelian semi-simple gauge groups
Relative to the following
Indeed, the modern point of view is that the operator of electric charge is the generator of a U(1) group. The charge quantization condition arises in models of unification if the electromagnetic subgroup is embedded into a semi-simple non-Abelian gauge group of higher rank. In this case, the electric charge generator forms nontrivial commutation relations with all other generators of the gauge group.
I have a few questions:
Could someone explain me why it is important that the gauge group $U(1)$ embedded in the a large gauge group should have non-trivial commutation relations to guarantee charge quantization. Isn't it enough that it should be compactly embedded?
Why does the group has to be semi-simple? Aren't the main Grand Unified Theories that are now considered simple group, like $SO(5)$ or $SO(10)$ ?
And finally: the group has to be non-abelian to embed the groups $SU(2)$ and $SU(3)$ describing the other forces?
• Looking up the definitions of these terms and thinking about them for a few minutes would give the answers. Have you checked wikipedia? Simple Lie Algebra, Non-Abelian Group – Flint72 May 23 '14 at 18:38
$\\$
Firstly, if the Lie brackek relations were trivial then you would not have any charge.
$\\$
For your second point, any simple algebra is semi-simple. Look up the definition of semi-simple. An algebra is semi-simple if it has non non-trivial ideals, and thus can be decomposed into a direct sum of simple components. Hence any simple algebra $\mathfrak{g}$ is trivially semi-simple, since it is a direct of itself and the 'zero algebra'
$$\mathfrak{g} = \mathfrak{g} \oplus \mathfrak{0}$$
$\\$
As for your third point, in order for a group $G$ to have a non-Abelian subgroup $H$ we must have that $G$ is non-Abelian.
I suggest you study group theory, Lie groups, Lie algebras and Representation theory. Fulton & Harris is a good place to start.
• @user41736: In addition to Flint's Fulton and Harris resource recommendation, you can quickly brush up on some group theory, etc. in Peskin and Schroeder's chapter before non-abelian gauge theory. – JamalS May 24 '14 at 9:11
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(1 point) Solve the problem PDE: Utt 25uI1, X < % < 0, t > 0 IC: u(x,0) 8e ut(z,0) = 0...
##### Lly(c)] = Y(s) olmak uzere, y(x) 'e gore bir baslangic deger problemine Laplace donusumu uygulandiginda asagida verilen denklem elde edilmistir: Buna gore y(0) degerini bulunuz [Let Lly(z)] = Y(s) the following equation has been obtained when Laplace transformation has been applied to an initial value problem in y(a) Find y(0)]: 49(5 82 Y(8) = s(82 + 49)Lutfen birini secin: -514951-49-50
Lly(c)] = Y(s) olmak uzere, y(x) 'e gore bir baslangic deger problemine Laplace donusumu uygulandiginda asagida verilen denklem elde edilmistir: Buna gore y(0) degerini bulunuz [Let Lly(z)] = Y(s) the following equation has been obtained when Laplace transformation has been applied to an initia...
##### 200180160140120100 PpM
200 180 160 140 120 100 PpM...
##### Sin(Tx) + sin( 87 lim T72 log(2 2c)
sin(Tx) + sin( 87 lim T72 log(2 2c)...
##### Solve the problem: 6) To what new value should f(1) be changed to remove the discontinuity? Explain using limits x2 + 4, X < 1 f(x) 37 X=1 X + 4, X>1 Show your work:
Solve the problem: 6) To what new value should f(1) be changed to remove the discontinuity? Explain using limits x2 + 4, X < 1 f(x) 37 X=1 X + 4, X>1 Show your work:...
##### T - X < 2 1 + 1, x> 2_Let f(x)T-{=(J = 2 +14. Find lim,_2' f(x) and lim,_2 f(x) _ Does lim _2 f(x) exist? If so, what is it? If not; why not? Find lim,-4 f(x) and lim,_4" f(x) . Does lim, .4 f(x) exist? If so, what is it? If not, why not?
T - X < 2 1 + 1, x> 2_ Let f(x) T-{=( J = 2 +1 4. Find lim,_2' f(x) and lim,_2 f(x) _ Does lim _2 f(x) exist? If so, what is it? If not; why not? Find lim,-4 f(x) and lim,_4" f(x) . Does lim, .4 f(x) exist? If so, what is it? If not, why not?...
##### Why are the tires for trucks carrying gasoline and other flammable fluids manufactured to be electricallyconducting?
Why are the tires for trucks carrying gasoline and other flammable fluids manufactured to be electrically conducting?...
##### The diagram below shows the first 5 energy levels in the Bohr model of the hydrogen atom. Suppose that the hydrogen electron occupies the N equals 2 energy level. The electron makes a transition from N equals 2 to 4, followed by another transition from N equals 4 to 1. Which of the following could possibly have occurred?a. The atom first absorbed a green photon and then admitted a UV photonb. The atom first emitted a blue photon and then absorbed an IR photon c. The atom first absorbed a green p
The diagram below shows the first 5 energy levels in the Bohr model of the hydrogen atom. Suppose that the hydrogen electron occupies the N equals 2 energy level. The electron makes a transition from N equals 2 to 4, followed by another transition from N equals 4 to 1. Which of the following could p...
##### 4.3.77Find s(t), where s(t) represents the position function, v(t) represents the velocity function, and a(t) represents the acceleration function_ a(t) = 6t + 2, with v(O) = 1 and s(O) = 8s(t)
4.3.77 Find s(t), where s(t) represents the position function, v(t) represents the velocity function, and a(t) represents the acceleration function_ a(t) = 6t + 2, with v(O) = 1 and s(O) = 8 s(t)...
##### Cytochrome € oxidase is 3 component of the electron transport chain (ETC) in humans: This enzyme passes electrons from the ETC to oxygen Cyanide is a poison that inhibits cytochrome € oxidase: If cyanide poisoning occurs, would you expect the hydrogen ion concentration of the matrix of mitochondria to increase (more aciclic) decrease (less acidic)?IncrcascDecrcase
Cytochrome € oxidase is 3 component of the electron transport chain (ETC) in humans: This enzyme passes electrons from the ETC to oxygen Cyanide is a poison that inhibits cytochrome € oxidase: If cyanide poisoning occurs, would you expect the hydrogen ion concentration of the matrix of m...
##### 18 - The binary operation "#" is defined as follows:P#QWhich of the following is not equivalent to (P # Q) ?I) not (P and Q) I) P = Q III) P => (not Q) I Q=> (not P)Note: "=>" is the implication symbol
18 - The binary operation "#" is defined as follows: P#Q Which of the following is not equivalent to (P # Q) ? I) not (P and Q) I) P = Q III) P => (not Q) I Q=> (not P) Note: "=>" is the implication symbol...
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# The velocity of electromagnetic waves in a dielectric medium with relative permittivity of 4 will
This question was previously asked in
SDSC (ISRO) Technical Assistant (Electronics): Previous Paper 2018 (Held On: 8 April 2018)
View all ISRO Technical Assistant Papers >
1. 3 × 108 m/s
2. 6 × 108 m/s
3. 1.5 × 108 m/s
4. 0.75 × 108 m/s
Option 3 : 1.5 × 108 m/s
Free
CT 1: Network Theory 1
12266
10 Questions 10 Marks 10 Mins
## Detailed Solution
Concept:
The velocity (Vp) of a plane electromagnetic wave is given as-
$${V_p} = \frac{1}{{\sqrt {μ \varepsilon } }}$$
Here, μ = μoμr
ε = εo εr
$${v_p} = \frac{1}{{\sqrt {{μ _o}{μ _r}{\varepsilon _o}{\varepsilon _r}} }}$$
$${v_p} = \frac{c}{{\sqrt {{μ _r}{\varepsilon _r}} }}$$-----(1)
$$c = \frac{1}{{\sqrt {{μ _o}{\varepsilon _o}} }} = 3 \times {10^8}\;m/sec$$
μ0 = permeability in free space 4π × 10-7 H/m
εo = permittivity in free space 8.854 × 10-12 C2/Nm2
Calculation:
Given:
μr = 4
From Equation (1):
$$v_p=\frac{3\times10^8}{\sqrt4}$$
vp = 1.5 x 108 m/sec
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# Tag Info
7
It is a good question, it is a gemstone hiding in the mud. I have searched the Cambridge database, all the bis-acetylacetonatonickel complexes which have four coordinate nickel centres are square planar. There are three common coordination geometries for nickel(II) which we need to consider. Tetrahedral, square plannar and octahedral. Here is a diagram (...
1
From a structure diagram alone it is hard to determine these interactions. The reason for this is that important structural parameters are stretched or shortened in order to flatten the molecule for a 2D drawing. In a first order approximation a molecular modelling kit based on balls and sticks could already be very helpful. It'll let you approximate the 3D ...
0
You may simply say that energy is needed to break a bond. Breaking a bond is endothermic. So the inverse is exothermic. Energy is released when a bond is formed. Apparently you want to just discuss what is happening when an electron approaches a proton from far away (x1 in your drawing) to a shorter distance (x2). The electron is supposed to have no ...
1
The Hamiltonian for the collection of charged particles (electrons and nuclei) comprising a pair of molecules or ions has terms describing the kinetic energy of individual particles and pair-potential terms describing Coulombic interactions, that is, all interactions between particles are Coulombic. Where a Pauli repulsion term can crop up is when ...
3
I found a nice figure and the relevant statement in a paper by Frenking and Krapp (Unicorns in the world of chemical bonding models, 2006, https://doi.org/10.1002/jcc.20543): The crucial term which is responsible for repulsive interactions in chemical bonds except in two‐electron systems such as H2 is the Pauli repulsion. The three terms (a) ...
1
Lattice energy Now, according to wikipedia, NaCl has a lattice energy of −756 kJ/mol. First, we have to understand the term lattice energy. Here is the textbook explanation (Fleming: Physical Chemistry): The lattice energy is the energy required to separate the ions in an ionic lattice so that they are at infinite distance (but still ions). This would be ...
2
As you are probably aware, generalizations like "$\ce{BCl3}$ is a stronger Lewis acid than $\ce{AlCl3}$" can be problematic, as the results can be dependent on the base used and the conditions (eg solvent choice). That said, a common context for this ranking is with respect to carbonyl bases, such as in a Friedel-Crafts acylation. For these bases, $\ce{... 0 I can't give you a definitive answer, since I've not studied this myself, but here is what my physical intiution says: Let's use the following labels: Oxygen atom in the same molecule as the hydrogen: O' Oxygen atom with which the H is hydrogen bonding, i.e., the oxygen on the other molecule: O As you know, a hydrogen bond forms because of the ... 0 It has direct mechanical analogy. If you pull a tree branch weakly, it remains more or less in its original direction. If you pull it strongly, it points towards you. By other words, if an object is pulled by 2 non colinear forces, it has tendency to move itself to make them colinear, as the net force makes it to do so. If one of forces is much weaker, ... 0 The number of complex compounds that are surrounded by water molecules is quite big , a few examples of water being a ligand include [Cu(H2O)4]2+ or [Co(H2O)6]2+. Anyway oxygen is indeed less eager to donate its electrons because its electronegativity is considerably high, nitrogen for example works better as a ligand (actually the first complex compounds to ... 4 Oxygen very much does form bonds in which both electrons come from the oxygen atom. Examples include: 1. The$\ce{H3O^+}$ion at the center of the solvated proton in aqueous acids, also available as salts of some of the strongest acids such as$\ce{(H3O)(ClO4)}$2. Carbon monoxide, with its triple rather than double bond. 3. Ozone, in which the oxygen ... -1 Right, the text you are mentioning is not clear enough. It is not correct to say that the molecule$\ce{Al_2O_3}$exists and is separated from the next$\ce{Al_2O_3}$. Such a molecule does not exist. A sample of$\ce{Al_2O_3}$is made of a superposition of$\ce{Al^{3+}}$ions and$\ce{O^{2-}}$ions. It is not made of a superposition of molecules$\ce{...
-3
The quote you give clearly summarises the situation. In a solid (or liquid) there has to be some significant interaction between individual molecules otherwise there would only be gasses. This inter-molecular interaction is not that forming chemical bonds, i.e. a solid is not a super big molecule, but is additional to these and is due to the nature of the ...
-1
It is because of the higher electronegativities of Br and Cl, than Hydrogen. So there is a higher repulsion of the bonding electrons.
2
The larger angles can simply be explained as a result of repulsion between the larger atoms of $\ce{Br}$ and $\ce{Cl}$. Hydrogen atoms in $\ce{PH3},$ as the are so small, experience less repulsion as compared to $\ce{Br}$ atoms in $\ce{PBr3}$ or $\ce{Cl}$ atoms in $\ce{PCl3},$ therefore the larger bond angles in $\ce{PBr3}$ and $\ce{PCl3}.$
1
Generalisations are impossible. There are many examples of strong and weak bonds in both ionic and covalent compounds There are a number of problems with the question. One is that there bonding is not an either/or concept: there is a something of a continuum between "pure" ionic and "pure" covalent bonding. There is also a lot of confusion about what sort ...
1
Part of the reason for many answers is that "stronger" can be different things. If we consider a typical two-electron bond between two atoms A and B, it can break in three ways: 1) 1 electron can stay with A and one with B: $\ce{A-B -> A. + B.}$ This is called homolytic bond cleavage. 2) Both electrons can stay with A: $\ce{A-B -> A- + B+}$ This is ...
0
Usually ionic bonds are stronger than covalent bonds. But there are exceptions. Quartz SiO2 for example is made of covalent bonds, and it melts at very high temperature (> 1400°C). I am afraid there are no general rules.
-4
Covalent bond is defined as the sharing of electrons between two atoms (non metals). Ionic bond is defined as the transfer of electrons from the valence shell of anion(s) to the valence shell of cation(s). On a covalent bond, there are two types of forces that held the atoms together; (1) intramolecular forces and (2) intermolecular forces. Intermolecular ...
0
Good question! Hydrogen bonding in the solid appears to be at least partially responsible for the planar (sp2) structure. ("Crystalline boric acid consists of layers of B(OH)3 molecules held together by hydrogen bonds of length 272 pm." https://en.wikipedia.org/wiki/Boric_acid) If the empty orbital on boron (which is partially filled by donation from the ...
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# Why do we use filters in telescopes for astronomical imaging?
I have read that if we image without a filter we get no information about the color or SED of objects. Can anyone elaborate the reasons for using filters for imaging/photometry? What happens if we image without a filter?
In general, the CCDs used to capture images do not register the energy (therefore, colour) of the incident photons on them - they just count the number of photons observed by each pixel (or a value proportional to the number of photons, as the are not 100% efficient). So, they essentially just show overall brightness variations across the image.
If you want to capture colour information you therefore have to use filters. I.e., if you want to get information on the number of red photons (e.g., the intensity of red light incident on each pixel of your image), you use a filter to block out all other light. You can do this with multiple different filters to build up a full colour image.
In most commercial digital colour cameras the CCDs have a filter mask over four pixel patches: two filtering for green light, one for blue light, and one for red light. The outputs from these pixel is used to build the full colour image.
As WDC pointed out in his comment, without filters, you simply get a recording of the received irradiance as a function of the sensor's spectral response function. In other words, a normal CCD that detects the light in a camera isn't capable of picking up every wavelength of light perfectly and the response function tells you how good that CCD is at picking up every wavelength of light.
Sometimes though, you don't want to take a picture and record every possible photon the CCD is capable of recording. Sometimes you want to record specific wavelengths. You do this by applying a filter, before the CCD, that let's in only specific wavelengths.
This has all sorts of uses. A simple example would be taking three pictures, one with a red filter to primarily let in red light, another with a green filter to primarily let in green light, and a third with a blue filter to primarily let in blue light. When you look at the individual images on your screen, the computer doesn't know what colors (i.e., wavelengths) of light the CCD saw, it just knows how many photons were observed so it can only show you a greyscale. However, you can then, in post-processing, tint your image with the red filter red, tint your green image green, etc. and then combine you red, green, and blue images into a single, colored picture to get a close-to-true, color image of your object. In fact, that's actually how digital cameras actually work to take colored pictures!
Besides using filters to get colored pictures, Astronomers use filters for a wide variety of science goals. It is very possible to create a special filter that only lets in a single wavelength (or as near to a single wavelength as one can get). Often single wavelengths of light are tied to specific physical processes. By that I mean, only specific physical processes can create that exact wavelength. So by looking at something with a filter for a specific wavelength, you're looking at the components of that object that created that wavelength of light.
A common single-wavelength filter people like to use is the H-alpha filter and a common target for observation is the Sun. Shown below, and taken from APOD, is a picture of the Sun using the H-alpha filter.
Or simlarly, the Solar Dynamics Observatory (SDO) spacecraft is constantly observing the sun in all sorts of filters. Note how different the sun looks at the different wavelengths!
Note: these are false color images for effect!
A group of students are browsing the internet before class.
One of the students asks, &ldquoDo you guys ever look at the Astronomy Picture of the Day website?&rdquo
• Cyle: &ldquoOf course. I love APoD, especially those colorful nebulae and supernova remnants. It&rsquos too bad there aren&rsquot any of those close by that you can see just with your eyes.&rdquo
• Donna: &ldquoThose aren&rsquot the real colors. They have to do things to those images, like color in the clouds and gasses.&rdquo
• Eric: &ldquoI thought they had to take different images of the same object and merge them together.&rdquo
• Fiona: &ldquoThen how do they know what colors x-rays are?&rdquo
Taking color pictures with optical telescopes such as Hubble, or any ground-based telescope with CCD detectors, is very different and much more complex than using film in a traditional camera. Electronic detectors do not read out information in color&mdashrather, the energies of the photons must be assigned colors in a computer process known as image processing. For visible light images, the color choices are sometimes assigned to try to faithfully reproduce what our eyes could see (if they could stare at the object for a long time without blinking or otherwise taking &ldquosnapshots&rdquo). Many full-color images are combinations of data taken in separate exposures of red, green, and blue visible light. When mixed together, these three colors of light can simulate almost any color of light that is visible to human eyes. That is how televisions, computer monitors, and video cameras recreate colors.
The standard colors that are mixed together on a television screen are called R, G, and B for red, green, and blue. This is done with a set of filters that pass light of wavelengths centered around 650, 520, and 450 nm, respectively. They block all other colors. Each filter is typically about 100 nm wide. While RGB filters are adequate for creating color images on a screen, these are not the filters that astronomers typically use.
Astronomical filters were not developed to create color images, though they can be used for that purpose. Rather, they were designed to study the physics of stars and other astrophysical objects. For instance, by comparing the brightness of a star in two filters, it is possible to determine its temperature. This is because the separate filters sample different points on the star&rsquos Planck spectrum. Because Planck curves with different temperatures are unique, these two points are sufficient to uniquely determine the shape of the curve, and thus, its temperature. But stars are not perfect Planck emitters. They have absorption lines. (Sometimes, they even have emission lines.) Filters can be designed to be especially sensitive to these absorption lines, and thus, provide the ability to distinguish one type of star from another by means of the absorption features. Filters allow this determination to be made by simple imaging techniques rather than more complicated spectral techniques, usually saving a lot of time at the telescope.
Standard photometric filter sets have been developed over the past 50 years. The most common set is called the Johnson/Cousin system. It was developed in the 1960s and uses U, B, V, R, and I filters, for ultraviolet, blue, visible, red, and infrared. These filters typically have widths of about 100 nm, give or take, and they are centered at 365, 445, 551, 658, and 806 nm, respectively. Additional filters have been developed that push farther into the near and mid-infrared, going out into the region between 1,000 and 5,000 nm (1 and 5 microns). Other filter sets have been developed as well, usually with some specific use in mind. For instance, both Hubble and the Sloan Digital Sky Survey developed special filter sets based on their instrumentation and science goals.
In addition to these broadband filters, there are narrowband ones that only pass light close to a particular wavelength. These narrowband filters typically have widths less than 10 nanometers and are centered on an emission line from hydrogen, oxygen, sulfur, etc. Many of the beautiful pictures we see of nebulae (gas clouds) are taken using several of these narrowband filters to highlight emission from different atomic species.
All astronomical observations are done using these (or other) standard filter sets. This standardization allows one set of observations to be easily compared to another, a very important ability to have when doing science. Whenever a new filter standard is created, a lot of effort goes into understanding how it is related to others so that the new observations can be compared to older ones. Of course, it would be much easier to always use the same sets of filters for all observations, but sometimes, the science goals make that impractical or impossible.
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### Seeing Planets in a Different Light Reveals More Detail
Earth's atmosphere is in constant fluctuation turbulent air currents blur fine surface detail on solar system objects viewed through a telescope. Faint contrasting areas blend together due to "irradiation" &mdash a distortion of the boundaries between light and dark surfaces.
When you employ a color filter to zero in on a narrow region of the spectrum, the scattering of interfering wavelengths is enormously reduced. Suddenly, the smeared, pale bands of Jupiter resolve into loops and festoons. Delicate markings appear on Saturn's globe, and the Cassini ring division darkens and solidifies. Mars' polar caps stand out like tiny pearls, and vague lunar rilles acquire greater depth and contrast. Bad "seeing" becomes acceptable good seeing becomes superb!
Because many planets have a characteristic color (e.g., Mars is reddish), you will dramatically increase detail by reducing the predominant hues, uncovering hidden contrast and surface markings. That's why "the Red Planet" is most effectively enhanced with a green filter.
Below is a guide to using color filters to best view the planets in our solar system.
Mercury
#25 Red will make the planet's disk stand out against a blue sky, permitting daytime or twilight viewing. Mercury is usually best observed just after sunset, when the sky is awash in orange light, so employ #21 Orange with high magnifications to see the planet's phases.
Venus
No matter what telescope aperture you use, Venus's excessive brightness usually causes a very "overexposed," roiling image. With a #47 Violet filter, or stacked #58 Green and #80A Medium Blue filters, you'll reduce the severe twinkling for a better view of the fascinating changing phases.
Mars
#25 Red passes the predominant reflections of surface plains and maria, and #21 Orange is good for reducing the intense glare to enhance detail and mottling. The polar caps stand out with #15 Deep Yellow and #80A Medium Blue examine the melt lines with #58 Green.
Jupiter
This great planet reveals its cloud bands, loops, festoons, ovals, and Red Spot with #11 Yellow-Green, #80A, #58, and #21. Go from seeing only two bands without a filter to seven or more with a filter! Try stacking filters to reduce heavy glare.
Saturn
Many subtle globular details are improved by #15 Deep Yellow. See the difference in brightness of the extremes of the rings with #25, #11, #58, or #80A. #15 helps sharpen Saturn's image in photographs, improving the resolutions of Cassini's division.
Moon
Reduce the Moon's glare with #80A Medium Blue, and enhance the contrast of lunar rilles and strata with #15 Deep Yellow.
Other Uses
You will improve black-and-white photographs by blocking UV light with #15 Deep Yellow. Refractor chromatic distortion is also reduced by #15, and by #80A. The #82A Pale Blue is great for stacking with other colors, and can adjust film color balance by absorbing excess yellow and red. #58 Green will block street light while passing much of the wavelength of doubly ionized oxygen in emission nebulae. Try #25 Red for long black-and-white exposures of the Omega or Rosette Nebulas.
## The Lake County Astronomical Society
Newbies to observational astronomy quickly learn that the full moon is a daunting adversary that blots out contrast and detail in the faint fuzzy objects we pursue. If you're getting your feet wet in astrophotography, you've probably noticed that the bright moon does an even nastier job on your photos. The light pollution in your sky is amplified by the sensitivity of the camera. Moisture in the air reflects the moonlight and gives a smoky look to the sky through the camera. And nasty gradients start harmlessly at one end of your image and increase until they fog the details of your image.
All in all, the moon is a much bigger enemy to the imager than the observer. But what if there was a way to neutralize the moon and even neutralize the local light pollution we're constantly bemoaning? Wouldn't you like to be able to image even during the 13 brightest days of the lunar month from the 9-day old moon to the 22-day old moon? Well of course there's something you can do and involves doing something that is totally the opposite of what we normally do in astronomy: You want to throw away light. Huh? Yes, throw away light. Whah? Yep, get rid of a LOT of light. Let me explain.
Conventional progress in astronomy is all about collecting lots of light: We get big lenses and mirrors - more surface area means more light collected. In photography, we use those big optical surfaces with cameras that can stare at the sky for a long time to pile up the incoming light. When we're collecting light from a nebula or stars or dust the goal for better pictures is to use bigger optics, more sensitive cameras, and longer exposures. That's the philosophy from light pollution-free areas and (relatively) moonless nights.
When there's light pollution and/or a bright moon, the rules change. Bigger optical systems amplify light pollution - man- and moon-made. Long exposures on sensitive cameras do the same thing. And what's worse is that the sky mess isn't uniform. The area of the sky closest to the terrestrial or lunar light source is brighter than the areas that are farther away. The camera can catch this gradual brightening and dimming - a "gradient" - much better than your eye can detect. So collecting a big bunch of "bad" light makes your photos much poorer instead of richer.
The solution many astrophotographers use is to be selective - VERY selective - in our light "shopping". Discard all the bad light, and keep only the very good light. We'll skip the physics lesson, but the most popular lights to collect are Sulphur II (SII), Oxygen III (OIII), Hydrogen beta (Hbeta) and the most popular of all, Hydrogen alpha (H-alpha). These specific wavelengths of light correspond to different atomic states of different elements. To collect only specific light requires a filter and a small detour to explain terminology.
Normal astronomical filters 1 , like an infrared blocker or primary color filters will allow a fairly large part of the optical spectrum to pass through the filter. The part (or "band") of the spectrum is wide, so these are called "broadband filters". On the other hand, when we want to be snobbish about the light we want to collect we choose a filter that only allows a thin part of the spectrum onto the camera. So these specialized filters are called "narrowband filters". Narrowband imaging is quite popular these days because it allows astrophotographers to take photos regardless of the phase of the moon and it can substantially eliminate the effects of your local light pollution. No more hiding out during bright phases of the moon and no more chagrin at your local light pollution.
In this article, I want to concentrate on H-alpha imaging, first because it's so popular (in the universe and among astrophotographers) and second because it's important to understand the limits of H-alpha imaging before you get too excited about it.
Picking your targets: So what objects benefit from H-alpha filtering? The H-alpha light comes from excited hydrogen. Certainly stars have a lot of that so stars show well through an H-alpha filter. And other types of energy can excite clouds of hydrogen gas, so clouds surrounding star creation and star destruction activity are also candidates. What we see in objects like the Eagle Nebula, the Orion Nebula and the Lagoon Nebula are clouds of gas surrounding locations that represent forming stars. The energy of these stars excites the gas that then emits light. So these nebulae are a called "emission nebulae". Emission nebulae are VERY GOOD candidates for H-alpha filtering. The light they send out is broadcast in one or a few very narrow parts of the spectrum.
On the other hand, objects that are merely reflecting light usually have light that is broadcast along much larger parts of the spectrum. Reflecting nebulae are poor candidates for H-alpha imaging. The Running Man Nebula (NGC1977) and The Pleiades (M45) are famous reflection nebulae.
There are composite objects like the parts of the Veil Nebula (including NGC6992 and NGC6940), and the Trifid Nebula (M20) that have an emission nebula that illuminates a reflection nebula. You can use H-alpha filtering to capture part of the object, but the reflection part of the nebula will need other techniques to be captured.
Some galaxies have strong areas of H-alpha emissions (like M82 and M33), but overall, galaxy images don't benefit from H-alpha photography. Like galaxies, there are some planetary nebulae with portions of H-alpha light, but again, the reflection part of the nebula will not be seen through the filter.
The moon and other solar system objects don't benefit from H-alpha imaging. H-alpha light is significant in solar photography, but YOU MUST USE SPECIALIZED SOLAR FILTERS OR YOU CAN DAMAGE YOUR EQUIPMENT OR CAUSE SERIOUS PERSONAL INJURY. In this article we're discussing nighttime H-alpha images (otherwise the phase of the moon wouldn't matter!).
So, the primary targets for H-alpha imaging are the emission nebulae and the open clusters that include an H-alpha gas cloud (for example, M16 - the Eagle Nebula).
Camera matters. Not all cameras are good at collecting H-alpha light. Many dedicated astrophotography cameras have good or excellent sensitivity to the H-alpha wavelength (6463 angstroms). Consumer digital cameras sometimes contain a chip that is sensitive to H-alpha (as well as infrared) light, but the manufacturers mount a blocking filter to help get a good color balance for everyday photos. So the normal pocket digital camera won't do the job in H-alpha.
The popular DSLR's (e.g., Canon 350XT and Nikon D-70) have the same problem, but there are home modifications that can be made to the camera to remove the manufacturer's filter and allow the H-alpha light onto the chip. Sometimes, this renders the camera unusable for routine photography and sometimes you can correct the colors using external filters or special settings and processing. One company, Hutech Astronomical Products, even sells fully modified Canon and Fuji cameras to permit H-alpha photography.
By far, the best candidate for H-alpha photography is a dedicated astronomical camera. Cameras from SBIG, Starlight Xpress, FLI, Meade, Orion will almost always have decent H-alpha sensitivity. The easiest way to find out if your camera has good H-alpha sensitivity is to find the spectral response specifications or just use Google to search for examples of imaging in H-alpha. The exact sensitivity varies with the camera models. For example, my SX MX716 has great H-alpha sensitivity while its cousin the MX916 has fairly poor sensitivity.
Mount requirements. So you've decided your camera will gladly accept H-alpha light, you like the choice of imaging objects and you think you're ready to go. But hold on - the mount you will use needs to do a little more work than you are accustomed to. Here's why: In normal, multi-color light imaging (for example with a DSLR), you collect light from a wide spectral band - about 400nm wide. H-alpha filters come in bandpass widths of 3nm to 13nm. So you are only collecting 1/30th to 1/130th the amount of light. This means you need to take longer exposures (or many, many short exposures) to compensate. That image you captured with 10 exposures of 30 seconds (5 minutes total exposure) will need 30 to 130 TIMES more exposure - 150 minutes to almost 11 hours exposure. There are processing techniques so that narrowband imagers don't necessarily endure these marathon exposures, but the bottom line is be prepared for many and long exposures.
This requirement puts a lot of pressure on your mount. Getting smooth tracking in a short exposure is (somewhat) easy. But in a 4, 5, 8, 10 minute exposure? Only the best mounts can do unguided images that long. If you have an autoguider setup you will get a lot of use out of it during H-alpha and other narrowband imaging. 2
Another mount-related aspect is the quality of your alignment. For sequences of short exposures any errors in your alignment won't cause too much field rotation. However when your exposure length get longer - 4 to 10 minutes - you might notice that the stars on the edge of the image seem to be more oblong compared to the round stars in the center. This is the effect of field rotation. A poor polar alignment (or a good alt/azimuth alignment) will show arcs of stars - small arcs in the middle and larger arcs as you get farther from the center.) If the polar alignment is good (but not perfect) you don't see the effect much in the middle of the image, but nearer the edge, the short arcs make the stars a little squat. The solution?? Put more effort into refining your polar alignment if you want the long exposures needed for good H-alpha (and other narrowband) images.
Filters. Narrowband filters, including H-alpha, are like typical colored filters. They come in 1.25" and 2" sizes (priced accordingly). The filters have standard threads to attach to most cameras. Like normal filters, narrowband filters can be used in color filter wheels and strips.
Each filter has a specific bandwidth and bandpass center. The H-alpha filters are centered on the 6563 3 angstroms or 656.3 nm. The bandwidth of the filters varies from 3nm to 13nm. The 3nm filters have a very, very narrow bandwidth and gets only the purest H-alpha wavelength and normally have a higher price. This bandwidth results in really small, sharp stars. On the other hand, the 13nm filter has a more "open" bandwidth and gives slightly larger (but still small!) stars and fractionally less contrast compared with the narrower filters.
The masters of narrowband imaging prefer the narrowest bandwidth forms. They feel that the precise exclusion of all other light makes for better photos. Of course, their images take about 4 times as long as images with the wider filter because they need to compensate for the narrower slice of the spectrum. Most of these experts have excellent mounts and automated controllers, so some extra hours of exposure aren't a burden to them.
One other consideration when selecting a bandwidth is "drift". The coatings used for narrowband imaging presume that your optical system operates within a specific range of f/-ratios. Often, the comfort zone is about f/4 to f/11. As you go outside that range, the filter's "center" will shift red-ward or blue-ward. If your filter has a narrow bandwidth and you use optics outside the comfort zone, the filter can totally miss the targeted light. The wider bandwidth filters have the same shift, but being wider they won't miss the goal light.
So if you plan to use optics outside the comfort zone (for example, very long focal length telescopes or very fast camera lenses), you should stick to the wider bandwidth filters.
Other filters, other applications - We only touched on the other kinds of filters for narrowband imaging. Each has a use for a specific kind of light that corresponds to specific astronomical environments and events such as star formation, star destruction, etc. You are welcome to explore the use of other filters via on-line resources, but I recommend starting your imaging with H-alpha to get the best choice of targets and ease of imaging.
Conclusion and Resources: So, do you want to get out and take images regardless of the phase of the moon or even if your sky isn't perfectly dark? Can you mount handle longer exposures? Can your camera record the fun parts of the spectrum, like H-alpha?
If so, buy a nice H-alpha filter and give narrowband imaging a try.
You can find great examples of narrowband imaging at Richard Crisp's website -
http://www.rdcrisp.darkhorizons.org/
The Starizona website has a good discussion of narrowband imaging located at -
There is a Yahoo! group dedicated to narrowband imaging located at -
http://groups.yahoo.com/group/narrowbandimaging/
You should also check with astrophotography groups for your specific camera to see what tips for narrowband imaging apply to your equipment.
1 Filters come in two forms: "pass" and "block". A pass filter allows specific light to pass. So a red filter allows red (but no other) light to pass. A block filter allows all light except a specific light to pass through the filter. An "IRB" blocks the infrared light, but allows all other wavelengths. Unfortunately, the common terminology for filters is not specific, so a reference to an "infrared filter" could mean a filter that blocks infrared light or a filter that passes only infrared light. For this article, the term "H-alpha filter" means a pass filter that only allows light near the H-alpha wavelength.
2 There is an upside to the laughably long exposure requirements for H-alpha imaging. When you kick off a 90-minute sequence of images (under computer control), you can use that time effective for other tasks. Some people do observing with a separate set of optics. Me? I take a very nice, short nap!
3 In New Mexico, most highways have a 3-digit designation, but there is one 4-digit highway: 6563 also known as the "Sunspot Highway". It leads to the National Solar Observatory at Sacramento Peak.
One of the simplest ways to boost the quality of views offered by your telescope is the addition of filters to your setup. Whether scrutinizing the ice caps of Mars or surfing the clouds of distant nebulae, looking through a telescope filtered for its target can radically improve the viewing experience. However, choosing the proper filter for the job can be a daunting task. Never fear! After reading this article, you will be prepared to outfit your gear no matter what your favorite celestial target is.
### Night Skies and Our Eyes
The quality of views offered by your telescope depends primarily upon three factors: resolving power, contrast, and sharpness. While resolving power depends largely upon the aperture of your scope, sharpness and contrast can be boosted by the application of filters. Both attributes are affected by the imperfections of human vision and what astronomer’s call seeing, the effects of light scattering in the atmosphere. Proper filtration can do wonders for both issues.
If you have ever forgotten your sunglasses on a trip to the beach, you know the benefits of filtering light before it reaches your eyes. Your retinas are loaded with light-sensitive cells and when they become exposed to large quantities of light it can be difficult, if not painful, to see. Viewing the sky at night poses the opposite problem—our eyes are under-stimulated by the light in their surroundings. It might seem counterintuitive at first that the solution to this problem involves the use of filters which, by nature, reduce the amount of light reaching your eyes. Without getting lost in the complexities of human vision, it is worth knowing a little about how our eyes work to understand why filters are an astronomer’s best friend.
You might remember from biology class that your eyes contain two kinds of photoreceptors: rods and cones. Three types of cone cells, each responding to a different wavelength of light (red, green, and blue) are responsible for our perception of color. They are fewer in number and require much brighter conditions than our rods to relay sensory information to our brains. Our cones take the reins during photopic (daylight) conditions. Scotopic (night) vision relies entirely upon our rods. Although you have about twenty times as many rods as cones, there is only one type of receptor at work, and it responds ideally to a wavelength of light situated between the blue and green spectra. If you have ever wondered why the world seems a little bluer at night, it is because your rods are working solo. Scientists call this the Purkinje Effect, named after the Czech neuroscientist, Jan Evangelista Purkinje. So, what does all of this mean for astronomers? Because scotopic vision is monochromatic, the determining factor for perception in the dark is contrast. This is where filters come into play.
Peak response curves for rods and cones
Telescope filters screw into the barrel of your eyepiece and are sized accordingly. All filters operate by reflecting some light and transmitting the rest. Their value to astronomers resides in their ability to let you pick and choose which wavelengths of light reach your eye. By controlling the type and quantity of light that your eyes perceive, you will be better able to distinguish differences in low-light conditions. By reflecting certain wavelengths, transmitted light appears much brighter, increasing contrast and improving your view. Since different celestial bodies reflect or emit different wavelengths of light, filters allow you to fine-tune your instruments, depending upon where you are looking in the sky.
### Turn off the Bright Lights
Unless you are planning to use your telescope thousands of miles from civilization, your overall viewing experience will be improved by the application of a light pollution reduction (LPR) filter. If you are looking for one filter to enhance your views across the board, add one of these to your bag. LPR filters reflect the wavelengths of light associated with sodium-vapor and mercury-vapor lamps, the most common types used for street illumination. If you live near or in a city, an LPR is as necessary as a tripod. While the elimination of all artificial light is impossible, LPRs will give you a noticeably darker sky to work with.
While most of this article is aimed at providing solutions for boosting our perception of light associated with distant subjects, one beloved night-time target is so bright that it requires dimming: the moon. The best options for teasing detail out of the moon are the use of polarizing and/or neutral density filters. If you have ever experimented with filters on your camera, you are probably familiar with these two photography staples. Neutral density filters (often marketed as “moon filters” for the astronomy crowd) reduce glare while leaving the colors transmitted to your eyes unaltered. Because neutral density filters uniformly reduce light across the spectrum, they will not increase contrast, but they will cut back the intensity of light reaching your eyes, allowing you to see otherwise invisible details. Polarizing filters cut back on reflection and have the added benefit of allowing you to manually adjust the filter strength. With the turn of a thumbscrew, you can choose the precise amount of filtration for optimal viewing.
### Neighborhood Watch
As we move further away from Earth, the variables that affect our view of the sky begin piling up. The atmospheric conditions and physical composition of each planet present unique challenges, depending upon where you are looking and what you are looking for. Consequently, there are countless ways to enhance your views for each planet. This is where color telescope filters come in handy. Several manufacturers offer “planetary sets” that consist of varying grades of red, yellow, and blue filters. As a bonus, most sets include a neutral density filter in their lineup. Color telescope filters are described using Wratten numbers, the same industry standard used to categorize camera lens filters. A comprehensive list of filter-planet combinations would triple the length of this article. However, a few general tips should get you started. Red filters help with daytime viewing of Mercury and Venus. Yellow filters boost contrast in Neptune and Uranus while teasing out detail in the belts of Jupiter and the surface of Mars. Blue filters are the most versatile of the group, revealing dust storms on Mars, the belts of Jupiter, and the rings of Saturn. Finally, if you are interested in the stormy skies of Venus, try a violet filter. It is never a bad idea to try a couple of different filters for each subject—you may be surprised by what you see!
### In a Galaxy Far, Far Away
It comes as no surprise that the most difficult (and often beautiful) celestial phenomena to observe are the ones furthest away. Narrowband and line filters add clarity to this otherwise cloudy subject. Narrowband filters block out all light except for small ranges of wavelengths associated with specific phenomena. Line filters are even more precise, blocking out all but one or two wavelengths of light. The most popular of these groups is the Oxygen III (OIII) filters, which reflect all but 496 and 501nm lines, associated with planetary and emission nebulae. Such extreme filtration provides a clean, black background for observation.
No matter what your favorite celestial subject may be, there is a filter out there that will help you get to know it better. Experimenting with the many available options is the best way to determine what works best for you. So, grab some filters and have a look!
*Note: The photographs in this article serve as simulations. Cameras register greater detail and more colors than the human eye is capable of perceiving when looking through a telescope.
## Moon filter
A neutral, grey or moon filter is used to lessen the intensity of bright moonlight and to slightly increase contrast. Anyone who has ever been to an observatory and looked at the Moon through a large telescope without a filter will vividly remember the experience and know why this filter is so important. Observing the moon without a filter will not cause any damage, but it is so bright that it really dazzles you. If you then turn away from the telescope and look into the darkness you will often still have a ghostly afterimage of the moon in the eye you observed with. Although this afterimage will gradually fade, it is still very irritating.
Of course these filters are available in different light reduction levels. They range from a light transmittance of about 8% up to 50%. The filters with a high transmittance are suitable for the smaller telescopes and those with a low transmittance are suitable for larger telescopes.
Adjustable polarizing filters are the luxury version of Moon filters. This is not just one, but two filter elements, which are connected to each other. Rotating one filter element relative to the other continuously adjusts the amount of darkening. Most polarizing filters allow light transmission levels from 1% to 40%. They can be used to set the optimal balance between light level and contrast for the size of telescope you are using.
## PLEASE PURCHASE DIRECTLY FROM THE MANUFACTURER AT THE NEW FARPOINTASTRO.COM STORE.
Astrodon is known for designing the best performing, most durable, premium filters for astronomical imaging and research.
For astrophotography, Astrodon LRGB filters simplify imaging by allowing you to take one exposure time for each color, only one corresponding dark exposure time and nearly equal color combine weights in post-processing. The resulting color balance is superb, which is why so many of the top imagers now use Astrodon LRGB filters. These designs eliminate halos around bright stars that detract from the beauty of the galaxy or nebula.
Astrodon NARROWBAND filters set the highest, consistent performance level, and are spectrally narrower than most other filters, leading to the best contrast and faintest structures in your nebula. Astrodon has a performance guarantee of >90%T at the emission line on every box.
UVBRI and Sloan PHOTOMETRIC filters are 100%-coated using no colored glass for long-term durability that is so critical for consistent, long-term research. They have the highest throughputs available for better signals and fainter objects and are becoming widely accepted in professional observatories in sizes up to 150mm. Some of these larger filters are used on the famous Palomar 200″ telescope, at the MacDonald Observatory, Las Cumbres Observatory of Global Telescopes, AAVSO and universities and research organizations worldwide.
All Astrodon astrophotography filters are manufactured in the U.S. with superb quality control using 100% hard-oxide sputtered coatings. Astrodon filters cost a little more because of the benefits that their high performance and great durability provides. Filters are a critical part of telescope systems. They are the “spark plugs” that make the “engine” go. Step up to Astrodons and see the difference.
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## Final Words
All in all, telescope filters are a must-have accessory for all telescope users. It is obvious that without a filter, the universe viewing experience is incomplete. You can begin by getting a moon filter because it is the most basic one. Every astronomer imagines what it must be like to go to the moon and look at the craters by yourself. That is a little impossible but you can get a taste of that experience by viewing the moon through a good telescope filter.
If you are more inclined towards viewing the planets, get the colored filter that works best to enhance the details of your favorite planet and get lost in the beauty of it. Or if there's a solar eclipse coming up, it is absolutely necessary and important to get a solar filter to be able to observe different phases of the eclipse. In other words, get filters for your telescope and look at the universe with a new perspective.
You can choose anyone from the above list according to your requirements and you will not be disappointed surely.
## Why do we use filters in telescopes for astronomical imaging? - Astronomy
The single biggest problem facing any observer wishing to undertake a programme of high resolution photography is the atmosphere. When a good quality, well collimated telescope is used the atmosphere is responsible for nearly all deterioration of the image quality delivered at focus. Astronomical seeing is a very well-documented phenomenon, but with the increasing number of observers employing large aperture telescopes for high resolution imaging, another not so well-known process can affect image quality far more than observers realise. Indeed until recently I had rather underestimated the effect of this phenomenon. This effect is atmospheric dispersion.
Atmospheric dispersion and its effects
The atmosphere imparts many deleterious effects on the light that passes through it. Astronomical seeing (the mixing of air of different temperatures) is undoubtedly the most destructive property when it comes to obtaining high resolution images, however atmospheric dispersion also imparts serious effects, especially when employing large aperture telescopes with the object of interest located well away from the zenith.
Dispersion is the ‘smearing out’ of light of different colours due to differential refraction as it passes through our atmosphere. The level of dispersion present is related to the wavelength of light and the filter passband. Shorter wavelengths / wider filters are more seriously affected than longer wavelengths / narrower filters. Effectively our atmosphere behaves as a prism, splitting white light into its spectrum of colours. Dispersion is worse the lower in the sky you observe, as the light is passing through more air. For example when observing an object at about 30° altitude you are looking through around twice as much air as you would be at the zenith – a considerable difference.
Pressure, temperature and humidity all affect the amount of dispersion that will occur for a given altitude but, for the typical amateur observer, these secondary effects are very small. The main culprit is the altitude of the object above the horizon, as shown in Figure 1.
Larger aperture telescopes are affected more than smaller ones because of their better resolving power, so the effect of dispersion becomes significant at a higher object altitude. For example, as a 16" (40cm) aperture delivers four times better theoretical resolution than a 4" (10cm) aperture it becomes clear that for this larger telescope to deliver performance to its maximum resolving potential, the object must be located very high above the horizon. Even at an altitude of 60°, around 0.7 arcseconds of dispersion is present from 400-650nm. Since many observers live at latitudes where the planets do not pass close to the zenith it soon becomes apparent that we are often imaging objects well away from the zenith where dispersion has serious potential to degrade image quality.
A typical 6" (15cm) telescope should achieve a performance not hindered by dispersion down to an altitude of around 40° in white light, although when we consider the wide spectral response of CCDs and the resolution possible under excellent seeing it becomes clear we should look at ways to try and overcome dispersion in order to maximise the potential of our telescopes.
Overcoming dispersion
There are ways in which we can overcome the effects of dispersion. The general consensus among observers is that the use of filters eliminates these effects, meaning typical RGB colour imaging ‘bypasses’ the effect as the filters are passing only a narrow band of wavelengths. However, in reality this is not the case.
As discussed above the amount of dispersion is dependent on the bandwidth used. Typical RGB filters cover around 100-150nm in bandwidth and a red filter of 100nm bandwidth will be less affected by dispersion than a blue filter of the same bandwidth. Dispersion does have less effect for filtered light compared to unfiltered light as shown in Figure 1. This figure also shows that white light is quite seriously affected by dispersion. For example, if we say the typical highest resolution attained on a planetary target by a 36cm telescope is around 0.25 arcseconds (which in practice is about right from my own imagery), then we can conclude that for a 36cm aperture, to maintain 0.25" resolution unaffected by dispersion, with different filters the altitude of the object above the horizon must be greater than the following:
UV/IR blocked white light: 77° Astronomik blue filter: 72° Astronomik green filter: 52° Astronomik red filter: 42°
It is therefore apparent that dispersion can play a major role in the attempt to obtain high resolution imagery of the planets even when using filters. From typical northern European latitudes the planets only rarely attain an altitude of 60° and, for much of the time, we must work at altitudes much lower than this. Therefore while filters can provide some relief from the effects of dispersion they certainly do not cure the problem. We must turn to another device for this purpose.
Dispersion correctors
Basic correctors that reduce the smearing effects of dispersion have been employed by visual planetary observers for many years. 2 The 19th century astronomer George Airy employed a set of wedge prisms to correct for the effects of dispersion during his observations. In use a prism was orientated so that its dispersion was opposite to that produced by Earth’s atmosphere.
Depending upon the altitude of the object more than one prism would be required to exactly nullify dispersion, but it is possible to use two wedge prisms that rotate with respect to one other to provide an adjustable corrector for almost any altitude. This is known as a Risley prism. This type of system is ideal for the observer as it offers an easily adjustable system without the need for multiple single prisms.
Single wedge prism correctors are typically specified as 2° or 4° prisms which will nullify dispersion in unfiltered light for a given altitude. For example a 2° prism will nullify dispersion across the visible spectrum at 65° altitude, while a 4° prism will work at 35° altitude. Adirondack Astronomy in the USA manufactured a set of such prisms which they marketed as Prismatic Atmospheric Dispersion Correctors 4 (PADCs) which could either be used alone or as a pair for adjustable correction. Sadly these have since been discontinued.
Fortunately, fully adjustable dispersion correctors are now available with a prism pair incorporated into a single convenient unit. Astro Systems Holland (ASH) manufactures such a device which is available to amateurs. 3 This unit is ideally suited to the task of high resolution imaging as it offers easily adjustable correction via a pair of prisms with levers extending out of the device barrel for quick and easy adjustment.
Typical prices for such correctors are not especially cheap coming in at around the £250 mark for the adjustable ASH corrector. Single prism correctors are less expensive, however I know of no current source for them.
Dispersion correctors in practice – are they worth it?
In theory a corrector sounds as if it should be an essential piece of equipment for the serious planetary observer, but what about in practice under the night sky? My own experience so far is an extremely positive one – so much so it has prompted me to compile this article. I began with an Adirondack 2° single prism which I still have. During the 2011 apparition of Saturn this device enabled me to obtain a notably higher level of image quality despite the mediocre altitude of the planet at just 37° at maximum. It enabled me to use unfiltered light to obtain sharp images, something which would have been impossible without the corrector in place at such an altitude. Even red light images showed a notable increase in sharpness. These positive results prompted me to obtain a fully adjustable dispersion corrector identical to the one detailed earlier.
One tricky problem faced by users of such a device is keeping the corrector aligned properly rotationally with regard to the direction of the dispersion. For example a planet’s position angle relative to the local horizon changes as it rises, culminates and sets. This means we must slowly adjust the corrector over time to keep it correctly aligned to counteract the direction of dispersion. This sounds complex but in practice is easily achieved if we know an object’s position angle relative to the horizon in our field of view. In practice the corrector needs to be adjusted every 30-60 minutes to keep the orientation of the device optimal for dispersion correction.
In truth there is no simple answer to the question ‘Are dispersion correctors worth acquiring?’ It depends upon a number of factors. Those using smaller telescopes would not really see much benefit apart from times when the planets are very low in the sky. For those using large apertures a dispersion corrector would appear to be essential equipment when seeking to obtain the best possible image quality.
For those fortunate enough to be located within the tropics it is likely that only a small benefit would be realised since for most of the time the planets are high enough in the sky to be well away from the worst effects of dispersion.
Many observers employ colour cameras for a single shot colour image. These are especially vulnerable to the effects of dispersion, and the figures quoted for white light apply for the amount of dispersion for a given altitude. I would consider a corrector essential for anyone using a colour camera for planetary imaging purposes. Simply re-aligning the colour channels back into line to remove colour fringing does not remove all of the dispersion affecting the image.
For those located in the northern hemisphere the years ahead, while very favourable for Jupiter, are not so good for Mars and Saturn, both of which are sinking lower in our skies. Obtaining good quality images of these planets will become increasingly difficult. A dispersion corrector such as those discussed in this article would help greatly to improve both image quality for CCD users and the view in the eyepiece for those observing visually.
For the casual observer the expense of a dispersion corrector may seem rather steep, however for more serious observers it is a very worthwhile investment, especially those employing large aperture telescopes for high resolution imaging or using colour CCD cameras.
Dispersion has been a largely forgotten issue from an amateur standpoint in recent years, however the use of dispersion correctors is on the increase, and in the age of very high resolution imaging many now consider these devices an essential piece of equipment to help coax the best out of their telescopes.
1. Prost J. P., ‘Atmospheric dispersion’, http://www.astrosurf.com/prostjp/Dispersion_en.html
2. Dall H. E., ‘Atmospheric dispersion’, J. Brit. Astron. Assoc.,71, 75-78 (1960 April)
3. Van Kranenburg A., ‘The atmospheric dispersion corrector’, http://www.astrosystems.nl/
4. Dobbins T. A., ‘AVA’s Dispersion Corrector’, Sky&Tel, 2005 June, 88-91
Article originally published in the JBAA 122, 4, 2012
[To search for planetary observations uploaded by BAA members, following this link to search the BAA Member Pages]
## XRISM telescope filter wheel, calibration system sent to Japan for assembly
SRON engineers wrap up the filter wheel for transport to the Japanese space agency JAXA. Credit: SRON
On June 9, SRON Netherlands Institute for Space Research sends its contributions to the XRISM X-ray telescope to Japan, where space agency JAXA will mount it on the satellite. SRON has been working on a filter wheel plus calibration system for the past few years. In 2023, XRISM will be launched into space, where it will observe phenomena such as black holes and supernovae.
The Earth's atmosphere blocks X-rays from space, much to the relief of people and animals, because it can be harmful to every living species. But because of this protective layer, astronomers miss out on a lot of information about, for example, black holes, the thin matter between clusters of galaxies, supernovae and cosmic particles. Space telescopes offer a solution. In 2023, the Japanese space agency will launch the X-ray satellite XRISM into orbit. Together with the University of Geneva, SRON contributes to XRISM with a filter wheel and the accessory calibration system.
On June 9, SRON sends the filter wheel plus a backup copy to Japan, where all XRISM components will be assembled. In September, an SRON team will fly over to carry out a number of tests on the filter wheel, which will be mounted in the telescope next year. "Everything has been delayed for a year and a half because of corona," says engineer Martin Grim, a member of the team that is traveling to Japan. "We actually wanted to carry out the instrument tests in Japan in May 2020 and XRISM was initially scheduled for launch in 2022."
The filter wheel puts several filters in front of XRISM's X-ray camera, allowing astronomers to filter out the brightness and wavelength of the cosmic rays as desired. For example, they will use the molybdenum neutral-density filter if a star or black hole emits too much X-ray radiation and they will select the beryllium or polyimide aluminum filter to block certain wavelengths. A low-radioactive iron-55 filter is part of the filter wheel to calibrate the camera. Iron-55 continuously emits a known X-ray spectrum serving as a reference point. The calibration system also includes Modulated X-ray Source (MXS) that provide a reference spectrum. The Dutch company Photonis has supplied these MXS units to SRON.
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1. Dec 29, 2007
### eagles_reciever
i am trying to solve this problem:
In the 1950s, an experimental train that had a mass of 2.60 X 10^4 kg was powered across a level track by a jet engine that produced a thrust of 5.25 X 10^5 N for a distance of 509 m. Assume that air resistance is negligible.
i am trying to find out the change in kinetic energy of the train and the final kinetic energy. however i cannot figure out what the velocity of the train is. Can anyone help please or provided an equation??
Thank you
2. Dec 29, 2007
### arildno
You don't need it.
You are to find the CHANGE in the kinetic energy; that can be calculated by the work done by the force given.
3. Dec 29, 2007
### eagles_reciever
im not understanding.. so find the change in work???
4. Dec 29, 2007
### eagles_reciever
Speed equation?
how would i find the speed of the train in this problem:
In the 1950s, an experimental train that had a mass of 2.60 104 kg was powered across a level track by a jet engine that produced a thrust of 5.25 105 N for a distance of 509 m. Assume that air resistance is negligible.
The equation is d/t but i do not know what the time is. can anyone help please????
Last edited by a moderator: Dec 29, 2007
5. Dec 29, 2007
### Staff: Mentor
6. Dec 29, 2007
### Erythro73
If I assume the acceleration to be uniform, I would suggest to use the kinematic equation :
$V_{f}^{2}=V_{i}^{2}+2*a*d$
as you know d and a.
7. Dec 29, 2007
### arildno
No, force times distance equals change in kinetic energy.
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Archives
On the Evolution of the Empirical Measure for the Hard-Sphere Dynamics
by Mario Pulvirenti S. Simonella
Vol. 10 No. 2 (2015) P.171~P.204
ABSTRACT
We prove that the evolution of marginals associated to the empirical measure of a finite system of hard spheres is driven by the BBGKY hierarchical expansion. The usual hierarchy of equations for $L^{1}$ measures is obtained as a corollary. We discuss the ambiguities arising in the corresponding notion of microscopic series solution to the Boltzmann-Enskog equation.
KEYWORDS
BBGKY hierarchy, hard sphere, empirical measure, marginal, Enskog equation
MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: 82C05, 82C22, 82C40, 35Q20.
MILESTONES
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# complex in the form a+ib
• March 15th 2013, 07:33 PM
sigma1
complex in the form a+ib
hello all, i have the following question which i am trying to solve. express ln1+ e^i in the form a + ib.
what i have done.
ln both sides therefore
i = ln (a+ib)
i= ln a + ln ib
equating real and imaginary parts
ln a = 0
a= 1
and
i = ln ib
1= ln b
b= e
ans= 1+ie
is this correct? if not how would i appraoch the problem. thanks.
• March 15th 2013, 08:51 PM
Prove It
Re: complex in the form a+ib
Quote:
Originally Posted by sigma1
hello all, i have the following question which i am trying to solve. express ln1+ e^i in the form a + ib.
what i have done.
ln both sides therefore
i = ln (a+ib)
i= ln a + ln ib
equating real and imaginary parts
ln a = 0
a= 1
and
i = ln ib
1= ln b
b= e
ans= 1+ie
is this correct? if not how would i appraoch the problem. thanks.
Some brackets would be nice. Are you asking to write $\displaystyle \ln{\left( 1 + e^i \right)}$ in the form $\displaystyle a + b\,i$?
• March 15th 2013, 09:27 PM
princeps
Re: complex in the form a+ib
Hint :
$e^{ix}=\cos(x)+i\sin(x)$
• March 16th 2013, 02:36 PM
sigma1
Re: complex in the form a+ib
Quote:
Originally Posted by Prove It
Some brackets would be nice. Are you asking to write $\displaystyle \ln{\left( 1 + e^i \right)}$ in the form $\displaystyle a + b\,i$?
well the question does not have any brackets in it. am wondering if its an error. but how would you attempt to solve it without the brackets. that is ln1 + e^i .
• March 16th 2013, 03:03 PM
Prove It
Re: complex in the form a+ib
Well ln(1) = 0, so you're really just trying to simplify e^i, or if you like, e^(1i). Use the hint Princeps gave you.
• March 16th 2013, 05:37 PM
sigma1
Re: complex in the form a+ib
Quote:
Originally Posted by Prove It
Well ln(1) = 0, so you're really just trying to simplify e^i, or if you like, e^(1i). Use the hint Princeps gave you.
cos1 + isin1 ?
• March 16th 2013, 05:38 PM
Prove It
Re: complex in the form a+ib
Correct
• March 16th 2013, 05:45 PM
sigma1
Re: complex in the form a+ib
Quote:
Originally Posted by Prove It
Correct
thanks alot.
• March 16th 2013, 07:21 PM
Prove It
Re: complex in the form a+ib
Quote:
Originally Posted by sigma1
thanks alot.
http://fc00.deviantart.net/fs70/f/20...ma-d3eafmb.png
The Alot says "You're Welcome" :)
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# How do you find the roots, real and imaginary, of y= (x-3)^2+2 using the quadratic formula?
Nov 29, 2015
Expand the expression on the right side into standard form, then apply the quadratic formula.
In this case there will be two imaginary roots $2 \pm i \sqrt{5}$
#### Explanation:
$y = {\left(x - 3\right)}^{2} + 2$
$\Rightarrow y = 1 {x}^{2} + \left(- 4\right) x + 9$
which is an example of quadratic standard form: $y = a {x}^{2} + b x + c$
for which the roots are given by the quadratic formula
$\textcolor{w h i t e}{\text{XXX}} x = \frac{- b + 1 \sqrt{{b}^{2} - 4 a c}}{- 2 a}$
Which for our example becomes:
$\textcolor{w h i t e}{\text{XXX}} x = \frac{- \left(- 4\right) \pm \sqrt{{\left(- 4\right)}^{2} - 4 \left(1\right) \left(9\right)}}{2 \left(1\right)}$
$\textcolor{w h i t e}{\text{XXXX}} = \frac{4 \pm \sqrt{16 - 36}}{2}$
$\textcolor{w h i t e}{\text{XXXX}} = \frac{4 \pm 2 \sqrt{- 5}}{2}$
$\textcolor{w h i t e}{\text{XXXX}} = 2 \pm i \sqrt{5}$
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