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https://nsdcudaan.com/kseeb-solutions-for-class-10-maths/
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# KSEEB Solutions for Class 10 Maths Karnataka State Syllabus
Last Updated on November 6, 2021
Expert Teachers at nsdcudaan.com has created Karnataka SSLC KSEEB Solutions for Class 10 Maths PDF Free Download in Kannada Medium and English Medium of KTBS Karnataka State 10th Standard Maths Textbook Solutions Answers Guide, Textbook Questions and Answers, Notes PDF, Model Question Papers with Answers, Study Material are part of KSEEB SSLC Class 10 Solutions.
Here we have given KSEEB Karnataka State Board Syllabus Class 10 Maths Textbook Solutions based on NCERT Syllabus. Students can also read Karnataka SSLC Maths Model Question Papers with Answers 2020-2021 PDF.
## Karnataka State Board Syllabus for Class 10 Maths Solutions
10th Standard Maths Textbook Solutions Karnataka State Syllabus in English Medium
KSEEB Solutions For Class 10 Maths Chapter 1 Arithmetic Progressions
• Chapter 1 Arithmetic Progressions Ex 1.1
• Chapter 1 Arithmetic Progressions Ex 1.2
• Chapter 1 Arithmetic Progressions Ex 1.3
• Chapter 1 Arithmetic Progressions Ex 1.4
• Chapter 1 Arithmetic Progressions Additional Questions
KSEEB 10th Maths Solutions Chapter 2 Triangles
• Chapter 2 Triangles Ex 2.1
• Chapter 2 Triangles Ex 2.2
• Chapter 2 Triangles Ex 2.3
• Chapter 2 Triangles Ex 2.4
• Chapter 2 Triangles Ex 2.5
• Chapter 2 Triangles Ex 2.6
• Chapter 2 Triangles Additional Questions
SSLC Maths Solutions Class 10 KSEEB Chapter 3 Pair of Linear Equations in Two Variables
• Chapter 3 Pair of Linear Equations in Two Variables Ex 3.1
• Chapter 3 Pair of Linear Equations in Two Variables Ex 3.2
• Chapter 3 Pair of Linear Equations in Two Variables Ex 3.3
• Chapter 3 Pair of Linear Equations in Two Variables Ex 3.4
• Chapter 3 Pair of Linear Equations in Two Variables Ex 3.5
• Chapter 3 Pair of Linear Equations in Two Variables Ex 3.6
• Chapter 3 Pair of Linear Equations in Two Variables Ex 3.7
• Chapter 3 Pair of Linear Equations in Two Variables Additional Questions
10th Standard Karnataka State Syllabus Maths Guide Chapter 4 Circles
• Chapter 4 Circles Ex 4.1
• Chapter 4 Circles Ex 4.2
• Chapter 4 Circles Additional Questions
Karnataka State Board 10th Maths Solutions Chapter 5 Areas Related to Circles
• Chapter 5 Areas Related to Circles Ex 5.1
• Chapter 5 Areas Related to Circles Ex 5.2
• Chapter 5 Areas Related to Circles Ex 5.3
• Chapter 5 Areas Related to Circles Additional Questions
SSLC Maths Solutions KSEEB Chapter 6 Constructions
• Chapter 6 Constructions Ex 6.1
• Chapter 6 Constructions Ex 6.2
• Chapter 6 Constructions Additional Questions
10th Maths In KannadaSolutions for Class 10 Chapter 7 Coordinate Geometry
• Chapter 7 Coordinate Geometry Ex 7.1
• Chapter 7 Coordinate Geometry Ex 7.2
• Chapter 7 Coordinate Geometry Ex 7.3
• Chapter 7 Coordinate Geometry Ex 7.4
• Chapter 7 Coordinate Geometry Additional Questions
KSEEB SSLC Class 10 Maths Solutions Chapter 8 Real Numbers
• Chapter 8 Real Numbers Ex 8.1
• Chapter 8 Real Numbers Ex 8.2
• Chapter 8 Real Numbers Ex 8.3
• Chapter 8 Real Numbers Ex 8.4
• Chapter 8 Real Numbers Additional Questions
10th Maths Syllabus State Board Karnataka Solutions Chapter 9 Polynomials
• Chapter 9 Polynomials Ex 9.1
• Chapter 9 Polynomials Ex 9.2
• Chapter 9 Polynomials Ex 9.3
• Chapter 9 Polynomials Ex 9.4
• Chapter 9 Polynomials Additional Questions
SSLC Mathematics Solution Part 2 KSEEB Chapter 10 Quadratic Equations
• Chapter 10 Quadratic Equations Ex 10.1
• Chapter 10 Quadratic Equations Ex 10.2
• Chapter 10 Quadratic Equations Ex 10.3
• Chapter 10 Quadratic Equations Ex 10.4
Maths Guide For Class 10 State Syllabus Karnataka Chapter 11 Introduction to Trigonometry
• Chapter 11 Introduction to Trigonometry Ex 11.1
• Chapter 11 Introduction to Trigonometry Ex 11.2
• Chapter 11 Introduction to Trigonometry Ex 11.3
• Chapter 11 Introduction to Trigonometry Ex 11.4
• Chapter 11 Introduction to Trigonometry Additional Questions
Maths Solutions for Class 10 State Syllabus Karnataka Chapter 12 Some Applications of Trigonometry
• Chapter 12 Some Applications of Trigonometry Ex 12.1
• Chapter 12 Some Applications of Trigonometry Additional Questions
Karnataka SSLC Maths Text Book Solutions Chapter 13 Statistics
• Chapter 13 Statistics Ex 13.1
• Chapter 13 Statistics Ex 13.2
• Chapter 13 Statistics Ex 13.3
• Chapter 13 Statistics Ex 13.4
• Chapter 13 Statistics Additional Questions
Karnataka State Board Syllabus for Class 10 Maths textbook Solutions Chapter 14 Probability
• Chapter 14 Probability Ex 14.1
• Chapter 14 Probability Ex 14.2
• Chapter 14 Probability Additional Questions
Karnataka SSLC Maths Notes PDF Chapter 15 Surface Areas and Volumes
• Chapter 15 Surface Areas and Volumes Ex 15.1
• Chapter 15 Surface Areas and Volumes Ex 15.2
• Chapter 15 Surface Areas and Volumes Ex 15.3
• Chapter 15 Surface Areas and Volumes Ex 15.4
• Chapter 15 Surface Areas and Volumes Ex 15.5
• Chapter 15 Surface Areas and Volumes Additional Questions
### Karnataka State Board Syllabus for Class 10 Maths Solutions in Kannada Medium
10th Standard Maths Textbook Solutions Karnataka State Syllabus in Kannada Medium
KSEEB Solutions For Class 10 Maths Chapter 1 Arithmetic Progressions
• Arithmetic Progressions Ex 1.1
• Arithmetic Progressions Ex 1.2
• Arithmetic Progressions Ex 1.3
KSEEB 10th Maths Solutions Chapter 2 Triangles
• Triangles Ex 2.1
• Triangles Ex 2.2
• Triangles Ex 2.3
• Triangles Ex 2.4
• Triangles Ex 2.5
SSLC Maths Solutions Class 10 KSEEB Chapter 3 Pair of Linear Equations in Two Variables
• Pair of Linear Equations in Two Variables Ex 3.1
• Pair of Linear Equations in Two Variables Ex 3.2
• Pair of Linear Equations in Two Variables Ex 3.3
• Pair of Linear Equations in Two Variables Ex 3.4
• Pair of Linear Equations in Two Variables Ex 3.5
• Pair of Linear Equations in Two Variables Ex 3.6
10th Standard Karnataka State Syllabus Maths Guide Chapter 4 Circles
• Circles Ex 4.1
• Circles Ex 4.2
Karnataka State Board 10th Maths Solutions Chapter 5 Areas Related to Circles
• Areas Related to Circles Ex 5.1
• Areas Related to Circles Ex 5.2
• Areas Related to Circles Ex 5.3
SSLC Maths Solutions KSEEB Chapter 6 Constructions
• Constructions Ex 6.1
• Constructions Ex 6.2
10th Maths In KannadaSolutions for Class 10 Chapter 7 Coordinate Geometry
• Coordinate Geometry Ex 7.1
• Coordinate Geometry Ex 7.2
• Coordinate Geometry Ex 7.3
KSEEB SSLC Class 10 Maths Solutions Chapter 8 Real Numbers
• Real Numbers Ex 8.1
• Real Numbers Ex 8.2
• Real Numbers Ex 8.3
• Real Numbers Ex 8.4
10th Maths Syllabus State Board Karnataka Solutions Chapter 9 Polynomials
• Polynomials Ex 9.1
• Polynomials Ex 9.2
• Polynomials Ex 9.3
SSLC Mathematics Solution Part 2 KSEEB Chapter 10 Quadratic Equations
Maths Guide For Class 10 State Syllabus Karnataka Chapter 11 Introduction to Trigonometry
• Introduction to Trigonometry Ex 11.1
• Introduction to Trigonometry Ex 11.2
• Introduction to Trigonometry Ex 11.3
• Introduction to Trigonometry Ex 11.4
Maths Solutions for Class 10 State Syllabus Karnataka Chapter 12 Some Applications of Trigonometry
• Some Applications of Trigonometry Ex 12.1
Karnataka SSLC Maths Text Book Solutions Chapter 13 Statistics
• Statistics Ex 13.1
• Statistics Ex 13.2
• Statistics Ex 13.3
• Statistics Ex 13.4
Karnataka State Board Syllabus for Class 10 Maths textbook Solutions Chapter 14 Probability
• Probability Ex 14.1
Karnataka SSLC Maths Notes PDF Chapter 15 Surface Areas and Volumes
• Surface Areas and Volumes Ex 15.1
• Surface Areas and Volumes Ex 15.2
• Surface Areas and Volumes Ex 15.3
• Surface Areas and Volumes Ex 15.4
We hope the given Karnataka SSLC KSEEB Solutions for Class 10 Maths PDF Free Download in English Medium and Kannada Medium of KTBS Karnataka State 10th Standard Maths Textbook Solutions Answers Guide, Textbook Questions and Answers, akub koyyur 10th maths Notes PDF, Model Question Papers with Answers, Study Material will help you.
If you have any queries regarding KSEEB Karnataka State Board Syllabus Class 10 Maths Solutions based on NCERT Syllabus, drop a comment below and we will get back to you at the earliest.
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2021-12-01 09:05:59
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http://crypto.stackexchange.com/tags/encryption/hot?filter=month
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# Tag Info
## Hot answers tagged encryption
5
I think the 10100 is a typo and should be $10^{100}$ as shown here The period would be something along the lines of how long until the byte stream repeats. For example if the byte stream were "ABCDABCDABCD" and so on, then the period would be 4. For security you want a large period so that you can encrypt large amounts of data.
5
A Diffie-Hellman exchange needs not be synchronous. In DH, each party has a private key (x) and a public key (gx mod p). If the sender knows the recipient's public key ga, then he can build his own key pair (b and gb), compute the shared secret (gab), and send both his public key (gb) and the encrypted message (symmetric key derived from gab) to the ...
5
There is a line between encryption and obfuscation, I would say this is on the latter side. If someone knows the method, and is able to correctly guess even 1 of the original values, simple math will reveal all the original values. Additionally, depending on the type of data in the field, guessing the floating point number may only take seconds even with no ...
3
In short: hybrid encryption. Encrypt the content with a random symmetric key. And encrypt this key with the public key of each recipient. Then all you need is a public key from all the people you want to share the content with. It doesn't even have to be the same public key cryptosystem for all the receivers. It might be they each come up with their ...
3
This is flawed: Cracking: The format leaks information on how often specific characters occur. For instance, if input message contains 6 o letters, there is likely much more 111 values than the most other values present. Such a small biases are sufficient for cryptoanalysis to break the message in many cases. Also, random.randint does not return random ...
2
This all depends on the IVs. If they are independently generated, the IVs will not only be different (so $IV_1 \neq IV_2$), but also their sequences will not overlap with overwhelming probability. In that case, then everything should be fine, so $C_2 = E(K,(nonce,IV_2))$, and $C_1 = M_1 \oplus E(K,(nonce,IV_1))$. However, if they are reused (so $IV_1 = ... 2 At its simplest, to encrypt a message for$n$different recipients, you could just make$n$copies of the message, encrypt each one with a different key, and join the encrypted messages together into a single long ciphertext. Of course, the disadvantage of this scheme is that the ciphertext length grows linearly with the number of recipients. To avoid ... 2 As Ricky Demer comments above, an obvious weakness of this scheme is that if you know (or can guess) the plaintext$M$corresponding to any ciphertext$C$, then you can immediately recover the key by calculating$K = C/M$. Modern ciphers are generally expected to resist such known-plaintext attacks, making your cipher insecure by any current standards. In ... 2 In order to tell for certain which algorithm is used from the cipher text alone, you have to break the key as well. It is however possible to make some educated guesses about the algorithm. Your strings are only 88 bits long. That rules out every asymmetrical algorithm that I know of. If it is correct, that all of those strings encrypt the same message, ... 2 In addition to what Richie has noted, here are some problems with the encryption method that you normally don't like to see in encryption. It preserves order (if a > b, then E(a) > E(b)). So, say the cells encode salary information. Even if it doesn't reveal the salary, it still reveals a lot of information. It is not semantically secure. In other words, I ... 2 Here is the answer for why a deterministic public-key encryption scheme cannot be CPA secure. For CPA security it is sufficient if an adversary can distinguish between encryptions of two messages$m_0$and$m_1$. That is, an adversary gets to see an encryption$c \gets \textsf{Enc}(pk,m_b)$for a random bit$b$together with the public key$pk$. Now in ... 2 None of Twofish, Serpent and AES are currently known as broken, so as far as security is concerned, you can use any of them. AES has a slight advantage because it's very widely used, so if it gets broken you're more likely to hear about it and get relevant software updates quickly. The Snowden postings haven't changed much as far as cryptography usage is ... 2 Yes, the same weaknesses apply. Text on computers is a bunch of numbers; a OTP encrypts a sequence of numbers modulo 2. 2 Could someone even recognize that the values are encrypted? Well, maybe, maybe not. You're correct that the values would all appear to be valid dates (this is known as format-preserving encryption, by the way), so they would not look obviously encrypted, the way, say, a random hex string would. If someone just saw a small number of such dates, with no ... 1 There is no out of the box tester which can tell you what you need, you need to do research. I'm having the same 'problem'. I'm doing an internship for a company which wants to protect their self-created protocol which works on top of TCP. So to tackle this I've create a plan of approach and defined my research parameters (quite broadly). Now i'm looking ... 1 Encryption & Decryption: It increases the messages secrecy. Sender encrypts the plain text using a secret Key and sends it(cipher text) to receiver. Then the receiver receives the cipher text and decrypt it using the same secret key. The secret key should be shared between the sender and receiver. Hashing: It ensures the integrity of a message (which ... 1 Is it theoretically possible? Yes. Is it practical? Absolutely not. In your scenario, there are two approaches, but both are pretty much theoretical only. First, there is fully homomorphic encryption (FHE). This means, you can do computations like addition and multiplication of ciphertexts and get another encrypted message. To actually have some of your ... 1 The fundamental difference between hash and encryption techniques is that hash is irreversible while encryption is reversible. Hash algorithms generate a digest of fixed length output cipher text for a given input plain text. The output text cannot be converted back to input text. The generated output will always be same for a given input plain text that ... 1 For a (fixed-length) cipher to meet your first condition, it needs to be the case that it's no easier to guess the plaintext if you have the ciphertext than it is to guess the plaintext without the ciphertext. Now, suppose I send a random 1024-bit string XORed with$G(k)$for some 128-bit$k$and computationally secure$G$. The probability that my message is ... 1 summmary In general, no. An attacker who has lots of ciphertext+plaintext pairs may never be able to reverse-engineer an algorithm from them. An attack may not even be able to distinguish which one of a large group of known encryption algorithms was used to generate those ciphertexts. However, various weaknesses in some algorithms and protocols are known. ... 1 Yes, key derivation with known parameters can add to security. For a start, in the scenario of the question (understood as a real-world scenario, with PDF encryption used in step 4), when key derivation is used, you can hand one encrypted PDF document and its derived key to a person, and that won't enable her to decipher other enciphered documents; this ... 1 To show that a family of functions is not a PRP, you have to either show that the functions are not permutations or that they do not behave pseudo-randomly. As it is already established that the functions are in fact permutation you need to show the latter. For a family of permutations to be a PRP means that it is computationally infeasible to distinguish a ... 1 If your nonce is 16 bytes, and your message pre-nonce is a multiple of 16 bytes (i.e. no padding is needed), sending the nonce in the clear opens you up to replay-ish attacks. Specifically, if an attacker captures one exchange with nonce$N$and response$R$(with$b$blocks$R_1$through$R_b$), and then impersonates the server and the client sends them ... 1 Malicious JavaScript could trigger calls against any SSL server. Although this JavaScript is not able to actually read the cookie for that server directly, the browser will send that cookie through the SSL tunnel. The JavaScript will then alter the URL, keeping the server but changing the path or request parameters. In that way it can inject data before ... 1 I generate a derived key for every document via KD=KDF∗KM∗filename where KDF and filename are known to the public. If I brute-force a document and get the correct KD, I know 3 out of 4 variables and therefore know KM. As a result, I’m able to derive every KD. The main problem here is, that you assume the existence of a function, which can calculate the ... 1 First of all the Additional Data (AD) is not a tag. It is data that is also authenticated by the authentication tag. This authentication tag is appended to the ciphertext by libsodium. The tag doesn't consist of separate portions for AD and the ciphertext (and IV), the AD is taken into account during calculation of the tag. The AD can be any data, including ... 1 Your scheme does mean that the server as a passive participant can't read the data, but if your threat model includes the server trying to get access to the data you'll need to do more. tylo mentions the server (or one of its agents) attempting to join a group, but it could probably also MITM the process of another user joining the group (by returning a ... 1 Your questions: if all the clients leave the group (at which moment they delete K from their local store), all the information on the server will be useless as nobody has K anymore if Client2 comes online for the first time and another group member is not online it will not be able to obtain K and will not be able to decrypt any of the data stored on ... 1 The entropy of a random password is given by the formula:$H = log_2 N^L = L\log_2 N$where$N$is the number of possible symbols, and$L$is the length of the password. Since you want to know the minimum length to achieve a determined level of security, then the answer to your question is$L = \lceil \frac{H}{log_2 N} \rceil$. In your case,$N=20\$, and ...
Only top voted, non community-wiki answers of a minimum length are eligible
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2015-03-03 22:57:32
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http://newspaperarchive.com/ca/manitoba/baldur/baldur-gazette/2011/11-08/
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Baldur Gazette, November 8, 2011 : Front Page
## Publication: Baldur Gazette November 8, 2011
Baldur Gazette (Newspaper) - November 8, 2011, Baldur, Manitoba Tuesday, November 8, 2011 Vol. 9 No. 2 \$ 1.00 WALLIS AGENCIES INSURANCE Local Flu Shot Clinics now underway Page 7 Baldur students host Halloween activities Page 10 CMCA AUDITED by Tracy Rimmer FREELANCE WRITER ! " # \$ % ! & ' ! ( ) ! ! * ( ! ! ! # ! ! ! ! " ( + ( ! ! " * , - ( ! ! \$ ! + ( * " + * " . # \$ % / 0 1 ! & ! ! ! ! ! ! # ! ! 2 3 ! 4 ! ! ! ! ! CONTINUED ON PAGE 3 1 5 6 ) 7 ) ! 7 8 9 8 6 5 8 : ! ; ! 5 8 6 5 5 ! ! 5 6 ) < 5 6 ) , ) ! ! , ! ! / , , 0 = > 5 ? ! 8 6 5 8 8 6 = < ! ! . ! ! ! ! 5 6 ) ! ! < 5 6 ) ) ( ! ! ! ! ! . ! , ! ! ! ! ! " # ( ! ! . ! ! ! ! 1 ! ! 5 6 ) ; ! 5 8 6 5 5 " \$ < < ( 5 6 ) . ! ! ! ! 1 ! ! 5 6 ) 3 ! 3 3 ! . 8 9 8 6 5 8 ( ! < ! ! ! * ! ! ( < \$ ; ! 5 8 6 5 5 8 9 8 6 5 8 ! ! . ! @ < ! 1 ! 5 6 ) ! / 8 6 = A B ) ! 0 @ . ! ! . ! ! CONTINUED ON PAGE 2 GLENBORO TRADE SHOW HAS PLENTY TO OFFER… On Saturday, November 5th approximately 32 vendors took part in the annual trade show held at the Glenboro Community Kinsmen Hall. Maxine Goodman, who sells the Lia Sophia jewellery line, was just one of the vendors taking part in the day’s event. \$ 1266.00 was raised from the event for the Glenboro Gaiety Theatre. GAZETTE PHOTO/ ED TORZ ;
#### Share Page
Publication: Baldur Gazette
Location: Baldur, Manitoba
Issue Date: November 8, 2011
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2016-09-27 00:28:12
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https://dsp.stackexchange.com/questions/22404/fourier-synthesis-with-varying-frequencies-over-time
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# Fourier Synthesis With Varying Frequencies Over Time
I'm doing some fourier synthesis by making bins of complex numbers, filling them out with the amplitude and phase information and using IFFT to turn that into time domain samples.
I'm able to make sine, saw, triangle and square waves, and other more exotic periodic wave forms.
My question is, how do you do fourier synth for sounds that have frequency amplitudes that change over time?
Do you do it in small windows and append them together? If so, how do you make sure the phase's line up for each frequency from window to window?
Thanks so much for any info!
You might want to have a look at the overlap-add (http://en.wikipedia.org/wiki/Overlap%E2%80%93add_method) and overlap-save (http://en.wikipedia.org/wiki/Overlap%E2%80%93save_method) methods. But, if all you are trying to do is additive synthesis, you don't absolutely have to use the IFFT to generate your signal.
You can set up a bank of $M \in {[1 \ldots N_{Osc}]}$ elementary oscilators of the form $x_m(n)=\alpha_m \times sin(\frac{2 \times \pi \times f_m \times n}{Fs}+\phi_m)$ where $m \in M, N_{Osc} \in \mathbb{N}$, $\alpha_m, \phi_m, F_s, f_m \in \mathbb{R}$ and final output $y(n) = \sum_{m \in M}{x_m(n)}$.
In this case, $\alpha_m, \phi_m$ are amplitude and phase coefficients for each oscillator at frequency $f_m$ and to make them variable in time, all you have to do is turn them to $\alpha_m(n), \phi_m(n)$, i.e. to make them dependent on time sample $n$. This can be an absolute definition, by actually defining a varying waveform or by using some form of interpolation to provide initial and final values for these quantities and let the intermediate values be calculated automatically.
This would work both for generating one sample at a time or a frame of samples at a time (by iteration) and would not have any problems with continuity.
When you search for Fourier transform pairs you can find quite a few functions in the time and frequency domain which are each others (inverse)Fourier transform. Some of these pairs also contain time domain functions, which are sinusoidal with changing amplitude over time. For example exponential decay can be written as,
$$\mathcal{L}\{e^{-at}x(t)\}=X(s+a),$$
where $\mathcal{L}\{x(t)\}=X(s)=\int_0^\infty x(t)e^{-st}dt$ and $x(t)=0$ for $t<0$. Another example would be multiplying by time,
$$\mathcal{L}\{tx(t)\}=-\frac{d}{ds}X(s),$$
by using the fact that $\mathcal{L}\{\alpha x_1(t)+\beta x_2(t)\}=\alpha X_1(s)+\beta X_2(s)$, then this can be used to construct a variable amplitude over time using a polynomial.
• I wish I could accept your answer as well as A_A's. Thanks so much for the info, I'm going to have to play with the info in both answers and see what is most practical in my usage case. Thanks!! – Alan Wolfe Mar 29 '15 at 0:11
• i surprized you accepted either. i don't think either got to the issue of using the iFFT to synthesize sound. better look into sinusoidal modeling and phase vocoder. (other references can be found.) especially to answer the question: "how do you make sure the phase's line up for each frequency from window to window?" and i gotta paper too that you can get. – robert bristow-johnson Mar 30 '15 at 2:36
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2021-03-02 05:17:58
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http://mathhelpforum.com/calculus/131806-average-value-function-help-print.html
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# Average value function help
Printable View
• March 3rd 2010, 01:45 AM
BadMonkey
Average value function help
Hey
Can you please tell me if the integration in my photo is correct
Thank you
Muchas gracias
• March 3rd 2010, 05:02 AM
Grandad
Hello BadMonkey
Welcome to Math Help Forum!
Quote:
Originally Posted by BadMonkey
Hey
Can you please tell me if the integration in my photo is correct
Thank you
Muchas gracias
The lower limit of the integral in each case is $0$, and the upper limit is $a$. This will give a positive sign in the answer.
Also, each result can be simplified, by dividing by $a$.
Grandad
• March 3rd 2010, 05:32 AM
BadMonkey
Hello grandad
Thank you for welcome
I think I have the correct answer now,from your kind help
g(x) functions answer would be a^2/(3), and the first part f(x) works out a/(2)
is this correct please?(Clapping)
• March 3rd 2010, 10:45 AM
Grandad
Hello BadMonkey
Quote:
Originally Posted by BadMonkey
Hello grandad
Thank you for welcome
I think I have the correct answer now,from your kind help
g(x) functions answer would be a^2/(3), and the first part f(x) works out a/(2)
is this correct please?(Clapping)
Yes, they're quite correct now.
Grandad
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2013-12-11 05:52:11
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http://math.stackexchange.com/questions/387199/you-are-johnny-depp-3
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# You are Johnny Depp 3!
An extension of this question.
As @Jared stated in his answer the solution is:
we assume that the head pirate chooses between multiple possible proposals that maximize his profit by rewarding seniority, then a proposal of $(G-1-\lfloor\frac{n}{2}\rfloor,0,1,2,0,1,0,1,\ldots,\frac{1+(-1)^n}{2})$ will be accepted. The proof is similar to the induction above.
For 9 pirates this gives $\left(G-5,0,1,2,0,1,0,1,0\right)$
The pirate captain has worked this out and knocks the lock off the looted chest to be confronted by a measly 3 gold doubloons!
-
Lets work our way up and start with just 4 pirates. No problem, a proposal of $(1,0,1,1)$ works based on the answer to the question linked above.
• 5 pirates: $(0,2,1,0,0)$ works since the two second mates both get more than they would have otherwise.
• 6 pirates: $(0,1,0,0,1,1)$ works, since again, all pirates who wouldn't die get more money than they would have.
• 7 pirates: $(0,1,0,1,1,0,0)$ works by the same argument.
• 8 pirates: This guy is dead. No allotment of coins will allow him to survive. There are 4 pirates getting 0 coins at 7, so there aren't enough coins around for the captain to bribe them.
• 9 pirates: $(0,0,0,1,0,0,1,1)$ works. The first-mate will vote in favor, because he doesn't want to die. The ones getting coins will also vote in favor, because they would get 0 coins if the captain and first-mate die.
• 10 pirates: The captain is once again a dead man. The first- and second-mate both know they will survive at 9 pirates, so they can vote to kill, unless the captain gives them each a coin, but there aren't enough coins to go around then - at best the captain can secure 4 votes in favor. Not a clear majority.
• 11 pirates: The captain and first mate will vote in favor, otherwise they die, plus three other pirates who each get a coin. The vote of 5 to 6 however means no one lives.
• 12 pirates: Almost! Now, the captain, first-mate and second-mate all vote in favor, otherwise they die, plus another three pirates who each get coins, but with a vote of 6:6, the captain doesn't have a clear majority, so he dies.
• 13 pirates: AH! The Captain can live once again with $(0,0,0,0,1,1,1,0,0,0,0,0,0)$ or similar. The captain and first three mates all vote for, since otherwise they die. The other three pirates would have received 0 coins at 9 pirates remaining. But now they each get 1 coin again, so they live.
The next number that allows the captain to live is 21 when pirates 14-20 all want to live, plus himself, and he can give 3 other pirates who would have received 0 coins at 13 pirates which gives a total of 11 pirates for, 10 pirates against.
For $n\ge 9$, the number of pirates which allow the captain to live are: $9,13,21,69,133,...$ $=2^{k}+5$
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2014-08-28 03:54:28
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https://web2.0calc.com/questions/help_78514
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+0
# help!!!
0
140
1
+794
Consider the geometric sequence $$\frac{16}{9}, \frac{8}{3}, 4, 6, 9, \ldots$$. What is the eighth term of the sequence? Express your answer as a common fraction.
Jul 1, 2020
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2021-01-17 00:59:17
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https://ask.sagemath.org/users/1214/babgen/?sort=recent
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2021-04-13 22:27:34 +0200 received badge ● Notable Question (source) 2016-04-10 16:42:54 +0200 received badge ● Popular Question (source) 2015-10-20 11:54:11 +0200 received badge ● Taxonomist 2014-11-17 08:54:16 +0200 asked a question All Ideals of Ring For given group G, by G.subgroups(), we can list all subgroups. now my question is : Is there something for subgroups() for finding all left ideals of a given ring. For instance, k = GF(5); M = MatrixSpace(k,2,2) How can I have all left ideals? 2014-01-15 15:09:25 +0200 commented answer Subgroups of Linear Groups It is not working on mu SAGE. I have following error: Traceback (most recent call last): File "", line 1, in File "_sage_input_26.py", line 10, in exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("RyA9IFNMKDIsR0YoMykpCmEgPSBHLmdlbnMoKVswXSA7IGEKSCA9IEcuc3ViZ3JvdXAoW2FdKSA7IEg="),globals())+"\\n"); execfile(os.path.abspath("___code___.py")) File "", line 1, in File "/private/var/folders/rp/rpRaJRnOEQyzp4ftzijESU+++TI/-Tmp-/tmpYB4kTc/___code___.py", line 5, in exec compile(u'H = G.subgroup([a]) ; H File "", line 1, in File "parent.pyx", line 620, in sage.structure.parent.Parent.__getattr__ (sage/structure/parent.c:5943) File "misc 2014-01-15 10:37:38 +0200 asked a question Subgroups of Linear Groups I want to find the subgroup which is generated by two elements. G = SL(2,GF(3)) D = G.subgroup([G[1],G[2]]) Why do I have this error: Traceback (most recent call last): File "", line 1, in File "_sage_input_50.py", line 10, in exec compile(u'open("___code___.py","w").write("# -- coding: utf-8 --\n" + >_support_.preparse_worksheet_cell(base64.b64decode("RyA9IFNMKDIsR0YoMykpCkQgPSBH>LnN1Ymdyb3VwKFtCWzFdLEJbMl1dKQ=="),globals())+"\n"); >execfile(os.path.abspath("___code___.py")) File "", line 1, in File "/private/var/folders/rp/rpRaJRnOEQyzp4ftzijESU+++TI/-Tmp-/tmpGFGoqT/___code___.py", line 4, in exec compile(u'D = G.subgroup([B[_sage_const_1 ],B[_sage_const_2 ]]) File "", line 1, in File "parent.pyx", line 620, in sage.structure.parent.Parent.__getattr__ >(sage/structure/parent.c:5943) File "misc.pyx", line 200, in sage.structure.misc.getattr_from_other_class >(sage/structure/misc.c:1427) AttributeError: 'SpecialLinearGroup_finite_field_with_category' object has no >attribute 'subgroup' 2014-01-02 02:26:09 +0200 commented question Error with Automorphism group @kcrisman If order() replace with Order(), then the error eliminates. 2014-01-01 13:46:23 +0200 received badge ● Student (source) 2014-01-01 13:00:14 +0200 asked a question Error with Automorphism group I use this for computing automorphism group: G = SymmetricGroup(3) H = libgap(G).AutomorphismGroup() but when I want to compute order of automorphism group, I cant compute and I have this error: Traceback (most recent call last): File "", line 1, in File "_sage_input_20.py", line 10, in exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("RyA9IFN5bW1ldHJpY0dyb3VwKDMpCkggPSBsaWJnYXAoRykuQXV0b21vcnBoaXNtR3JvdXAoKQpILm9yZGVyKCk="),globals())+"\\n"); execfile(os.path.abspath("___code___.py")) File "", line 1, in File "/tmp/tmpm9uLkv/___code___.py", line 5, in exec compile(u'H.order() File "", line 1, in File "element.pyx", line 1867, in sage.structure.element.RingElement.order (sage/structure/element.c:15911) File "element.pyx", line 1873, in sage.structure.element.RingElement.additive_order (sage/structure/element.c:15968) NotImplementedError 2013-12-29 02:19:04 +0200 commented question Automorphism Group of a Group @kcrisman we need automorphism group, as our research continues in graph theory and we use graph tools of sage and GAP doesnt have such tools. 2013-12-28 04:59:48 +0200 asked a question Automorphism Group of a Group I used to find the automorphism group of a group with GAP. Now, I would like to know that Is the way which I can compute automorphism group of a group with SAGE without using GAP? 2013-04-21 16:54:11 +0200 commented question affine variety in che soali akhe? 2013-02-17 01:47:54 +0200 commented answer Locally-Dihedral 2-group I check but I can't find any thing. 2013-02-14 15:09:54 +0200 asked a question Locally-Dihedral 2-group Let $Z_{2^{\infty}}$ be 2-prufer group. I want to define $G=Z_{2^{\infty}}$semi-direct $b$ =$<$$a$$,b|b^2=1,b^{-1}ab=a^{-1}$, for $a$ $\in A>$ in sage. I think this is possible in GAP but I don't know how to define? 2013-02-06 14:58:23 +0200 marked best answer One-dimension subspace You need to take care of the zero element of the subspace. for x in i will iterate over all the vectors including the zero vector. 2013-02-05 07:34:43 +0200 commented answer One-dimension subspace how can I take care of the zero element? 2013-02-05 03:33:19 +0200 asked a question One-dimension subspace I am looking for one-dimension subspace $x$ such that for every $v\in$$\langle x\rangle$ we have $v.v=0$. I try this: V = VectorSpace(GF(5),2) for i in V.subspaces(1): for x in i: if (x*x==0): print(i) but I have trouble with it: sage gives me subspace "Vector space of degree 2 and dimension 1 over Finite Field of size 5 Basis matrix: [1 0]" this subspace is not satisfied my condition 2013-01-27 07:29:16 +0200 asked a question permutation representation Dear all Does sage support groups such as Frobenius groups or general affine group. How can I find the permutation representation of an arbitrary group. Best regrad 2013-01-19 13:41:45 +0200 received badge ● Scholar (source) 2013-01-19 13:41:45 +0200 marked best answer Product of Elements in Group Use D(x)*D(y) instead of D(x*y): sage: D = SymmetricGroup(4) sage: H=D.subgroups() sage: A=Set(D(x)*D(y) for x in H[18] for y in H[6]) sage: print(A) {(1,4,3,2), (), (1,3,4), (2,4), (1,3,2,4), (1,2,3), (1,4,2,3), (1,2)(3,4)} sage: A=Set(D(x)*D(y) for x in H[18] for y in H[26]) sage: print(A) {(1,4), (1,3)(2,4), (1,3,4,2), (2,3), (3,4), (1,4,2,3), (), (2,3,4), (1,3,2,4), (1,4)(2,3), (1,4,3), (1,2,4), (1,2), (1,2)(3,4), (1,3,2), (1,2,4,3)} 2013-01-19 01:34:05 +0200 asked a question Product of Elements in Group I have problem with product of two elements. it runs for H[18] and H[6] and it has no error but it doesn't run for H[18] and H[26]. I can't figure out this problem. I was wondering if you could help me. D = SymmetricGroup(4) H=D.subgroups() A=Set(D(x*y) for x in H[18] for y in H[6]) print(A) {(1,4,3,2), (), (1,3,4), (2,4), (1,3,2,4), (1,2,3), (1,4,2,3), (1,2)(3,4)} // D = SymmetricGroup(4) H=D.subgroups() A=Set(D(x*y) for x in H[18] for y in H[26]) print(A) Traceback (most recent call last): File "", line 1, in File "_sage_input_54.py", line 10, in exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("RCA9IFN5bW1ldHJpY0dyb3VwKDQpCkg9RC5zdWJncm91cHMoKQpBPVNldChEKHgqeSkgZm9yIHggaW4gSFsxOF0gZm9yIHkgaW4gSFsyNl0pCnByaW50KEEp"),globals())+"\\n"); execfile(os.path.abspath("___code___.py")) File "", line 1, in File "/private/var/folders/rp/rpRaJRnOEQyzp4ftzijESU+++TI/-Tmp-/tmpcoW8YW/___code___.py", line 5, in A=Set(D(x*y) for x in H[_sage_const_18 ] for y in H[_sage_const_26 ]) File "/Applications/sage/local/lib/python2.7/site-packages/sage/sets/set.py", line 129, in Set return Set_object_enumerated(list(X)) File "/private/var/folders/rp/rpRaJRnOEQyzp4ftzijESU+++TI/-Tmp-/tmpcoW8YW/___code___.py", line 5, in A=Set(D(x*y) for x in H[_sage_const_18 ] for y in H[_sage_const_26 ]) File "element.pyx", line 1388, in sage.structure.element.MonoidElement.__mul__ (sage/structure/element.c:11883) File "element.pyx", line 1382, in sage.structure.element.MonoidElement.__mul__ (sage/structure/element.c:11742) File "coerce.pyx", line 797, in sage.structure.coerce.CoercionModel_cache_maps.bin_op (sage/structure/coerce.c:7672) TypeError: unsupported operand parent(s) for '*': 'Permutation Group with generators [(1,2)(3,4), (1,3,2,4)]' and 'Permutation Group with generators [(1,2)(3,4), (1,3)(2,4), (1,4)]' 2012-10-30 16:33:45 +0200 received badge ● Editor (source) 2012-10-30 16:32:49 +0200 asked a question Finding subgroup summand Hi everyone, Let (G,.) be group with two subgroup H and K, I want to know when HK=G? $HK={h.k | h\in H ,k\in K}$ I try this D= CyclicPermutationGroup(9) H=D.subgroups() for i in H: for j in H: i*j=K if(K==G) I have no idea how define " * " 2012-10-17 12:17:15 +0200 asked a question Create graph I want define new graph in sage. Let G be finite group. The graph's vertices are subgroup and two vertices are adjacent if and only if sum of two subgroup is G. I have trouble with define this graph in sage. Any suggestion? I have idea in gap but I don't have idea what can I change in sage? Summands := function(G) local n, i, sgl, l, A, B, D; obtain a list of all subgroups sgl := List(LatticeSubgroups(G)!.conjugacyClassesSubgroups, Representative); n is the number of divisors of |G| n := Size(DivisorsInt(Size(G))); D := []; if IsOddInt(n) then l := QuoInt(n + 1, 2); else l := QuoInt(n, 2); fi; for i in [1..l] do for A in Filtered(sgl, function(g) return Size(g) = DivisorsInt(Size(G))[i]; end) do for B in Filtered(sgl, function(g) return Size(g) = DivisorsInt(Size(G))[n+1-i]; end) do Add(D, [A, B]); od; od; od; return D; end;
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2022-09-28 15:40:02
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https://nebusresearch.wordpress.com/tag/toy/
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## The Box Drops
So the last piece I need for figuring out whether it’s easier to tip a box over by pushing on the middle of an edge or along one corner is to know the amount of torque applied by pushing with, presumably, the same force in both locations. Well, that’s almost the last bit. I also need to know how the torque and the moment of inertia connect together to say how fast an angular acceleration I can give the box.
## A Third Thought About Falling
I’m surprised my father let me get away with it. I assume that he was just being courteous and letting me get to my next points in studying whether a box is easier to tip over by pushing from the center of one of its edges or by pushing from its corner. Or he’s figured it’s too much bother to write a response; he’s been living his computer life on an iPod for a long while now and I can’t figure how he types at any length on that.
## A Second Way To Fall Over
I admit not being perfectly satisfied with my answer, about whether a box is easier to tip over by pushing on the middle of one of its top edges or by pushing on its corner, just by looking at it from the energy both approaches need to raise the box’s center of mass above the pivot. It’s straightforward enough, but I don’t do this sort of calculation often, so maybe I’m looking at the wrong things. Can I find another, independent, line of argument? If I can, does that get to the same answer? If it does, good. If it doesn’t, then I get to wonder which line of argument I believe in more. So here’s one.
## One Way To Fall Over
[ Huh. My statistics page says that someone came to me yesterday looking for the “mathematics behind rap music”. I don’t doubt there is such mathematics, but I’ve never written anything approaching it. I admit that despite the long intertwining of mathematics and music, and my own childhood of playing a three-quarter size violin in a way that must be characterized as “technically playing”, I don’t know anything nontrivial about the mathematics of any music. So, whoever was searching for that, I’m sorry to have disappointed you. ]
Now, let me try my first guess at saying whether it’s easier to tip the cube over by pushing along the middle of the edge or by pushing at the corner. I laid out the ground rules, and particularly, the symbols used for the size of the box (it’s of length $a$) and how far the center of mass (the dead center of the box) is from the edges and the corners last time around. Here’s my first thought about what has to be done to tip the box over: we have to make the box pivot on some point — along one edge, if we’re pushing on the edge; along one corner, if we’re pushing on the corner — and so make it start to roll. If we can raise the center of mass above the pivot then we can drop the box back down with some other face to the floor, which has to count as tipping the box over. If we don’t raise the center of mass we aren’t tipping the box at all, we’re just shoving it.
## Tipping The Toy
My brother phoned to remind me how much more generally nervous I should be about things, as well as to ask my opinion in an utterly pointless dispute he was having with his significant other. The dispute was over no stakes whatsoever and had no consequences of any practical value so I can see why it’d call for an outside expert. It’s more one of physics, but I did major in physics long ago, and it’s easier to treat mathematically anyway, and it was interesting enough that I spent the rest of the night working it out and I’m still not positive I’m unambiguously right. I could probably find out for certain with some simple experiments, but that would be precariously near trying, and so is right out. Let me set up the problem, though, since it’s interesting and should offer room for people to argue I’m completely wrong.
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2022-08-12 23:53:30
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https://math.eretrandre.org/tetrationforum/archive/index.php?thread-1164.html
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Tetration Forum
Full Version: Hyper-volume by integration
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Hi, everyone!
My dream is to get a formula to get the n-dimensional hyper-volume of an n-dimensional function in cartesian AND polar coordinates, too! So the length of f(x), the area of f(x,y), the volume of f(x,y,z) ... etc. According to the other existing formulas I have created an own in cartesian coordinate system:
$V_N = \int ... \int_{V_N} \sqrt{1+\sum_{k=1}^N {}{df \over dx_k}} dx_1 ... dx_N$
1st question: Do you find it correct?
2nd: How could it look in polar coordinate system?
(My final goal is to use these formulas to determine a few things about the base units of the hyperdimensional and interdimensional spaces from its derivatives and its existences. But for it, I need these formulas!)
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2020-01-18 14:19:05
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https://math.stackexchange.com/questions/1689377/inequalities-using-the-am-gm-inequality-the-cauchy-schwarz-inequality-and-brut
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# Inequalities Using the AM-GM inequality, the Cauchy-Schwarz Inequality, and brute force
Prove that if $x,y,z$ are positive real numbers then the following inequality holds $$\frac {x+y}{x^2+y^2} + \frac {y+z}{y^2+z^2} + \frac {z+x}{z^2+x^2} \leq \frac 1x + \frac 1y + \frac 1z .$$
I have tried everything to listed in the title like the AM-GM inequality and the Cauchy-Schwarz Inequality but it seems to be getting me nowhere. I also tried brute force but that really just led me to going in circles.
We have $a^2+b^2\ge 2ab$ for all $a,b\in\mathbb R$, because this is equivalent to $(a-b)^2\ge 0$, which is true.
Since $x,y,z>0$, we get:
$$\frac {x+y}{x^2+y^2} + \frac {y+z}{y^2+z^2} + \frac {z+x}{z^2+x^2}$$
$$\leq \frac{x+y}{2xy}+\frac{y+z}{2yz}+\frac{z+x}{2zx}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$$
Equality holds if and only if $x=y=z$.
• All too easy. +1 – Mark Viola Mar 9 '16 at 2:32
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2019-08-22 04:33:08
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https://www.geteasysolution.com/x/x=100/110
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# x/x=100/110
## Simple and best practice solution for x/x=100/110 equation. Check how easy it is, and learn it for the future. Our solution is simple, and easy to understand, so dont hesitate to use it as a solution of your homework.
If it's not what You are looking for type in the equation solver your own equation and let us solve it.
## Solution for x/x=100/110 equation:
x/x=100/110
We move all terms to the left:
x/x-(100/110)=0
Domain of the equation: x!=0
x∈R
We add all the numbers together, and all the variables
x/x-(+100/110)=0
We get rid of parentheses
x/x-100/110=0
Fractions to decimals
1-100/110=0
We multiply all the terms by the denominator
-100+1*110=0
We add all the numbers together, and all the variables
10!=0
There is no solution for this equation
`
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2020-02-20 03:29:04
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https://isabelle.in.tum.de/repos/isabelle/file/d97a944c6ea3/doc-src/Logics/preface.tex
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doc-src/Logics/preface.tex
author wenzelm Wed, 20 Jan 1999 18:07:34 +0100 changeset 6148 d97a944c6ea3 parent 6120 f40d61cd6b32 child 6582 75f31d45fb8b permissions -rw-r--r--
isabelle.in.tum.de;
%% $Id$
\chapter*{Preface}
Several logics come with Isabelle. Many of them are sufficiently developed
to serve as comfortable reasoning environments. They are also good
starting points for defining new logics. Each logic is distributed with
sample proofs, some of which are described in this document.
The logics \texttt{FOL} (first-order logic) and \texttt{ZF} (axiomatic set
theory) are described in a separate manual~\cite{isabelle-ZF}. Here are the
others:
\begin{ttdescription}
\item[\thydx{CCL}] is Martin Coen's Classical Computational Logic,
which is the basis of a preliminary method for deriving programs from
proofs~\cite{coen92}. It is built upon classical~\FOL{}.
\item[\thydx{LCF}] is a version of Scott's Logic for Computable
Functions, which is also implemented by the~{\sc lcf}
system~\cite{paulson87}. It is built upon classical~\FOL{}.
\item[\thydx{HOL}] is the higher-order logic of Church~\cite{church40},
which is also implemented by Gordon's~{\sc hol} system~\cite{mgordon-hol}.
This object-logic should not be confused with Isabelle's meta-logic, which is
also a form of higher-order logic.
\item[\thydx{HOLCF}] is a version of {\sc lcf}, defined as an
extension of \texttt{HOL}\@.
\item[\thydx{CTT}] is a version of Martin-L\"of's Constructive Type
Theory~\cite{nordstrom90}, with extensional equality. Universes are not
included.
\item[\thydx{Cube}] is Barendregt's $\lambda$-cube.
\end{ttdescription}
The directory \texttt{Sequents} contains several logics based
upon the sequent calculus. Sequents have the form $A@1,\ldots,A@m\turn B@1,\ldots,B@n$; rules are applied using associative matching.
\begin{ttdescription}
\item[\thydx{LK}] is classical first-order logic as a sequent calculus.
\item[\thydx{Modal}] implements the modal logics $T$, $S4$, and~$S43$.
\item[\thydx{ILL}] implements intuitionistic linear logic.
\end{ttdescription}
The logics \texttt{CCL}, \texttt{LCF}, \texttt{HOLCF}, \texttt{Modal}, \texttt{ILL} and {\tt
Cube} are undocumented. All object-logics' sources are
distributed with Isabelle (see the directory \texttt{src}). They are
also available for browsing on the WWW at
\begin{ttbox}
http://www.cl.cam.ac.uk/Research/HVG/Isabelle/library/
http://isabelle.in.tum.de/library/
\end{ttbox}
Note that this is not necessarily consistent with your local sources!
Isabelle} and performing some Isabelle proofs. Consult the {\em Reference
%%% Local Variables:
%%% mode: latex
%%% TeX-master: "logics"
%%% End:
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2021-04-12 20:26:29
|
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|
http://tex.stackexchange.com/questions/74688/autofit-for-inline-equations/74690
|
# Autofit for inline equations
I am trying to use an inline equation to list bunch o tuples. However, I it does not automatically go to next line. I am doing it manually, but this time it looks ugly. By the way I am writing in a two column template, so I have limited space at each line. Here is what I do:
${x_i,\;y_i }$= ${{60,45},\; {60,90},\;{60,135},\;{60,180},\;{60,225},\; \ {100,90},\; {100,160},\;{100,225}}$
Is there a way to automatically do this?
-
You have to distinguish which commas are eligible for breaks; so I suggest a command for it:
\documentclass[twocolumn,12pt]{article}
\newcommand{\bc}{,\penalty0 \;}
\begin{document}
$x_i,y_i=60,45\bc 60,90\bc 60,135\bc 60,180\bc 60,225\bc 100,90\bc 100,160\bc 100,225$
\end{document}
Don't overuse \; (it's wrong between x_i and y_i) and also braces: the ones you have in your example are all useless.
-
the \; before the penalty will leave a space at the end of the line if it's broken there. it would be better to put the penalty before the space. (tested; space at the beginning of line is ignored after break.) – barbara beeton Sep 30 '12 at 13:03
@barbarabeeton You're right, of course. – egreg Sep 30 '12 at 13:05
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2014-04-24 23:25:54
|
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https://www.physicsoverflow.org/40344/why-are-topological-properties-described-by-surface-terms
|
# Why are topological properties described by surface terms?
+ 4 like - 0 dislike
189 views
An example are the anomalies in abelian and non-abelian gauge quantum field theories.
For example, the abelian anomaly is $\tilde {F}_{\mu\nu}F^{\mu\nu}$ and the integral over this quantity is a topological invariant which measures a topological characteristic of the gauge field $A_\mu$.
All such quantities can be rewritten as total derivatives and then, using Gauss' law transformed into a surface integral.
**What's the intuitive reason that quantities which describe topological properties can always be written as surface integrals?**
Formulated a bit differently: Why are topological properties always completely encoded in the boundary of the system?
Locally (i.e., in the bulk), topology cannot enter as all manifolds are locally diffeomorphic to the unit ball.
Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor) Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsOve$\varnothing$flowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.
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2020-08-04 08:57:55
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https://support.bioconductor.org/p/76583/
|
error while running Basic4CSeq
1
0
Entering edit mode
@sinhashruti-9407
Last seen 5.9 years ago
Hi,
I am trying to run Basic4Cseq, but I am always getting the error
Error in if (tempPlus < 0) { : argument is of length zero
Execution halted.
Can anyone help to solve the issue?
Thanks
Best Regards,
shruti
Basic4Cseq • 1.0k views
0
Entering edit mode
Hi Shruti,
Did you solved this problem?
I am having the same one while using Basic4Cseq on my 4Cseq data.
Thanks,
Tal
0
Entering edit mode
@carolin-walter-7897
Last seen 4.4 years ago
Germany
Hi Shruti,
sorry for the delay in response, and sorry for the problem that you encountered.
The "tempPlus < 0" error is usually caused by a mismatch of chromosome annotations, e.g. "1,2,...X,Y" in your reference genome and "chr1, chr2...chrX, chrY" in your virtual fragment library. The function "createVirtualFragmentLibrary" has the option "useOnlyIndex", which is per default set to FALSE, so if your reference includes "chr"s, then you might either want to recreate the virtual fragment library with "useOnlyIndex = TRUE", or add the missing "chr" via R to the virtual fragment library. The format of this library is pretty simple, basically just a tab-separated file that should be easy to manipulate.
If this doesn't work, please ask again (either here or per mail; I'm the package's maintainer); to help me track down the problem I'd also ask you to specify both the type of reference genome (like "mm10" or whatever) and the restriction enzymes that you use - this allows me to create the basic virtual fragment library myself and check it for any strange occurrences.
Best wishes, and thanks for trying out the package,
Carolin
0
Entering edit mode
Hi Carolin,
I am also having this kind of problem. I created virtual fragment library with useOnlyIndex=TRUE and also with useOnlyIndex=FALSE, and it didn't solve the problem. I tried to run also the package's files and I got the same error.
Can you think about possible explanation?
Thanks,
Tal
0
Entering edit mode
Hi Tal,
sorry for the delay (again), my e-mail notification seems to miss questions/comments when the package name is misspelled (in this case, Basic4CSeq instead of Basic4cseq).
As said before, if everything seems to be fine (1...Y or chr1...chrY everywhere, etc), please feel free to contact me directly via e-mail - I'm the package's maintainer.
Best,
Carolin
0
Entering edit mode
Hi Carolin,
After testing with different options, I noticed that the problem occurs due to the chrM label. I use chr[123...XYM] labeling in my bam file and Hsapiens hg19 from the BSGenome package uses the same, as well as additional chr names and contigs etc.
With default options in Basic4Cseq, i.e. useOnlyIndex=FALSE and useAllData=TRUE, it runs successfully, However, if I select useAllData=FALSE, it gives the above error whether the other option is TRUE or FALSE, because it does not recognize the chrM, as in the manual of your package it says it can rezognize chrMT instead:
useAllData
Variable that indicates if all data of a BSgenome package is to be used. If FALSE, chromosome names including a "_" are removed, reducing the set of chromosomes to (1 ... 19, X, Y, MT) for the mouse genome or (1 ... 22, X, Y, MT) for the human genome
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2021-11-28 06:16:58
|
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|
https://worldbuilding.stackexchange.com/questions/19535/how-can-i-get-a-very-large-air-bubble-floating-under-europas-ice
|
# How can I get a very large air bubble floating under Europa's ice?
I hope this isn't a stupid question ("just add air").
I would like my intrepid explorers to have drilled through the ice of Europa and into the water without creating a geyser (they plugged the hole behind them). This is assuming a liquid subsurface ocean.
Once under the ice, they encountered a very large, stable cavern of air between the water and ice. Their submersible 'surfaces, and when they look around they see water, and when they look up they see ice. It measures several kilometers wide and at least one km high to the ice. I am not married to those dimensions, though.
It doesn't necessarily need to be breatheable, but it should be enough that they can exit their vehicle with scuba-like gear (ie - breathing and warming/cooling apparatus). However, bonus if they surface and the pressure, temperature, and composition are suitable, however shortly.
I really hope this isn't as straightforward as it seems.
• When I get time later I'll try to answer. Until then, I'll leave you with this. I don't think it'd be that hard to form ...**but**... I don't think it'd be stable for geologic periods of time either. – Jim2B Jun 22 '15 at 19:04
• I don't think this is straightforward. :) You have to (at least) worry about the bubble caving in or spreading out to much or becoming highly pressurised by the tons of water and ice all around. I hope you get some good answers for this – Amziraro Jun 22 '15 at 19:05
• See also The Frozen Sky by Jeff Carlson. – JDługosz Jun 23 '15 at 6:54
## 5 Answers
It seems you can find everything you need in Wikipedia:
Plumes
Europa may have periodically occurring plumes of water 200 km (120 mi) high, or more than 20 times the height of Mt. Everest. These plumes appear when Europa is at its farthest point from Jupiter, and are not seen when Europa is at its closest point to Jupiter, in agreement with tidal force modeling predictions. The tidal forces are about 1,000 times stronger than the Moon's effect on Earth. The only other moon in the Solar System exhibiting water vapor plumes is Enceladus. The estimated eruption rate at Europa is about 7000 kg/s compared to about 200 kg/s for the plumes of Enceladus.
In other words, Europa may have cryovolcanoes/geysers. These cryovolcanoes would, presumably have the possibility of becoming extinct. When operational, a cryovolcano would draw whatever gases are dissolved in Europa's ocean (CO2, N2, CH4*) to the surface, where they would come out of solution. It's therefore not only possible, but quite likely, that if there are extinct cryovolcanoes on Europa, they have gas pockets beneath them. It also gets round the issue of the thickness of the ice, which as a best estimate seems to be several miles, but could be much thinner under an extinct cryovolcano. Even an active one might do, 7000kg/s is not a particularly high flow rate, and according to the theory these cryovolcanoes are only active once per orbit (3.5 days.)
*Normally it is difficult for O2 to exist because it is so chemically reactive and tends to combine with other elements over a geological timescale. But Europa apparently has a mechanism for splitting H2O into H2 + O2 and ejecting the H2 into space:
Unlike the oxygen in Earth's atmosphere, Europa's is not of biological origin. The surface-bounded atmosphere forms through radiolysis, the dissociation of molecules through radiation. Solar ultraviolet radiation and charged particles (ions and electrons) from the Jovian magnetospheric environment collide with Europa's icy surface, splitting water into oxygen and hydrogen constituents..... Molecular hydrogen never reaches the surface, as it is light enough to escape Europa's surface gravity.
Some of this may make it into the oceans:
Observations of the surface have revealed that some of the molecular oxygen produced by radiolysis is not ejected from the surface. Because the surface may interact with the subsurface ocean (considering the geological discussion above), this molecular oxygen may make its way to the ocean, where it could aid in biological processes.
• Wow I never knew that. +1 – rodolphito Jun 22 '15 at 21:47
• Fantastic stuff, particularly as the OP can give the geyser as a reason for landing and boring down at that point; the ice is thinner there. – Phil H Jun 23 '15 at 16:04
Introduction
You want beans with that?
I can think of two natural processes that might generate a very large bubble of gas under Europa's ice.
Air under arctic ice
and no, flatulence isn't one of them.
Precipitation
The first would be an ocean completely saturated with gases (you can pick your gases but some of them are not as suited to this as others) encountering conditions (lower pressure / higher temperature) which causes some of those gases to precipitate / out gas.
If the process needs to be abiotic, then the most likely composition of this gas would be the volcanic gases (shown below). But if you have a lively biosphere, then it could be something like $O_2$ or other biological gases.
The bubble would be created when the water conditions suddenly changed. For instance, if the layer of ice above it suddenly thinned and the temperature increased (say due to ice crustal deformation).
Out gassing
The second would be volcanic fissures releasing gases beyond what the ocean can absorb (another case of a saturated liquid). On Earth the volcanic gases most likely to be encountered are:
$H_2O$ Water
$CO_2$ Carbon dioxide
$SO_2$ Sulfur dioxide
$N_2$ molecular Nitrogen
$Ar$ Argon
$He$ Helium
$Ne$ Neon
$CH_4$ Methane
$CO$ Carbon Monoxide
$H_2$ molecular Hydrogen
The problem
The problem is this condition is not stable. The gas is MUCH lighter than ice and will flow through any cracks to reach the surface. I wouldn't expect it to last long. Since I have no background in this subject I couldn't tell you how long is "not long".
It could fail by gradual (or sudden!) deflation, gradual reabsorption by the water, or most spectacularly by the sudden break-up of overlying ice.
I hope a temporary bubble is good enough for your purposes.
Size
Since the ice above this air has no support, there are mechanical limits to the amount of air in this space. Under great pressures (like a mile or more of overlying ice), ice tends to flow like plastic.
This is another reason I think the bubble would be a temporary feature. The larger the bubble the shorter the time it could exist.
• What's the relevance of the photo? – Samuel Jun 22 '15 at 23:50
• I was hoping to find something bigger but I ran out of time. I posted what I could find quickly. I hope to find an image or reference describing the largest "bubbles" found under ice flows here on Earth as a ballpark for the larges we might find under Europan ice. That project may need to wait until tomorrow. – Jim2B Jun 23 '15 at 1:24
• Clarification on the picture - what is that guy doing? Skin-diving, on a line, in the arctic ocean, under ice, and without scuba gear? – Mikey Jun 23 '15 at 20:07
Not only are you dealing with a creating a pocket of air under the ice you have to deal with the pressures. Some say the Ice could be 62 miles deep. That is a lot of Ice over your head and putting a lot of pressure on the 'air' in the space. It would have to be at the same pressure as the water because if it was less, the water would push up into the cavity until they reached equilibrium.
Now can this happen? quite likely there are bubble trapped in the ice in different places through the planet. This could come from chemical reactions deep in side, but it is unlikely that it would be 'livable' and quite likely that it is highly toxic and/or caustic.
• The wikipedia article says 62 miles of H2O, with the preferred "thick ice" hypothesis indicating 6-9 miles of this is ice (which is still a lot) and the less favoured hypothesis 200m (with a surface gravity 1/7 of the earth, this would be equivalent to diving 30m deep on earth.) – Level River St Jun 22 '15 at 21:20
• @steveverrill thanks! I was trying to find that info, missed it! Though one article I read said that some thought that ice might be close to 60 miles deep with only a small part liquid. I like the wiki answer better! – bowlturner Jun 22 '15 at 21:22
• Reading the question I kind-of assumed the OP wasn't married to the official ice depth either, but maybe I was wrong? – o0'. Jun 22 '15 at 21:22
• Well it's likely pretty deep either way, a bit of pressure. – bowlturner Jun 22 '15 at 21:24
• Good point. I doubt if you could just poke a hole and stick your head in. More likely you and your scientist friends would be doing summersaults in the air from the resulting explosion of ice and air. Like popping a balloon, except you're on the surface of the balloon as a gnat. :) – Neil Jun 23 '15 at 7:02
It actually may just be that simple.
The concern is presumably one of structural integrity of the ice layer above and the ability for a gaseous atmosphere to form (and not precipitate immediately). I think there are not specific data on this, but it appears that a cave of the size you mention is feasible for the ice of Europa. Actually, I think the least plausible part of the layout is that there would be a liquid surface. There would very likely be a layer of ice on the water inside the cave. It might be broken through in many places from ice falling from the cave ceiling above, but I don't imagine it would stay liquid. The air inside is presumably warmer than the ice though, so it may help that it's slightly melting the surrounding cavern which, upon refreezing, would create an airtight seal. The stability is likely only in the thousands of years range though, the tidal forces from Jupiter will likely cause such a structure to eventually fail.
Think about the scale of things here. The small squared off section is way way bigger than the area you're asking for. If you were to add about a 100 million cubic meters of pressurized air to that space it would provide the bubble you're looking for.
The Cave
The Wikipedia article you linked states that the very cold surface temperature of Europa makes the surface ice "as hard as granite". Granite is a pretty tough stone. There are some very large cave systems on Earth which are made of limestone, which is about 50% stronger than granite. However, gravity on Europa is also 7.5 times less than on Earth, so larger structures would be able to form.
The Air
Europa has a weak atmosphere of oxygen. Though it's difficult to hypothesize a way for very much to get under the surface ice. A more feasible idea might be that the gas is a byproduct from organic life or a chemical process related to the hydrothermal vents. It would strain plausibility for it to be breathable.
• "... made of limestone, which is about 50% stronger than granite ...". Is this really what you meant to write? The context suggests "half the strength of". – Ville Niemi Jun 22 '15 at 23:43
• @VilleNiemi Limestone is stronger, so yeah I wrote that right. I thought the context was more towards "granite is weaker, however, gravity is less so it's ok". But if you have a suggestion to clarify I'd appreciate it. – Samuel Jun 22 '15 at 23:48
• The context was you talking about granite being pretty tough stone... Maybe remove that sentence. Also the ice being as hard as granite actually says nothing of its structural strength. Granite and ice have pretty different structures. Although I think the "error" would be in the right direction, they used to drive tanks over ice in the WW2... – Ville Niemi Jun 23 '15 at 0:03
• – Neil Jun 23 '15 at 7:30
I suspect that due to the cold and huge depth of the water column, gasses dissolved in the oceans of Europa will stay dissolved, so cracks in the ice "crust" will fill with water rather than gasses. This may partially explain the cryovolcanoes observed on Europa; if the ice cracks open enough to expose the water to vacuum, the gasses will explosively come out of solution.
For your intrepid explorers, they might be able to use shockwaves or other means to try and trigger the release of dissolved gasses inside a crevasse under the ice, but will need to be very careful. The released gasses will temporarily displace the water and create the cavern you are looking for, but as noted on other posts the temporary cavern will not have breathable air, and the gasses will either be reabsorbed by the water or leak through the ice towards the surface. Calculating how long this will take depends on a multitude of factors, but the cavern will probably be "dry" for only few years at most before the gasses are absorbed, leak out or "geological" processes cause the ice crevasse to close up on its own.
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2020-02-18 23:25:32
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http://tim.hibal.org/blog/tile-based-game-overview/
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# Tile Based Game Overview
When it comes to making games there are several standard ways to implement physics. Games like Braids and N+ use vector-based physics engines with complicated geometry, whereas games like the GameBoy Pokemon titles use far more simple tile-based approaches. There are, of course, a wide range of games that do something in-between.
The original Legend of Zelda had a tile-based map but allowed the player to move arbitrarily between tiles. This allowed for fine tuned control in combat while reaping the benefit of an easily maintained tile-based map format.
Original LOZ Screenshot
A tile based map is when the chambers or areas of the game are subdivided into a grid of tiled images, allowing many areas to be created from a limited subset of images. This greatly reduces the amount of custom art required for a project, reduces the disk space requirements for all of those images, and makes creating new areas relatively easy. The drawback is that you are more or less stuck with placing things on the grid, which gives a blocky look to your game.
Mystic Dave is a puzzle game, and one in which I do not want to implement combat. As such I have decided to break from the free-range nature of the original LOZ and restrict motion to the grid as well, GameBoy Pokemon style.
Old Mystic Dave Screenhot
In this old screenshot you can see that Dave is misaligned with the grid. Back then the map was defined on a grid, but the character could move around freely. This caused a lot of headache when it came to implementing physics and properly handling collisions. I could do it, but was it worth it?
As stated, I have moved to where all entities are restricted to the grid, thus making movement and collision detection far easier. Consider the following:
Mystic Dave is located in tile (1,1). He is facing East, towards position (2,1), which has a block in the way. If the player hits the right arrow key, all I need to do is check whether that tile to the right contains anything that would block Dave from entering. If the target tile is free, then Dave can be moved to the next tile.
In code, this is roughly implemented as follows:
int dx = 0, dy = 0; if ( keysPressed[DOWN] ) { dy = 1; } elseif ( keysPressed[UP] ) { dy = -1; } elseif( keysPressed[RIGHT]) { dx = 1; } elseif( keysPressed[LEFT]) { dx = -1; }
int desDir = DirFromDelta( dx, dy ); int curx = player->GetPos()->GetTileX(); int cury = player->GetPos()->GetTileY(); int tx = curx + dx; int ty = cury + dy;
//check if target is free if ( curChamber->CanTileBeEntered( tx, ty ) ) { curChamber->RegisterTileEntityInTile( player, tx, ty ); player->MoveDir( desDir, curx, cury ); }
The code simply determines the tile we are trying to move to, checks whether the tile can be entered, and then moves the player over. Note that movement doesn’t happen instantly. Moving the player over instantly would look terrible and play poorly.
Here we can see Dave moving from a tile to the one to his right. He is undergoing a walking animation while simultaneously translating to the right. It takes 24 ticks ( 24/60 sec) for Dave to complete his motion so that everything looks nice and smooth and game play progresses at a good pace.
However, this causes a new issue to arise. Isn’t Dave in two tiles at once?
Well yes, during his motion Dave is in both tiles at once. In the code segment above Dave is registered in the destination tile, but note that this does not free him from the tile he starts off in. Thus, Dave is occupying both tiles, preventing other entities from walking into either while his motion is taking place. Once Dave’s motion completes it will automatically free him from the tile he came from, allowing other entities to potentially move into it.
This system for registering and freeing tiles makes collision detection easy, as you never have two collideable objects occupying the same tile(s) in the grid space. This makes movement logic far easier and prevents a lot of the headaches that come with multiple things moving into the same space simultaneously.
Developed this week:
I implemented smooth transitions between chambers! Woot!
Also did a little bit of work under the hood, changing how TileEntity positions are represented. Instead of using a vector in pixel space, position is now represented by an object containing a vector with coordinates in tile space and a pixel-space vector used as a delta off of that. This makes it much easier to tell which tile you are in while preserving the ability to plot anywhere in tile space.
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2022-05-29 09:03:44
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http://cstheory.stackexchange.com/questions/21270/why-is-linearizability-a-safety-property-and-why-are-safety-properties-closed-se
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# Why is linearizability a safety property and why are safety properties closed sets?
In Chapter 13 "Atomic Objects" of the book "Distributed Algorithms" by Nancy Lynch, linearizability (also known as atomicity) is proved to be a safety property. That is to say, its corresponding trace property is nonempty, prefix-closed, and limit-closed, as defined in Section 8.5.3. Informally, a safety property is often interpreted as saying that some particular "bad" thing never happens.
Based on this, my first problem is as follows:
What are the advantages of linearizability as a safety property? Are there some results based on this fact in the literature?
In the study of the classification of safety property and liveness property, it is well-known that safety property can be characterized as the closed set in an appropriate topology. In the paper "The Safety-Progress Classification" @1993 by Amir Pnueli et al., a metric topology is adopted. More specifically, a property $\Phi$ is a set of (finite or infinite) words over the alphabet $\Sigma$. The property $A(\Phi)$ consists of all infinite words $\sigma$ such that all prefixes of $\sigma$ belong to $\Phi$. For example, if $\Phi = a^{+}b^{\ast}$, then $A(\Phi) = a^{\omega} + a^{+}b^{\omega}$. An infinitary property $\Pi$ is defined to be a safety property if $\Pi = A(\Phi)$ for some finitary property $\Phi$. The metric $d(\sigma, \sigma')$ between infinite words $\sigma$ and $\sigma'$ is defined to be 0 if they are identical, and $d(\sigma, \sigma') = 2^{-j}$ otherwise, where $j$ is the length of the longest common prefix on which they agree. With this metric, the safety property can be characterized as closed sets topologically.
Here comes my second problem:
How to characterize linearizablity as closed sets topologically? In particular, what is the underlying set and what is the topology?
-
What are the advantages of linearizability as a safety property? Are there some results based on this fact in the literature?
Suppose that you've implemented a shared memory machine $M$ that only satisfies eventual linearization, defined as follows: in every run $\alpha$ of $M$, there exists some point in time $T_\alpha$, such that linearization holds from time $T_\alpha$ on. Note that there is no upper bound on $T$. (*) (This is an artificial liveness counterpart of the standard safety property definition of linearizability.)
Such a shared memory implementation wouldn't be very useful to the programmer: Note that if only eventual linearizability holds, there are no guarantees whatsoever on the consistency of read/write operations in any "early" prefix of a run (before the unknown time $T$). Or, in other words, whatever has happened until now, you can still extend the current prefix of a run to one that satisfies eventual linearizability.
(*) If there was such an upper bound, then eventual linearizability would become a safety property.
How to characterize linearizablity as closed sets topologically? In particular, what is the underlying set and what is the topology?
We can define a metric topology on the set $ASYNC$, which is the set of all possible runs of a distributed algorithms. Note that each run $\alpha \in ASYNC$ corresponds to an infinite sequence of state transitions. For $\alpha, \beta \in ASYNC$, $\alpha \ne \beta$, we define $$d(\alpha,\beta) := 2^{-N}$$where $N$ is the earliest index where the state transitions in $\alpha$ and $\beta$ differ; otherwise, if $\alpha = \beta$, we define $d(\alpha,\beta) = 0$.
We first argue that $d$ is a metric on $ASYNC$. By definition, $d$ is nonnegative and $\forall \alpha,\beta \in ASYNC$ we have $d(\alpha,\beta)=d(\beta,\alpha)$. For $\alpha,\beta,\gamma \in ASYNC$, the triangle-inequality $d(\alpha,\beta) \le d(\alpha,\gamma) + d(\gamma,\beta)$ trivially holds if $\gamma=\alpha$ or $\gamma=\beta$. Now consider the case that $d(\alpha,\gamma) \ge d(\gamma,\beta) > 0$, i.e., $d(\alpha,\gamma)=2^{-n_1}$ and $d(\gamma,\beta)=2^{-n_2}$, for some indices $n_1\le n_2$. Since $\gamma$ shares a common prefix of length $n_2-1$ with $\beta$ but only a prefix of length $n_1-1$ with $\alpha$, it follows that $\alpha$ and $\beta$ differ at index $n_1$, and thus $d(\alpha,\beta) = d(\alpha,\gamma)$ and the triangle-inequality follows. The case where $0<d(\alpha,\gamma) < d(\gamma,\beta)$ follows analogously.
The metric $d$ induces a topology (e.g., page 119 of [1]) where the $\epsilon$-balls $B_\varepsilon(\alpha) = \{ \beta \in ASYNC \mid d(\alpha,\beta) < \varepsilon \}$ are the basic open sets. We will now argue why safety properties correspond to closed sets: If an execution $\alpha$ does not satisfy a safety property $S\subseteq ASYNC$, i.e.\ $\alpha \notin S$, then there is an index $N$ where all runs $\beta$ that share a prefix longer than $N$ with $\alpha$ are not in $S$. This closely matches intuition, since once a safety property is violated in a prefix of an execution, it makes no difference how this prefix is extended! Formally speaking, suppose that $\alpha \notin S$. There exists an $N\geq 0$ such that, if some $\beta \in ASYNC$ has $d(\alpha,\beta) < {2^{-N}}\text{,}$ i.e., $\alpha$ and $\beta$ share a prefix of length $\ge N$, then $\beta \notin S$. Thus, the set of runs $S$ is closed, since its complement is open.
[1] James Munkres. Topology.
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Thanks for your answer. I have to ponder over it. By the way, are you referring to the book "Topology" by James R. Munkres when you says that The metric d induces a topology (e.g., page~119 of [1]) where the ϵ-balls...? – hengxin Feb 27 at 13:53
Yes, I've added the reference. – Peter Feb 27 at 16:38
I noticed that you have suggested a modification of the title of this post (if I have made a mistake, please ignore this comment). First of all, I agree that the two subproblems should be reflected in the title. However, I am not asking about "why is linearizability a safety property?". I am asking about the consequences of this fact. I am not sure how to modify the title appropriately and I have skipped this modification. Please let me know if you have other comments or ideas. – hengxin Feb 28 at 9:03
I realized that the characterization (proof) of linearizability as closed set basically has nothing to do with the notion of linearization points. It seems like a more general proof which characterizes any safety property as closed set. Did I miss something? – hengxin Feb 28 at 9:54
Yes, all safety properties are closed sets, while liveness properties are dense sets in this topology. In fact, every property (i.e. set of runs) can be expressed as a conjunction (i.e. intersection) of safety and liveness properties. – Peter Mar 3 at 1:40
Regarding your first question - safety properties are, in a way, the "easiest" properties to handle, with respect to problems such as model-checking and synthesis.
The basic reason for this is that in the automata-theoretic approach to formal methods, reasoning about safety properties reduces to reasoning about finite traces, which is easier than the standard infinite-trace setting.
See the work of Orna Kupferman here as a starting point.
-
When expressed in linear temporal logic, the safety properties can be captured and checked against some special class of B$\ddot{u}$chi automata. However, I have not realized any material on how to express and check linearizability in terms of automata. Therefore, such advantage may be not shared by linearizability (as a safety property). What do you think of that? – hengxin Feb 27 at 13:46
I'm pretty sure Iv'e seen papers that deal with linearizability via LTL, at least in specific cases. If I find them, I'll comment. – Shaull Feb 27 at 14:07
That will be great. I am always curious about how to deal with linearizability via LTL, especially with the notion of linearization points. Following your hint, I find the paper Proving linearizability with temporal logic. I will try to read it in these days. However, I not sure about its quality. Looking forward to your comments. – hengxin Feb 28 at 2:02
Perhaps this will be of use. Judging by the authors, this is a serious paper. I'm not sure how tight the connection to LTL is, though. – Shaull Feb 28 at 5:53
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2014-09-02 11:35:12
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http://cms.math.ca/10.4153/CMB-2008-052-5
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Canadian Mathematical Society www.cms.math.ca
Abstract view
# The Effective Cone of the Kontsevich Moduli Space
Published:2008-12-01
Printed: Dec 2008
• Izzet Coskun
• Joe Harris
• Jason Starr
Features coming soon:
Citations (via CrossRef) Tools: Search Google Scholar:
Format: HTML LaTeX MathJax PDF PostScript
## Abstract
In this paper we prove that the cone of effective divisors on the Kontsevich moduli spaces of stable maps, $\Kgnb{0,0}(\PP^r,d)$, stabilize when $r \geq d$. We give a complete characterization of the effective divisors on $\Kgnb{0,0}(\PP^d,d)$. They are non-negative linear combinations of boundary divisors and the divisor of maps with degenerate image.
MSC Classifications: 14D20 - Algebraic moduli problems, moduli of vector bundles {For analytic moduli problems, see 32G13} 14E99 - None of the above, but in this section 14H10 - Families, moduli (algebraic)
© Canadian Mathematical Society, 2013 : http://www.cms.math.ca/
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2013-05-22 02:25:58
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http://www.gamedev.net/index.php?app=forums&module=extras§ion=postHistory&pid=4979452
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• Create Account
### #Actualdmatter
Posted 12 September 2012 - 03:02 PM
Of those two options I would go with templates, mrbastard gave some good reasons why.
In terms of making it "reusable as possible" I could be tempted to follow the principles used by the standard library and have the search accept the path-cost function, the admissible goal-distance function and a node-expander function instead of assuming they are all members of the node type (so now a node could simply be a pointer into a 2d array, etc), e.g.
template <typename NodeT, typename PathCostFunctionT, typename GoalDistanceFunctionT, typename NodeExpanderFunctionT>
void search(NodeT start, NodeT goal, PathCostFunctionT pathCostScorer, GoalDistanceFunctionT admissibleGoalDistanceEstimator, NodeExpanderFunctionT nodeExpander);
(Obviously what this signature is lacking is away to emit the discovered path, assuming the full path is what you're trying to discover. I'll leave that up to you, you could return a collection of nodes, or perhaps take an iterator to write out nodes in the path - and then pass it a back_insert_iterator).
### #1dmatter
Posted 12 September 2012 - 03:02 PM
Of those two options I would go with templates, mrbastard gave some good reasons why.
In terms of making it "reusable as possible" I could be tempted to follow the principles used by the standard library and have the search accept the past-cost function, the admissible goal-distance function and a node-expander function instead of assuming they are all members of the node type (so now a node could simply be a pointer into a 2d array, etc), e.g.
template <typename NodeT, typename PathCostFunctionT, typename GoalDistanceFunctionT, typename NodeExpanderFunctionT>
void search(NodeT start, NodeT goal, PathCostFunctionT pathCostScorer, GoalDistanceFunctionT admissibleGoalDistanceEstimator, NodeExpanderFunctionT nodeExpander);
(Obviously what this signature is lacking is away to emit the discovered path, assuming the full path is what you're trying to discover. I'll leave that up to you, you could return a collection of nodes, or perhaps take an iterator to write out nodes in the path - and then pass it a back_insert_iterator).
PARTNERS
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2014-04-23 17:11:57
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https://gsebsolutions.com/gseb-solutions-class-12-english-unit-2-read-2/
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# GSEB Solutions Class 12 English Unit 2 Read 2 Shaper Shaped
Gujarat Board GSEB Solutions Class 12 English Second Language Unit 2 Read 2 Shaper Shaped Textbook Exercise Questions and Answers.
## GSEB Std 12 English Textbook Solutions Unit 2 Read 2 Shaper Shaped (2nd Language)
### GSEB Class 12 English Shaper Shaped Text Book Questions and Answers
Comprehension
Question 1.
The poet has ceased to be a potter and learnt to be the clay. What would you like to become? Why?
I would like to become a cloud. fortning various shapes enlivening the imagination of the cowherds in the terrain of the countryside. I would cast my shadow on the land shielding the heat.
Question 2.
Complete the table.
No. Journey Who Acts Reason 1. shaper shaped potter clay moulded the clay to patterns of wheel Wisdom won and pride died 2. shaper shaped poet song to win the hearts of men Through new-got knowledge ceased to be a poet. 3. shaper shaped fashioner of swords sword gleamed on battlefields Brimming with silence of the Lord, learned to be a sword. 4. shaper shaped dreamer dream who would hurl insolence of emrald and pearl Kneeling at the feet of the Supreme, learned to be a dream.
3. Answer the following questions in two or three sentences each:
Question 1.
What is the significance of the title of the poem ?
In life, mostly we adopt a trend with which we are familiar. We shape our life accordingly. But in many cases circumstances, ideas mould our life and compel us to pursue another course in life. So the changes occur in our lifestyle.
Question 2.
How does the poet contrast his former life with the present life ?
The poet says that in his bygone days, he was just ignorant about many realities and considered himself to be a creator of many a thing and strut with great pride. But later on, he realised that the creator of everything on the earth is Supreme power, i.e., God and he is nothing but a thing to be shaped in His hands.
Question 3.
What examples does the poet give to forewarn the human beings who are obsessed with their own selves ?
The examples that the poet gives of the people who are obsessed with their own selves are as under :
(1) Previously the poet was a potter and thought that he could create any shape from yielding clay, but later on he realised that he was not the shaper but merely ‘clay’ to be shaped.
(2) In his bygone days, the poet was a great poet and could compose innumerable songs or poems but later he realised that he was merely a ‘song’ to be composed.
(3) In another example, the poet had thought that he could produce gleaming swords, but his later realisation taught him that he was merely a ‘sword’ to be prepared.
(4) In the last example, the poet says that once upon a time, he thought that he was a dreamer possessing costly stones like emrald and pearl but later he realised that he was not more than merely a ‘dream’ and never a dreamer.
Question 4.
How does the poet cite the example of potter to refer to his awakening of the soul ?
In bygone days, the poet thought that he was a skilful potter and could prepare wonderful patterns from yielding clay, but his lately won wisdom awakened his soul and made him realise that he was not a potter – a creator – but merely the clay which needed to be shaped.
Question 5.
Why did the poet write innumerable songs in his earlier years ?
The poet wrote innumerable songs in his earlier years as he was vainly proud that his pen could compose wonderful poems which could win the hearts of people. That time he did not know that the power or potential he had got was not of his own but of some Supreme power.
Question 6.
What does the phrase ‘fashioner of swords’ refer to ?
The phrase ‘fashioner of swords’ refers to the maker of swords. The word ‘fashioner’ refers to the ‘maker of variety of swords’ to be used on the battlefield.
Question 7.
How did the poet behave in bygone times when he was a dreamer?
When the poet was a dreamer in bygone times, he behaved very arrogantly as he possessed great wealth. His pride had made him insolent. He was a great dreamer who never knew that he himself was merely a dream at the feet of the Supreme.
Question 8.
What happens when the poet surrenders at the feet of the Almighty?
When the poet surrenders at the feet of the Almighty, he has the sudden realisation that he is no more a dreamer. He has no right to feel proud or be arrogant about the things that he claims to be his own. The wealth he dreams of does not belong to him at all. Then he ceases to be a dreamer and realises that he is merely a dream to get realised.
4. (a) Lines 2 and 4 end with words ‘feel-wheel’. Study the last word of each line and make a list of other rhyming words in the poem. Work in pairs.
Stanza 1: Rhyming words
Lines : 2. 4 — Feel, wheels
Lines: 3. 6. 8 — clay, away. clay
Stanza 2:
Lines: 2, 4 — pen. men
Lines : 6, 8 — long, song
Stanza 3:
Lines: 1, 3 — swords, battlefields
Lines : 2, 4 — gone. shone
Lines : 6. 8 — Lord, sword
(b) Select any three pairs of rhyming words from the poem and find one more rhyming word for each pair.
Example : feel – wheel – kneel
Pairs of rhyming words are :
(1) clay – away – astray
(2) pen – men – den
(3) long – song – gong
(4) hurl – pearl – swirl
(c) Prepare a list of words that are used more than twice in the poem.
Words used more than twice are : used to be, in days, on, but, now, ceased, had, sword.
(a) self-realization of the poet.
(b) advantages of getting materialistic things.
(c) benefits of writing beautiful songs.
(d) gain from fighting battles.
(a) self-realization of the poet.
(2) The poet wishes to cease to be a potter because …
(a) he has become master of making pots.
(b) he has realized the harm of declaring himself to be a great potter.
(c) he has acquired fame of a skilled potter.
(d) he wishes to adopt some other profession.
(b) he has realized the harm of declaring himself to be a great potter.
(3) The poet wishes to surrender to God because …
(a) he wants to acquire emerald and pearls.
(b) he has learnt the truth of mortality of living beings.
(c) he wants to win hundred battles.
(d) he wishes to impress others.
(b) he has learnt the truth of mortality of living beings.
Question 1.
Identify the Figures of Speech in the following line:
‘His fingers mould the yielding clay’
The Figure of Speech used in the line ‘His fingers mould the yielding clay’ is Synecdoche
Question 2.
What is the meaning of the line – ‘Innumerable songs would come To win the hearts of men’.
The meaning is that the poet would compose a number of songs which would be popular among people and win their acclaim.
Question 3.
Identify the Figure of Speech in the line -‘Glittered and gleamed and shone.’
Answer:The Figure of Speech in the line – ‘Glittered and gleamed and shone’ is Climax.
Question 4.
Explain –
‘In bygone times I used to be A dreamer who would hurl On every side an insolence of emerald and pearl.
These lines suggest that in past the poet was a wealthy man and felt greatly proud of his ; abundant wealth.
(1) In days gone by I used to be
A potter who would feel
His fingers mould the yielding clay
To patterns on his wheels;
But now through wisdom lately won,
That pride has died away;
I have ceased to be the potter
And have learned to be the clay.
Questions:
(1) What pride was the poet as a potter bearing formerly?
(2) What did the poet as a potter learn with the course of time?
(1) As a potter the poet felt proud that he could mould clay and make different patterns.
(2) With the course of time, the poet became wise, shook off his pride and felt that he had ceased to be the potter and learned to be the clay.
(2) In other days I used to be
A poet through whose pen
Innumerable songs would come
To win the hearts of men;
But now through new-got knowledge
I have ceased to be the poet
And have learned to be the song.
Questions:
(1) Describe the poet’s compositions.
(2) What was the poet’s realisation on a later stage ?
(1) The poet composed innumerable songs which could win the hearts of men.
(2) On a later stage, with the help of newly got-knowledge, the poet ceased to be the poet and learned to be the song.
OR
Questions:
(1) In other days what would come through poet’s pen ?
(2) Why has the poet ceased to be a poet ?
(3) Why does the poet write the poems ?
(1) In other days, innumerable songs would come through the poet’s pen.
(2) The poet has ceased to be a poet as he has got new knowledge.
(3) The poet writes the poems to win the hearts of men.
OR
Questions:
(1) What was the poet in the past ?
(2) What was the effect of poet’ song ?
(3) The poet was a but now he wants to become
(1) The poet was a poet in the past.
(2) The poet’s song could win the hearts of people.
(3) The poet was a poet but now he wants to become a song.
OR
Questions :
(1) Why did the poet write innumerable songs in his earlier years ?
(2) What knowledge had the poet got ?
(3) Find out the word from the stanza for ‘the things can’t be counted’.
(1) The poet wrote innumerable songs in his earlier years to win the hearts of men.
(2) Afterwards the poet came to know that he was no longer a poet, but was the song.
(3) The word for ‘the things can’t be counted’ is ‘innumerable’.
(3) I was a fashioner of swords
In days that now are gone
Which on a hundred battlefields,
Glittered and gleamed and shone;
But now that I am brimming with
The silence of the Lord,
I have ceased to be sword-maker
And have learned to be the sword.
Questions :
(1) What do the words ‘fashioner of swords’ in the stanza mean ?
(2) What does the poet mean by ‘I am brimming with the silence of the Lord’ ?
(1) The words ‘fashioner of swords’ mean ‘maker of swords’.
(2) The words ‘I am brimming with the silence of the Lord’, the poet means that now he has grown wise enough to understand the futility of war and ‘peace’ is the ultimate need of the world.
(4) In bygone times I used to be
A dreamer who would hurl
On every side an insolence
Of emerald and pearl;
But now that I am kneeling
At the feet of the Supreme
I have ceased to be the dreamer
And have learned to be the dream.
Questions :
(1) What did the poet dream of in his bygone days ?
(2) What is the latest realisation of the poet ?
(1) In his bygone days, the poet dreamt of being very wealthy lying by the side of gems.
(2) Lately the poet realised that whatever wealth he collected was futile. The real solace he could find submitting himself to God.
OR
Questions:
(1) What was the poet in his past ?
(2) What changed the poet’s thoughts ?
(3) The poet was a but ……… now he wants to become ………
(Fill in the gaps with proper words.)
(1) In his past, the poet was a dreamer.
(2) The poet surrendered himself to God and in the submission he found his thoughts changed.
(3) The poet was a dreamer but now he wants to become a dream.
(5) In days gone by I used to be
A potter who would feel
His fingers mould the yielding clay
To patterns on his wheels;
But now through wisdom lately won,
That pride has died away;
I have ceased to be the potter
And have learned to be the clay.
Questions :
(1) What was the potter proud of ?
(2) What is the potter’s realisation after winning wisdom ?
(3) Identify and explain the Figure of Speech in the line ‘That pride has died away’.
(1) The potter was proud that he could mould the clay on his wheels to any patterns he wished to create.
(2) After winning wisdom, the potter realised that his pride was useless. He ceased to be the potter and learned to be the clay.
(3) The Figure of Speech in the line ‘That pride has died away’ is personification because the human quality of ‘dying’ is attributed to ‘pride’.
Vocabulary
1. Use the correct forms of the words given in the brackets.
A pioneer of the American animation (animate) industry, Walter Elias ‘Walt’ Disney introduced (introduce) several developments in the production (product) of cartoons. Walt developed (develop) the character Mickey Mouse in 1928 which was his first highly (high) popular success. He also provided (provide) the voice for his creation (create) in his early years. In the 1950s, Disney expanded (expand) into the amusement park industry and in 1955, he opened (open) Disneyland. Disney was a heavy smoker throughout his life and his death (die) occurred due to lung cancer in 1966 before the park was completed. His film work continued (continue) to be shown (show) and adapted. His studio maintains high standards in its production (product) of popular entertainment. He is considered to be a national cultural (culture) icon and remains an important figure in the history of animation.
2. Replace the underlined words with the words opposite in meaning from the brackets.
(brilliant, forget, success, borrowed, imagination, adding, flopped, celebrated)
Question 1.
Walt was a genius who had an extraordinary skill of removing imagination to any story or idea.
Walt was a genius who had an extraordinary skill of adding imagination to any story or idea.
Question 2.
Walt started a little animated-cartoon company in Kansas city which succeeded miserably.
Walt started a little animated-cartoon company in Kansas city which flopped miserably.
Question 3.
Walt lent $500 from an uncle to start a cartoon series called Alice in Wonderland. Answer: Walt borrowed$ 500 from an uncle to start a cartoon series called Alice in Wonderland.
Question 4.
The 40th birthday of Walt’s mouse, Mickey was condemned in 1968.
The 40th birthday of Walt’s mouse, Mickey – was celebrated in 1968.
Question 5.
Mickey was the first successful outcome of Walt’s brilliant reality.
Mickey was the first successful outcome of Walt’s brilliant imagination.
Question 6.
Walt’s ‘crazy idea’ of starting a Disneyland ! turned out to be a great failure.
Walt’s ‘crazy idea’ of starting a Disneyland turned out to be a great success.
Question 7.
Walt’s graphic telling of the fairy tale made the young Roy Edward remember all about – his measles.
Walt’s graphic telling of the fairy tale ( made the young Roy Edward forget all about his measles.
Question 8.
Walt’s idea of starting an amusement park seemed to be a silly idea.
Walt’s idea of starting an amusement park seemed to be a brilliant idea.
3. Fill in the blanks with proper words from those given in the brackets.
(riveted, grinned, splash, hell-bent, nickel, extravagance, dissent, snuggling)
(1) Aditya is stubbornly determined to achieve his goals. He is ……… on fulfilling his dreams.
(2) Sonali drew close to her mother to receive; comfort and affection. She was …………
(3) Riya gave a coin of five cents to her friend. The coin was a …………
(4) The Alice cartoons could not display things much prominently. They could not create a …………
(5) Everyone firmly fixed their eyes on the trick that the magician was performing on the stage. Their eyes were ………… on the magician.
(6) The children smiled broadly when they were given ice-cream. They at the sight of the ice-cream.
(1) hell-bent
(2) snuggling
(3) nickel
(4) splash
(5) riveted
(6) grinned
4. Select the proper word from the brackets and use it at the proper place.
Example:
The town is surrounded by and has annually a large number of visitors. (nickels / orchards)
The town is surrounded by orchards and has annually a large number of visitors.
Question 1.
After school some students always around the building to talk to their friends, (linger / splash)
After school some students always linger around the building to talk to their friends.
Question 2.
He shows great for his work. (grin / enthusiasm)
He shows great enthusiasm for his work.
Question 3.
Tom won the lottery, but went a year later. (bankrupt / industrious)
Tom won the lottery, but went bankrupt a year later.
Question 4.
Tom’s cat against his leg. (snuggled / cranked)
Tom’s cat snuggled against his leg.
Question 5.
The play was a total, and was shut down after three days, (dissent/flop)
The play was a total flop, and was shut down after three days.
Question 6.
She at me when she came into the room. (grinned / riveted)
She grinned at me when she came into the room.
5 Find out a word from the text to answer the question. Write the word and use it in a sentence of your own.
Example:
Question 1.
Which qualities are required in a person in order to become successful ?
Industrious : The industrious worker was able to finish the difficult task in just a few hours.
Question 2.
What happens when a person spends more than his earnings ?
Bankrupt: A person becomes bankrupt when he spends more than his earnings.
Question 3.
What do you do when you are afraid ?
Desperate: When we are afraid, we became desperate to jet rid of the danger.
Question 4.
What will people think about you if you consistently disagree with others?
Stubborn : If we consistently disagree with others, people will take us as stubborn.
Question 5.
Where do you find a lot of fruits and vegetables ?
Farm: We find a huge lot of fruits and vegetables in a farm.
Question 6.
How will you react if you see your / favourite dish on your plate ?
Gleam: If I see my favourite dish on my plate, at once my face would gleam.
Question 7.
If a film does not make good earning, how do the film critics react?
Criticism: If a film does not make a good earning, the film critics trigger criticism at it.
6. Unscramble the words in Part ‘A’ and match it with the description.
No. Part ‘A’ Description Ex. 1. UHLR (HURL) -F (A) end 2. ENELK (B) previous 3. EASEC (C) shine 4. EBONYG (D) rudeness 5. TPTERNA (E) go down on one’s knees 6. ITTERLG (F) shout abuse 7. CINOLENES (G) arrangement
(1) HURL – shout abuse — (F)
(2) KNEEL – go down on one’s knees — (E)
(3) CEASE – end — (A)
(4) BYGONE-previous — (B)
(5) PATTERN – arrangement — (G)
(6) GLITTER-shine — (C)
(7) INSOLENCE – rudeness — (D)
Answer the questions using words from the brackets.
(wise, proud, insolent, silence, dreamer, patterns, glittering, battlefield)
Question 1.
Where do you find the things like swords, daggers and spears ?
We find swords, daggers and speare on battlefield.
Question 2.
How does a teacher feel when his student s wins a prize ?
The teacher feels proud when his student wins a prize.
Question 3.
Why does Rohan behave disrespectfully with his elders ?
Rohan behaves disrespectfully because he is insolent.
Question 4.
When can you concentrate and meditate properly ?
We can concentrate and meditate in silence.
Question 5.
What are the different kinds of design called ?
Different kinds of design are called patterns.
Question 6.
Why does Aditya behave humbly with everyone?
Aditya behaves humbly because he is wise.
Question 7.
Who thinks of becoming a successful person in future ?
A dreamer thinks of becoming a successful person.
Question 8.
Why are the diamonds looking amazing ?
Diamonds look amazing because of their glittering.
8. Find antonyms of these words from the text. Frame sentences of your own using the antonyms.
recent, numerable, dull, began, stupidity, respect, illiterate
Example : recent x bygone
We always remember our bygone days.
(1) numerable x innumerable
Innumerable ants feasted on the dead body of the snake lying near the ant hill.
(2) dull x fillip
The advertising company gave much-needed fillip to the sales.
(3) began x finished
The campers finished their assignment after a long struggle.
(4) stupidity x wisdom
His handling of the meeting displayed his wisdom.
(5) respect x insolence
The teacher informed the parents that their word’s insolence was intolerable.
Function
1. Read these conversations and try to understand the use of the underlined words:
(a) Mr Khan : What do you want as your birthday gift, beta?
Zoya : Whatever you want to give.
Mr Khan: I want to buy that smart phone for you.
Zoya: But it may be too costly for you to buy.
Mr Khan: Don’t worry, I have money enough to get it for you, dear.
Zoya: That mobile is too large to look beautiful. (The smaller, the better.)
Mr Khan: Hmmm. Look at the one next to it. It seems delicate enough for you ? to have it.
Zoya: Yaaah. But, it’s too fragile for me to handle as I do a lot of commuting for my study. Papa, see that on the left. This one looks sturdy enough to keep with.
Mr Khan : Ok, then. I have read the price on the tag. It isn’t that costly. Shall we go for it?
Zoya : Thank you, pa.
Mr Khan : You’re smarter than that phone.
(b) Manan : Hey, buddy. Did you watch the IPL match yesterday?
Shreyansh : Yes. I am cricket crazy, you know. Kohli played the game marvellously enough to make his team win.
Manan : It was a glorious game indeed. It was too difficult for any bowler to send him to the pavilion.
Shreyansh: Gayle also made a rocking half century. This time, he was calm enough to play sensibly.
Manan: Oh yes, I liked his game too. The bowlers made the ball too short-pitched to control the batsmen.
Shreyansh: Earlier the team’s run-rate was too slow to chase but later in the game, this duo turned the tables.
Manan: As a captain, his strategy was well- thought-out enough to lead the team towards victory.
(c) Mrs Mehta : Where are you going now, beta ?
It’s time to study.
Salil: I have told you many times not to ask every now and then.
Mrs Mehta: But, you see, it’s already too late to begin for preparation of exams.
Salil: I have done my preparation enough to get good score in exams, mom.
Mrs Mehta: And you also waste your time using your cell phone and all. Don’t be too distracted at this juncture of time.
Salil: Ohh, mom. You are too over-caring to understand me.
Mrs Mehta: Don’t blame on me. Due to your wandering with your friends, you have become overconfident enough to ignore all our advices.
Salil: Not at all, mama. I need to go now.
Mrs Mehta: Come back early. Take care.
(d) Read the conversation and fill in the gaps with appropriate options given in the brackets.
[ you also need to be cool enough to convince your parents nicely, your parents are well-experienced enough to guide you, you are also too possessive to understand him fully, you are too stubborn to accept the reality ]
(After counsellors’ meeting with Mr and Mrs Mehta and Salil individually)
Counsellor : Hi, buddy. What’s going on ?
Salil: Fine, Sir.
Mrs Mehta: I am fine too.
Counsellor: Dear Salil, you are too stubborn to accept the reality.
Salil: Ohh, is it so ? Am I not mature enough to take decisions by my own ?
Counsellor : Yes, you are but your parents are well-experienced enough to guide you in proper direction.
Mr Mehta: You see, Salil. You are too immature to think beyond the limits. We want you to be responsible now.
Counsellor: See, your parents are right. But Mrs and Mr Mehta, you are also too possessive to understand him fully.
Mrs Mehta: What! Really ? What should we do now?
Counsellor: You must be patient enough to listen to your kid’s problems. You should suggest him later. Beta, you also need to be cool enough to convince your parents nicely.
2. Read examples and notice the true and false statements as meanings given below them.
Ex. 1 (a)
This mobile phone is too costly for an ordinary rickshaw driver to buy.
(1) It is very cheap. – False
(2) It is very costly but he can buy it. – False
(3) It is very costly and he cannot buy it. – True
Ex. 1 (b)
This mobile phone is not costly enough for a middle class worker to buy.
(1) He can afford it. – True
(2) It is very costly but he can buy it. – False
(3) It is very cheap and he cannot buy it. – False
Ex. 2 (a)
This mobile phone looks delicate enough for a young girl to have it.
(1) It looks very much delicate so that she cannot have it. – False
(2) It looks very much delicate so that she can have it. – True
(3) It does not look very delicate so she can’t have it. – False
(4) The girl has sufficient money to buy this phone. – True
Ex. 2 (b)
This mobile phone looks too delicate for a truck driver to have it.
(1) It looks very much delicate so that he will have it. – False
(2) It looks very much delicate so that he will not have it. – True
(3) It does not look very delicate so he can’t have it. – False
(4) Drivers don’t like/can’t buy such delicate phones. – True
Now, read the sentences taken from F. 1.
Write True for the options with similar meanings.
(1) Kohli was calm enough to play sensibly.
(a) He was very much calm so he played sensibly. – True
(b) He was calm but he didn’t play sensibly.
(c) He was not calm so he played sensibly.
(d) He was quite sensible therefore he played calmly.
(2) The team’s run rate was too slow to chase the winning score.
(a) The team’s run rate was very slow so that they could chase the winning score.
(b) The team’s run rate was not very slow to chase the winning score.
(c) The team’s run rate was very slow so they could not chase the winning score. – True
(d) The team won the match due to their fast play.
(3) Mother is too over-caring to understand Salil.
(a) She is not over-caring to understand him.
(b) She is so over-caring that she doesn’t understand him. – True
(c) She is so over-caring that she understands him.
(d) Due to pampering by his mother, Salil feels that he is not understood well by her.
(4) Salil also needs to be cool enough to convince his parents nicely.
(a) He must be cool so that he can’t convince his parents nicely.
(b) He must be cool so that he can convince his parents nicely. – True
(c) He must not be cool to convince his parents nicely.
(d) He must lose his temper to make his parents understand him.
3. Match ‘A’ with ‘B’Tick mark the sentence similar in meaning.
‘A’ ‘B’ ‘ (a) Due to his poor result, John was too sad to say a word. (1) He was so sad that he didn’t/couldn’t speak a word. (2) He was so sad that he said many words. (b) Vedant is too aggressive to listen to his parents. (1) He is so aggressive that he doesn’t like to listen to his parents. (2) He is so aggressive that he always listens to his parents. (c) Samir’s dance performance was impressive enough to win the audience’s hearts. (1) His performance could not win the audience’s hearts. (2) His performance was so impressive that he could win the audience’s hearts.
(a) → (1) (b) → (1) (c) → (2)
Write sentences without changing the meaning and without using ‘too’ or ‘enough’:
Question 1.
The good fortune was too good to last.
The good fortune was so good that it did not last.
Question 2.
Every kid isn’t fortunate enough to have l a bicycle of his own.
Every kid isn’t so fortunate that it can have a bicycle of his own.
4. Read the situation and the two options. Give your advice by writing ‘Yes’ or ‘No’ with a reason using ‘too’ or ‘enough’. An example is given for you.
Example : Mayank earns ₹ 7000 per month,
(a) Should he buy a mobile phone worth ₹ 10,000 ? (No)
Reason: The mobile phone is too costly for him to buy.
(b) Can he pay ₹ 200 as his birthday party bill ? (Yes)
Reason: Mayank has money enough to pay his bill.
(1) Ilyas has scored 52 percent in HSC Science Stream with B Group.
(a) Can he get admission in medical field ? (No)
Reason : His merit is too low. (low)
(b) Can he get admission in B.Sc. ? (Yes)
Reason: He has secured marks enough high to get admission in B.Sc. (high)
(2) Vedant is sitting in a local train to reach Mumbai urgently.
(a) Will he reach there as early as expected ? (No)
Reason : He will not reach there as early as expected because train is too slow, (slow)
(b) Should he take a superfast train to reach there earlier ? (Yes)
Reason: He should take a train fast enough to reach there earlier. (fast)
(3) Vinayak books cabs, tickets and pays bills through online banking.
(a) Is it advisable enough for him to write the password in his diary? (No)
Reason: It is not safe enough for him to write the password in his diary, (safe)
(b) Should he motivate his friends for online banking ? (Yes)
Reason: He should motivate his friends for online banking because it is convenient enough, (convenient)
5. Complete the paragraph using proper words / phrases from the brackets. Use capital letters where needed :
(excellent, they, he, to excel, too, has won, that, clever, enough, she, his, her, such)
Heer is my student ………… is clever ………….. to answer any question. She is an extraordinarily clever student. On the contrary,………. brother Neel is not ………. a ………. student like her. ………. is ………. dull ………. in any exam. But he is a good sports person. He is so ………. in sports ………. he ………. many competitions ………. attitude remains positive about health and life. Though they are good at different activities,………. help each other.
Heer is my student. She is clever enough to answer any question. She is an extraordinarily clever student. On the contrary, her brother Neel is not such a clever student as she is He is too dull to excel in any exam. But he is a good sports person. He is so excellent in sports that he has won many competitions. His attitude remains positive about health and life. Though they are good at different activities, they help each other.
We went for a long walk in the evening. After a while, Jenish was too tired to walk further. He had to hire a cab. Unfortunately, none had any money and we had forgotten to bring any ATM cards with us. We became too frustrated to think of any option. Luckily, one of our friends – Muneer appeared and he gave us money. Jenish was too happy to express his feelings. He was too overjoyed not to hug our friend – Muneer.
Rewrite the above paragraph without using ‘too’ and ‘enough’. See that the meaning doesn’t change.
We went for a long walk in the evening. After a while Jenish was so tired that he could not walk any further. He had to hire a cab. Unfortunately, none had any money and we had forgotten to bring any ATM cards with us. We became so frustrated that we could not think of any option. Luckily one of our friends – Muneer appeared and he gave us money. Jenish was so happy that he could not express his feelings. He was so overjoyed that he hugged our friend – Muneer.
7. Read this dialogue. Pay attention to the underlined words.
Ami: Hi, dear. I was waiting for you the whole day. You didn’t come online then.
Ravi : Sorry, yaar. I was so busy with the project that I could not spare time.
Ami: Oh, really. Good excuse. You are so smart that you can convince anybody.
Ravi: Is it so ? Thanks then.
Ami: But …. You can’t convince me as I am not so innocent that I will accept what you say.
Ravi: Ok Baba. Will be online tonight.
This time, use ‘too’ and ‘enough’ and rewrite the statements without changing the meaning.
Ami: Hi dear. I was waiting for you the whole day. You didn’t come online then.
Ravi: Sorry, yaar. I was too busy with the project to spare time.
Ami: Oh, really. Good excuse. You are smart enough to convince anybody.
Ravi: Is it so ? Thanks then.
Ami: But… you can’t convince me as I am not innocent enough to accept what you say.
Ravi: Ok Baba. Will be online tonight.
(1) Jenish was so tired that he could not walk further.
Jenish was such a tired boy that he could not walk further.
(2) The park is so beautiful that people visit it every now and then.
It is such a beautiful park that people visit it every now and then.
Write the sentences using ‘such … that’ You can add a suitable noun.
(1) You are too young a boy to cast your vote.
You are such a young boy that you cannot cast your vote.
(2) He is so old a person that the driving license cannot be issued.
You are such an old person that the driving licence cannot be issued.
(3) Meera was bold enough to slap that eve-teaser.
Meera was such a bold person that she slapped the eve-teaser.
(4) His result was so nice that he got admission in that college.
He got such a nice result that he will get admission in that college.
9. Read the questions and complete the answers using the word/s given in the brackets using ‘too’, ‘enough’, ‘so… that’.
Example:
Anwer:
No, I am busy enough not to join you. (busy)
1. Did the children play on that road ?
No, that road was too dusty to play, (dusty)
2. Should Medhavi work hard for the next exam ?
Yes, her result in the first exam was enough dull, (dull)
3. Can that machine be operated by this operator ?
No, the machine is too rusty to be operated. (rusty)
4. Should Vivek ride his bike slowly ?
Yes, there is too much traffic not to get hurt, (traffic, hurt)
5. Do you visit banks for money transactions ?
Generally I don’t, as online banking is convenient enough, (convenient)
10. Compose a paragraph on each situation using ‘too’, ‘enough’, ‘so … that’, ‘such … that’, etc.
(a) Shashank wants to sell his old vehicle on OLX. He has to describe his vehicle for getting better selling price.
I have an old car. Its brand name is Fiat. It is too strong not to bear weight on its top. It is spacious enough to accommodate five persons. The engine is in so good condition that one can travel 400 km without stopping.
The tyres are strong enough to run long distance without getting heated. The focus of the head light is too bright not to clear darkness for at least 1 km. Its sky blue colour is too good to miss attracting one’s attention. It is such a good car that one would not regret spending money on it.
(b) Two friends converse praising their pet animals. Write a dialogue.
Dhruv: Rocky is my pet dog. It is too strong. It is courageous enough to keep thieves away.
Ruthvij : Harry my pet dog is too small to keep thieves away. But it is playful enough by nature.
Dhruv: Dogs are meant to watch the house. They should be ferocious enough to keep the strangers away.
Ruthvij : My dog is too attractive not to draw the attention of a passer-by.
Dhruv: My dog keeps the unwanted people away.
11. (A) Work in groups of six or seven. Match A with B and write at least four sentences for each item as shown in the example. Mark your time to find out which group completed it first.
Example : smooth-hilly to walk
(1) This road was smooth enough to walk.
(2) The road was so smooth that they could walk properly.
(3) It was such a smooth road that they could walk properly.
(4) This road was too hilly for them to walk comfortably
(5) This road was so hilly that they could not walk comfortably.
(6) It was such a hilly road that they could not walk comfortably.
‘A’ ‘B’ 1. rich-poor to carry 2. tall-short to run fast 3. dull-clever to catch a bus 4. fat-thin to answer the questions 5. heavy-light to reach the shelf 6. slow-fast to pay fess
1. rich-poor – to pay fees
(1) Bhalchandra is too rich not to pay his fees.
(2) Gurprit is so rich that he cannot pay his fees.
(3) Bhalchandra is rich enough to pay the fees.
(4) For Gurprit it was such a difficult task that he could not take care of his fees.
2. tall-short – to reach the shelf
(1) He is not so tall that he can reach the shelf.
(2) He is too short to reach the shelf.
(3) The shelf was not so high for the tall boy that he could not reach easily.
(4) The boy was tall enough to reach the shelf.
3. dull-clever – to answer the questions
(1) He was too dull to answer the questions.
(2) He was so dull that he could not answer the questions.
(3) The student was clever enough to answer 7 the questions.
(4) He was such a clever student that he could answer questions easily.
4. fat-thin – to run fast
(1) The competitor was too fat to run fast.
(2) The competitor was thin enough to run fast.
(3) The competitor was so fat that he could not run fast.
(4) The competitor was such a thin person that he could easily run fast.
5. heavy-light – to carry
(1) The luggage was too heavy for the traveller to carry.
(2) The bag was so light that the student could easily carry it.
(3) The victim of the accident was light enough to be carried to the hospital.
(4) It was such a heavy coffin that even six persons could not carry it.
6. slow-fast – to catch a bus
(1) The bus was running too fast for the boy to catch it.
(2) The thief was so slow that he could not catch the bus to escape.
(3) The boy ran fast enough to catch the bus.
(4) He ran with such a slow speed that he was not able to catch the bus.
(B) Read the sentences. Fill in the blanks with names from your own class.
(a) Suresh / Sidhi is so friendly that he / she can make new friends easily.
(b) Gaurav is kind enough to help even a stranger.
(c) Nirmal is too introvert to communicate with any.
(d) Gautam / Geeta has such a beautiful handwriting that he/she is liked by all teachers.
You must be having some students with the qualities mentioned here. Identify those students and frame sentences as given in F. 11 (B).
(clever, helpful, extrovert, tall, strong, proactive, understanding, punctual, polite, adjusting, co-operative and more)
(1) Narendra is too clever to indulge in foolish activities.
(2) Pandurang is helpful enough to look after the poor.
(3) Naina is too extrovert not to give a stage performance.
(4) Shakuntala is such a tall girl that she is unable to make herself comfortable in the class.
(5) Madhavi is so strong that she can perform any task assigned to her.
(6) Navneet is proactive enough to find a solution to the rising problems.
(7) Neetish is too understanding not to help the needy.
(8) Bansidhar is such a punctual person || that one can depend on him.
(9) Pushpa is so polite that elderly people like her.
(10) Paritosh has enough adjusting nature to accommodate any new comer.
(11) Hitesh is too cooperative not to make his classmates like him.
12 Read the pairs and tick mark if they are similar in meaning. If not, rewrite the second sentence to make it similar.
Question 1.
Dhanpura is too small as a village for a bank to open its branch here.
Dhanpura is such a small village that any bank will open its branch here.
Dhanpura is such a small village that no bank will open its branch here.
Question 2.
The maintenance of this bike is low enough to attract more buyers.
This maintenance-free bike doesn’t attract any buyers.
This maintenance-free bike attracts more buyers.
Question 3.
She is not fast enough to win the race in Khel Mahakumbh.
Hemangini is such a fast runner that she will win the race in Khel Mahakumbh.
Hemangini is fast enough to win the race in Khel Mahakumbh.
Question 4.
APJ Abdul Kalam had such an effective personality that people still love him. People still love Dr Kalam for his charismatic personality.
People still love Dr Kalam for his charismatic personality.
Writing
Question 1.
Describe your favourite cartoon series in about 100 words. You can use these points.
(Name of the show-main character – creator of the show – storyline / theme-Why do you like it?)
‘Mickey Mouse’ is not only my favourite show but also of many other viewers. The main character of the show is a mouse. This show and its character was the creation of Walt Disney. He was highly imaginative and industrious. In spite of failures he never gave up efforts to innovate that could entertain the children and the grownups.
The main contents of the show always portrays how the mouse is clever enough to find for itself. It also escapes many attempts made by its enemies to kill him. The show is liked by one and all because it creates humour and fun as a result of clever activities. It is not ordinary. It is very active and clever. Walt Disney creation of the series was appreciated by all.
Question 2.
Draw or Paste an image of your favourite Walt Disney cartoon character, and write a note answering these questions.
• When were you introduced to this character ?
• Through which medium (book or movies) were you familiarized with this character and who introduced it to you?
• Do you find any similarity in traits between that character and yourself or people you know around you ? Describe one incident how it is similar.
• I was introduced to this character at a very early age.
• I was familiarized to this character through books. I was introduced to it by a childhood friend of mine.
• I like the adventures performed by this character.
• Since guests had come, mother asked us (brothers) to avoid feasting on a piece of cheese.
• One of my brothers was interested in having the piece for himself as I wanted. Both of us tried to get the piece and finally I succeeded in having the piece for myself.
Question 3.
As a newspaper reporter if you want to ask five interview questions to Walt Disney, what would they be ?
Example : How old is Mickey Mouse this year ?
(1) Why did you think of a mouse and not any other creature to work on ?
(2) What made you think that you will get such an overwhealming success by creating the character of Mickey Mouse ?
(3) Had you conceived Disneyland as your dream-project or just a tryout?
(4) Suppose you are asked to prepare another dream-project, what would you like to work on ?
(5) Do you think that Mickey Mouse is presented in various media as you had conceived home
Activity
1. Visit some websites to know more about Disney and his world of imagination
(a) http://thewaltdisneycompany.com
(b) www.waltdisney.org
Find out interesting things which are not mentioned in Read – 1. Present them before the class using powerpoint presentation / chart.
Note: Students will collect this information browsing on net and prepare the presentation. They will present the presentation / chart using powerpoint in their classroom.
### Shaper Shaped Summary in Gujarati
પરિચય કવિશ્રી હરીન્દ્રનાથ ચટ્ટોપાધ્યાય દ્વારા રચાયેલું આ કાવ્ય આપણી કેટલીક શરૂઆતની, અણસમજણની માન્યતાઓ અને કાળક્રમે સાચી વાતનું ભાન થતાં તેમાં આવેલા બદલાવની સૂચક વાત કરે છે. આવી ચાર ઘટનાઓ અહીં નિરૂપાઈ છે.
• એક સમયે ‘હું માટીને ધારું તેમ ઘડી શકું એવો વહેમ ધરાવતો કુંભાર સમય જતાં સ્વીકારે છે કે હું એ ‘ઘડવૈયો નથી, પણ માત્ર ઘડાનાર માટી’ જ છું.
• એક સમયે ‘મારા શબ્દ સામર્થ્યથી માનવ-મનને સ્પર્શી જનાર અગણિત ગીત (કાવ્ય) રચી શકું’ એવા નશામાં મહાલતા કવિને નવું જ્ઞાન લાધતાં એ સ્વીકારવા મજબૂર થઈ જાય છે કે હું ગીત (કાવ્ય)નો રચનાર નહિ, પરંતુ રચાનાર ગીત પોતે જ છું.
• ‘હું સેંકડો યુદ્ધભૂમિમાં ઝળહળતી ફતેહનું કારણ બને એવી તલવારો બનાવી શકું છું.’ એવા વ્યર્થ અભિમાનમાં રાચતો તલવારનો ઘડનાર ઈશ્વરના મૌનનો સાક્ષી બને છે ત્યારે માનવા લાગે છે કે “તલવાર ઘડનાર હું નથી, હું તો ઘડાનાર તલવાર માત્ર છું.’
• વીતેલાં વર્ષોમાં હીરામોતીનાં સપનાંમાં આળોટનારને ઈશ્વરના સાનિધ્યમાં એ વાત સમજાઈ ચૂકી છે. મારાં સપનાં એ માત્ર સપનાં જ છે, સપનાં જોનાર માટે એમનું સાકાર થવું લગભગ અશક્ય છે. આમ, આ કાવ્યમાં ઈશ્વરની વિરાટ શક્તિની સામે ગર્વમાં રાચનાર માનવ કોઈ જ મહત્ત્વ ધરાવતો નથી એવી સભાનતા આણવાનો પ્રયાસ થયો છે.
Glossary (શબ્દાર્થ)
Phrase
brim with (બ્રિમ વિથ) overflow with something – છલકાવું, ઊભરાવું
ભાષાંતર
ભૂતકાળમાં, હું
એક કુંભકાર હતો (અને) માનતો હતો કે
મારી આંગળીઓ લચીલી માટીને
ચાકડા પર જુદા જુદા ઘાટ આપી શકે છે;
પરંતુ હવે, પાછળથી પ્રાપ્ત ડહાપણથી (જાણ્યું કે),
એ ગર્વ ઊતરી ગયો છે;
હું હવે ઘડવૈયો મટી ગયો છું
અને માટી બનવાનું શીખી લીધું છે.
ગત દિવસોમાં,
હું એક કવિ હતો જેની
કલમમાંથી અસંખ્ય કાવ્યો (ગીતો) ઊતરતાં રહેતાં
કે જેમનાથી માનવોનાં દિલ જીતી લેવાતાં;
પણ હવે, નવા લાધેલા જ્ઞાનથી,
કે જે અત્યાર સુધી લાધ્યું નહોતું,
હું કવિ મટી ગયો છું
અને કાવ્ય (ગીત) બનવાનું શીખી લીધું છે.
હું એ સમયે એક કુશળ તલવાર ઘડનાર હતો |
(જોકે) હવે એ (દિવસો) વહી ગયા છે
જે (તલવારો) સેંકડો યુદ્ધભૂમિ પર,
ઝબકી છે, ચમકી છે (વિજયપ્રાપ્તિનું સાધન બની છે);
પણ હવે મારામાં છલકાય છે.
ઈશ્વરની (ભેટરૂપ) એ શાંતિ, કે (આજે)
હું તલવાર ઘડનાર મટી ચૂક્યો છું અને (મું)
શીખી લીધું છે તલવાર બનવાનું.
વીતેલાં (એ) વર્ષોમાં, હું સ્વપ્નોમાં રાચતો રહેતો અને મારો
(પેલાં) નીલમ અને મોતીઓના માલિક હોવાનો
ઘમંડ સતત ઊભરાતો રહેતો;
પણ હવે હું ઘૂંટણિયે પડું છું
એ સર્વશક્તિમાનનાં ચરણોમાં
હું (હવે) એ સ્વપ્નોમાં રાચનાર મટી ગયો છું.
અને (મું) સ્વપ્ન બનવાનું શીખી લીધું છે. -– હરીન્દ્રનાથ ચટ્ટોપાધ્યાય
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2023-03-22 17:04:44
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|
https://root-forum.cern.ch/t/library-error-problem-solved/4568
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Library error -- PROBLEM SOLVED
Hi
I just installed Root 5.14 to Macos 10.4.9. I updated my .rootrc file. But when i compile my root application i got following errors (I put only some of it in here).
TSQLRow::Streamer(TBuffer&) referenced from libTree expected to be defined in /Users/brun/root/lib/libNet.dylib
typeinfo for TSQLRowreferenced from libTree expected to be defined in /Users/brun/root/lib/libNet.dylib
vtable for TSQLRowreferenced from libTree expected to be defined in /Users/brun/root/lib/libNet.dylib
TSQLResult::ShowMembers(TMemberInspector&, char*) referenced from libTree expected to be defined in /Users/brun/root/lib/libNet.dylib
TBufferFile::ShowMembers(TMemberInspector&, char*) referenced from libTree expected to be defined in /Users/brun/root/lib/libRIO.dylib
TFileCacheRead::ShowMembers(TMemberInspector&, char*) referenced from libTree expected to be defined in /Users/brun/root/lib/libRIO.dylib
TKey::ShowMembers(TMemberInspector&, char*) referenced from libTree expected to be defined in /Users/brun/root/lib/libRIO.dylib
TKey::TKey(TKey const&)referenced from libTree expected to be defined in /Users/brun/root/lib/libRIO.dylib
TSQLRow::ShowMembers(TMemberInspector&, char*) referenced from libTree expected to be defined in /Users/brun/root/lib/libNet.dylib
collect2: ld returned 1 exit status
make: *** [CZT_Analysis] Error 1
My roo installtion is trying to look /Users/brun directory. But i don’t know how to fix it.
Thanks for the help
Problem solved when i install root 5.14. Something most be wrong with the development version.
( In my presious message root version should be 5.15. )
Thanks,
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2022-12-05 15:53:19
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https://www.math.tamu.edu/Calendar/listday.php?date=20210225
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# Events for 02/25/2021 from all calendars
## Mathematical Physics and Harmonic Analysis Seminar
Time: 11:00AM - 12:00PM
Location: Zoom
Speaker: Salma Lahbabi, ENSEM, UHII/MSDA, UM6P
Title: A mean-field model for disordered crystals
Abstract: In this talk, we consider disordered quantum crystals in the reduced Hartree- Fock (rHF) framework. The nuclei are supposed to be classical particles ar- ranged around a reference periodic configuration. In particular, we consider a family of nuclear distributions μ(ω, ·), where ω spans a probability space Ω. Under some assumptions on the nuclear distribution μ, the average energy per unit volume admits a minimizer, which is a solution of the rHF equations with short-range Yukawa interaction [2, 1]. We obtain partial results for Coulomb interacting systems. We also study localization properties of the mean-field Hamiltonian numerically [3]. References [1] Éric Cancès, Salma Lahbabi, and Mathieu Lewin. Mean-field electronic structure models for disordered materials. In Proceeding of the International Congress on Mathematical Physics, Aalborg (Denmark), August 2012. [2] Éric Cancès, Salma Lahbabi, and Mathieu Lewin. Mean-field models for disordered crystals. J. math. pures appl., 100(2):241274, 2013. [3] Salma Lahbabi. Étude mathématique de modèles quantiques et classiques pour les matériaux aléatoires à l'échelle atomique. PhD thesis, Université de Cergy-Pontoise, 2013.
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2021-05-16 12:52:58
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https://gateoverflow.in/blog/10343/bug-fixes-on-go-site
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Hi,
We just had a major server upgrade and so some bugs are expected. Please list down any issue you are facing since today. You can also add any other bug which was already present.
Also, there are some new themes to try out if you are interested.
1. Muffin Theme which has both Light and Dark Versions
2. Frapuchino – for Desktop
3. Aven
7
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Love
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Haha
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Wow
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Angry
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Missing smiley issue is fixed now
Maybe trivial but I can't find my draft (last month's) blog anywhere
You mean it was there but currently got missing?
Sir my GO points usually don't get updated after an upvote even when the posts are not deleted :/ I haven't checked it very recently but this is what I have noticed even a week before or so.
Oh. For upvote on answer?
For comments I'm sure it wasn't getting updated but for answers I didn't notice..
1
Okay. Comment upvotes were given points by a plugin. Will check that.
The answers section is completely missing for me. Only the question is visible for all questions. I am using MBP 2017, Google Chrome 81.0.4044.138. Double checked in Safari and I am experiencing the same issue.
Can you see now?
1
Yes! Thanks.
sir my GO points have changed! (by -620)
It should be back soon. There was a change in point calculation but as it is messing up the monthly topper rankings, it is rolled back as of now.
(Sorry if I've misunderstood some features as bugs)
1.Donut theme desktop: comment upvote / downvote buttons are bigger, like the answers one.
2. Activity on nav bar is renamed to Q&A.
Will add more soon, as I encounter them.
sir how to change the theme of go to dark
@toxicdesire vote buttons are now fine?
@PRANAVCOOL it is available as an option in Muffin theme.
I means how to apply on your website where is the option
You have to go to your account page and choose the Muffin theme
Are you using a third party to build and maintain the site or is it just you? Please consider exposing the sites source code and making it open source. You will have a lot of support from loyal users to collaborate and help contribute to the site. :)
1
It is an open source CMS Q2A -- you can see link in footer. Waiting for your contribution :)
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Another few,
2. Notifications take 10+ seconds to load, every time.
3. I don't know if it's intentional, but gateoverflow.in/updates doesn't include blog comments/activity ?
1
Thanks. Issue 1 is fixed.
2. Had noticed that. Will be fixed soon.
3. Yes, Updates only includes the Question/answer part. We shall integrate exam/blog updates but might take time.
How's issue 2 now?
1. Thank you sir.
2. It's been fixed now, thank you. However, "Exams taken" section on gateoverflow.in/exams also takes 15+ seconds to load, after clicking on the exam that was previously taken.
3. Okay :)
2
Exam results page should be loading faster now. Thanks for reporting 👍
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Why are my points decreased by 530 :(
The point calculation formula is changed. We were waiting for the month start so that monthly points are not affected. Everyone's point would be affected.
My Verification is showing as "Not verified Yet" , but it was already verified a year ago.
Verification will automatically expire after 400 days unless you accumulate 5000 points.
Sir,
Unable to see bookmarked/favorite posts in "My favorites" section.
You mean favorited blogs or questions? I dont think favorite blogs work - you can use bookmark of browser for it.
@gatecse
You mean favorited blogs or questions?
Yes the start button on blog posts. Since it does record if I have marked it or not, I thought its section is missing in "my favorite" tab.
I dont think favorite blogs work - you can use bookmark of browser for it.
Thank you for suggestion! As of now, using browser bookmarks.
Yes, ideally it should have. Will fix it soon.
1
There's a bug in testimonials' section: GATE 2020 entires aren't displayed...
1
Font color of dates is very light, It's hard to see.
There's same kinda issue with font color in dark theme, font color of dark and light theme seems to be same at some places.
Icon to see the result(under the pointer) is almost invisible.
1
I kinda had to zoom in order even find there was text in that image. lol
Testimonial and date color should be fine now. Dark theme may not work well for Exam pages.
The main page isn't working as it supposed to be...
why this happens?? some questions are visible but some show like this.
That link is working for me. Can you tell exactly which links are the issue?
sir actually, this link gives me such screen: https://gateoverflow.in/87191/gate-cse-2006-question-77,
I face this issue with some questions but now it is working properly.
Probably you double clicked flag button - thats the only error I see.
1
I would like to inform about a bug I found in the profile section of GateOverflow: Those who have the section called Network Sites (eg. I had answered an Aptitude q in GO Mechanical, so my ‘Networking Site” section is active and ‘answers’ section is absent) in their profile section, their “Answer” section is missing. I hope it can be corrected. @Arjun Sir.
Thank you for reporting. Will fix by next week 👍
A category should be added for reporting bugs and errors on the site so that if anyone wants to report anything, they can properly frame their problem with the correct category. This post is not pinned on the main page or anywhere, so we have to search for it. It would be good if we could have a proper visible link for such problem reporting. @Arjun sir.
It is now added under "Others/Site Issues" now.
The pages of : My Circle, Received Circle Requests and Sent Circle Requests under "My Profile" Section is not proper. The avatars seem to cover the whole page and names of the people can't be read. This needs fixing.
Pip box does not detect latex code properly. remains as dollar sign.
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2023-03-23 17:45:48
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http://www.pa3cor.nl/electronics/test/
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# L-meter
The usage of inductors in RF circuits is often intrinsic to its operation. Unfortunately, the measurement of small values of inductance is often difficult without specialized equipment. If, you have the ability to measure frequency, either directly with a frequency counter or indirect with an oscilloscope, this burden can easily be overcome.
The circuit diagram of the L-meter is shown above. It can be broken down into three sections. The circuit around Q1 from a Colpitts oscillator. The main frequency determining elements are C1, C2 and Lx together with stray capacitance in the circuit.
Q2 is the buffer stage. It isolates the oscillator from the load of the frequency counter or oscilloscope. The gate of Q2 is lifted to half the supply voltage. In this way the source has more headroom and the FET will be prevented from conduction on the positive half of the cycles even with larger input voltages.
Voltage regulator VR1 guarantees a stable voltage for the oscillator circuit so that fluctuations in battery voltage cannot change the oscillator frequency. VR1 has a drop down voltage of 3V, so the minimum required voltage at the input is 8V. Shunt regulator D1 (TL431) is used as voltage monitor in this circuit. The voltage at the reference terminal is compared with a 2.5V internal reference. If the voltage at the input comes above this value the regulator will start to conduct and will turn on the LED D2. This will happen at a voltage of 2.5V * (R9 + R8 + R6)/ R6 = 8.1V.
Calibration can be done in a number of ways. You can borrow an inductor with a known (measured) value from a fellow electronics enthusiast. Plug it into the circuit measure the frequency and calculate the in-circuit capacitance.
Another approach could be to buy some a couple of inductors from an electronics shop with a known value. Repeat the process above a couple of times and average the outcome.
A final approach might be to use a capacitor with a known value and inductor with a unknown value. This might sound counter-intuitive and illogical but actually works quit well !
Procedure
1. Connect the inductor Lx (with the unknown value) to the L-meter and measure the frequency. Let’s call this f1
2. Connect the inductor Lx(with the unknown value) to the L-meter, together with the capacitor (with the known value, let’s call it C1) and measure the frequency. Let’s call this f2
The value of inductor Lx can now be calculated with the following formula:
$L_x = { 1 \over{ C_1 4 \pi^2}} * ({ 1 \over {f_2^2}} – { 1 \over {f_1^2}} ) \tag {a}$ Cx, the capacitance in the circuit, can then be calculated with the following formula: $Cx = {1 \over { L_x 4 \pi^2 f_1^2} } \tag {b}$
For the calibration of my device I used a coil I had within hand’s reach. The coil form was 19mm PVC conduit with a length of 35mm. I put 20 turns of 0.8mm enameled copper wire over a length of 20mm. According to the calculator at hamwaves.com this should yield an inductance of 5.0uH.
The next item was MLCC capacitor marked with 100pF. Measured on digital multimeter, the actual capacitance was 102pF. Finally the procedure as described above was carried out which gave the following results:
\begin{align} C_1 & =102 pF\\ f_1 &= 3043 kHz\\ f_2 & = 2800 kHz \end{align} If we plug the numbers above in (a) we get a value for $$L_x =4.86 uH$$ Pretty close to the calculated value! If we the use formula (b) we get a value for $$C_x = 559pF$$.
The mathematical derivation for the calibration procedure outlined above is described below:
$f_1 ={ 1 \over {2 \pi \sqrt{LC_x} } } => C_x = {1 \over {L_x (2 \pi f_1)^2 }} \tag{1}$ $f_2 ={ 1 \over {2 \pi \sqrt{LC_2} } } => C_2 = {1 \over {L_x (2 \pi f_2)^2 }} \tag {2}$ $C_2 = C_x + C_1$ \begin{align}C_x + C_1 & = { 1 \over { L_x 4 \pi^2}} {1 \over {} f_2^2} \tag {3}\\ \\ C_x & = { 1 \over { L_x 4 \pi^2}} {1 \over {} f_1^2} \tag {4} \end{align} If we substract (4) from (3) we get : $C_1 = { 1 \over{ L_x 4 \pi^2}} * ({ 1 \over {f_2^2}} – { 1 \over {f_1^2}} ) \tag {5}$ Now we solve the equation for $$L_x$$ $L_x = { 1 \over{ C_1 4 \pi^2}} * ({ 1 \over {f_2^2}} – { 1 \over {f_1^2}} ) \tag {6}$ Now that we have $$L_x$$ we can use equation (4) to calculate $$C_x$$
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2022-06-24 21:53:32
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https://docs.astar.network/docs/xcm/building-with-xcm/native-transactions/
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# Native Transactions
Let’s look into how we can transfer native relay-chain assets (DOT or KSM) to our Astar or Shiden.
This will be a more low-level demonstration since we’ll be using the Polkadot.js portal. Standard users will interact with this via our portal where all the complexity will be hidden, so don’t worry! 🙂
For this demonstration, we will use Shiden Network and KSM tokens.
## Initial State
Native relay chain asset is represented on Astar or Shiden via asset Id 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF or 340282366920938463463374607431768211455 in decimal.
Initially, we have no KSM assets on Shiden.
Let’s now prepare and bring some assets from Kusama over to Shiden.
This will be an example of VMP protocol usage or DMP to be more specific. We will make use of the reserve_transfer_asset functionality. This consists of moving an asset on one blockchain to another blockchain via an intermediary Sovereign account. Origin chain asset won’t be destroyed, instead, the Sovereign will hold it. The target chain will mint a wrapped asset and deposit it to the target address.
The Sovereign account ensures that the same assets cannot simultaneously be used on both chains. It also guarantees that the wrapped asset is interchangeable with the asset on the origin chain.
Alice wants to send some tokens to an account on Shiden Network so she prepares reserve_transfer_asset call on the relay chain. Parameters look like this:
We specify:
• Destination
• Using the MultiLocation notation, we describe the target chain, which is Shiden, as /Parachain(2007). The notation resembles filesystem paths where the relay chain is considered to be root.
• Beneficiary
• Once again using the MultiLocation notation, we describe the beneficiary of the asset transfer. Take note that this is the context of the destination chain. We’re transferring to an account whose address is Zcvndjciib1X92KeyVBZMv4oUE99Ut1xRCA1e9xMdqXfP27 on Shiden. But instead of specifying the Shiden address, we’ll specify the account ID (or public key) associated with this address: 0xa2ee1f41b5bd08934f178a12e2b0169af05630a212e89d1bb644f67d2192f475
• Assets being transferred
• Finally, we describe the assets being transferred. Since we’re transferring native relay chain assets, we specify these are Concrete assets and their location is Here in the context of the relay chain. The asset is fungible, therefore we need to specify the amount being transferred, in this case, a total of 4200000000 tokens.
After initiating the transfer, we can observe what happens on the relay chain:
Execution of an XCM message was attempted and funds were transferred from Alice to the Sovereign account.
On the Shiden side, we can observe the following:
XCM message has been received and instructions executed. As a result, an asset with Id 340282366920938463463374607431768211455 has been minted, a total of 4196000000 tokens. These tokens have been deposited into the designation account.
Note that that received amount is less than what was originally sent. This is because 4000000 has been used to pay the transaction fee on the destination chain.
We’ve successfully transferred assets from Kusama over to Shiden!
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2023-01-30 15:39:49
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https://docs.zhinst.cn/uhfqa_user_manual/tutorial_monitor.html
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# Generate and Acquire a Test Signal
This tutorial is applicable to all UHFQA Instruments.
## Goals and Requirements
This tutorial explains how to generate and measure a simple pulsed signal with the AWG and the Monitoring Scope in the Quantum Analyzer Input tab.
The measurements in this tutorial can be performed using simple loop back connections.
## Preparation
Connect the cables as illustrated below. Make sure that the UHF unit is powered on and connected by USB to your host computer or by Ethernet to your local area network (LAN) where the host computer resides. After starting LabOne the default web browser opens with the LabOne graphical user interface.
Figure 1. UHF connections for the Input Monitor tutorial
The tutorial can be started with the default instrument configuration (e.g. after a power cycle) and the default user interface settings (e.g. as is after pressing F5 in the browser).
## Test Signal Generation
First, we enable both outputs of the UHF instrument.
Table 1. Settings: enable the Signal Outputs
Tab Sub-tab Section # Label Setting / Value / State
In / Out
Signal Outputs
1
On
ON
In / Out
Signal Outputs
2
On
ON
Copy the following code into the Sequence Editor in the AWG tab.
const LENGTH = 4096;
wave w = gauss(LENGTH, LENGTH/2, LENGTH/8);
var loop_cnt = getUserReg(0);
var wait_time = 0;
repeat (loop_cnt) {
playWave(w, -w);
startQA(QA_INT_NONE, true);
playZero(LENGTH);
}
Upload this sequence program to the AWG of the UHFQA by clicking on "Save" or "To Device". This program will generate a series of dual-channel Gaussian pulses. The repetition number is defined by the integer variable loop_cnt. To make it possible to control the repetition number through the user interface, we use one of the User Registers rather than to write this number into the program.
Apply the settings in the following table in order to configure the AWG output as well as the User Register.
Table 2. Settings: configure the AWG output
Tab Sub-tab Section # Label Setting / Value / State
AWG
Control
Rerun
OFF
AWG
Control
Output 1
Amplitude (FS)
1.0
AWG
Control
Output 1
Mode
Plain
AWG
Control
Output 2
Amplitude (FS)
1.0
AWG
Control
Output 2
Mode
Plain
AWG
Control
User Registers
Register 1
64
## Configure the QA setup tab
In the Quantum Analyzer Setup tab, apply the setting in the table below.
Table 3. Settings: configure the integration delay in the QA Setup tab
Tab Sub-tab Section # Label Setting / Value / State
QA Setup
Deskew
Delay (sample)
200
This setting configures the trigger delay of the monitor, so that the generated signal is correctly aligned with the acquisition window.
## Configure the Input Monitor
In the Quantum Analyzer Input tab, apply the settings in the table below.
Table 4. Settings: configure the Input Monitor
Tab Sub-tab Section # Label Setting / Value / State
QA Input
Control
Input Monitor
Length
4096
QA Input
Control
Input Monitor
Averages
64
QA Input
Control
Run / Stop
ON
In the AWG tab, click on "Start/Stop" in order to run the AWG. The line startQA(QA_INT_NONE, true); triggers the Input Monitor acquisition. See Architecture and Signalling for an overview of the different functional blocks and internal trigger lines. 64 consecutive dual-channel pulses are acquired, averaged, and displayed. The figure below shows the signal as displayed in the QA Input tab. We denote the raw input signals (without averaging) as V1(t) and V2(t).
The Input Monitor Averages setting must agree with the number of Input Monitor triggers generated by the AWG in one measurement burst (here this is determined by the User Register 1 equal to the sequencer variable loop_cnt. If the AWG generates more triggers than that, which is e.g. the case when AWG Rerun would be enabled, the Input Monitor may not be able to process and transmit all data in time, and may deliver corrupted data to the computer.
Figure 2. Test signal as measured in the Quantum Analyzer Input tab
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2023-02-02 08:45:37
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https://direct.mit.edu/evco/article/20/3/395/934/Runtime-Analysis-of-an-Evolutionary-Algorithm-for
|
Abstract
For stochastic multi-objective combinatorial optimization (SMOCO) problems, the adaptive Pareto sampling (APS) framework has been proposed, which is based on sampling and on the solution of deterministic multi-objective subproblems. We show that when plugging in the well-known simple evolutionary multi-objective optimizer (SEMO) as a subprocedure into APS, ε-dominance has to be used to achieve fast convergence to the Pareto front. Two general theorems are presented indicating how runtime complexity results for APS can be derived from corresponding results for SEMO. This may be a starting point for the runtime analysis of evolutionary SMOCO algorithms.
1. Introduction
In recent years, both the fields of multi-objective combinatorial optimization (MOCO) and of stochastic combinatorial optimization (SCO) have made considerable progress, and numerous articles have appeared in these areas. Authors in the MOCO field argue that most real-world problems are characterized by multiple objectives, and authors from SCO emphasize that in real-world situations, decision makers typically face a smaller or larger degree of uncertainty on parameters on which the decisions have to be based. If we join these arguments, we arrive at the conjecture that in a very broad range of applications of decision technology, realistic quantitative models will be characterized by the presence of both multiple objectives and uncertainty (the latter represented by stochastic models). This makes it desirable that, in particular, the toolkit for combinatorial optimization is extended to methods solving stochastic multi-objective combinatorial optimization (SMOCO) problems. Indeed, problems of this type have been described in diverse areas such as financial engineering (Spronk et al., 2005), production planning (Hnainen et al., 2010), transportation logistics (Hassan-Pour et al., 2009), supply chain management (Amodeo et al., 2009), health care management (Baesler and Sepúlveda, 2001), and project management (Gutjahr and Reiter, 2010).
A general methodology for tackling SMOCO problems has begun to develop only recently. Both in the MOCO and in the SCO field, evolutionary algorithms (EAs) play an important role, as seen, for example, from the surveys for MOCO (Coello Coello, 2006) and for SCO (Jin and Branke, 2005; Bianchi et al., 2008). This suggests that EAs could also play an important role in the SMOCO field.
Let us give a short review of available approaches for the solution of SMOCO problems. Concerning the multi-objective part, we focus on the solution concept of determining Pareto-optimal (i.e., efficient) solutions, which appears to be best suited for the purposes of a decision maker who does not want (or is not able) to weigh the different objectives against each other a priori. Caballero et al. (2004) deal with stochastic multi-objective continuous optimization problems, which differ from the stochastic multi-objective combinatorial optimization problems investigated in this paper. Nevertheless, their work is also interesting in our context since they outline the two basic alternative pathways along which a stochastic multi-objective problem can be attacked: Either it is reduced, in a first step, to a deterministic counterpart problem, which is then still multi-objective and can be solved by suitable techniques of multi-objective optimization, or it is first reduced to a single-objective problem, which is then still stochastic and can be solved by techniques of stochastic optimization. In the present work, we choose the first alternative—called the multi-objective approach in Caballero et al. (2004)—which seems to be the more common approach in the literature, but let us mention that also the second alternative—called the stochastic approach in Caballero et al. (2004)—has been applied in the SMOCO area, for example, by adopting the quality indicator concept for scalarizing the given multi-objective problem (see Basseur and Zitzler, 2006; Liefooghe et al., 2007).
The most frequent approach to the solution of a SMOCO problem consists of using a fixed sample of random scenarios1 in order to reduce the stochastic (multi-objective) problem to a deterministic (multi-objective) one, and then to apply some exact or metaheuristic technique to obtain the set of Pareto-optimal solutions w.r.t. the average objective function values over the sample. If an exact technique is chosen, the authors typically apply the so-called ε-constraint method for further reducing the multi-objective problem version to a series of single-objective problems and solve the resulting single-objective combinatorial optimization problems by mathematical programming (MP) methods. (Examples for this solution strategy can be found in Guillén et al., 2005; Cardona-Valdés et al., 2011; Franca et al., 2010.) Alternatively, it is also possible to apply a metaheuristic to deal with the multi-objective problem. For an example, see Claro et al. (2010a, 2010b), where a multi-objective heuristic hybridizing Tabu search and variable neighborhood search is used for this purpose.
Fixed-sample approaches have the limitation that the probability model specified by the drawn sample scenarios is only an approximation of the original probability model given by the problem formulation. As a consequence, finally, a changed problem is solved. The changed problem may considerably differ from the original one especially in a situation where the high computational effort for solving the resulting MOCO problem only allows for the consideration of a relatively small sample of scenarios, which bears the risk that practically relevant scenarios are not represented among the chosen scenarios. In the case where the probability model includes events that are rare but connected with high losses, this may be particularly detrimental. In the area of single-objective sampling-based stochastic optimization, variable-sample methods have been used as a possible alternative to fixed-sample methods. Contrary to techniques working with a fixed randomly selected sample, the application of a variable-sample method can be shown to converge, under suitable conditions, to the solution of the original stochastic optimization problem (see, e.g., Homem-de-Mello, 2003).
In the SMOCO field, some papers use variable-sample approaches. Typically, variable-sample techniques for SMOCO rely on modifications of evolutionary algorithms, especially multi-objective genetic algorithms such as the NSGA-II algorithm by Deb et al. (2002). Whereas the first articles of this kind (e.g., Hughes, 2001; Teich, 2001; Gutjahr, 2005) pursue the variable-sample idea in a rather empirical way, it has been shown (Gutjahr, 2009; Gutjahr and Reiter, 2010) that by a suitable iterative procedure called adaptive Pareto sampling (APS) that exchanges random samples from iteration to iteration, convergence of the currently proposed solution set to the exact set of Pareto-optimal solutions with probability one can be ensured under mild conditions. However, the mentioned articles do not provide analytical results on the speed of convergence. This topic is the subject of the present article.
We study an APS version where the simple evolutionary multi-objective optimizer (SEMO) by Laumanns et al. (2002b) is used for the solution of the occurring deterministic multi-objective subproblems. As a performance measure, we use the expected runtime until the exact Pareto front of the SMOCO problem has been found. It will turn out that for obtaining an efficient algorithmic variant, SEMO has to be modified by including the concept of ε-dominance, using a sufficiently small value of ε. Our theorems show how available runtime results for SEMO on MOCO problems can be translated to runtime results for our APS version on corresponding SMOCO problems. For a recent survey on theoretical analysis of multi-objective evolutionary algorithms in general and SEMO in particular, the reader is referred to Brockhoff (2011).
The plan of this article is as follows: Section 2 presents the investigated APS algorithm. Section 3 introduces the analyzed class of SMOCO problems and explains the sampling technique. In Section 4, the way that SEMO is modified by an ε-dominance concept and how it is incorporated into APS is described. Section 5 derives the general results in expected runtime and provides examples. Finally, Section 6 gives concluding remarks.
We focus on the case of two objective functions. However, the algorithm APS described below can be generalized to more than two objectives in a straightforward way.2 Let a stochastic bi-objective combinatorial optimization problem
1
with (i=1, 2) be given, where X is a finite decision space, and denotes the influence of randomness.3 Maximizing the vector of the two objective functions is understood in the sense of determining the Pareto front of the problem.
We make use of the following customary definitions: For zi=Fi(x) and , dominates x if for i=1, 2, and for at least one i. In this case, we write . Furthermore, x is nondominated by some if there is no such that dominates x, and x is efficient (i.e., Pareto-optimal) if it is nondominated by X. For abbreviation, we shall write F(x) for the image point in objective space to which solution x is mapped by the objective functions. The set of all efficient solutions is the efficient set; its image in objective space is the Pareto front .
The APS framework algorithm was proposed in Gutjahr (2009) and further investigated in Gutjahr and Reiter (2010). Its purpose is to enable the solution of Equation (1) in the frequently occurring case where the expectation cannot be computed directly (or only at high computational costs), but has to be estimated by sampling instead. For getting an estimate of , APS draws s random scenarios independently from each other. Then, the sample average estimate of Fi(x) is given by
2
Evidently, the sample average estimate is an unbiased estimator for Fi(x). An approximation to the solution of the given problem in Equation (1) can be computed by solving a related problem where the two expectations forming the objective functions are replaced by the corresponding sample average estimates under a sample of size s. In this way, we obtain the following deterministic bi-objective problem:
3
We call the problem in Equation (3) the bi-objective sample average approximation (BSAA) problem corresponding to the original problem in Equation (1). The vector will be written in abbreviated form as .
In Algorithm 1, we present the pseudocode of the APS algorithm. The algorithm is iterative and works with a current solution set L(k) which is updated from iteration to iteration. In each of these iterations, first of all a corresponding deterministic BSAA problem is solved in order to obtain a proposal for the solution set. After that, the elements of the solution of the BSAA problem are merged with those contained in L(k - 1), the elements in the union of both sets are evaluated based on independent samples for each solution and each objective function, and dominated elements (w.r.t. the evaluation results) are eliminated. This yields the new solution set. The sample sizes are controlled by sequences (sk) and of positive integers ().
The determination of the efficient set S(k) in part (a) of APS can either be performed by a (multi-objective) exact algorithm, for example, the adaptive -constraint algorithm by Laumanns et al. (2006), or alternatively by a (multi-objective) metaheuristic.
As shown in Gutjahr (2009), APS converges under mild conditions with probability 1 to the efficient set. This extends even to the situation where the deterministic subproblem is only solved heuristically, provided that the used heuristic possesses a suitable convergence property itself. The most essential condition is that the sample size for the solution evaluation part is increased sufficiently fast; linear growth is sufficient. In particular cases, convergence can be obtained for a scheme where the sample size sk for the solution proposal part is kept constant, but to ensure this, certain problem-specific conditions have to be verified. The problem-specific check can be dropped and replaced by a mild (random-search type) convergence property of the heuristic for the solution of the deterministic subproblem, if a scheme is used where sk is also increased (see Gutjahr and Reiter, 2010). In the following, we choose the scheme
4
with an integer constant .
To give an example of an application, let us consider a location problem where out of a set of n possible locations (nodes) for public facilities, a subset of nodes has to be selected. In each selected node j, a capacitated facility is to be built (opened) at a random cost Cj with known distribution function . The facilities provide service for r population nodes. The demand in population node is a random variable with a known distribution function . Suppose that costs and demands in different nodes are independent. From independent random numbers uniformly distributed on [0,1], realizations of Cj and can be computed by and , respectively. We can consider as a particular scenario and write and . The decision is given by the vector where the variable xj takes the value 1 if a facility in node j is opened and 0 otherwise. As the objective functions, let us consider the expectations F1(x) and F2(x) of the negative total cost and of a public benefit function , respectively, where G builds on the distances of customers to opened facilities that can serve them under the given capacity restrictions. If a procedure computing the function G and a random number generator is available, APS can be applied to solve this problem, based on the BSAA estimates in Equation (2). Note that although in this example, the expectation F1(x) could also be computed analytically, that is, without sampling, this may be very difficult or impossible for the expectation F2(x) since G is typically a complex nonlinear function.
3. Analyzed Problems and Scenario Sampling
3.1. Considered Problem Class
From now on, the investigation will be restricted to a somewhat more special case of the general class of problems of the form in Equation (1). As the search space X, the set of binary strings of length n will be considered. (Of course, the assumption that further constraints are absent, which is also made in many other works on the analysis of evolutionary algorithms, is rather strong. It should be noted, however, that by the use of penalty functions, constrained problems can be included in the present framework as well.) By , we denote the efficient set. Concerning the influence of randomness, we assume that the noise terms
5
are independent. We start with the assumption that the noise terms are normally distributed with fixed variance :
6
We will then also proceed to situations where the noise terms follow other distributions. The assumptions above are less restrictive than they look: Using binary problem encodings is standard in applications of evolutionary algorithms, which justifies the special choice of X. The assumption of independence of the noise terms is w.l.o.g., since dependent noise terms would leave the optimization problem in Equation (1) unchanged.4 If the variances of the noise terms are not equal, we may replace them by their maximum value , which gives a conservative estimate as long as upper runtime bounds are investigated. The assumption that the noise distribution is Gaussian, finally, is suggested by the observation that in the application of APS to a problem of the above kind, the noise only enters through sample averages over i.i.d. random variables, and if the number of iterations is large, the sample size k is large as well during later iterations, which, by the central limit theorem, shows that we do not obtain a completely different situation if we replace the given noise distribution by a normal distribution.
3.2. Sampling Procedure
Mathematically, the sampling procedure in the solution proposal part of APS can be imagined as the random choice of a sample of sk=k independent scenarios where each scenario can also be represented by a matrix
7
of noise terms assigned to the elements of . It is clear that in the computer implementation, not all line vectors of the matrix above have to be generated at once when the th sample is created: the only line vectors of interest are those that correspond to the single solutions x; these are examined in the current iteration by the (possibly heuristic) solution algorithm for the deterministic subproblem. These line vectors are generated in order as they are needed. For example, if we would apply random search with 10 random solutions to get an approximate solution to the BSAA problem, we would choose 10 random elements x, corresponding to 10 lines, and generate the noise terms only for these 10 lines within each of the k matrices.5 The scenarios in the solution evaluation part of APS can be represented by analogous matrices.
The objective functions of the BSAA problem are then
8
with
9
For a given sample , the functions are well-defined. In the sequel, the sample error terms are composed of the vector .
4. Incorporating SEMO into APS
4.1. Basic Version
As mentioned in Section 2, any exact or heuristic algorithm for multi-objective combinatorial optimization problems can be plugged into an APS for the solution of the BSAA subproblem. In the following, we shall discuss the use of the well-known SEMO algorithm (Laumanns et al., 2002b) for this purpose. The pseudocode of SEMO is presented in Algorithm 2. Therein, f(x)=(f1(x), f2(x)) is the vector of objective function values of solution , and is the dominance relation in objective space defined in Section 2. SEMO uses an archive which is initialized by a random solution and successively updated by the addition of nondominated mutants and the removal of elements dominated by mutants. In the pseudocode, the termination criterion is left unspecified; later, we shall use a maximum number of iterations as a termination criterion. When SEMO is terminated, the current is taken as the proposed approximation to the efficient set.
Although the algorithm APS/SEMO obtained by applying SEMO as a subprocedure of APS works, its efficiency is usually rather poor. The reasons are explained by the following example.
Example 1:
Consider the following stochastic extension of the bi-objective LOTZ (leading ones, trailing zeros) problem introduced in Laumanns et al. (2002b). For , problem LOTZ is given by
10
The first and second objective functions are the number of leading ones and the number of trailing zeros, respectively, in the binary string x. We extend LOTZ to the stochastic bi-objective problem SLOTZ by adding Gaussian noise: The objectives of SLOTZ are the expectations F1(x) and F2(x) of the random functions
11
with independent noise variables and distributed according to Equation (6). Obviously, F(x)= LOTZ(x). Applying a black box consideration, it is assumed that the solution algorithm does not know this but has to work based only on the observation of the noisy realization vectors .
An application of APS/SEMO to SLOTZ is inefficient for the following two reasons. In the case where the sample error vectors are very small,
• different solutions x and with the same image in objective space (e.g., 10110100 and 10100100) usually have different sample average evaluations, which lets the archive grow too fast, and
• it may easily happen that although x is dominated by (but the points in objective space are identical in one component), the sample average evaluations of x and are incomparable, which misleads the search by SEMO.
In order to modify APS/SEMO to an algorithm overcoming these difficulties, we adopt the idea of ε-dominance, which has also turned out to be useful in deterministic multi-objective optimization.
4.2. A Variant Based on ε-Dominance
The key idea is to replace—during the execution of the algorithm—the standard dominance relation by an ε-tolerant version , but to do this in such a way that for small ε, the efficient set is nevertheless determined based on the original relation . The concept of ε-dominance has frequently been applied in the multi-objective optimization literature (e.g., see Laumanns et al., 2002a; Grosan, 2004; Schuetze et al., 2007; Horoba and Neumann, 2008). In this article, the following definition will be used (we give reasons below):
Definition 1:
Let . For two points z and in the objective space , we say that
• weakly ε-dominates z, written as , if and only if (i=1, 2).
• is ε-equal to z, written as , if and only if (i=1, 2).
• ε-dominates z, written as , if and only if and not . In other words, ε-dominates z exactly if (i=1, 2) and.
For two points x and in the solution space, ε-dominance or ε-equality holds if it does so for their respective image points z and in the objective space. For fixed , we define:
Furthermore, we set Figure 1 illustrates the sets defined above.
Let us discuss the connections of Definition 1 to the definitions in the literature. The relation is an additive (or absolute) version of the multiplicative (or relative) definition of ε-dominance given in Laumanns et al. (2002a), where is said to ε-dominate z if and only if for all i. In our context, it is more natural to choose the additive version because noise is also assumed to be additive. Except for a marginal modification, such an additive version has also been used in Schuetze et al. (2007). Note that the relation relaxes the ordinary definition of dominance. This is no longer true for , that is, in the case where is excluded. In this latter case, we obtain an indented dominance region which is similar (but not identical) to that of the ε-dominance relation used in Grosan (2004). At the end of this section, we see that the exemption of is useful in our context, because it removes one possible source for a too rapid growth of the archive in the case of noisy evaluations.
Figure 1:
Illustration of the sets , , , and related to Definition 1.
Figure 1:
Illustration of the sets , , , and related to Definition 1.
Definition 2:
Let be a finite set of cardinality larger than one. By
we denote the minimal nonzero distance between two points in in one of the two coordinates. By the finiteness of , we have .
In our proofs, we shall use the fact that for ,
12
This follows immediately from the observation that each point in , except z itself, has in at least one coordinate a strictly positive distance to z that is smaller or equal to , and can therefore not belong to by definition of . An immediate consequence of Equation (12) is that for , also
13
since .
It is not difficult to verify that for small enough, ε-dominance coincides with dominance, and ε-equality coincides with equality:6
Proposition 1:
For and ,
Proof:
Easy.
The modified overall algorithm, which we shall call APS/ε-SEMO, differs from APS/SEMO by the following simple changes: Each comparison of objective function values via is replaced by a comparison via , and each comparison of objective function values via = is replaced by a comparison via . More precisely, the following two modifications have to be performed: First, in the solution proposal part of APS, SEMO is replaced by its ε-tolerant version, the algorithm ε-SEMO shown in Algorithm 3. Secondly, also in the solution evaluation part of APS, the trimming, that is, the statement “obtain L(k) as the set of nondominated solutions in according to the objective function estimates just determined,” is now implemented in an ε-tolerant way by the procedure ε-TRIM described in Algorithm 4. Therein, for abbreviation, is denoted by X+, and L(k) is denoted by X-. In both procedures, for F(x), the sample average approximation of F(x) has to be inserted when they are used within the APS framework.
Returning to Example 1, we observe that the disadvantages (a) and (b) of APS/SEMO mentioned there no longer occur with APS/ε-SEMO once the noise becomes small enough. First, for small noise, sample average evaluations of two different solutions x and with are considered as ε-equal, with the consequence that only one of the two solutions is kept in the archive. Secondly, if x dominates but F(x) and are equal in one component, then for small noise (and small ε), the sample average evaluation of x still ε-dominates that of (cf. Figure 1).
5. Runtime Analysis
In this section, we turn to the analysis of the expected runtime that is needed until APS/ε-SEMO proposes a solution set whose image (in objective space) is the entire Pareto front of problem in Equation (1), that is, until equals . This runtime is called the optimization time. The expected optimization time according to the definition above has been used in several investigations of MOCO algorithms (see, e.g., Neumann and Wegener, 2006). Note that for the stopping criterion it is not required to find all Pareto-optimal solutions; it is sufficient to find, for each z on the Pareto front, at least one solution x with F(x)=z. When this has been achieved, we say in short that the Pareto front has been found. This does not mean that the current (disturbed) objective function value estimates F(x) coincide with the points on the Pareto front, but only that the Pareto front is covered with respect to the undisturbed evaluations F(x) of the proposed solutions.
In the theory of evolutionary algorithms, it is customary to use the number of fitness evaluations as a measure of the runtime required by an algorithm. Essentially, we follow this convention, but in our context where fitness approximations are based on random samples, it is necessary to extend the definition in order to obtain a meaningful measure, since an evaluation based on a sample of size, say, one million, will certainly require much more computation time than an evaluation based on a sample of size one. For this reason, the following runtime unit definition will be applied. A runtime measure is an approximate objective function evaluation in Equation (2) based on sample size s that is assumed to require s time units.
Essentially, Theorem 1 below (which is proven with the help of three lemmas) states that by passing from the deterministic to the noisy version of the problem, the optimization time is only increased by a certain factor; this factor is polynomially bounded in important special cases. Before precisely formulating the result and giving a rigorous proof, let us shortly outline the key proof ideas.
First, it is shown (Lemma 1) that for ε small enough, function evaluations disturbed by less than will induce an ε-dominance relation between image points that is equivalent to the original dominance relation between the image points for undisturbed function evaluations, and analogously for the ε-equality relation. Thus, if the function evaluation vectors obtained from the sample averages fall into sufficiently small squares around the exact objective function vectors, the ε-dominance relation on the noisy problem behaves precisely like the ordinary dominance relation on the deterministic problem.
Secondly, the following consequence of the increasing sample size used in APS/ε-SEMO is shown (Lemma 2 and Corollaries 2–4): The probability that in a given iteration, all function vector estimates fall into squares of a given size around the exact objective function vectors, can be bounded from below by certain explicit expressions (tending to one with increasing iteration index). Note that increasing the sample size in a sample average estimate decreases the probability of large deviations from the true value.
Furthermore, it is also necessary to show (Lemma 3) that in the solution evaluation part of APS/ε-SEMO, previously found efficient solutions are confirmed, provided that the disturbances by the sample average estimates are small enough.
In the proof of Theorem 1, finally, the obtained expressions are used to bound the probability of the sample success event, that is, the event that in a certain iteration, the noisy evaluations do not lead to a wrong judgement about mutual dominance relations between the currently considered solutions. The desired bound on the expected runtime is then obtained by adding (i) the time required to reach an iteration such that in all subsequent iterations, the sample success event has become likely enough, and (ii) the expected optimization time after this iteration.
In the following, denotes the sup-norm, and as usual, the image set is denoted by F(x).
Lemma 1:
Let and with and , where with according to Definition 2. Then
Proof:
(a) We start by showing the second equivalence. If , then
and hence . Conversely, let . Then, according to the second equivalence of Proposition 1, for . Hence there is an i with . Since
this implies , and therefore .
(b) Now we show the first equivalence. Let . Since and cannot hold simultaneously, implies and hence also by the already shown second equivalence of the lemma. We show by contradiction: Assume does not hold. Because of , this means that either , or z and are incomparable. In both cases, there is a coordinate i such that . By definition of and because of , even must hold in this case. It follows that
However, entails , which contradicts the just derived inequality.
Conversely, let . If and , it follows that (i=1, 2) and thus (i=1, 2) analogously as above, so . Now assume that there is an i such that ; w.l.o.g., let i=1. From we conclude and thus in particular . It follows that . Furthermore,
In total, we have .
Corollary 1:
Suppose that for the noise terms in a fixed iteration k of the part solution proposal of APS, it holds that
Then ε-SEMO, applied to the problem with objective function vector , produces the same sequence of mutated solutions and the same sequence of current solution sets as SEMO, applied to the problem with objective function vector F, provided that the same (pseudo-)random numbers for the choice of the initial solutions and the bit mutations are used. In particular, the runtime of ε-SEMO is then just k times the runtime of SEMO.
Proof:
As seen by comparing ε-SEMO with SEMO, it suffices to show that for two solutions x(1) and x(2),
14
According to Equation (8), we have , so by assumption, with for all . Thus, with , , and , the conditions of Lemma 1 are satisfied. The first and the second equivalence of the Lemma translate then immediately into the first and the second equivalence of Equation (14), respectively. The statement about the runtime follows directly from the equivalent behavior of SEMO and ε-SEMO in the considered iteration and from our chosen runtime definition, since in iteration k, the sample size k is applied.
For the sequel, it is convenient to introduce the notation
for the open square with half side length r>0 around z. Moreover, let 0=(0, 0).
Lemma 2 below is a technical auxiliary result, providing a bound for the probability that sample averages of Gaussian random variables do not deviate too much from their means. We shall apply it later (in Corollaries 2 to 4) to the sample average estimates used in APS.
Lemma 2:
For fixed r>0, i.i.d. random variables , and with
15
the probability
is bounded below by
16
for each .
Proof:
With , we find
where denotes the cumulative distribution function of the standard normal distribution. Since with for x<0, one obtains
For , the bound on the r.h.s. is positive, since then
17
Therefore,
From the last inequality, the statement of the Lemma follows by .
Let us now apply Lemma 2 to the solution proposal part of APS by choosing the variables as the noise terms , such that becomes equal to the error term of the sample average estimate as given by Equation (9).
Corollary 2:
For , the probability
that for the solution proposal part of iteration k of APS, the error terms of the sample average estimates are in Qr(0) for all , is bounded below by
Proof:
In the Appendix.
Corollary 3:
Let denote the probability that not only for the solution proposal part of iteration k of APS, but also for the solution evaluation part of this iteration, the error terms of the sample average estimates are in Qr(0) for all . Then, for ,
Proof:
In the Appendix.
Corollary 4:
For as in Corollary 3, , and with
it holds that
Proof:
In the Appendix.
Lemma 3:
If holds for the efficient set , and if for the error terms of the sample average estimates in the solution evaluation part of iteration k of APS, the inequalities with hold, then the application of ε-TRIM(X+) in this iteration produces an X- with .
Proof:
We have to show that after the execution of ε-TRIM, (i) for every , an element x with F(x)=z is contained in X-, and (ii) no element is contained in X-. The proof uses Lemma 1 in an analogous way as it is used in the proof of Corollary 1.
To verify (i), let . Because of , there is an element with F(x)=z, so and there is an iteration of ε-TRIM in which x is the current element of X+ to be checked for possible insertion. Because of Lemma 1, for some other solution y would entail in contradiction to the efficiency of x. Again by Lemma 1, entails F(y)=F(x). Therefore, if there is an element y in the current set X- with , then F(y)=F(x). As a consequence, either x is added to X- in this iteration, or X+ already contains a y with F(y)=F(x)=z. In both cases, F(X-) contains z after the current insertion trial. It remains to be shown that the element of X- with image z (let us call it x for simplicity) will not be removed from X- in any later iteration. A removal of x could only take place if another element with were added. However, by Lemma 1, this would entail and thus contradict the efficiency of x.
To verify (ii), let . Then there is a such that . Because of , there is an iteration where an element x with F(x)=z is the current element to be checked for possible insertion. If in the beginning of this iteration y is contained in X-, it will be removed from X-, because by Lemma 1, and x will be inserted instead. In a later iteration, y can no longer be added to X-, because x or an element with is already in X-, which prevents the insertion of y.
Now we are in the position to prove our main results, as shown in Theorems 1 and 2 below. For a complete specification of APS/ε-SEMO, we still have to define how many iterations of ε-SEMO are to be performed in each iteration of APS. In agreement with our general assumption that we can base our analysis on the corresponding results for SEMO on the deterministic (i.e., noise-free) limiting case of the problem under consideration, we make the mentioned decision dependent on an upper bound hn for the expected number of function evaluations required by SEMO to find the Pareto front of a noise-free problem instance of size n: In each iteration of APS, exactly 2hn iterations of ε-SEMO are executed.
Theorem 1:
Let the assumptions in Section 3 be satisfied, let hn be an upper bound for the expected optimization time of SEMO on noise-free problems of instance size n, let denote the minimum of the values (given by Definition 2) over all problem instances of size n, let denote the maximum of the noise standard deviations over all problem instances of size n, and let
18
where . Then
is an upper bound for the expected optimization time of APS/ε-SEMO in an implementation where each call of ε-SEMO performs 2hn iterations.
Proof:
We set in Corollaries 2–4, such that for the error terms , the condition with required in Corollary 1 as well as in Lemma 3 is satisfied. Furthermore, in Corollary 4, we choose the constant C as . By this special choice, Corollary 4 yields
19
with
Therein, the first argument of max is smaller than the second argument and can therefore be omitted. Moreover, can obviously be replaced by an upper bound without making Equation (19) invalid. Therefore, can be replaced by k0 as defined in Equation (18). Hence, Equation (19) says that in each iteration of APS, with a probability of at least 1/2, samples are drawn for which the error terms both in the solution proposal part and in the solution evaluation part all lie within the square . We call the event that this happens the sample success event. Let us denote the ensemble of samples used for the solution proposal in a fixed iteration k by sample 1, and the ensemble of samples used for the solution evaluation in this iteration by sample 2.
Let us now use Corollary 1. It ensures that in the sample success event, the procedure ε-SEMO applied to the disturbed problem according to sample 1, behaves equivalently to the procedure SEMO applied to the noise-free problem. In particular, if given an unlimited number of iterations, ε-SEMO finds then the Pareto front in an expected runtime with upper bound hn. Denoting the runtime until has been found by T, the well-known Markov inequality yields
such that within 2hn iterations, the Pareto front will be found with probability of at least 1/2. Observe that as soon as SEMO has found for the first time in the sense that the current satisfies , it preserves this current solution during the remaining iterations until iteration 2hn, and again by Corollary 1, in the sample success event, the same holds for ε-SEMO.
By Lemma 3, in the sample success event, the solution evaluation part of APS, applied to the disturbed problem according to sample 2, confirms the with image if the last has been found by ε-SEMO in the solution proposal part.
In total, we find that in each iteration , with a probability larger or equal to , the solution set L(k) proposed by APS at the end of this iteration has image . Note that the random numbers used for obtaining the two samples and those used for the mutations during the run of ε-SEMO are independent, and that in the sample success event, the error terms no longer matter. Moreover, for future use, it should be noted that the lower overall success probability bound 1/4 for an iteration holds independently from the events in previous iterations.
We now compute the runtime required by iteration k of APS/ε-SEMO. In this iteration, ε-SEMO performs 2hn iterations, that is, at most 2hn new solutions are evaluated in addition to the initial solution. Because of sk=k, this yields a runtime smaller or equal to (2hn+1) k for the solution proposal part. The set of solutions to be evaluated in the solution evaluation part contains at most elements, because in each APS iteration, at most 2hn+1 new solutions can be proposed. Considering , this gives an upper runtime bound of (2hn+1)mk2 for the solution evaluation part in iteration k. Therefore, the cost of the entire iteration is bounded from above by with R=(2hn+1)(m+1).
The total runtime in iterations 1 to k0−1 of APS is therefore bounded from above by
20
Finally, we turn to the expected total runtime in iterations k0, k0+1, and so on, until has been identified. We use the lower bound 1/4 on the total success probability in an iteration and assume pessimistically that the total success probability is equal to 1/4 instead of larger or equal. This gives the following upper bound for the expected optimization time after the start of iteration k0:
with . By using the well-known summation formulas for the series , and , we can evaluate the last expression as R(k20+6k0+21). Addition to Equation (20) and insertion of the expression for R yields the result. Remark 1: From an application point of view, Theorem 1 raises the question of how ε and the runtime given to ε-SEMO in each call of this procedure should be chosen in an implementation of APS/ε-SEMO if and/or the bound hn are not known. A pragmatic answer is to tune both parameters experimentally for the considered class of problem instances, such as is usually done for other parameters of a metaheuristic. Obviously, an upper bound for the expected optimization time under the choice of these two parameters as described in Theorem 1 is also an upper bound for the expected optimization time under the choice of the (experimentally found) best combination of these parameters. A more sophisticated way of handling the issue would be to gradually decrease during the run of the APS framework algorithm, and to gradually increase the runtime given to ε-SEMO. For all iterations where has fallen below and the runtime of ε-SEMO has exceeded 2hn, analogous estimations as in the proof of Theorem 1 become applicable. Possibly, this will allow an extension of Theorem 1 to the described self-adapting variant. Remark 2: Our presentation of the SEMO algorithm has been based on the original version of SEMO as introduced in Laumanns et al. (2002b). In the literature, also a slightly modified version of SEMO has been analyzed (see, e.g., Neumann and Wegener, 2006). This version, which we shall denote by SEMO’, differs from the original one by the fact that if to an element considered for insertion, there exists already an element with the same objective function values, then instead of keeping y and disregarding , the new is inserted into , and y is omitted. More formally, the two last statements in the loop of SEMO (Algorithm 2) are replaced by
By obvious analogous replacements, we obtain corresponding procedures ε-SEMO’ and ε-TRIM’ from ε-SEMO and ε-TRIM, respectively. Going through our proofs, it is easily seen that Theorem 1 remains valid if in APS/ε-SEMO, subprocedure ε-SEMO is replaced by ε-SEMO’, and subprocedure ε-TRIM is replaced by ε-TRIM’.
We immediately obtain from Theorem 1:
Corollary 5:
If and , the expected optimization time of APS/ε-SEMO grows at most by a factor of order O(n3) faster in n than that of SEMO applied to the noise-free problem.
Example 2:
(Continuation of Example 1 from Section 4.1.) For the SLOTZ problem introduced in Section 4.1, we have . A well-known result by Laumanns et al. (2002b) states that the expected optimization time required by SEMO to find the efficient set for LOTZ is on the order of , with an upper bound of hn=(1/2)n3−(1/2)n2. By applying our Theorem 1, we derive from this result that the expected optimization time for finding the efficient set for SLOTZ is at most on the order of O(n6).
Example 3:
The article Neumann and Wegener (2006) analyzes the expected optimization time of SEMO’ (cf. Remark 2) for a bi-objective minimum spanning tree (MST) problem. In an edge-weighted graph with r nodes and n edges, to each subset x (representable as a binary string) of the set of edges, two objective functions are assigned: The first objective c(x) is the number of connected components of the subgraph defined by x, the second objective w(x) is the total weight of the edges in x. Both objectives are to be minimized. Weights are assumed to be integers with a maximum value of wmax. Neumann and Wegener show that the expected optimization time of SEMO’ is on the order of O(nr(log r+log wmax)+nr2).
In a first noise model, we extend the problem to the stochastic case in an analogous way as was done for SLOTZ by adding random Gaussian noise to each objective:
with independent noise variables and distributed according to Equation (6). For the standard deviations, we assume . Similarly as in the SLOTZ example, it is supposed that the solution algorithm is blind with respect to the underlying MST structure of the problem. Because of the integrality of the objectives, we have . Let us assume that , and that the relation between the number of nodes and the number of edges satisfies with some . (The case represents dense graphs.) Then , and Corollary 5 yields an expected optimization time on the order of for the stochastic problem.
As an alternative, let us also consider a second noise model where the noise results from imprecise measurement of the weights of the edges on the assumption that an estimate of w(x) is obtained by summing up the individual weight estimates of the edges contained in x. Suppose that the individual noise terms for the measurement of the edges are independent and normally distributed with a variance smaller than or equal to some constant independent of n, and that the number c(x) of components can be determined without noise. Then , and again on the assumption that and , Theorem 1 yields an expected optimization time on the order of .
Corollary 5 shows that in many cases, the transition from a MOCO problem to a SMOCO problem only generates a polynomial-time overhead factor. Although this is a helpful insight, the bounds presented in Corollary 5 and the SLOTZ example are still rather weak. We conjecture that stronger bounds hold, and below we shall outline a situation where stronger bounds can indeed be rigorously derived. Before turning to this issue, a generalization of Theorem 1 will be demonstrated. We start with a definition.
Definition 3:
(From Birnbaum, 1984.) For two random variables and , the variable is called more peaked about zero than the variable if
21
Corollary 6:
The assertion of Theorem 1 remains valid if the -distributed noise is replaced by independent noise following some other distribution (which is allowed to depend on i and x), provided that for all and i=1, 2,
• the density of is continuous, symmetric (i.e., the density in −z is equal to the density in z) and nonincreasing for positive arguments z, and
• is more peaked about zero than a random variable .
Proof:
Theorem 1 in Birnbaum (1984) states that under the indicated conditions, the corresponding sample averages inherit the more peaked relation, that is, is more peaked about zero than with . A reinspection of the proof of our Theorem 1 above and the preceding lemmas (especially Lemma 2 and Corollaries 2–4) shows that this property is all that is needed to make the used upper bound estimations still valid for the case where the are replaced by the .
Corollary 6 above allows for the case of bounded noise to be included, for example, noise following a triangular distribution or a truncated normal distribution.7 Our next result shows that in the case of bounded noise, a considerably stronger runtime bound than that of Theorem 1 can be derived.
Theorem 2:
Let the conditions of Corollary 6 be satisfied, and let for all n,
• ,
• .
Then the expected optimization time of APS/ε-SEMO on an instance of size n in an implementation where each call of ε-SEMO performs 2hn iterations is bounded from above by
with ko as in Theorem 1.
Proof:
First of all, we compute the maximal length of an ε-antichain in a square with side length . An ε-antichain is a set of points with the property that two different points are ε-incomparable, that is, that neither nor nor . In particular, this means that two elements z and in an ε-antichain differ by at least ε in each of their two coordinates. By projection to one of the two coordinate axes, one immediately sees that in the considered square, not more than mutually ε-incomparable elements can exist. We choose as in Theorem 1. Then we have that an ε-antichain can contain not more than elements.
In iteration k, both the solution S(k) delivered by ε-SEMO and the solution L(k-1) delivered in the previous iteration by ε-TRIM are ε-antichains. Since by assumption, the modulus of the noise is smaller or equal to cn, also
holds, and therefore both the sample average estimates used for the elements of S(k) in the call of ε-SEMO in iteration k and those that have been used for the elements of L(k-1) in the solution evaluation part of iteration k−1 all lie in the square , that is, in a square with side length . Using the bound above, we conclude that and , so .
Now we go through the proof of Theorem 1, with respect to Corollary 6 which allows a generalization to noise that is more peaked than Gaussian noise. In these proofs, we used an upper bound of for the number of elements in that have to be re-evaluated in the solution evaluation part of the APS iteration. On the current premises, this set can contain at most elements. For iteration k of APS, this yields an upper runtime bound of with
Therefore, the total runtime in iterations 1 to k0−1 of APS is bounded from above by
and the expected total runtime in iterations k0, k0+1, and so on, until has been identified is bounded from above by
with . Hence is an upper bound for the total expected optimization time. Insertion for gives the result.
Example 4:
(Further Continuation of Example 1 from Section 4.1.) Replace in the SLOTZ problem introduced in Section 4.1 the Gaussian noise by noise , where the distribution of is more peaked about zero than , and where for all with some constant c>0.8 Note that for an=0, bn=n, and cn=c, the conditions of Theorem 2 are satisfied, and that . Then Theorem 2 yields an upper bound for the expected optimization time of
22
which is only by a factor of order O(n2) worse than the expected optimization time in the deterministic boundary case of LOTZ. With a view at the proof of Theorem 2, it can also be seen from Equation (22) that now, the term on the order of referring to the solution evaluation part of APS is negligible compared to the term on the order of O(n3) referring to the solution proposal part. This is a typical situation also for applications to more complex problems, since the current approximation of the efficient set is usually rather small, such that the solution evaluation part is computationally unproblematic.
Remark 3: Obviously, the O(n2) bound for the ratio of expected optimization times between stochastic and deterministic problem in the bounded-noise case can be generalized from SLOTZ to a broader class of problems, analogous to the case in Corollary 5. However, it is not quite general, as the following example shows.
Example 5:
(Continuation of Example 3 from Section 4.1.) For the stochastic bi-objective MST problem of Example 3, let us perform a similar replacement of the Gaussian noise by noise with a distribution of more peaked about zero than as in Example 4 above. Let all other assumptions be the same as before. For , Theorem 2 yields in this case an exponential bound. However, if with some , Theorem 2 (with an=0, bn=wmax, and cn=c) gives a bound for the expected optimization time on the order of .
Remark 4: Also for a normal distribution, in the case of sufficiently large cn>0, absolute noise values larger than cn are very unlikely. Therefore, it can be conjectured that for Gaussian noise, the expected optimization time is bounded by an order that conforms to the often stronger bound of Theorem 2 instead of only to the bound of Theorem 1. We leave a rigorous proof of this conjecture as an open question. Remark 5: We do not claim that the sample size increment scheme and is optimal for the expected optimization time of APS/ε-SEMO. Concerning the upper bound for the expected optimization time, it can even be shown that it becomes better the faster the sample size is increased: Consider a scheme where with some . For the sake of simplicity, let us restrict the discussion to the SLOTZ special case. Then by analogous derivations to those in Theorems 1 and 2, we find an expected optimization time bound on the order of O(n4+2/p) and of O(n4+1/p) in the case of Gaussian and of bounded noise, respectively, which decreases for increasing p. Nevertheless, it should be kept in mind that these are only upper bounds. It is easily possible that the actual expected optimization time of the linear scheme is better than , since in our proofs, we used rather coarse bound estimations, assuming that the efficient set has to be identified as a whole in one of the APS iterations, whereas APS can also find the elements of the efficient set one after another or subset by subset, which considerably speeds up the process. This gradual identification of the efficient set, however, is prevented if the sample size is increased too fast.9
6. Conclusions
In this paper, an algorithm for stochastic multi-objective combinatorial optimization (SMOCO) has been analyzed. The algorithm is obtained by inserting the simple evolutionary multi-objective optimizer (SEMO) as a subprocedure into adaptive Pareto sampling (APS). Normally distributed noise as well as more general noise distributions have been investigated. It has been shown that if SEMO applies a specific ε-dominance relation instead of the ordinary dominance relation, the expected optimization time of the overall algorithm can be bounded from above, provided that ε is chosen small enough. The bound depends on the expected optimization time of SEMO on the corresponding deterministic counterpart problem. An explicit expression for this dependence has been given. In the special case where the variance of the noise is bounded from above by a constant independent of the problem size n, and ε is bounded from below by a constant independent of n, the ratio between the expected optimization times for the stochastic and the corresponding deterministic problem, respectively, is on the order of O(n3). For bounded noise, the ratio reduces to O(n2).
Since the underlying deterministic single- or multi-objective optimization problems of practically relevant SMOCO problems are often NP-hard, such that the optimization time for them must be anticipated to grow exponentially fast in the instance size, the results show that in typical cases, stochastic MOCO problems are not essentially harder than their deterministic counterparts.
Future research should address several topics. A question of practical relevance is whether the results presented here can also be transferred from the use of SEMO to the use of a more elaborate multi-objective metaheuristic such as NSGA-II or SPEA2. This question is especially important in view of well-known drawbacks of SEMO, for example, the fact that SEMO can end up with prohibitively large populations. Furthermore, a generalization of the results presented here to the case of more than two objective functions would be of interest.
Expected optimization time is not the only relevant criterion for the performance of an evolutionary optimization algorithm; often, the immediate goal is not to find the exact solution (in the multi-objective case: the efficient set), but rather to identify a good approximate solution within a short time. For investigating this question in the multi-objective situation, the behavior of the hypervolume or of another quality indicator during the run of a combined algorithm of the type considered here could be studied.
The bounds derived here are rather coarse, and it is possible that tighter bounds may be found if the way that APS gradually adapts the currently proposed solution set is taken into more detailed consideration. Moreover, in the present work, we focused on upper bounds for the expected optimization time. A most interesting question would be how good lower bounds look like and whether there are cases where it can be shown that their order matches that of the found upper bounds. Based on these results, the question of how the sample size scheme in APS can be organized in order to obtain the fastest possible convergence to the Pareto front could also be addressed.
Acknowledgments
The author thanks Lothar Thiele for a discussion on ε-dominance concepts during the Dagstuhl Seminar 2010 on theory of evolutionary algorithms, and three anonymous reviewers for their very helpful comments on the first version of this article.
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Appendix
Proof of Corollary 2:
From Equation (17), we see that for , the lower bound in Equation (16) is nonnegative. Considering Equation (9), Equation (6), and the independence of the random variables for different , and observing that the power function up (peven) is nondecreasing as long as , we obtain from Lemma 2:
From this inequality, the statement follows by means of the inequality .
Proof of Corollary 3:
If the size of a sample of i.i.d. random noise variables is increased, the variance of the sample average decreases. As a consequence, the probability that the error terms of the sample average estimates over a sample of size with m>1 all lie in Qr(0) is larger or equal to the corresponding probability for the case m=1. Therefore, the choice m=1 is conservative w.r.t. the lower bound on . For m=1, the only difference from Corollary 2 is that now, not only in 2n, but in 2n+2n=2n+1 random experiments, the error terms have to be sufficiently small. This immediately yields the indicated formula.
Proof of Corollary 4:
By Corollary 3,
for each . The assertion follows since if in addition to , then the argument of exp in the rightmost expression above is smaller or equal to −C.
Notes
1
In the stochastic programming literature, a scenario is a specific outcome of a random experiment (see Birge and Louveaux, 1997, p. 50).
2
Presumably, our runtime results in Section 5 can also be extended to more than two objectives by a rather straightforward generalization of the proofs, but this topic is beyond the scope of the present article.
3
Formally, for each x and i, the function is a random variable on a probability space , and is an element of the sample space . Each determines a specific realization of the random variables , that is, a scenario; hence the notation . As is customary in the stochastic programming literature, we shall identify with the scenario determined by it (cf. Birge and Louveaux, 1997).
4
Of course, dependence could make a difference in the way the sampling is done in the solution proposal part of APS. However, in order to stay within the premises of the following analysis, we may always replace dependent sampling by independent sampling, drawing a specific, independent scenario for each x and each i, as is done in the solution evaluation part. This may deteriorate the performance of the algorithm, but as we are interested in upper runtime bounds, this is not a problem.
5
In particular, the described procedure implies that if the solution algorithm for the deterministic subproblem examines one and the same solution x more than once, the same noise term realizations should always be used. For this purpose, the algorithm has to store the sample average estimates for the already examined solutions in a look-up table. Such an implementation may be less efficient than one where, if a solution is examined a second time, the noise is evaluated by a new independent simulation. For large n, however, the case where an evolutionary search algorithm returns in a later iteration to a solution that has already been examined before is comparably rare, such that it can be expected that in terms of the quality evolution of the successively examined solutions, the second implementation variant will behave very similarly to the variant investigated in this article.
6
Observe that contrary to , the relation of Definition 1 does not coincide with ordinary dominance even for small ε because z does not dominate itself.
7
Strictly speaking, in the last case, the truncation must be defined in such a way that the density function remains continuous, which can of course approximate a noncontinuously truncated normal distribution to any desired degree of accuracy.
8
The special case of large c and a continuously truncated normal distribution produces a very close approximation to the Gaussian model.
9
In the case where instead of SEMO, a solver based on mathematical programming is used for the solution of the deterministic subproblem, as has been done in Gutjahr (2009), a rapid increase of the sample size is especially detrimental, since the runtime of such a solver typically increases exponentially fast in the sample size, such that high sample sizes have to be avoided at any cost.
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2021-04-10 19:55:07
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http://www.lofoya.com/Solved/1243/a-pump-can-be-used-either-to-fill-or-to-empty-a-tank-the-capacity-of
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Moderate Time and Work Solved QuestionAptitude Discussion
Q. A pump can be used either to fill or to empty a tank. The capacity of the tank is (3600text{ m}^3). The emptying capacity of the pump is (10 text{ m}^3/text{min}) higher than its filling capacity. What is the emptying capacity of the pump if the pump needs 12 more minutes to fill the tank than to empty it?
✖ A. $10 \text{ m}^3/\text{min}$ ✔ B. $60 \text{ m}^3/\text{min}$ ✖ C. $45 \text{ m}^3/\text{min}$ ✖ D. $90 \text{ m}^3/\text{min}$
Solution:
Option(B) is correct
Let f $\text m^3/\text{min}$ be the filling capacity of the pump.
Therefore, the emptying capacity of the pump will be = $f+10$ $\text m^3/\text{min}$
The time taken to fill the tank will be = $\dfrac{3600}{f}$ minutes
And the time taken to empty the tank will be = $\dfrac{3600}{f+10}$
We know that it takes 12 more minutes to fill the tank than to empty it
$\dfrac{3600}{f}-\dfrac{3600}{f+10}=12$
$3600f+36000-3600f=12(f^2+10f)$
$f^2+10f-3000=0$
Solving for positive value of $f$ we get, $f=50$
Therefore, the emptying capacity of the pump = $50+10=60$ $\text m^3/\text{min}$.
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2017-09-26 00:18:27
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https://socratic.org/questions/what-is-0-12-divided-by-1
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What is 0.12 divided by 1?
May 4, 2018
$0.12$
Explanation:
Every number divided by $1$ remains unchanged!
You may justify this using the fact that $1$ is the neutral element for multiplication, id est
$a \setminus \cdot 1 = a$ for every number $a$
For example, $5 \setminus \cdot 1 = 5$
Now, in general, we know how to invert multiplication and turn them into divisions:
$5 \setminus \cdot 3 = 15 \setminus \implies 15 \setminus \div 3 = 5$
So, with one, it works like this:
$a \setminus \cdot 1 = a \setminus \implies a \setminus \div 1 = a$
May 4, 2018
$0.12$
Explanation:
Anything divided by $1$ will equal to itself, because multiply anything by $1$ will give out the exact same result.
$\therefore 0.12 \div 1 = 0.12$
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2019-01-20 01:24:38
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https://math.stackexchange.com/questions/2819667/expected-length-of-longest-stick
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Expected length of longest stick
The problem is the same as here.
A stick of 1m is divided into three pieces by two random points. Find the average length of the largest segment.
I tried solving it in a different way, and the logic seems fine, however I get a different result to $\frac{11}{18}$.
Here is my solution. Please let me know what I did wrong.
Let $X$ be the length of the stick from the beginning to the first cut. $Y$ be the length of the stick between the first and second cut and $1-X-Y$ the length between the second cut and the end of the stick.
We want to find the CDF of the following random variable: $Z=\max(X,Y,1-X-Y)$. (I believe that if anything is wrong, this might be it).
$$\begin{split} F_Z(z) = P(Z\leq z) & = P(\max(X,Y,1-X-Y) \leq z)\\ & = P(X\leq z, Y\leq z, 1-X-Y\leq z)\\ &= P(1-Y-z\leq X \leq z, Y\leq z) \end{split}$$
Since we have $1-Y-z\leq z$ we deduce that $Y\geq 1-2z$. Hence: $$\begin{split} F_Z(z) &= \int_{1-2z}^z\int_{1-y-z}^z 1 dx dy = \int_{1-2z}^z (z-1+y+z) dy\\ &= (2z-1)(z-1+2z) + \left. \frac{y^2}{2}\right|_{y=1-2z}^{y=z} \\ &=(2z-1)(3z-1) + \frac{1}{2}(z^2- (2z-1)^2) \\ & = (2z-1)(3z-1) +\frac{1}{2}(-3z^2 + 4z -1) \\ & = \frac{1}{2}(3z-1)^2 \end{split}$$ Now, the pdf of $Z$ is : $$f_Z(z) = \frac{d}{dz}F_Z(z) = 9z-3$$
And now, in order to find the expected value of the largest length, we need to integrate over $(\frac{1}{3},1)$ as the largest piece needs to be greater than $\frac{1}{3}$. Hence
$$\begin{split} E[Z] = \int_{\frac{1}{3}}^{1} z f_Z(z) dz = \int_{\frac{1}{3}}^{1} z (9z-3) dz = \frac{14}{9} \end{split}$$ The result is obviously wrong as it needs to be something between $0$ and $1$, however after going over the solution multiple times, and checking the calculations with Wolfram, I cannot seem to figure out what went wrong.
• Offhand, it looks like your setup allows $1-X-Y$ to be negative. – Barry Cipra Jun 14 '18 at 15:54
• What are $X,Y$ supposed to represent? If they're the lengths of two of the parts, those are not uniformly distributed. I would set $X,Y$ to the coordinates of the two points, and then find the expectation of $\max(\min(X,Y), |X-Y|, 1-\max(X,Y))$. – Daniel Schepler Jun 14 '18 at 15:54
• Note that $F_Z(1)=\frac 12\times 4=2>1$. For that matter $F_Z(0)=\frac 12$ which is already absurd. As a suggestion, walk through the calculation of $F_Z(0)$ to see where you go awry. – lulu Jun 14 '18 at 15:55
• @DanielSchepler $X$ is the length of the segment from $0$ to the first cut, and $Y$ is the length from the first cut to the second cut. – Andrei Crisan Jun 14 '18 at 15:58
• @lulu we must have $z\geq \frac{1}{3}$, as it is the largest part – Andrei Crisan Jun 14 '18 at 16:01
This is just a suggestion not an answer but could you try solving it keeping the distance of first division as x and second as y making the lengths of the segments as x, y-x ,1-y
• This makes a lot of sense. – Andrei Crisan Jun 14 '18 at 15:57
The first integral is wrong, because it assumes that $X,Y$ are uniform and independent on $[0,1]^2$. They are not (for one thing, $X \le Y$).
Here is how I would do it.
Lets define $x$ to be the short stick, $y$ to be the medium stick and $z$ to be the long stick.
$x\le y\le z\\ z = 1-x-y\\ x\le y \le \frac {1-x}{2}\\ x\le \frac 13$
$$\bar z = \frac {\displaystyle\int_0^\frac 13\int_x^{\frac {1-x}{2}} 1-x-y\ dy\ dx}{\displaystyle\int_0^\frac 13\int_x^{\frac {1-x}{2}} 1\ dy\ dx}$$
• Would you care to explain the numerator and denominator of your fraction please? – Andrei Crisan Jun 15 '18 at 10:07
• I understand the surface over which you are integrating, however I do not get the existence of the $dz$ terms, given that you only have a double integral. The bottom would simply be the CDF of the $\min(X,Y)$? – Andrei Crisan Jun 15 '18 at 10:13
• Thanks, only integrating over 2 dimensions (and fixed above). $z = 1-x-y$ The numerator gives the average value of z per unit of surface area, and the denominator is the surface such that $x<y<z.$ – Doug M Jun 15 '18 at 16:10
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2019-12-13 08:32:23
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http://www.hrwiki.org/w/index.php?title=HRWiki:Sandbox&diff=prev&oldid=685887
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HRWiki:Sandbox
(Difference between revisions)
Revision as of 02:48, 13 November 2009 (edit) (add disclaimer or not)← Older edit Revision as of 03:24, 13 November 2009 (edit) (undo)Newer edit → Line 3: Line 3: Really? I'll have to check that out tonight, I love that song Really? I'll have to check that out tonight, I love that song :How did you realize that? That's awesome! {{User:MichaelXX2/sig}} 02:48, 13 November 2009 (UTC) :How did you realize that? That's awesome! {{User:MichaelXX2/sig}} 02:48, 13 November 2009 (UTC) - + ::Some People can count the time signature in songs, for example, I counted it in Metallicas nothing else matters, listen to the rhythm Guitar part, count it. you will see, or more hear, it is in 3/4. [[User:McArbys|The McArby!]] $Lion's Mouth+Decemberween present \div peer pressure \times height =$TRIPLE SALCHOW! $Lion's Mouth+Decemberween present \div peer pressure \times height =$TRIPLE SALCHOW!
Revision as of 03:24, 13 November 2009
This probably doesn't mean anything to you, but I just realized that "Sgt. Peppers Lonely Hearts Club Band (song)" by The Beatles is in $2 \over 4$ time.
Really? I'll have to check that out tonight, I love that song
How did you realize that? That's awesome! 02:48, 13 November 2009 (UTC)
Some People can count the time signature in songs, for example, I counted it in Metallicas nothing else matters, listen to the rhythm Guitar part, count it. you will see, or more hear, it is in 3/4. The McArby!
$Lion's Mouth+Decemberween present \div peer pressure \times height =$TRIPLE SALCHOW!
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2019-09-17 15:12:15
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https://zbmath.org/authors/?q=ai%3Amccullough.darryl
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McCullough, Darryl
Compute Distance To:
Author ID: mccullough.darryl Published as: McCullough, Darryl; McCullough, D. External Links: MGP
Documents Indexed: 67 Publications since 1980, including 5 Books Reviewing Activity: 202 Reviews Co-Authors: 20 Co-Authors with 44 Joint Publications 273 Co-Co-Authors
all top 5
Co-Authors
23 single-authored 8 Cho, Sangbum 8 Miller, Andy 6 Kalliongis, John E. 5 Hong, Sungbok 3 Rubin, Leonard Roy 3 Wanderley, Marcus 2 Canary, Richard D. 2 Feighn, Mark E. 2 Hatcher, Allen Edward 2 Zimmermann, Bruno P. 1 Aebischer, Beat 1 Anderson, James W. 1 Costa Gonzáles, Antonio Félix 1 Hendriks, Harrie 1 Johnson, Jesse Edward 1 Rajeevsarathy, Kashyap 1 Rubinstein, J. Hyam 1 Seo, Arim 1 Soma, Teruhiko 1 Wade, Elizabeth
all top 5
Serials
5 Proceedings of the American Mathematical Society 5 Topology and its Applications 4 Journal of Pure and Applied Algebra 3 Indiana University Mathematics Journal 3 Journal of the London Mathematical Society. Second Series 3 Memoirs of the American Mathematical Society 3 Pacific Journal of Mathematics 3 Transactions of the American Mathematical Society 2 American Journal of Mathematics 2 Fundamenta Mathematicae 2 Journal of Differential Geometry 2 Proceedings of the London Mathematical Society. Third Series 2 Topology 2 Geometry & Topology 2 Algebraic & Geometric Topology 1 Communications in Algebra 1 Israel Journal of Mathematics 1 Mathematics Magazine 1 Archiv der Mathematik 1 Duke Mathematical Journal 1 Geometriae Dedicata 1 Glasgow Mathematical Journal 1 Illinois Journal of Mathematics 1 Journal of the Mathematical Society of Japan 1 Journal für die Reine und Angewandte Mathematik 1 Mathematische Zeitschrift 1 Michigan Mathematical Journal 1 The Quarterly Journal of Mathematics. Oxford Second Series 1 Tôhoku Mathematical Journal. Second Series 1 Bulletin of the American Mathematical Society. New Series 1 Annals of Mathematics. Second Series 1 Lecture Notes in Mathematics 1 Lecture Notes Series, Seoul 1 The College Mathematics Journal
all top 5
Fields
58 Manifolds and cell complexes (57-XX) 18 Group theory and generalizations (20-XX) 14 Algebraic topology (55-XX) 8 Functions of a complex variable (30-XX) 8 Global analysis, analysis on manifolds (58-XX) 3 General topology (54-XX) 2 Number theory (11-XX) 1 Topological groups, Lie groups (22-XX)
Citations contained in zbMATH Open
58 Publications have been cited 427 times in 293 Documents Cited by Year
Virtually geometrically finite mapping class groups of 3-manifolds. Zbl 0721.57008
McCullough, Darryl
1991
Compact submanifolds of 3-manifolds with boundary. Zbl 0628.57008
McCullough, Darryl
1986
Homotopy equivalences of 3-manifolds and deformation theory of Kleinian groups. Zbl 1062.57027
Canary, Richard D.; McCullough, Darryl
2004
Group actions on handlebodies. Zbl 0638.57017
McCullough, Darryl; Miller, Andy; Zimmermann, Bruno
1989
The genus 2 Torelli group is not finitely generated. Zbl 0579.57007
McCullough, Darryl; Miller, Andy
1986
Homeomorphisms of 3-manifolds with compressible boundary. Zbl 0602.57011
McCullough, Darryl; Miller, Andy
1986
Minimal genus of abelian actions on Klein surfaces with boundary. Zbl 0755.57005
McCullough, Darryl
1990
Isometries of elliptic 3-manifolds. Zbl 1012.57023
McCullough, Darryl
2002
Roots of Dehn twists. Zbl 1219.57019
McCullough, Darryl; Rajeevsarathy, Kashyap
2011
The tree of knot tunnels. Zbl 1191.57005
Cho, Sangbum; McCullough, Darryl
2009
The topology of deformation spaces of Kleinian groups. Zbl 0976.57016
Anderson, James W.; Canary, Richard D.; McCullough, Darryl
2000
Symmetric automorphisms of free products. Zbl 0860.20029
McCullough, Darryl; Miller, Andy
1996
Uniqueness of cores of noncompact 3-manifolds. Zbl 0556.57009
McCullough, D.; Miller, A.; Swarup, G. A.
1985
Finiteness of classifying spaces of relative diffeomorphism groups of 3-manifolds. Zbl 0885.57008
Hatcher, Allen; McCullough, Darryl
1997
The stable genus increment for group actions on closed 2-manifolds. Zbl 0760.57004
McCullough, Darryl; Miller, Andy
1992
Cabling sequences of tunnels of torus knots. Zbl 1170.57005
Cho, Sangbum; McCullough, Darryl
2009
Tunnel leveling, depth, and bridge numbers. Zbl 1210.57004
Cho, Sangbum; McCullough, Darryl
2011
Automorphisms of punctured-surface bundles. Zbl 0607.57009
McCullough, Darryl
1987
Constructing knot tunnels using giant steps. Zbl 1192.57004
Cho, Sangbum; McCullough, Darryl
2010
Diffeomorphisms of elliptic 3-manifolds. Zbl 1262.57001
Hong, Sungbok; Kalliongis, John; McCullough, Darryl; Rubinstein, J. Hyam
2012
$$\pi_1$$-injective mappings of compact 3-manifolds. Zbl 0616.57004
Kalliongis, John; McCullough, Darryl
1986
On the diffeomorphism group of a reducible 3-manifold. Zbl 0624.57014
Hendriks, Harrie; McCullough, Darryl
1987
Mappings or reducible 3-manifolds. Zbl 0637.57009
McCullough, Darryl
1986
Twist groups of compact 3-manifolds. Zbl 0579.57010
McCullough, Darryl
1985
Nielsen equivalence of generating pairs of $$\mathrm{SL}(2,q)$$. Zbl 1284.20031
McCullough, Darryl; Wanderley, Marcus
2013
Homeomorphisms which are Dehn twists on the boundary. Zbl 1135.57011
McCullough, Darryl
2006
Intersections of separators and essential submanifolds of $$I^ N$$. Zbl 0552.55003
McCullough, Darryl; Rubin, Leonard R.
1983
Some m-dimensional compacta admitting a dense set of imbeddings into $$R^{2m}$$. Zbl 0715.54026
McCullough, Darryl; Rubin, Leonard R.
1989
Height and excess of Pythagorean triples. Zbl 1293.11055
McCullough, Darryl
2005
Manifold covers of 3-orbifolds with geometric pieces. Zbl 0673.57014
McCullough, Darryl; Miller, Andy
1989
Orientation-reversing involutions on handlebodies. Zbl 0861.57019
Kalliongis, John; McCullough, Darryl
1996
Isotopies of 3-manifolds. Zbl 0869.57010
Kalliongis, John; McCullough, Darryl
1996
Free actions on handlebodies. Zbl 1022.57010
McCullough, Darryl; Wanderley, Marcus
2003
Homeotopy groups of irreducible 3-manifolds which may contain two-sided projective planes. Zbl 0769.57013
Kalliongis, John; McCullough, Darryl
1992
Group actions on nonclosed 2-manifolds. Zbl 0704.57005
McCullough, Darryl; Miller, Andy; Zimmermann, Bruno
1990
Connected sums of aspherical manifolds. Zbl 0427.55004
McCullough, Darryl
1981
Semisimple tunnels. Zbl 1248.57005
Cho, Sangbum; McCullough, Darryl
2012
Topological and algebraic automorphisms of 3-manifolds. Zbl 0713.57009
McCullough, Darryl
1990
Mapping class groups of 3-manifolds, then and now. Zbl 1283.57023
Hong, Sungbok; McCullough, Darryl
2013
Finite aspherical complexes with infinitely-generated groups of self- homotopy-equivalences. Zbl 0434.57011
McCullough, Darryl
1980
Finiteness conditions for 3-manifolds with boundary. Zbl 0642.57001
Feighn, Mark; McCullough, Darryl
1987
Virtual cohomological dimension of mapping class groups of 3-manifolds. Zbl 0657.57007
McCullough, Darryl
1988
Maps inducing isomorphisms on fundamental groups of compact 3-manifolds. Zbl 0585.57010
Kalliongis, John; McCullough, Darryl
1987
The space of Heegaard splittings. Zbl 1281.57008
Johnson, Jesse; McCullough, Darryl
2013
Recursive enumeration of Pythagorean triples. Zbl 1278.11039
2003
Arc distance equals level number. Zbl 1219.57004
Cho, Sangbum; McCullough, Darryl; Seo, Arim
2009
Homotopy equivalences of punctured manifolds. Zbl 0528.57028
McCullough, Darryl
1982
Writing elements of $$\mathrm{PSL}(2,q)$$ as commutators. Zbl 1214.20046
McCullough, Darryl; Wanderley, Marcus
2011
Weak concentration points for Möbius groups. Zbl 0805.20041
McCullough, Darryl
1994
Recurrent geodesics and controlled concentration points. Zbl 0810.30034
Aebischer, Beat; Hong, Sungbok; McCullough, Darryl
1994
Concentration points for Fuchsian groups. Zbl 0959.20046
Hong, Sungbok; McCullough, Darryl
2000
3-Manifolds and their mappings. Zbl 0874.57015
McCullough, Darryl
1995
The group of homotopy equivalences for a connected sum of closed aspherical manifolds. Zbl 0432.55004
McCullough, Darryl
1981
Homotopy groups of the space of self-homotopy-equivalences. Zbl 0471.55004
McCullough, Darryl
1981
Errata: The group of homotopy equivalences for a connected sum of closed aspherical manifolds. Zbl 0562.55007
McCullough, Darryl
1985
Middle tunnels by splitting. Zbl 1262.57010
Cho, Sangbum; McCullough, Darryl
2012
Iterated splitting and the classification of knot tunnels. Zbl 1270.57019
Cho, Sangbum; McCullough, Darryl
2013
Orientation-reversing free actions on handlebodies. Zbl 1085.57015
Costa, Antonio F.; McCullough, Darryl
2006
Nielsen equivalence of generating pairs of $$\mathrm{SL}(2,q)$$. Zbl 1284.20031
McCullough, Darryl; Wanderley, Marcus
2013
Mapping class groups of 3-manifolds, then and now. Zbl 1283.57023
Hong, Sungbok; McCullough, Darryl
2013
The space of Heegaard splittings. Zbl 1281.57008
Johnson, Jesse; McCullough, Darryl
2013
Iterated splitting and the classification of knot tunnels. Zbl 1270.57019
Cho, Sangbum; McCullough, Darryl
2013
Diffeomorphisms of elliptic 3-manifolds. Zbl 1262.57001
Hong, Sungbok; Kalliongis, John; McCullough, Darryl; Rubinstein, J. Hyam
2012
Semisimple tunnels. Zbl 1248.57005
Cho, Sangbum; McCullough, Darryl
2012
Middle tunnels by splitting. Zbl 1262.57010
Cho, Sangbum; McCullough, Darryl
2012
Roots of Dehn twists. Zbl 1219.57019
McCullough, Darryl; Rajeevsarathy, Kashyap
2011
Tunnel leveling, depth, and bridge numbers. Zbl 1210.57004
Cho, Sangbum; McCullough, Darryl
2011
Writing elements of $$\mathrm{PSL}(2,q)$$ as commutators. Zbl 1214.20046
McCullough, Darryl; Wanderley, Marcus
2011
Constructing knot tunnels using giant steps. Zbl 1192.57004
Cho, Sangbum; McCullough, Darryl
2010
The tree of knot tunnels. Zbl 1191.57005
Cho, Sangbum; McCullough, Darryl
2009
Cabling sequences of tunnels of torus knots. Zbl 1170.57005
Cho, Sangbum; McCullough, Darryl
2009
Arc distance equals level number. Zbl 1219.57004
Cho, Sangbum; McCullough, Darryl; Seo, Arim
2009
Homeomorphisms which are Dehn twists on the boundary. Zbl 1135.57011
McCullough, Darryl
2006
Orientation-reversing free actions on handlebodies. Zbl 1085.57015
Costa, Antonio F.; McCullough, Darryl
2006
Height and excess of Pythagorean triples. Zbl 1293.11055
McCullough, Darryl
2005
Homotopy equivalences of 3-manifolds and deformation theory of Kleinian groups. Zbl 1062.57027
Canary, Richard D.; McCullough, Darryl
2004
Free actions on handlebodies. Zbl 1022.57010
McCullough, Darryl; Wanderley, Marcus
2003
Recursive enumeration of Pythagorean triples. Zbl 1278.11039
2003
Isometries of elliptic 3-manifolds. Zbl 1012.57023
McCullough, Darryl
2002
The topology of deformation spaces of Kleinian groups. Zbl 0976.57016
Anderson, James W.; Canary, Richard D.; McCullough, Darryl
2000
Concentration points for Fuchsian groups. Zbl 0959.20046
Hong, Sungbok; McCullough, Darryl
2000
Finiteness of classifying spaces of relative diffeomorphism groups of 3-manifolds. Zbl 0885.57008
Hatcher, Allen; McCullough, Darryl
1997
Symmetric automorphisms of free products. Zbl 0860.20029
McCullough, Darryl; Miller, Andy
1996
Orientation-reversing involutions on handlebodies. Zbl 0861.57019
Kalliongis, John; McCullough, Darryl
1996
Isotopies of 3-manifolds. Zbl 0869.57010
Kalliongis, John; McCullough, Darryl
1996
3-Manifolds and their mappings. Zbl 0874.57015
McCullough, Darryl
1995
Weak concentration points for Möbius groups. Zbl 0805.20041
McCullough, Darryl
1994
Recurrent geodesics and controlled concentration points. Zbl 0810.30034
Aebischer, Beat; Hong, Sungbok; McCullough, Darryl
1994
The stable genus increment for group actions on closed 2-manifolds. Zbl 0760.57004
McCullough, Darryl; Miller, Andy
1992
Homeotopy groups of irreducible 3-manifolds which may contain two-sided projective planes. Zbl 0769.57013
Kalliongis, John; McCullough, Darryl
1992
Virtually geometrically finite mapping class groups of 3-manifolds. Zbl 0721.57008
McCullough, Darryl
1991
Minimal genus of abelian actions on Klein surfaces with boundary. Zbl 0755.57005
McCullough, Darryl
1990
Group actions on nonclosed 2-manifolds. Zbl 0704.57005
McCullough, Darryl; Miller, Andy; Zimmermann, Bruno
1990
Topological and algebraic automorphisms of 3-manifolds. Zbl 0713.57009
McCullough, Darryl
1990
Group actions on handlebodies. Zbl 0638.57017
McCullough, Darryl; Miller, Andy; Zimmermann, Bruno
1989
Some m-dimensional compacta admitting a dense set of imbeddings into $$R^{2m}$$. Zbl 0715.54026
McCullough, Darryl; Rubin, Leonard R.
1989
Manifold covers of 3-orbifolds with geometric pieces. Zbl 0673.57014
McCullough, Darryl; Miller, Andy
1989
Virtual cohomological dimension of mapping class groups of 3-manifolds. Zbl 0657.57007
McCullough, Darryl
1988
Automorphisms of punctured-surface bundles. Zbl 0607.57009
McCullough, Darryl
1987
On the diffeomorphism group of a reducible 3-manifold. Zbl 0624.57014
Hendriks, Harrie; McCullough, Darryl
1987
Finiteness conditions for 3-manifolds with boundary. Zbl 0642.57001
Feighn, Mark; McCullough, Darryl
1987
Maps inducing isomorphisms on fundamental groups of compact 3-manifolds. Zbl 0585.57010
Kalliongis, John; McCullough, Darryl
1987
Compact submanifolds of 3-manifolds with boundary. Zbl 0628.57008
McCullough, Darryl
1986
The genus 2 Torelli group is not finitely generated. Zbl 0579.57007
McCullough, Darryl; Miller, Andy
1986
Homeomorphisms of 3-manifolds with compressible boundary. Zbl 0602.57011
McCullough, Darryl; Miller, Andy
1986
$$\pi_1$$-injective mappings of compact 3-manifolds. Zbl 0616.57004
Kalliongis, John; McCullough, Darryl
1986
Mappings or reducible 3-manifolds. Zbl 0637.57009
McCullough, Darryl
1986
Uniqueness of cores of noncompact 3-manifolds. Zbl 0556.57009
McCullough, D.; Miller, A.; Swarup, G. A.
1985
Twist groups of compact 3-manifolds. Zbl 0579.57010
McCullough, Darryl
1985
Errata: The group of homotopy equivalences for a connected sum of closed aspherical manifolds. Zbl 0562.55007
McCullough, Darryl
1985
Intersections of separators and essential submanifolds of $$I^ N$$. Zbl 0552.55003
McCullough, Darryl; Rubin, Leonard R.
1983
Homotopy equivalences of punctured manifolds. Zbl 0528.57028
McCullough, Darryl
1982
Connected sums of aspherical manifolds. Zbl 0427.55004
McCullough, Darryl
1981
The group of homotopy equivalences for a connected sum of closed aspherical manifolds. Zbl 0432.55004
McCullough, Darryl
1981
Homotopy groups of the space of self-homotopy-equivalences. Zbl 0471.55004
McCullough, Darryl
1981
Finite aspherical complexes with infinitely-generated groups of self- homotopy-equivalences. Zbl 0434.57011
McCullough, Darryl
1980
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Cited by 287 Authors
23 McCullough, Darryl 15 Zimmermann, Bruno P. 13 Cho, Sangbum 10 Kalliongis, John E. 9 Koda, Yuya 9 May, Coy L. 9 Minsky, Yair N. 9 Ohshika, Ken’ichi 9 Rajeevsarathy, Kashyap 8 Putman, Andrew 8 Wang, Shi Cheng 7 Canary, Richard D. 6 Brock, Jeffrey F. 6 Miller, Andy 5 Bromberg, Kenneth W. 5 Wang, Chao 5 Zhang, Yi Mu 4 Dranishnikov, Alexander Nikolaevich 4 Farb, Benson 4 Hamenstädt, Ursula 4 Hensel, Sebastian Wolfgang 4 Jensen, Craig A. 4 Kim, Jungsoo 4 Margalit, Dan 4 Schleimer, Saul 3 Brendle, Tara E. 3 Dhanwani, Neeraj K. 3 E., Qiang 3 Giulini, Domenico J. W. 3 Hidalgo, Rubén Antonio 3 Lecuire, Cyril 3 Lee, Michelle 3 Maclachlan, Colin 3 Magid, Aaron D. 3 McCammond, Jon 3 Meier, John 3 Ohashi, Ryo 3 Reid, Alan W. 3 Souto, Juan 2 Agol, Ian 2 Anderson, James W. 2 Balm, Cheryl Jaeger 2 Bestvina, Mladen 2 Biss, Daniel K. 2 Bujalance, Emilio 2 Bux, Kai-Uwe 2 Charitos, Charalampos 2 Chen, Binglong 2 Dicks, Warren 2 Ershov, Mikhail V. 2 Etayo Gordejuela, José Javier 2 Evans, Richard Allen 2 Gaifullin, Alexander A. 2 Gogolev, Andrey 2 Gromadzki, Grzegorz 2 Hatcher, Allen Edward 2 Hirose, Susumu 2 Huang, Xiantao 2 Ito, Kentaro 2 Jeon, Woojin 2 Jiménez Rolland, Rita 2 Kent, Autumn Exum 2 Kim, Inkang 2 Leininger, Christopher J. 2 Lock, Michael T. 2 Mann, Kathryn 2 Martinez, Ernesto C. 2 Mecchia, Mattia 2 Mj, Mahan 2 Monden, Naoyuki 2 Moriah, Yoav 2 Namazi, Hossein 2 Nariman, Sam 2 Neofytidis, Christoforos 2 Papadoperakis, Ioannis 2 Parlak, Anna 2 Parsad, Shiv 2 Repovš, Dušan D. 2 Shchepin, Evgenij V. 2 Stukow, Michał 2 Sun, Hongbin 2 Taylor, Edward C. 2 Taylor, Scott A. 2 Tsapogas, Georgios 2 Viaclovsky, Jeff A. 2 Wahl, Nathalie 2 Wanderley, Marcus 2 Zimmerman, Jay J. 1 Abikoff, William 1 Aebischer, Beat 1 Agarwal, Nikita 1 Agler, Jim 1 Akbulut, Selman 1 Akita, Toshiyuki 1 Arvanitakis, Alex S. 1 Aßelmeyer-Maluga, Torsten 1 Atkinson, Christopher K. 1 Austin, Jathan W. 1 Babajee, Diyashvir Kreetee Rajiv 1 Bachman, David ...and 187 more Authors
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Cited in 95 Serials
32 Topology and its Applications 16 Geometry & Topology 15 Transactions of the American Mathematical Society 14 Algebraic & Geometric Topology 13 Duke Mathematical Journal 11 Proceedings of the American Mathematical Society 11 Journal of Knot Theory and its Ramifications 10 Geometriae Dedicata 10 Mathematische Annalen 8 Mathematical Proceedings of the Cambridge Philosophical Society 8 Glasgow Mathematical Journal 6 Journal of Pure and Applied Algebra 5 Inventiones Mathematicae 5 Journal of Algebra 5 Journal of the American Mathematical Society 5 Groups, Geometry, and Dynamics 4 Rocky Mountain Journal of Mathematics 4 Mathematische Zeitschrift 4 Rendiconti dell’Istituto di Matematica dell’Università di Trieste 3 Archiv der Mathematik 3 Journal of the Mathematical Society of Japan 3 Michigan Mathematical Journal 3 Osaka Journal of Mathematics 3 Geometric and Functional Analysis. GAFA 3 Annals of Mathematics. Second Series 3 Journal of Algebra and its Applications 2 Bulletin of the Australian Mathematical Society 2 Communications in Mathematical Physics 2 General Relativity and Gravitation 2 Israel Journal of Mathematics 2 Commentarii Mathematici Helvetici 2 Monatshefte für Mathematik 2 Siberian Mathematical Journal 2 Tôhoku Mathematical Journal. Second Series 2 Ergodic Theory and Dynamical Systems 2 Sugaku Expositions 2 Conformal Geometry and Dynamics 2 Journal of the European Mathematical Society (JEMS) 2 Comptes Rendus. Mathématique. Académie des Sciences, Paris 2 Journal of Topology 2 Journal of Topology and Analysis 2 Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A: Matemáticas. RACSAM 1 International Journal of Modern Physics A 1 Houston Journal of Mathematics 1 International Journal of Mathematical Education in Science and Technology 1 International Journal of Theoretical Physics 1 Journal d’Analyse Mathématique 1 Nuclear Physics. B 1 Periodica Mathematica Hungarica 1 Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 1 Acta Mathematica 1 Advances in Mathematics 1 Annales de l’Institut Fourier 1 Annales Scientifiques de l’École Normale Supérieure. Quatrième Série 1 Canadian Mathematical Bulletin 1 Collectanea Mathematica 1 Commentationes Mathematicae Universitatis Carolinae 1 Compositio Mathematica 1 Illinois Journal of Mathematics 1 Publications Mathématiques 1 Journal of Combinatorial Theory. Series A 1 Journal of Number Theory 1 Journal für die Reine und Angewandte Mathematik 1 Kodai Mathematical Journal 1 Kyungpook Mathematical Journal 1 Memoirs of the American Mathematical Society 1 Proceedings of the Edinburgh Mathematical Society. Series II 1 Proceedings of the Japan Academy. Series A 1 Chinese Annals of Mathematics. Series B 1 Revista Matemática Iberoamericana 1 Differential Geometry and its Applications 1 The Journal of Geometric Analysis 1 Linear Algebra and its Applications 1 Bulletin of the American Mathematical Society. New Series 1 Indagationes Mathematicae. New Series 1 Experimental Mathematics 1 Journal of Algebraic Combinatorics 1 Journal of Mathematical Sciences (New York) 1 Boletín de la Sociedad Matemática Mexicana. Third Series 1 Izvestiya: Mathematics 1 Discrete and Continuous Dynamical Systems 1 Transformation Groups 1 Annales Mathematicae Silesianae 1 Journal of the Australian Mathematical Society 1 Moscow Mathematical Journal 1 Missouri Journal of Mathematical Sciences 1 Proceedings of the Steklov Institute of Mathematics 1 Annali dell’Università di Ferrara. Sezione VII. Scienze Matematiche 1 Frontiers of Mathematics in China 1 Journal of Modern Dynamics 1 Journal of Homotopy and Related Structures 1 Science China. Mathematics 1 Forum of Mathematics, Pi 1 Journal of Mathematics 1 AIMS Mathematics
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Cited in 28 Fields
239 Manifolds and cell complexes (57-XX) 80 Group theory and generalizations (20-XX) 64 Functions of a complex variable (30-XX) 25 Algebraic topology (55-XX) 18 Differential geometry (53-XX) 11 Number theory (11-XX) 11 Dynamical systems and ergodic theory (37-XX) 11 Global analysis, analysis on manifolds (58-XX) 8 Algebraic geometry (14-XX) 7 Combinatorics (05-XX) 7 General topology (54-XX) 5 Several complex variables and analytic spaces (32-XX) 4 Topological groups, Lie groups (22-XX) 4 Relativity and gravitational theory (83-XX) 3 Quantum theory (81-XX) 2 $$K$$-theory (19-XX) 2 Geometry (51-XX) 1 Order, lattices, ordered algebraic structures (06-XX) 1 Field theory and polynomials (12-XX) 1 Linear and multilinear algebra; matrix theory (15-XX) 1 Nonassociative rings and algebras (17-XX) 1 Measure and integration (28-XX) 1 Operator theory (47-XX) 1 Calculus of variations and optimal control; optimization (49-XX) 1 Convex and discrete geometry (52-XX) 1 Probability theory and stochastic processes (60-XX) 1 Fluid mechanics (76-XX) 1 Mathematics education (97-XX)
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2022-10-05 10:08:26
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http://consorziovalledelbelice.it/stepper-motor-torque-equation.html
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# Stepper Motor Torque Equation
The number of coils will differ based on type of stepper motor , but for now just understand that in a stepper motor the rotor consists of metal poles and each pole will be attracted by a set of coil. Variation torque function of the speed The motor is supplied with a current about 0. When the motor is unexcited the permanent magnet, and hybrid stepper motor develops a Detent torque confining the rotation of the rotor. 28BYJ-48 5V Stepper Motor Components datasheet pdf data sheet FREE from Datasheet4U. Power is measured in watts (W) and is calculated as volts (V) × current (A). • Stepper motors. They draw the most current when they are doing no work at all. The VR and PM stepper motors are the most common type of stepper motors. These offer the advantage of faster and more precise operation compared to a conventional vacuum (pneumatic) operated system. When a stepper motor is accelerating, it has to produce torque to overcome its own rotor inertia and the mass of the load it is driving. At it’s rated current a stepper motor can be quite hot. 8mH ± 20%(1KHz)Frame Size: 57 x 57mmBody Length: 113mmWeight: 1. Torque curves may be extended to greater speeds if the stator poles can be reversed more quickly, the limiting factor being the winding inductance. The motor has no torque (this is why he mis. Stepper Motors. stepper motors as oppose to unipolar (i. Brushless DC motors are more efficient than brushed motor, as they are able to continuously achieve maximum rotational force/torque. MICROSTEPPING Single stepping, or turning a stepping motor at its rated step size, results in jerky movement. DC servo motors, however, have a higher torque *during rotation* than steppers and a much higher RPM. The L293D can provide bidirectional drive currents of up to 600-mA at voltages from 4. 5 V to 36 V while L293 can provide up to 1A at same voltages. 2A motor being driven at rated torque, and less issues. Stepper motors generate their highest torque when they are at 0 RPM. The 28BYJ-48 is a small stepper motor suitable for a large range of applications. Terminals 3 through 6 are where you connect the wires from the stepper motors to the stepper driver. Most stepper manufacturers publish torque/speed curves for specified bus voltage. Torque equation. Characteristics of DC generators Internal (E/Ia), external (V/I L) and open circuit characteristics. 4 we saw that for an excited rotor motor with rotor saliency, the reluctance torque acting alone caused the unloaded rotor to come to rest with the rotor direct axis aligned with stator m. These characteristics are depicted in two curves on the motor’s torque-speed chart, and it’s important to understand the differences between these curves and what each one means for the motor’s operation. Open Loop Positioning – Stepper motors move in quantified increments or steps. Multi-phase stepper motors with many phases tend to have much lower levels of vibration. 28BYJ-48 5V Stepper Motor Components datasheet pdf data sheet FREE from Datasheet4U. Describe the working of a 3 stack stepper motor having 12 poles in stator and in rotor. For an application operating at low speed, a stepper motor may provide a more economical solution. 9 degree increments. Each phase is composed of a single coil and a "center tap" or "common wire". Hybrid stepper motor. A step motor operated at a fixed voltage has a decreasing torque curve as the frequency or step rate increases. Torque Equation for Shunt Motor The shunt type diagram is explained below. In reality this system is damped, so the oscillation comes to an end after some time. But BLDC motors are better suited to controlling the force. These will directly… Read More ». Figure 4: Torque Deflection. motor torque/speed curve is a very good approximation. Sine commutation allows for almost fully resonance-free operation of the motor. + 3 x Nema23 2. I would assume the extruder motor is also 10 Ohms. In bipolar mode, the same motor has 800 gcm of torque. T inlb = torque (in lb f) P hp = horsepower delivered by the electric motor (hp) n = revolution per minute (rpm) Alternatively. In the conventional DC motor commutation takes place mechanically through the commutator-and-brush system. Driving the stepper motor in open loop is the reason we have stepper motors in the first place! Decreasing current while not running, however, is pretty much the right thing to do. It has two motor windings (phase A and B) that must be controlled. T = torque (ft lb f) Example - Moment created by a Rotating Motor. Stepper Motor Driver info. If the equation for the motor torque/speed curve is known, then equation [2] (with ωo=0) can be solved analitically for ω. Measure and record the motor voltage, V1, and currents, I1 and I2, and the developed starting torque. You can use them where you need high torque at high speed. If a stepper motor is operated no load over the entire frequency range, one or more natural oscillating resonance points may be detected, either audibly or by vibration sensors. Half Step Mode: This method combines One Phase On and Two Phase On modes to energize one phase, then two, then one, etc. The motor will have full rated torque. If a stepper motor is operated no load over the entire frequency range, one or more natural oscillating resonance points may be detected, either audibly or by vibration sensors. The mechanical power develops at the shaft of the DC motor is always less than the armature power due to friction and windage losses. 32 * (√mH inductance) = Power Supply Voltage If your motor has 2mH of inductance, the equation would look as follows. And with a stepper motor, holding the position of a structure such as a robot arm would require a relatively large and continuous current. This means motor torque is the inverse of motor speed. Over a motor's normal load range, the torque's slope is approximately linear or proportional to slip because the value of rotor resistance divided by slip, ′ / , dominates torque in a linear manner. The commutation of brushless motors. 2: A typical closed-loop control system for a three-phase BLDC motor includes a controller, driver, and power transistor half-bridge H. Detent torque is beneficial for stepping the motor. A brake torque is set on the stepper motor shaft. 2 V, allowing for a holding torque of 19 kg-cm. with the experimental data for the tested motor. I would assume the extruder motor is also 10 Ohms. It is quite common to have gear ratios of 40, 50, or 100. B=viscous friction (will be identified by experiment) T m =motor torque. This quantity does not change with the number of drive wheels. The bipolar motor produces more torque AN235 8/23 Doc ID 1679 Rev 2 3 The bipolar motor produces more torque The torque of the stepper motor is proportional to the magnetic field intensity of the stator windings, which is proportional to the number of turns and the current in the winding, so torque is proportional to N*I. Step angle of the stepper motor is defined as the angle traversed by the motor in one step. State and explain static and dynamic characteristics of a stepper motor. 5º Stepper Motors Page 19 Series 26M024B Holding Torque: mN•m/oz-in Unipolar 4. Stepper Motors - The stepper motor is an electromagnetic device that converts digital pulses into mechanical shaft rotation. which has negligible moment of inertia and elasticity. If you set the timer to overflow (like i did in my above code), all these equations will. For a particular DC Motor, the number of poles (P) and the number of Thus, from the above equation (5) it is clear that the torque produced in the armature is directly. autocommutation, torque ripple depends on the motor torque function vs rotor position and on the way the phases are energized. h-- holding torque of composite. Pull-out torque The stepper motor Pull-Out torque is measured by accelerating the motor to the desired speed and then increasing the torque loading until the motor stalls or "pulls Out of synchronism. NEMA 23 is a stepper motor with a 2. The equation above actually represents a linear motor, in adapting this to an angular rotating motor we consider the flux to be constant at its full value. Due to the high holding torque, a gearbox is not required in most cases. One is that high torque motors also have high rotor inertia, so by running the motor at very much less than its rated current you reduce the available acceleration. A stepper motor or step motor or stepping motor is a brushless DC electric motor that divides a full rotation into a number of equal steps. 2 Operational Principle of a VR Spherical Motor The VR spherical motor discussed here has a structure similar to that of a spherical stepper [6]. But in the running condition net running torque means only the hysteresis torque. Performance characteristic of stepper motors The torque Vs Speed graph is shown below Fig. 1 Introduction 7. The 5V version of the 20BYJ46 has a resistance of 60ohm. 9ohmsInductance: 3. This is a hard way of achieving exactly the principle of the DC motor. Holding Torque The amount of torque the motor produces at rest when it has rated current flowing through its windings. Torque Moment calculation For a movement, we have to perform a follow up of positions between a Start and a Stop position. Multi-phase stepper motors with many phases tend to have much lower levels of vibration. Solving Equation 3 for time gives the results shown in Equation 4. For an application operating at low speed, a stepper motor may provide a more economical solution. Theoretical Motor Regulation Using Motor Constants Equation 4b. A loss of 80% of the rated torque at 90% of the Stepper motors are better suited for lower acceleration, high holding-torque applications. CHARACTERISTICS OF STEPPER MOTOR Stepper motors are constant power devices. Because of the number of variables, the process is iterative. The difference Between Stepper motors, Stepper Motors, Servos, and RC Servos. “If you don’t need the speed, you can get much higher torque in the same envelope at lower speeds so there are a lot of advantages for the right application in. This information is intended to provide basic information to technicians about a kind of motor ( known as step motor , stepping motor or stepper motor ) widely applied in precision machines, the industry, in minilabs to develop photography, printers, in sewing machines, etc. Figure 6 : 35PP Speed : Dynamic torque. The mechanical power develops at the shaft of the DC motor is always less than the armature power due to friction and windage losses. With this implementation, the system is tuned to use only the necessary amount of current, decreasing motor vibrations. Use a 5V version when using the Arduino power supply. In this case, the variables in the line equation represent the following. The PUL signal is PWM with the duty cycle depends on the current required by the motor. It is quite common to have gear ratios of 40, 50, or 100. This is the absolute time. The motor's position can then be commanded to move and hold at one of these steps without any feedback sensor (an open-loop controller), as long as the motor is carefully sized to the application in respect to torque and speed. This ratio (measurement) determines how quickly a stepper motor can. The torque (T) developed by the motor is a function of the holding torque (T H. The Maximum Tractive Torque represents the maximum amount of torque that can be. And what are the main causes of stall in a stepper motor system? The stall condition is what happens when the load torque on a motor exceeds the motors pull-out torque. As motor speed increases, torque decreases. T inlb = torque (in lb f) P hp = horsepower delivered by the electric motor (hp) n = revolution per minute (rpm) Alternatively. Author: Reston Condit Microchip Technology Inc. Multi-phase stepper motors with many phases tend to have much lower levels of vibration. (4) Other type stepper motor •Single phase Stepper Motor •Disc Magnet(DM)Stepper motor •Claw Tooth or Can Stack permanent Magnet Stepper Motor 28. ABS melts at 105-120 °C but softens at 80 °C. The curve is then approximated by connecting these two points with a line, whose equation can be written in terms of torque or angular velocity as equations 3) and 4): The linear model of a D. Close suggestions Search Search. 1 Force production 6. Using this equation for the torque-speed curve, we can find the motor's torque at a given speed. Choosing a compatible step angle motor and micro-step driver is a critical part of this equation. The motor's position can then be commanded to move and hold at one of these steps without any position sensor for feedback (an open-loop controller), as long as the motor is carefully sized to the application in respect to torque and speed. If a screw of 4 mm is run at 1500 RPM, we obtain a displacement speed of 1500*4mm=6000mm/min or 6 m/min. Well, the stepper motor still has friction torque due to its bearings and it has a detent torque (in addition to other harmonic distortions). A more detailed discussion about stepper motors can be found in our dedicated article, but making a long story short, like any DC motor, these motors rotate when the coils are energized however, if the coils are continuously energized in the same way, the movement will stop when opposite magnetic poles are aligned, e. As we have seen that in half mode, the number of steps taken by the motor to complete one revolution gets doubled, so step angle reduces to half. The selection of a stepping motor is not just selecting a motor. 3 Torque balance equation 6. 1Background The stepper motor is a device generating a stepwise rotational motion, which has an advantage in applications where a precise position control is of importance [1]. The particularity of the ironless DC motor is such that the speed and torque function are linear. For example: a 3. In order to lift a 5-lb weight using a string on a 2-inch diameter spool, you will need to use a gear reduction of approximately 22 to 1; that is a secondary gear that has 22 times as many teeth on it as the existing gear. The main difference between this usage and the usual stepper usage is that in an FFB rig, the motor mostly never moves; it is mainly used to create torque to oppose the user’s own force. inductance motor). The Maximum Tractive Torque represents the maximum amount of torque that can be. With a 5mm steel under-carriage, 4mm heated bed as well as a 4mm removable top plate there is plenty of inertia to this assembly and the motor is pretty small. It has two motor windings (phase A and B) that must be controlled. Motor rated torque: 98. This will depend on the type of stepper motor you select. Practical implementation As a simple solution for autocommutation a stepper motor with encoder and a standard stepper. which has negligible moment of inertia and elasticity. Multi-phase stepper motors with many phases tend to have much lower levels of vibration. The Nema17 stepper motor is a stepper motor with an end face size of 1. Step motors must be carefully sized for the system in which they are intended. You can check that video out below. Stepper motors are so named because each pulse of electricity turns the motor one step. 8° step angle and 0. The formulation is discussed in conjunc tion with the goal of designing a spherical motor with uniform torque characteristics. 3A, 1288 oz-in holding torque, 1. − Calculation and reconstruction of torque-displacement curves. 55 A and 64 µpas/pas. Stepper Motor Driver info. The torque equation of a normal motor is given as Te α φ. " The torque T m is the motor torque, and the inertia J eff is the e ective inertia \felt" by the motor. 5A or less, it makes about the same torque as the 1. x-- equilibrium position. The electrical system consists of armature and field circuit but for analysis purpose, only field circuit is considered because the armature is excited by a constant voltage. Data Dosen Program Studi Agribisnis. NEMA 23 is a stepper motor with a 2. Shaft Torque of DC Motor. In the alternative case, a larger gear driving a smaller one will result in a faster spinning, but less torque output. The stepper motor is a two-phase permanent magnet motor with 7. Therefore we need Stepper Driver Module like A4988 or. at the motor drive shaft of the mechanism. Special Electrical Machines, Torque Equation of SyRM,electrical machines, electrical engineering, DC motor,DC generator, emf equation, advanced electrical ma. Theoretical Motor Regulation Using Motor Constants Equation 4b. This type of motor also produces a lot of torque given it size which is why it has found itself in a number of industrial applications. These offer the advantage of faster and more precise operation compared to a conventional vacuum (pneumatic) operated system. Stepper motor drivers can also use microstepping, which modulates the current through the coils. “If you don’t need the speed, you can get much higher torque in the same envelope at lower speeds so there are a lot of advantages for the right application in. The advantages of the Hybrid Stepper Motor are as follows:- The length of the step is smaller. These are the motor parameters that are needed: Motor voltage constant Ke(volts-sec/rad) Motor torque constant KT(lb-in/amp) Motor resistance Ra(ohms) Motor inductance La(Henries) Motor inertia Jm(lb-in-sec^2) Load inertia reflected to the motor armature shaft Jload(lb-in-sec^2) Total inertia=Jm+JloadJtotal(lb-in-sec^2). The torque generated by the stepping motor can then be written as the vector product of the resulting current vector is and the permanent magnet rotor flux R: Tmotor is ΦR (1) For a hybrid stepping motor, this equation can be rewritten as: Tmotor CT (δ). Torque curves may be extended to greater speeds if the stator poles can be reversed more quickly, the limiting factor being the winding inductance. Hey guys! I recently found this group and i want to share this topic. It is not necessary to use an encoder on a stepper motor unless you're concerned about the motor losing count of the steps over a long period of time or in high torque situations. Micro-stepping is not only a good way to reduce resonance, it can also be used to increase stepper motor positon accuracy. Thank you for downloading Introduction to Stepper Motors. (most motors exhibit maximum torque when stationary, however the torque of a motor when stationary 'holding torque' defines the ability of the motor to maintain a desired position while under external load). So the higher the applied voltage, the faster the current rises, the faster the motor can get to its next step, the faster it can accelerate/go, and the greater torque it can generate at any high speed. The motor torque required at constant speed is the sum of the torque needed to drive the load, the preload torque of the screw assembly, and the torque due to friction of the support bearings and seals. To resume, the stepper motors drivers dosen't comes calibrated. Each phase draws 2. Hybrid Stepper Motor. A more accurate calculation of power supply voltage is to find your motor’s inductance, and put it into the following equation. B=viscous friction (will be identified by experiment) T m =motor torque. The stepper motor Pull-Out torque is measured by accelerating the motor to the desired speed and then increasing the torque loading until the motor stalls or "pulls Out of synchronism" with the step frequency. For example, NEMA 17 is a stepper motor with a 1. Increasingly, speed control is being used with A. Motor Speed 0 20 40 60 80 100 120 0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 speed value is calculated using Equation 1 for each timer update. (I=V/R) (4. ) DC Drives – Torque Control: To control DC motors torque, a DC drive will regulate armature current. The stepper motor is usually designed as a torque motor with a limited speed range. L293D and L293 are dual H-bridge motor drivers. All stepper motor logic takes place within the TMC5160 itself – there’s no need for software when driving NEMA17 up to NEMA34 and bigger motors. 9° (100 – 400 steps per revolution). Your stepper motor's holding torque must equal or exceed this value in Nm. The servos with better torque must be preferred. 4 Kg m/sec2 x 15 cm x (1 m / 100 cm) = 1. A practical mathematical model for identification is derived. Motor—Speed RegulationTorQue and Speed Of a D. Stepper Motor Glossary of Terms. It is not necessary to use an encoder on a stepper motor unless you're concerned about the motor losing count of the steps over a long period of time or in high torque situations. We have a gr of 3:1. Example a motor that is rated at 6mH per phase 1000 * SQRT (0. (4) Other type stepper motor •Single phase Stepper Motor •Disc Magnet(DM)Stepper motor •Claw Tooth or Can Stack permanent Magnet Stepper Motor 28. Torque equation 29. Stepper motors are broadly available motors commonly used for positioning. When driving a stepper motor, regulating the current can work to the user ’s advantage in two ways. Motor speed-torque curves are essential when selecting and sizing servos. The rpm sensor. The speed control system is an electro-mechanical control system. AND, 8hp is about 6kW+, which is quite a lot of power to switch and control. The widespread. Here’s how to change the wiring in order to make it a bipolar stepper motor:. Torque in DC Motor This is an active graphic. [3] While they are more expensive, they do have a higher power density and with the appropriate drive electronics are often better suited to the application. RESIDUAL TORQUE The non-energized detent torque of a permanent magnet stepper motor is called residual torque. Stepper motors, with their ability to produce high torque at low speed while minimizing vibration, are ideal for applications requiring quick positioning over a short distance. Therefore, this mode provides twice the resolution. Do this many times for many mounting points then take average. As a result, less energy will be produced to move the rotor (i. As the motor tempera-ture increases and the curve becomes steeper, shaft speed will drop more for the same increase in load. (Of course you will have to use an extension cable between the motors and the stepper drivers. 5º Stepper Motors Page 19 Series 26M024B Holding Torque: mN•m/oz-in Unipolar 4. motor torque/speed curve is a very good approximation. Stepper-motor problems often begin when a customer approaches a stepper-motor supplier with insufficient or inaccurate data. The Wheel Torque calculated in Step Five is the total wheel torque. Keywords —Stepper motors, Permanent magnet motors, Finite-el ement method, Magnetostatics, Torque, Movement simulation, Position measurement, Velocity measurement. Mophorn Nema 17 Geared Stepper Motor 48mm Body 14:1 Planetary Gearbox Low Speed High Torque 3. 1 Force production 6. 3A Switching CNC Power Supply ; if you need a 110V ac fan. + 3 x Nema23 2. The transfer function from the input voltage to the resulting motor torque is found by combining equations (1. 44) Applying Equation 5. It is also called as torque. Jan 25, 2021 - All about Electrical and Electronics Engineering. Bipolar stepper motors can be driven with the A4988 driver. The larger the outer diameter of the motor, the greater the torque and the greater the maximum rated current passed. The output torque and power from a stepper motor are functions of the motor size, motor heat sinking, working duty cycle, motor winding, and the type of drive used. We sincerely hope to establish business relationships and cooperate with you. com Datasheet (data sheet) search for integrated circuits (ic), semiconductors and other electronic components such as resistors, capacitors, transistors and diodes. 0316 Vrad/s. , so that the motor moves in half step increments. It is not necessary to use an encoder on a stepper motor unless you're concerned about the motor losing count of the steps over a long period of time or in high torque situations. The motor torque Tm is proportional to the phase current. Figure 6 : 35PP Speed : Dynamic torque. c) Draw the free-body diagrams of the motor+pulley) system and the platform. Uploaded by. The Maximum Tractive Torque represents the maximum amount of torque that can be. The stepper motor Pull-Out torque is measured by accelerating the motor to the desired speed and then increasing the torque loading until the motor stalls or "pulls Out of synchronism" with the step frequency. For example: a 3. At low speeds the stepper motor can synchronise itself with an applied step frequency, and this Pull-In torque must overcome friction and inertia. Stepper Motors. stepper and servo control. The speed-torque characteristics of a hysteresis motor is shown below. We're using a shutter from Edmond optical. Figure 1: Speed reversal of a stepper motor the University of Utah. By energizing the coils of the motor in sequence through many of these steps, the direction of rotation, number of rotations, and exact position of the motor shaft can be easily controlled. RESIDUAL TORQUE The non-energized detent torque of a permanent magnet stepper motor is called residual torque. For example, a motor rated at 2 volts will perform best, without stalling or losing steps, with a 40 volt power supply. All stepper motor logic takes place within the TMC5160 itself – there’s no need for software when driving NEMA17 up to NEMA34 and bigger motors. Available in single and dual shaft, 1. A drive system consists of the 3-phase stepper motor and the appropri-ate drive. This means that the rotor’s stable stop position is in syn-chronization with the stator flux. To overcome the inductance and switch the windings quickly, one must increase the drive voltage. First two of the above equations determine the current in motor's windings (phase A and B), while the two latter correspond to Newton's 2nd Law of motion. 32 * (√mH inductance) = Power Supply Voltage If your motor has 2mH of inductance, the equation would look as follows. K t =motor torque constant=0. 3, the torque generation is. The motor was specified to have a max current of 350mA so that it could be driven easily with an Adafruit motor shield for Arduino (or other motor driver) and a wall adapter or lead-acid battery. Losses in a DC machine Copper losses, iron losses (core losses), mechanical losses, power flow diagram. Stepper Motor Glossary of Terms. 2 phase (4 wire) 📢 Make sure to read this before running any motor! Before running any BLDC motor with the Simple FOC library please make sure your hardware can handle the currents your motor requires. These peripherals include two on-chip comparators and an Enhanced Capture Compare PWM (ECCP) module. F²MC-16FX Family, Stepper Motor Controller www. Thus the load inertia may be reduced. A stepper motor or step motor or stepping motor is a brushless DC electric motor that divides a full rotation into a number of equal steps. This is determined by the ratio of inductance to resistance (L/R) of the motor and driver as illustrated in Fig 8-2 (a). Smaller step angles can be obtained trough. 81 times max value shown by the scale in kilograms. Figure 6 : 35PP Speed : Dynamic torque. Deciding wether to use a servo motor or stepper motor is based on the needed holding torque (steppers) versus torque while in motion (servo). Motor Speed 0 20 40 60 80 100 120 0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 speed value is calculated using Equation 1 for each timer update. Torque Moment calculation For a movement, we have to perform a follow up of positions between a Start and a Stop position. Highly recognized for High Performance and Quality Products, MOONS' offers a vast array of motion control & smart LED control products or solutions. Typical applications include. We sincerely hope to establish business relationships and cooperate with you. The motor has no torque (this is why he mis. motor torque/speed curve is a very good approximation. com Datasheet (data sheet) search for integrated circuits (ic), semiconductors and other electronic components such as resistors, capacitors, transistors and diodes. 08 ftlbs torque add a driving pulley 3 inch and a driven pulley 5 inch to this and through reduction we will increase torque. This type of motor provides a combination of the best features available on both the PM and VR types of stepper motors. The basic motor equation is T=Kt*I, where T is torque, Kt is the motor torque constant, and I is the current. The voltage equation for a motor in motion is: V = KEώ + iR + L (di/dt) A PWM driver will increase the voltage applied to the stepper motor to keep the current and torque constant. 73 Volts MAXIMUM. The load inertia has been reduced by a factor of n2 due to the gear train. Why a stepper motor? The stepper motors are used to precisely control movements, whether in cars, robots, 3D printers or CNC machines. As the motor tempera-ture increases and the curve becomes steeper, shaft speed will drop more for the same increase in load. The formula: Resolution is steps per inch or steps per milimeter I will go over this using steps/inch: steps = motor steps x driver microstepping inch = the amount of travel with one full stepper motor rotation In the case of our 1/2" 5 start 10 TPI lead screw, the axis will travel. If you don’t need LEDs, you only need 2 digital I/Os out of available 14. The name that you give to the stepper motor will be used later to set the speed, position, and acceleration for that particular motor. Why using a stepper motor? Stepper motors do not operate as DC or brushless motors. You need to calculate requirement of torque in your application and than choose best motor for that. 6 inches long, has a phase current of 1. Stepper Motor Page 16 Quick Reference Chart Stepper Motors Page 17 Series 15M020D Holding Torque: mN•m/oz-in 3. I have two printers, the aluminium geeetech i3 and the old trustworthy wooden makerbot replicator. P - power (W) n m. 83 Nm The motor toque T in N. at the motor drive shaft of the mechanism. Highly recognized for High Performance and Quality Products, MOONS' offers a vast array of motion control & smart LED control products or solutions. 0mH ±20% Rated Current/phase 2. The Variable Reluctance Stepper Motor Learners solve word problems involving basic mechanical principles such as torque, horsepower, work, and power. Motor Principle—Comparison of Generator and Motor Ac- lion—Significance of the Bad emf—Voltage EQuation of a Motor—Conditions for Maximum ture TorQue Of a Motor—Shaft Of a D. Picture a pulley attached to the stepper motor’s shaft that’s 1. 68A motor only makes rated torque at 1. Driving a stepper motor is a bit more complicated than driving a regular brushed DC motor. In induction motor too we have torque to be function of , the mutual inductance between the rotor and the stator windings. 8° step angle (200 steps/revolution). Unlike DC motors, applying current to the motor windings generates a torque which resists rotation (the holding torque). Describe the working of a 3 stack stepper motor having 12 poles in stator and in rotor. The L293D can provide bidirectional drive currents of up to 600-mA at voltages from 4. Thank you for downloading Introduction to Stepper Motors. 8 deg step motor where it can rotate up to 200. 1V 500mAh) and controlled with Arduino. The size of torque angle increments depends on encoder resolution. The formula: Resolution is steps per inch or steps per milimeter I will go over this using steps/inch: steps = motor steps x driver microstepping inch = the amount of travel with one full stepper motor rotation In the case of our 1/2" 5 start 10 TPI lead screw, the axis will travel. For every motor, you will see a torque constant listed, with units of newton-meters per ampere, or pound-inches per ampere. The sophisticated controllers that achieve higher stepping speeds (1000 rpm or more) and torque are so costly that they don't compete with the cheaper servo controllers. Load Independent – Stepper motors will turn at a set speed regardless of load as long as the load does not exceed the torque rating for the motor. Advantages of step motors are low cost, high reliability, high torque at low speeds and a simple, rugged construction that operates in almost any environment. It's the most basic setup but still I can't. This information is intended to provide basic information to technicians about a kind of motor ( known as step motor , stepping motor or stepper motor ) widely applied in precision machines, the industry, in minilabs to develop photography, printers, in sewing machines, etc. motors on applications such as pumps where it is. This vibration can become very bad at some speeds and can cause the motor to lose torque. Stepper Motor. Speed is the measure that gives the estimate that how fast the servo attains a position. A simple pulley and pull spring scale also can be used, but is difficult to read at low and high step rates. This holds true for any given motor and controller combination. Like other DC motors, torque is highest when the motor is stopped and reduces as the motor speeds up, but remember that a stepper motor does not rotate continuously. A more detailed discussion about stepper motors can be found in our dedicated article, but making a long story short, like any DC motor, these motors rotate when the coils are energized however, if the coils are continuously energized in the same way, the movement will stop when opposite magnetic poles are aligned, e. The stepper motor Pull-Out torque is measured by accelerating the motor to the desired speed and then increasing the torque loading until the motor stalls or "pulls Out of synchronism" with the step frequency. So the higher the applied voltage, the faster the current rises, the faster the motor can get to its next step, the faster it can accelerate/go, and the greater torque it can generate at any high speed. Torque Moment calculation For a movement, we have to perform a follow up of positions between a Start and a Stop position. T ftlb = P hp 5252 / n (1b) where. Stepper Motor Glossary of Terms. The maximum torque at which an energized step motor will start and run in synchronism, without losing steps, at constant speed. For an application operating at low speed, a stepper motor may provide a more economical solution. It is a simple 2 phases hybrid stepper motor. They complement the scene at their respective lower power ranges (Refs 1, 2). Because of this, they tend to run hot. The graph presented in figure 4 shows the relation generator torque (rotational force) vs speed. 5 inches with one stepper motor rotation. Using this equation for the torque-speed curve, we can find the motor's torque at a given speed. There are a number of relatively inexpensive stepper driver boards and chips available that work in the voltage range you want for small stepper motors. Terminals 3 through 6 are where you connect the wires from the stepper motors to the stepper driver. 08 ftlbs torque add a driving pulley 3 inch and a driven pulley 5 inch to this and through reduction we will increase torque. The VR and PM stepper motors are the most common type of stepper motors. motor's output torque. 67A) with the a4988 driver (. Increasingly, speed control is being used with A. While the motor torque output varies on alternate steps, this is more than offset by the need to step through only half the angle. A standard bipolar stepper-motor usually will have 1. Each terminal also has its function silk-screened onto the aluminum case. 75 in-lb torque motor 38. NEMA 23 is a stepper motor with a 2. This will depend on the type of stepper motor you select. There are a number of relatively inexpensive stepper driver boards and chips available that work in the voltage range you want for small stepper motors. 68a spec of your current motor, or at least is in that ballpark. DC Motor Operating Principles: Torque on a Coil:. For an example of how torque affects your creations, we can look at another field where torque is important, the automotive sector. If I increase the speed (or simply I try to execute the stepper motor exerciser from the wiki), the motor skips steps, and at higher speeds it simply stops, and make weird noise. Static characteristics of stepper motor Torque versus step angle and torque versus current are called statics characteristics of stepper motor. 1V 500mAh) and controlled with Arduino. Control Engineering - A stepper motor divides a full rotation into a number of equal steps, important for many industrial motor and motion control applications. For example, do you want speed, or do you want torque?. of the computer, and thenrerun the recorded G-code to make the stepper motor work without the computer any more. The stepper motor Pull-Out torque is measured by accelerating the motor to the desired speed and then increasing the torque loading until the motor stalls or "pulls Out of synchronism" with the step frequency. And what are the main causes of stall in a stepper motor system? The stall condition is what happens when the load torque on a motor exceeds the motors pull-out torque. The size of torque angle increments depends on encoder resolution. Servo motors are capable of delivering more power. I would assume the extruder motor is also 10 Ohms. They complement the scene at their respective lower power ranges (Refs 1, 2). Following are the useful equations and AC motors formulas while designing and analyzing synchronous motors, stepper motors and other related AC machines. 1Background The stepper motor is a device generating a stepwise rotational motion, which has an advantage in applications where a precise position control is of importance [1]. Slapping a long body 1. As we have seen that in half mode, the number of steps taken by the motor to complete one revolution gets doubled, so step angle reduces to half. Mach3, EMC2, KCAM4, etc. Because of the driver output is related to the speed versus torque of stepper motor. For example, do you want speed, or do you want torque?. We sincerely hope to establish business relationships and cooperate with you. stepper motors as oppose to unipolar (i. The output torque and power from a stepper motor are functions of the motor size, motor heat sinking, working duty cycle, motor winding, and the type of drive used. Motor speed (n): Motor speed in RPM. A motor with its maximum torque at low speeds is the stepper motor. We're using a shutter from Edmond optical. In the conventional DC motor commutation takes place mechanically through the commutator-and-brush system. This formula is quite general, but it offers little in the way of guidance for how to select appropriate values of the current through the two windings of the motor. It features industries’ most advanced stepper motor driver with simple Step/Dir interface. (3) (4) For the original problem setup and the derivation of the above equations, please refer to the DC Motor Position: System Modeling page. The stepper motor will play a factor in the above equation. DC Motor Operating Principles: Torque on a Coil:. Consider the following DC motor governing equations: where is the motor speed, is the motor terminal voltage, is the winding current, is the back-EMF constant of the motor, is the torque constant, is the terminal resistance, is the terminal inductance, is the motor and load inertia, and is the disturbance torque. A method for damping vibrations in a stepper motor with micro-stepping control which comprises the steps of identifying the force amplitudes and phase shifts of multiple harmonic detent torques of the stepper motor, such as the first, second and fourth harmonic detent torques, and tuning the stepper motor with different current commands until minimum friction and resistive torque are obtained. It features 3. Starting torque is the maximum torque that can be produced by a motor in order to start rotational movement of the load. The VR and PM stepper motors are the most common type of stepper motors. How to find the running torque of stepper motor? I know it will be less that Holding torque, but what percentage? It is unclear whether the equation that you found is the appropriate equation for calculating the holding torque. Gear motors can also be used on vehicles with smaller wheels if a slow top speed and a high amount of torque for uphill climbing or driving through sand or mud is required. If I increase the speed (or simply I try to execute the stepper motor exerciser from the wiki), the motor skips steps, and at higher speeds it simply stops, and make weird noise. So, lets assume you have a pair of suitable stepper motors for your micromouse. Stepper-motor problems often begin when a customer approaches a stepper-motor supplier with insufficient or inaccurate data. Special Electrical Machines, Torque Equation of SyRM,electrical machines, electrical engineering, DC motor,DC generator, emf equation, advanced electrical ma. The rated voltage of driver circuit must be five to ten times of stepper motor rating voltage. DC servo motors, however, have a higher torque *during rotation* than steppers and a much higher RPM. A stepper motor is a “doubly salient” motor and would have high cogging torque since it has teeth on the rotor and the stator. NEMA23 425 oz/in 2. If in full step commutation, controlling the maximum current level transfers into motor ’s shaft torque control. After calculating the required torque, a stepper motor can be used in low-speed applications. 1000 * SQRT (0. 5 reduction = 135. Question is completely wrong because Motor will produce whatever torque it was designed for. Equation 4a. This is determined by the ratio of inductance to resistance (L/R) of the motor and driver as illustrated in Fig 8-2 (a). Both phases are always on. 3, the torque generation is. 1) A lack of torque / power from the motor. Check the current with the formula: U=IxR. I went in a lot of troubles before gettings perfect quality prints in the I3, but i never been able to print more than 45 mm/s, because. Hybrid stepper motor. Stepper Motor. The design concept of a spherical stepper motor capable of three-degrees-of-freedom (DOF) motion in a single joint is presented. As motor speed increases, torque decreases. A drive system consists of the 3-phase stepper motor and the appropri-ate drive. Acceleration of stepper motors. The main advantage of using the stepper motor is the position control. If the weight is coaxial to the motor, use the formula. Using this equation for the torque-speed curve, we can find the motor's torque at a given speed. As a stepper motor turns, the inductance of each winding fluctuates through peaks and valleys a number of times per revolution. So, we can say, the armature torque is only proportional to the armature current. 0A Type Bipolar 4 wire Table 2: Specifications of stepper motor used for the tests Stepping Mode Full Step Speed 144rpm Decay Mode Slow Current 1A/phase Operation Continuous Supply Voltage 24V Ambient Temp. 9° step angle, round and cut shafts, and can be customized. 1Background The stepper motor is a device generating a stepwise rotational motion, which has an advantage in applications where a precise position control is of importance [1]. Look at a servomotor catalog. This information is intended to provide basic information to technicians about a kind of motor ( known as step motor , stepping motor or stepper motor ) widely applied in precision machines, the industry, in minilabs to develop photography, printers, in sewing machines, etc. 9 degree increments. If the weight is coaxial to the motor, use the formula. Torque equation 29. As with any speed changing system, the load inertia reflected back to the motor is a squared function of the speed ratio. 8 degHolding Torque: 3. For example: a 3. The Arduino generates the control signals to control the driver (M542T). There are a number of relatively inexpensive stepper driver boards and chips available that work in the voltage range you want for small stepper motors. If a screw of 4 mm is run at 1500 RPM, we obtain a displacement speed of 1500*4mm=6000mm/min or 6 m/min. These motors work with direct current (DC) and produce their rotation force (torque) by switching the current mechanically or electronically. Stepper motors offer inherently more torque at low speeds than do servo motors. In bipolar mode, the same motor has 800 gcm of torque. Since it is a simple 1DOF arm with a rotary joint, the gravitational torque is τ g = m g l c o s (θ). A stepper motor, also known as step motor or stepping motor, is a brushless DC electric motor that divides a full rotation into a number of equal steps. You’ll have to “wind up” enough incremental torque to overcome the bearing friction. If I increase the speed (or simply I try to execute the stepper motor exerciser from the wiki), the motor skips steps, and at higher speeds it simply stops, and make weird noise. Close suggestions Search Search. Stepper motors are controlled by a driver, which Reading a Speed-Torque Curve. Commonly used stepper motors are 42 and 57, these refer to the outer diameter of the motor. 1Background The stepper motor is a device generating a stepwise rotational motion, which has an advantage in applications where a precise position control is of importance [1]. Smaller step angles can be obtained trough. (1) (2) The above has the form of a standard set of state-space equations as described below. Stepper motors have fairly small sizes and are characterized by their ability for precise incremental position or speed adjustment with electric pulses Stepper motors are characterized by their moving torque and holding torque. For example, rapidly reversing the direction of an axis on a CNC machine or jumping a quadruped robot into the air. Therefore you probably can't run your steppers at their full rated torque without melting your plastic motor mounts. I need NEMA17 and NEMA23 stepper motors with max available holding torque (reasonably priced), best at 12V power supply (will do tests with both sizes to see which one to use). 68A motor only makes rated torque at 1. 0316 Vrad/s. Pull-out torque The stepper motor Pull-Out torque is measured by accelerating the motor to the desired speed and then increasing the torque loading until the motor stalls or "pulls Out of synchronism. Highly recognized for High Performance and Quality Products, MOONS' offers a vast array of motion control & smart LED control products or solutions. If the weight is coaxial to the motor, use the formula. Stepper motors need around 20 times their rated voltage to perform at their best. 8 deg step motor where it can rotate up to 200. For a brushed motor to achieve the same torque as a brushless motor, it would require a larger magnet. The torque depends on the motor and the voltage/current used to drive the motor. The resulting torque is T d = 2. TDL stepper motor has a wide range of applications, with low noise, strong stability, high accuracy, small size, high torque, low price, simple assembly and other characteristics. This vibration can become very bad at some speeds and can cause the motor to lose torque. The most rudimentary form of a SRM has the lowest construction cost of any electric motor because of its simple structure, and even industrial motors may have some cost reduction due to the lack of rotor windings or permanent magnets. Permanent Magnet Stepper Motor. m is calculated with the following equation: $$T=\displaystyle \frac{60}{2\pi}\cdot \displaystyle \frac {P \cdot 1000}{n}$$ Parameters: Motor rating (P): Motor rating in kW. We believe in quality and reliability. The reference voltage at the test points is related to the motor drive current by the following formula: Vref = Motor Current * 1. Picture a pulley attached to the stepper motor’s shaft that’s 1. For example, a motor that puts out 10 lb. Such terms are Pull-in Torque and Pull-out Torque. Open navigation menu. m Rated phase current Rotor inertia 38 0. The motor will have full rated torque. See full list on motioncontroltips. A stepper motor is a constant output power transducer, where power is defined as torque multiplied by speed. Control Engineering - A stepper motor divides a full rotation into a number of equal steps, important for many industrial motor and motion control applications. The field controlled DC motor speed control. The motor's position can then be commanded to move and hold at one of these steps without any feedback sensor (an open-loop controller), as long as the motor is carefully sized to the application in respect to torque and speed. 3 pcs NEMA23 425 oz/in 2. The driver is a mini-ULN2003. The motors on most V2's will have a phase resistance of ~10 Ohms. The motor speed for AC motors can be caculated with the Motor speed calculator. Home; Profil. This formula is quite general, but it offers little in the way of guidance for how to select appropriate values of the current through the two windings of the motor. Stepper motors are known for a very accurate motion, and for that reason are used in applications such as a floppy disk drive. 2 Torque development by 2 fields CHAPTER 7: MOTOR DESIGN 7. After considering some other possibilities I have decided to try using a stepper-type micro motor. I am developing an stepper motor position control board in the devicenet network. This property tends to reduce motor torque, so there must be a corresponding rise in motor current to keep torque levels up. The Maximum Tractive Torque represents the maximum amount of torque that can be. 3 pcs NEMA23 425 oz/in 2. Hybrid Stepper Motor Engineering Hybrid stepper motors provide excellent performance in areas of torque, speed, and step resolution. (3) (4) For the original problem setup and the derivation of the above equations, please refer to the DC Motor Position: System Modeling page. The most rudimentary form of a SRM has the lowest construction cost of any electric motor because of its simple structure, and even industrial motors may have some cost reduction due to the lack of rotor windings or permanent magnets. The 28BYJ-48 is a 5-wire unipolar stepper motor that runs on 5 volts and doesn’t require any driver. The output torque and power from a stepper motor are functions of the motor size, motor heat sinking, working duty cycle, motor winding, and the type of drive used. A more detailed discussion about stepper motors can be found in our dedicated article, but making a long story short, like any DC motor, these motors rotate when the coils are energized however, if the coils are continuously energized in the same way, the movement will stop when opposite magnetic poles are aligned, e. 0A Type Bipolar 4 wire Table 2: Specifications of stepper motor used for the tests Stepping Mode Full Step Speed 144rpm Decay Mode Slow Current 1A/phase Operation Continuous Supply Voltage 24V Ambient Temp. The motor sizing process involves a number of mathematical equations, which are most certainly documented, but not necessarily with the motor sizing process in mind. 3 Voltage Regulation The final method for stepper motor speed control is through voltage regulation. conventional stepper motor, this work illustrates the fundamentals of the spherical stepper design for robotic applications. These offer the advantage of faster and more precise operation compared to a conventional vacuum (pneumatic) operated system. -feet of torque used with a gear reducer with a 10:1 gear ratio will give you 100 lb. How to find the running torque of stepper motor? I know it will be less that Holding torque, but what percentage? It is unclear whether the equation that you found is the appropriate equation for calculating the holding torque. If you know the magnitude of the force (in Newtons) and the distance (in meters), you can solve for the torque, expressed in newton-meters (N∙m). Figure 6 shows an example of torque/speed character-istics. This type of motor provides a combination of the best features available on both the PM and VR types of stepper motors. single windings) type stepper motors for reasons of compactness and higher power efficiency. You need to calculate requirement of torque in your application and than choose best motor for that. This motor specified to have a max. MOONS', founded in 1994, is a global Top 3 stepper motor manufacturer with the yearly shippment of more than 10 million hybrid stepper motors. Basically two signals are being driven on the MT542T: Direction (DIR) and Speed (PUL). To resume, the stepper motors drivers dosen't comes calibrated. This is the most common equation used to calculate acceleration torque for all types of motors. 12 • Consider a multistack VR stepper motor with N=3 and 8 rotor teeth being fed from a single transistor. 8) That would end up with over 4 amps flowing and that motor would be toast pretty quick. Holding Torque 59Ncm Phase Inductance 3. If you know the magnitude of the force (in Newtons) and the distance (in meters), you can solve for the torque, expressed in newton-meters (N∙m). In dc shunt motor, the field flux is approximately constant. K t =motor torque constant=0. Commonly used stepper motors are 42 and 57, these refer to the outer diameter of the motor. This is important to know because if you exceed either, your motor could. 0" 50" 100" 150" 200" 250" 300" 350". 3, the torque generation is. 22°C Heat sink on PCB none. Motor Regulation Using Performance Curve Data R m = 9. Stepper Motor Basics. In fact, I’ve only ever seen a handful of VR motors for sale. In other words, if you drive it at 1. T ftlb = torque (lb f ft). 625 in-lb torque reduction Since the Gear has a 1” PD, the line of force is 1/2” from the center of the shaft to the gear rack. Force is 9. Stepper motors are suitable for applications where compact and robust solutions are required. Stepper motors need around 20 times their rated voltage to perform at their best. An accurate formula includes many variables (available bus voltage, inductance, dI/dt and peak current which depends on accel and load, Back EMF etc. 9° step angle, round and cut shafts, and can be customized. Like some kind of clock but where the clock hand is 2 meter and 10gkg. The output torque of a stepper motor and drive can best be measured by using a bridge type strain gage coupled to a magnetic particle brake load. Example a motor that is rated at 6mH per phase 1000 * SQRT (0. We are manufacturer of Hybrid Stepper Motors in China, if you want to buy Linear Stepper Motors, Step-Servo Motors, Brushless Dc Motor 24V, please contact us. Stepper motors and servo motors can use a different formula since they deal with pulse speed (Hz). SureStep stepper motor, NEMA 34 frame, IP40, single shaft, 6. 2: A typical closed-loop control system for a three-phase BLDC motor includes a controller, driver, and power transistor half-bridge H. As the motor tempera-ture increases and the curve becomes steeper, shaft speed will drop more for the same increase in load. Stepper motors have fairly small sizes and are characterized by their ability for precise incremental position or speed adjustment with electric pulses Stepper motors are characterized by their moving torque and holding torque. Figure 4: Torque Deflection. Operation of a 2 Phase Stepper motor Full Step with 2 Phases Active Full step drive (two phases on) This is the usual method for full step driving the motor. Hybrid Stepper Motor. Equation (7) is important|it is a torque equation in \motor coordinates. 8 degree steps. Torque calculation = 8. Provides Detent Torque with the de-energized. 2 Operational Principle of a VR Spherical Motor The VR spherical motor discussed here has a structure similar to that of a spherical stepper [6]. Author: Reston Condit Microchip Technology Inc. 1 Introduction 7. The motor torque Tm is proportional to the phase current. motor torque/speed curve is a very good approximation. The driver is a mini-ULN2003. Another factor affecting motor performance is the back-EMF (electro-motive force) produced by the rotor rotating in a magnetic field. The residual torque in a permanent magnet or hybrid stepping motor is frequently referred to as the cogging torque or detent torque of the motor because a naive observer will frequently guess that there is a detent mechanism of some kind inside the motor. (Image: Texas Instruments). This equation is helpful in determining how many steps your stepper motor requires to move one full revolution. 8° Higher torque per unit volume which is more than in case of variable reluctance motor; Due to permanent magnet, the motor has some detent torque which is absent in variable reluctance motor. To compute the maximum voltage that you should use depending on the inductance of the motor use this formula. The back emf, e, is related to the rotational velocity by the following equations: In SI units (which we will use), Kt (armature constant) is equal to Ke (motor constant). They are often controlled in open-loop, although the potential loss of syn-. High torque dc motor for robots in India is easily available at our stores with nominal charges. When a stepper motor is accelerating, it has to produce torque to overcome its own rotor inertia and the mass of the load it is driving. Stepper motors. 1 ohm sense resistor, the R100 resistor closest to the big black square in the middle) powered by a lipo (3S 11. 5º Stepper Motors Page 19 Series 26M024B Holding Torque: mN•m/oz-in Unipolar 4. It features 3. Home; Profil. If in full step commutation, controlling the maximum current level transfers into motor ’s shaft torque control. The feature-rich peripherals of Microchip’s PIC16F1776/9 allows the two H-Bridge switches to control different driving tech-niques for high- and low-power stepper motor, constant or high-torque microstepping, current limiting, motor step rate setting and motor Fault event detection. MISUMI offers free CAD download, short lead times, competitive pricing, and no minimum order quantity. Torque in Imperial units can be calculated as.
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2021-04-21 05:55:45
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http://openstudy.com/updates/55f0c570e4b0dedaa9ee0c1f
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## Rizags one year ago Is the following equation fully simplified? Equation Below:
1. Rizags
$f(x)= \sqrt{(\sin^2(x)-2)^2}-\sqrt{(\sin^2(x)+1)^2}$
2. jdoe0001
hmmm what do you think?
3. Rizags
I think it is because i can't just cancel the radical and the square (because of the possiblity that the inside is negative
4. Rizags
I was puzzled because of something like this: $\sqrt{(-2)^2}\neq-2$
5. jdoe0001
well... hmm kinda, yes but one could say $$\sqrt{(-2)^2}\to \begin{cases} -2\\ +2 \end{cases}$$
6. jdoe0001
if you use the +/- part but, for an expression, it depends on context in this expression, I'd get rid of the radical and exponent though and yes you're correct, the radical IS a constraint on the function but the question is, can it be simplified further and I think it can, you can have a simplified version, keeping in mind the constraints
7. Rizags
but if i complete that simplification, i get $f(x)=-3$ which fails for all x
8. jdoe0001
hmm
9. jdoe0001
let's try one value $$\bf f\left( \frac{\pi }{2} \right)= \sqrt{[\sin^2\left( \frac{\pi }{2} \right)-2]^2}-\sqrt{[\sin^2\left( \frac{\pi }{2} \right)+1]^2}$$ what would that give?
10. Rizags
-1
11. jdoe0001
hmm -1?
12. Rizags
ahh, i believe I've solved it. Since the first radicand will always be negative before it is squared, I will simplify take its absolute value, by flipping the terms, to get $\sqrt{(\sin^2(x)-2)^2}=\sqrt{(2-\sin^2(x))^2}=2-\sin^2(x)$ from here, I can simplify the second radicand easily because it is always positive and thus is simply $\sqrt{(\sin^2(x)+1)^2}=\sin^2(x)+1$ putting it all together, i get $f(x)=2-\sin^2(x)-(\sin^2(x)+1)=1-2\sin^2(x)$ which equals, finally, $\cos(2x)$ does that work?
13. jdoe0001
hmmm the first radicand, actually both, are raised to the 2nd power, and thus, even if you get a negative value from the sine addition, the exponent will make it positive anyway -* - = +
14. Rizags
but with respect to eliminating the radical entirely, doesn't the flipping work?
15. jdoe0001
ahemm nope 2-x $$\ne$$ x-2
16. jdoe0001
hmm actually... shoot got a mistake there.....
17. Rizags
wheres the mistake?
18. jdoe0001
hmm well. my -3 is... mistaken for one
19. jdoe0001
hmmm
20. jdoe0001
either way.. .you can't simply flip the terms in the radicand though, unless you multiply them by a -1
21. Rizags
Look at this, attempted with x=pi/2$\sqrt{(\sin^2(\pi/2)-2)^2}=1$ but also,$\sqrt{(2-\sin^2(\pi/2))^2}=1$ So if i rewrite it in the second way, I can remove the radical.
22. jdoe0001
hmmm somehow... I do see the difference, expression wise and yes, the 2nd power exponent, will make, whatever value inside, negative or not, positive
23. jdoe0001
$$\bf f\left( \frac{\pi }{2} \right)= \sqrt{[\sin^2\left( \frac{\pi }{2} \right)-2]^2}-\sqrt{[\sin^2\left( \frac{\pi }{2} \right)+1]^2} \\ \quad \\ f\left( \frac{\pi }{2} \right)=\sqrt{[1^2-2]^2}-\sqrt{[1^2+1]^2}\implies f\left( \frac{\pi }{2} \right)=\sqrt{[-1]^2}-\sqrt{[2]^2} \\ \quad \\ f\left( \frac{\pi }{2} \right)=-1-2\implies f\left( \frac{\pi }{2} \right)=-3 \\ \quad \\ f(0)=\sqrt{[sin^2(0)-2]^2}-\sqrt{[sin^2(0)+1]^2} \\ \quad \\ f(0)=\sqrt{[0-2]^2}-\sqrt{[0+1]^2}\implies f(0)=-2-1\implies -3$$ so.... I"d think the -3 kinda checks out though
24. jdoe0001
but yes, one could say, it CAN be simplified further by getting rid of the radical and exponent BUT the simplified version will still maintain the constraints of the original one
25. jdoe0001
and yes, I'm aware that $$\bf \sqrt{(-2)^2}\ne -2\qquad or\qquad \sqrt{(-1)^2}\ne -1$$ but all that, depends on the context at hand
26. jdoe0001
I mean... the same can be said... say. of an inverse trigonometric function their range is $$\textit{Inverse Trigonometric Identities} \\ \quad \\ \begin{array}{cccl} Function&{\color{brown}{ Domain}}&{\color{blue}{ Range}}\\ \hline\\ {\color{blue}{ y}}=sin^{-1}({\color{brown}{ \theta}})&-1\le {\color{brown}{ \theta}} \le 1&-\frac{\pi}{2}\le {\color{blue}{ y}}\le \frac{\pi}{2} \\ \quad \\ {\color{blue}{ y}}=cos^{-1}({\color{brown}{ \theta}})&-1\le {\color{brown}{ \theta}} \le 1& 0 \le {\color{blue}{ y}}\le \pi \\ \quad \\ {\color{blue}{ y}}=tan^{-1}({\color{brown}{ \theta}})&-\infty\le {\color{brown}{ \theta}} \le +\infty &-\frac{\pi}{2}\le {\color{blue}{ y}}\le \frac{\pi}{2} \end{array}$$ but depending on context, you can take the arcSine or arcCosine of a value and get just a "reference angle", and use it on 2nd, 3rd, or 4th quadrants even though the inverse function, may not extend to it
27. Rizags
Wow, i thought there was just a solid rule explaining $\sqrt{(-2)^2}$ I learned that a radical can never be negative no matter what the conditions inside it, so i don't know, I might just put cos(2x), because it checks out on all my tries both positive and negative
28. jdoe0001
hehe bear in mind that notations and formulas, are only phenomena representations some results in math are called "math fallacies" or "extraneous" because the procedure is correct, bu the answer is ODD so, they're correct mathematically, but wrong logically, and the logic throws them out in this case, is an extraneous case is correct mathematically, logicaly it makes no much sense, you're correct thus is "extraneous" but depending on the context they're used, extraneous results can be valid or not in this case, we don't have an explicit context or phenomena for this expression thus the extraneous result is ok
29. Rizags
ok, thank you
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2017-01-18 18:58:39
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https://blog.keiruaprod.fr/2020/05/13/delayed_job-crons.html
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# KeiruaProd
## Listing DelayedJob cron jobs and schedules
A small helper that I sometimes have to run on different machines. Out of all the jobs, it finds those registered as cron tasks, and displays their schedule in yaml (the y function).
cron_jobs = Delayed::Job.where('cron is not null')
y (cron_jobs.map do |c| [c.handler.match("job_class: (.*)\n")[1], c.cron] end)
---
- - SomeJob
- 0 6 * * *
- - SomeOtherJob
- 0 0 * * *
- - AgainAnotherJob
- "* * * * *"
- - YetAnotherJob
- 0 0 1 * *
- 0 10 * * 1,2,3,4,5,6
Not the most robust code out there for sure. Yet robustness and clarity are not what you always need, and these 2 lines can be pasted in a rails console quite easily.
Another option when you want to mess with this jobs in the console is to parse the job_class using the Yaml component:
Delayed::Job.pluck(:handler).map { |h| YAML.load(h).job_data["job_class"] }.group_by{|x| x}.map {|k, v| [k, v.count] }
Aaaaand another useful command is to list the error count. Here is a quick and dirty way: it takes the first line of the stack trace, strips the numbers (ids) and sorts the errors by count:
y Delayed::Job.where.not(last_error: nil).pluck(:last_error).map {|e| e&.lines[0].tr('0123456789', '')}.tally.sort_by(&:last)
Also, sometimes you want the objects that lead to these errors:
y Delayed::Job.where("last_error ilike '%worksheet%'").pluck(:handler).map { |h| YAML.load(h).job_data["arguments"] }
---
- - _aj_globalid: gid://app/SomeModel/23648
- - _aj_globalid: gid://app/SomeModel/2364
Finally, in order not to lose some jobs, it can be useful to have dedicated queues:
$bin/delayed_job --queue=mailers -n 4 start delayed_job.0: process with pid 227532 started. delayed_job.1: process with pid 227534 started. delayed_job.2: process with pid 227539 started. delayed_job.3: process with pid 227543 started.$ bin/delayed_job --pool=active_storage_analysis:6 --pool=mailers:2 start
See a typo ? You can suggest a modification on Github.
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2021-01-21 17:30:43
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https://johncarlosbaez.wordpress.com/2011/10/06/chaitins-theorem-and-the-surprise-examination-paradox/
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Chaitin’s Theorem and the Surprise Examination Paradox
If you followed the business about Edward Nelson’s claim to have proved the inconsistency of arithmetic, you might have heard people mention this paper:
• Shira Kritchman and Ran Raz, The surprise examination paradox and the second incompleteness theorem, AMS Notices 57 (December 2010), 1454–1458.
It’s got a great idea in it, which I’d like to explain in a really sloppy way so everyone can understand it. Logic is cool, but most people never get to the cool part because they can’t fight their way through the rather dry technicalities.
You all know the surprise examination paradox, right? The teacher says one day he’ll give a quiz and it will be a surprise. So the kids think “well, it can’t be on the last day then—we’d know it was coming.” And then they think “well, so it can’t be on the day before the last day, either!—we’d know it was coming.” And so on… and they convince themselves it can’t happen at all.
But then the teacher gives it the very next day, and they’re completely surprised.
People still argue a lot about how to settle this paradox. But anyway, Kritchman and Raz use a rigorous version of this paradox together with Chaitin’s incompleteness theorem to demonstrate Gödel’s second incompleteness theorem—which says, very roughly, that:
If math can prove itself consistent, it’s not.
If you’re a logician, I bet this sort of sloppy statement really annoys you. Yeah? Does it? Take a chill pill, dude: this post isn’t for you—it’s for ordinary mortals. I said I want to summarize Kritchman and Raz’s argument in a really sloppy way. If you want precision and details, read their paper.
Okay, here we go:
Chaitin’s theorem, which is already astounding, says there’s some length L such that you can’t prove any particular string of bits needs a program longer than L to print it out. At least, this is so if math is consistent. If it’s not consistent, you can prove anything!
On the other hand, there’s some finite number of programs of length ≤ L. So if you take a list of more numbers than that, say 1, 2, …, N, there’s got to be at least one that needs a program longer than L to print it out.
Okay: there’s got to be at least one. How many? Suppose just one. Then we can go through all programs of length ≤ L, find those that print all the other numbers on our list… and thus, by a process of elimination, find the culprit.
But that means we’ve proved that this culprit is a number can only be computed by a program of length > L. But Chaitin’s theorem says that’s impossible! At least, not if math is consistent.
So there can’t be just one. At least, not if math is consistent.
Okay: suppose there are just two. Well, we can pull the same trick and find out who they are! So there can’t be just two, either. At least, not if math is consistent.
We can keep playing this game until we rule out all the possibilities… and then we’re really stuck. We get a contradiction. At least, if math is consistent.
So if we could prove math is consistent, we’d know it’s not!
17 Responses to Chaitin’s Theorem and the Surprise Examination Paradox
1. Roger Witte says:
I haven’t read the paper yet (which, by the way, I couldn’t reach from your link) but informally your argument above depends on what kind of logic you are using to prove Chaitlin’s theorem. If you are using intuitionist logic, there exists a number ‘L’ gives you a number and the rest argument works. However, if Chaitlin’s theorem is non-constructive, then the argument only shows you cannot construct L (ie it is impossible to know the value of L).
By the way the url where I did find the paper was
http://www.ams.org/notices/201011/rtx101101454p.pdf
• John Baez says:
I left the “http” out of the URL—thanks for catching that. I fixed it now.
Kritchman and Raz are using classical logic, and their proof of Gödel’s incompleteness theorem applies to any recursively axiomatizable theory T extending, say, Peano arithmetic (though as Nelson notes one can get away with less, e.g. Robinson arithmetic). Chaitin’s theorem also applies in this context. Given such a theory and a choice of universal programming language, Chaitin’s theorem allows you to explicitly construct a number L such that T is unable to prove any particular string of bits needs a program longer than L to print it out. The sketch of a proof of Chaitin’s theorem I linked to makes this clear.
2. Roger Witte says:
By the way, a second conclusion from the article would be that Eugenia Cheng has a way with words :)
• John Baez says:
Every blog needs ‘in jokes’ to make its readers feel like privileged members of an exclusive club. But yeah.
3. Bruce Smith says:
So roughly how big is L, for at least one reasonable specific choice of universal programming language? (Since you say L can be explicitly constructed, I presume that there must be a way to estimate an upper bound on its size, though perhaps a very non-tight bound.)
• Actually, any explicit upper bound for whatever you’re calling L is itself a perfectly good explicit unprovable complexity: if for no x can T prove that K(x)>L, then a fortiori for no x can it prove K(x) > L+1.
That is, Chaitin’s theorem doesn’t need to find the minimal unprovable complexity to give explicit true non-theorems!
• John Baez says:
Hi, Bruce! I keep asking people to estimate $L$ for me, and nobody ever does. It’s possible Chaitin has, because he’s written a book that works everything out very explicitly for a particular programming language. But I don’t know.
To be precise: I want someone to give me a constant $L$ such that, say, Peano arithmetic can’t prove any natural number requires a C program more than $L$ characters long to print it out.
Since you’re a programmer, maybe you can guess if I tell you this:
Any number $L$ with
$U + \log_n(L) + C < L$
will do the job. Here:
$n$ is the base in which we write numbers in our programming language.
$U$ is the length of a program where you input a natural number $i$ and it tries to go through all proofs in Peano arithmetic until it finds one that proves some natural number has Kolmogorov complexity $\ge i$; then it outputs this number. (Of course, for the value of $i$ we care about, it will not halt.)
$C$ is a small overhead cost: the length of the extra 'glue' to create a bigger program that takes the number $L$, written out in our chosen base, and feeds it into the program described above.
This bigger program thus has length
$U + \log_n(L) + C,$
and for the proof to work we need this to be smaller than $L.$
The bulk of the problem is getting an upper bound on $U.$ The number $L$ is just a bit bigger, surely smaller than $2 U.$
Anyone want to take a crack at it?
• Roger Witte says:
I don’t think the program can be done.
There is a slight problem here.
The Kolmogrov Complexity is not a computable function:
3.1 Incomputability of Kolmogorov complexity, Wikipedia.
So will any particular such program run into issues such as halting problem?
So although I am convinced that for every n and s such that PA proves K(n)=s, s<L I suspect that there may be some n for which PA cannot prove the values of K(n),
So rather than giving you an actual value L it only shows it would be contradictory for there not to be such a value (Since our reasoning is classical this is not a distinction … isn't it weird that I now find this difficult to swallow when, 30 years ago during my BSc Mathematics course, I wouldn't have had difficulty percieving the distinction?).
• John Baez says:
As the Wikipedia article and I explained, you can explicitly compute a constant L such that Peano arithmetic can’t prove any natural number requires a C program more than L characters long to print it out.
It may be impossible to compute the smallest such constant L—that would be an interesting result which I haven’t seen. However, as ‘some guy in the street’ pointed out, if some constant does the job then so does any larger constant. I’m merely asserting that some such constant does the job and can be explicitly computed.
The halting problem and the incomputability of Kolmogorov complexity are no obstacle here. Indeed, they underlie everything we’re discussing.
Chaitin’s theorem says that not only is Kolmogorov complexity uncomputable, there’s no way to prove it takes any value bigger than L, even though it does so in all but finitely many cases. This is an extremely strong form of uncomputability!
• John Baez says:
From my description here, I’m sure that L = 109 works.
But a good programmer who has studied some logic (like Bruce) could probably guess whether L = 106, 105 or even some smaller value works!
• Bruce Smith says:
I don’t think it can be done in C, since C semantics are not well-defined unless you specify a particular finite machine size. (Since C programs can do things like convert pointers to integers and back, tell you the size of any datatype, and convert data of any specified datatype to bytes and back.) On a finite machine of N bits, all programs either finish in time less than about 2**N or take forever.
But if you take “C without size-specific operations”, or a higher level language like Python, or for that matter a different sort of low-level language like a Turing machine, then that’s not an issue — you can define a precise semantics that allows it to run a program for an arbitrarily long time and allocate an arbitrary number of objects in memory which contain pointers to each other. (To stick with the spirit of the question, for whatever language you choose, you’d want to disallow use of any large external batch of information like a “standard library”, except for whatever is so basic that you think of it as part of the native language. This is not a serious handicap for this problem.)
The main things that the program ‘U’ (I’d rather call the program itself ‘U’ than call its length ‘U’) needs to do are:
– recognize a syntactically correct statement or proof;
– check the validity of a purported proof;
– recognize certain statements as saying or implying “The Kolmogorov complexity of n is more than i” for some n and i. (It’s not necessary to recognize all such statements, just at least one for each n and i; so it can just recognize a statement that consists of some template with specific values of n and i inserted into it at certain places.)
Assuming that U expresses the proofs it wants to check in a practical proof language (which will be more like what a practical theorem-prover like Coq uses than like what a traditional logician would recognize as “straight Peano arithmetic”, but which will not be excessively powerful in the spirit of this question), I’d estimate that the most complex part is checking proof validity, but that that can still be expressed in at most a few dozen syntactic rules, each expressible in a few lines of code. (The authors of a system like Coq, which includes code to actually do that, would know better, as long as they remember that the vast majority of their system’s actual code is not needed for this problem.)
This makes me think that even without trying to compact it much, in a reasonable language we could write U in a few hundred lines of code, or (after a bit of simple compression) a few thousand bytes. (And perhaps much less if we tried hard to compact the whole program in clever ways.)
So L will also be “a few thousand” (bytes or digits), or perhaps less, rather than some number you can never possibly count to.
For comparison:
– Bill Gates’ first commercial success was an implementation of a useful version of BASIC in about 4000 bytes;
– 4k programs (4096 bytes or less) can produce graphical animations like this one http://www.youtube.com/watch?v=FWmv1ykGzis (to be fair, the complexity of some of the OS, graphics drivers, and hardware should be included, but this is a lot less than you might think if you imagine rewriting it purely for compactness rather than for speed, and only including what this sort of program needs to produce output);
– the complete genetic code of an organism can be as short as a few hundred thousand bytes, and that has to be encoded in a way that doesn’t allow for highly clever compression schemes.
So this answers my question, as precisely as I cared about. (The key point I didn’t know when I asked it was that it was basically just asking how complex a program like U needs to be.) In fact, given your description of Chaitin’s book, I’d be surprised if he doesn’t include a concrete version of a program that could be U, and therefore directly come up with a specific legal value for L.
4. Giampiero Campa says:
People in-the-know already know this, but for everybody else, there is a nice post by Terry Tao about this.
5. Peter says:
I’m no expert on these matters but regarding this paragraph:
“Okay: there’s got to be at least one. How many? Suppose just one. Then we can go through all programs of length ≤ L, find those that print all the other numbers on our list… and thus, by a process of elimination, find the culprit.”
Doesn’t the halting problem prevent you from doing this? Some of these programs will run forever, and some of them will run for a very long time, and it’s impossible to tell the difference. So you wouldn’t end up with just one program.
Or have I misunderstood?
• John Baez says:
The uncomputability of the halting problem is crucial; it’s the real reason for Chaitin’s theorem and thus this spinoff. We must constantly bear it in mind. But it doesn’t cause problems here.
At this stage in the proof we’re assuming that only one number on our list is not printed out by a program of length ≤ L. So, we can list all programs of length ≤ L and do the following trick:
let the 1st run one step, let the 2nd run one step,… let the last run one step,
let the 1st run one step, let the 2nd run one step,… let the last run one step,
and so on. Some will never halt, and some will halt and print out numbers that aren’t on our list. But by our assumption, all but one of the numbers on our list will eventually get printed out… so then we know the ‘culprit’.
I didn’t explain this very clearly before—thanks for pushing me to do so.
6. […] John Baez: The Inconsistency of Arithmetic, Chaitin’s Theorem and the Surprise Examination Paradox […]
7. More on Chaitin’s incompleteness theorem.
8. Yale Dikes says:
I think the reasoning has some interesting qualities which are time sensitive, language sensitive, but not paradoxical in the way it seems at first. To define terms, consider what it means to be surprised. There is clearly the issue of when something is a surprise. Let’s say the professor offered the test on Friday week prior, it would be a surprise until the end of class on Thursday. But let’s tighten up surprise — there would be 24 hours where you knew for certain there would be a test on Thursday, assuming at least one day of the week will definitely have a test. The professor says he will stipulate that that is not “really” a surprise, even though from the point of view one has on Monday, it surely would be. It would only be after class on Wed, that you would be sure the only option would be Thursday, not to experience this 24 hour window of knowledge going from Thursday to the Friday of the test. But on Monday, you would know it could be Monday, Tues or Wed, without the 24 hour window, going forward. On Tues, you know it could be that day or Wed. On Wed, you would know it had to be that day, or there would be a potential 24 hour period of not being surprised. If there is a potential, disallowed by the professor of never being aware, even for a moment of the test before it occurs, then you know the next day is a no go if the test doesn’t hit you that day, and we are back in the pickle. Tuesday must be the day, but we have the window. Uh oh. If the Professor lightens up and says on Monday you will not be able to guess when the test is, if the test hasn’t been given by Thursday you will have a one day heads up, then there is no predicting the test, until and unless the end of class on Thursday, but not on Monday or earlier( there is a Monty Hall issue, but that’s another story). Otherwise, the Professor is selling a bill of goods ( or he’s buying into wily George’s redefinition of surprise unwittingly), and can’t promise the test will be given on one of 5 days, and will never allow for a twenty four hour window.
So the Professor relents, and sees that looking forward until Thursday he will be able to tally surprise people, but not on Thursday for Friday — that like picking a marble out of jars, one marble, five jars, pick four empty jars you know that the fifth will have it. The paradox comes from a nutty stipulation of “surprise”, and the fallacy ignoratio elenchi , or redefinition thatGeorge introduces. It’s logically impossible to avoid the 24 window, and if that standard is utilized, it generates a nutty result. Just as if I said the same thing about the jars: “You will be surprised. You will never know which jar has the marble.” So you pick till 4, and realize it must be five = fail. Translate: can’t be 5, so 4 is the highest, number. So you get to three jars and know it must be 4 = fail. Translate: can’t be 4 so three is the highest number. And so on. What is occurring is just a product of a defective definition of “surprise” lurking that can’t be met given the other conditions. ( The contradiction argument is not a “proof,” but a disproof: A–> P. not P —> Contra—->invalid —->Plug in A, A is invalid). Change the def to what we normally mean as third dimensional creatures when we say surprise, meaning at the beginning of the sequence not foreclosing the possibility of this changing going forward. And we are back to being surprised, at least till Thursday. But on Monday, since we know it’s not foreclosed by the potential 24 hour window, it would still be a surprise to us if the test were to occur on Friday.
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2015-08-29 06:02:26
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https://codereview.stackexchange.com/questions/166911/paycheck-calculator-software
|
# PayCheck calculator software
I was participating in a job selection process and one of the tasks I was asked to do was to develop a take home pay application to let users to calculate their paychecks based on their location. Today I received the feedback, and it stated that:
• Taxes poorly factored
• Low OO knowledge
The software requirements are stated below:
Write a program which can be used to calculate the take home pay for employees based on their location. The application should be a console app which prompts the user for the hourly rate and hours worked for the employee.
Please enter the hours worked: Please enter the hourly rate: Please enter the employee’s location: The program will output their gross along with a list of deductions along with a net amount.
Employee location: Ireland
Gross Amount: €x
Less deductions
Income Tax : €x Universal Social Charge: €x Pension: €x Net Amount: €x
Requirements
As a payroll user I would like to see a gross amount calculation for an employee’s salary.
Given the employee is paid €10 per hour, when the employee works 40 hours, the Gross amount is €400
As a payroll user I would like to see deductions charged for employees located in Ireland
Given the employee is located in Ireland, income tax at a rate of 25% for the first €600 and 40% thereafter Given the employee is located in Ireland, a Universal social charge of 7% is applied for the first €500 euro and 8% thereafter Given the employee is located in Ireland, a compulsory pension contribution of 4% is applied
As a payroll user I would like to see deductions charged for employees located in Italy
Given the employee is located in Italy, income tax at a flat rate of 25% is applied Given the employee is located in Italy, an INPS contribution of is applied based on gross salary. This is charged at 9% for the first €500 and increases by .1% over every €100 thereafter.
As a payroll user I would like to see deductions charged for employees located in Germany
Given the employee is located in Germany, income tax at a rate of 25% is applied on the first €400 and 32% thereafter Given the employee is located in Germany, a compulsory pension contribution of 2% is applied
I really cannot understand why my code was so badly evaluated, so it would be wonderful to have some of you checking out what could have went so wrong.
Just to make it clear, I would like to have some of you guys checking out the code to try to find out some design flaws that could have motivated the bad evaluation I received from the tech team.
The code I wrote can be found, along with the requirements, here.
Here follows some important parts of the code:
Base class PayCheckCalculator
using Domain.Entities;
namespace Domain.Interface
{
public abstract class PayCheckCalculator
{
protected void CalculateGrossAmount(PayCheck payCheck, int hoursWorked, decimal hourlyRate)
{
payCheck.GrossSalary = hoursWorked * hourlyRate;
}
protected abstract void CalculateDeductions(PayCheck payCheck);
protected void CalculateNetAmount(PayCheck payCheck)
{
decimal netAmount = payCheck.GrossSalary;
foreach (var deduction in payCheck.Deductions)
{
netAmount -= deduction.Amount;
}
payCheck.NetSalary = netAmount;
}
public PayCheck Calculate(int hoursWorked, decimal hourlyRate)
{
PayCheck payCheck = new PayCheck();
CalculateGrossAmount(payCheck, hoursWorked, hourlyRate);
CalculateDeductions(payCheck);
CalculateNetAmount(payCheck);
return payCheck;
}
}
}
Concrete class GermanyPayCheckCalculator
using Domain.Entities;
using Domain.Interface;
namespace PayCheckCalculators.Calculators
{
public class GermanyPayCheckCalculator: PayCheckCalculator
{
protected override void CalculateDeductions(PayCheck payCheck)
{
}
private Deduction CalculateIncomeDeduction(PayCheck payCheck)
{
string deductionDesc = "Income tax";
decimal amount = 0.0m;
decimal surplus = payCheck.GrossSalary - 400;
if (surplus <= 0)
{
amount = payCheck.GrossSalary * 0.25m;
}
else
{
amount = 400m * 0.25m;
amount += surplus * 0.32m;
}
return new Deduction(deductionDesc, amount, Deduction.DeductionType.INCOME_DEDUCTION);
}
private Deduction CalculateCompulsoryPensionDeduction(PayCheck payCheck)
{
string deductionDesc = "Pension";
return new Deduction(deductionDesc, payCheck.GrossSalary * 0.02m, Deduction.DeductionType.PENSION_DEDUCTION);
}
}
}
Concrete class IrelandPayCheckCalculator
using Domain.Interface;
using Domain.Entities;
namespace PayCheckCalculators.Calculators
{
public class IrelandPayCheckCalculator : PayCheckCalculator
{
protected override void CalculateDeductions(PayCheck payCheck)
{
}
private Deduction CalculateIncomeDeduction(PayCheck payCheck)
{
string deductionDesc = "Income tax";
decimal amount = 0.0m;
decimal surplus = payCheck.GrossSalary - 600;
if (surplus <= 0)
{
amount = payCheck.GrossSalary * 0.25m;
}
else
{
amount = 600m * 0.25m;
amount += surplus * 0.4m;
}
return new Deduction(deductionDesc, amount, Deduction.DeductionType.INCOME_DEDUCTION);
}
private Deduction CalculateUniversalSocialChargeDeduction(PayCheck payCheck)
{
string deductionDesc = "Universal social charge";
decimal amount = 0.0m;
decimal surplus = payCheck.GrossSalary - 500;
if (surplus <= 0)
{
amount = payCheck.GrossSalary * 0.07m;
}
else
{
amount = 500m * 0.07m;
amount += surplus * 0.08m;
}
return new Deduction(deductionDesc, amount, Deduction.DeductionType.UNIVERSAL_SOCIAL_DEDUCTION); ;
}
private Deduction CalculateCompulsoryPensionDeduction(PayCheck payCheck)
{
string deductionDesc = "Pension";
return new Deduction(deductionDesc, payCheck.GrossSalary * 0.04m, Deduction.DeductionType.PENSION_DEDUCTION);
}
}
}
Concrete class ItalyPayCheckCalculator
using Domain.Interface;
using Domain.Entities;
namespace PayCheckCalculators.Calculators
{
public class ItalyPayCheckCalculator : PayCheckCalculator
{
protected override void CalculateDeductions(PayCheck payCheck)
{
}
private Deduction CalculateIncomeDeduction(PayCheck payCheck)
{
string deductionDesc = "Income tax";
return new Deduction(deductionDesc, payCheck.GrossSalary * 0.25m, Deduction.DeductionType.INCOME_DEDUCTION); ;
}
private Deduction CalculateInpsDeduction(PayCheck payCheck)
{
string deductionDesc = "INPS";
decimal amount = 0.0m;
decimal surplus = payCheck.GrossSalary - 500m;
if (surplus <= 0)
{
amount = payCheck.GrossSalary * 0.09m;
}
else
{
amount = 500 * 0.09m;
while (surplus >= 100)
{
amount += 1;
surplus -= 100;
}
}
return new Deduction(deductionDesc, amount, Deduction.DeductionType.INPS_DEDUCTION); ;
}
}
}
Abstract factory class
namespace Domain.Interface
{
public abstract class AbstractPayCheckCalculatorFactory<T>
{
public abstract T CreatePayCheckCalculator(string location);
}
}
Concrete factory class
using Domain.Interface;
using PayCheckCalculators.Calculators;
using System;
namespace PayCheckCalculators.Factory
{
public class PayCheckCalculatorFactory: AbstractPayCheckCalculatorFactory<PayCheckCalculator>
{
public override PayCheckCalculator CreatePayCheckCalculator(string location)
{
PayCheckCalculator calculator;
if (location == null)
{
throw new ArgumentNullException("Location cannot be null.");
}
if (location.Length == 0)
{
throw new ArgumentException("Location cannot be empty.");
}
switch (location.ToLower())
{
case "ireland": calculator = new IrelandPayCheckCalculator(); break;
case "italy": calculator = new ItalyPayCheckCalculator(); break;
case "germany": calculator = new GermanyPayCheckCalculator(); break;
default: throw new NotSupportedException("The provided country is not yet supported.");
}
return calculator;
}
}
}
Employee class
using System;
using System.Collections.Generic;
using System.Linq;
using Domain.Validation;
namespace Domain.Entities
{
public class Employee
{
public string Location { get; private set; }
public int HoursWorked { get; private set; }
public decimal HourlyRate { get; private set; }
public Employee(string location, int hoursWorked = 0, decimal hourlyRate = 0.0m)
{
Location = location;
HoursWorked = hoursWorked;
HourlyRate = hourlyRate;
var errors = Validate();
if (errors.Count > 0)
{
var except = new ArgumentException();
throw except;
}
}
private ICollection<ValidationError> Validate()
{
var errors = new List<ValidationError>();
return errors;
}
//TODO: I don't check for an upper limit for now because it must be discussed further with the PO.
private ICollection<ValidationError> ValidateHoursWorked()
{
var errors = new List<ValidationError>();
if (HoursWorked < 0)
{
String description = "Error: Hours Worked cannot be negative.";
}
return errors;
}
//TODO: I don't check for an upper limit for now because it must be discussed further with the PO.
private ICollection<ValidationError> ValidateHourlyRate()
{
var errors = new List<ValidationError>();
if (HourlyRate < 0)
{
String description = "Error: Hourly Rate cannot be negative.";
}
return errors;
}
private ICollection<ValidationError> ValidateLocation()
{
var errors = new List<ValidationError>();
String description = "";
if (Location == null)
{
description = "Error: Location cannot be null.";
}
else if (Location.Length == 0)
{
description = "Error: Location cannot be empty.";
}
else if (!Location.All(c => char.IsLetter(c)))
{
description = "Error: Location can only contain letters.";
}
return errors;
}
}
}
Program class
using Domain.Entities;
using Domain.Interface;
using Domain.Validation;
using PayCheckApp.View;
using PayCheckCalculators.Factory;
using System;
using System.Collections.Generic;
namespace PayCheckApp
{
public class Program
{
public void Run(AbstractPayCheckCalculatorFactory<PayCheckCalculator> calculatorFactory)
{
bool tryAgain = false;
do
{
var myUserInterface = new PayCheckCalculatorView();
Employee employee;
try
{
employee = myUserInterface.GetEmployee();
myUserInterface.ShowResult(calculatorFactory.CreatePayCheckCalculator(employee.Location).Calculate(employee.HoursWorked, employee.HourlyRate));
}
catch (ArgumentException ex)
{
if (ex.Data.Contains("errors"))
{
var errors = ex.Data["errors"] as ICollection<ValidationError>;
if (errors != null)
{
myUserInterface.ShowErrors(errors);
}
}
else
{
myUserInterface.ShowError(ex.Message);
}
}
catch (Exception ex)
{
myUserInterface.ShowError(ex.Message);
}
finally
{
tryAgain = myUserInterface.IsConcluded();
}
}
while (tryAgain);
}
static void Main(string[] args)
{
(new Program()).Run(new PayCheckCalculatorFactory());
}
}
}
So, let me explain the software architecture as I see. First, I splited the codebase in 4 different visual studio projects, as listed below:
1. Domain (Class Library):
Contains the domain entities, as Employee, Paycheck and Deduction. It also contains a Validation class that is used to validade the state of different classes in the system and two abstract classes used as superclasses for the calculators and calculator factories.
2. PayCheckCalculators (Class Library):
Contains the paycheck calculators. As the requirements state, each country have its own sort of deductions. However, the gross and net salary calculation is universal. Gross salary is the simple multiplication of HoursWorked and hourlyRate. The net salary is simply gross salary minus the summation of all deductions.
Consequently, the calculation of gross and net salary is done in the abstract class PayCheckCalculator, while the deductions are calculated in the appropriate child classes: ItalyPayCheckCalculator, GermanyPayCheckCalculator and IrelandPayCheckCalculator.
The superclass PayCheckCalculator resides in the Domain project, while the specializations reside in the PayCheckCalculators project. Here I believe I could improve the architecture creating an Interface containing all public methods of the PayCheckCalculator abstract class, and I would make this abstract class to extend the interface. So, I could put the interface in the domain project and the abstract class in the calculators project, avoiding Paycheck calculation implementation details in the domain project. Although it would be an improvement, I don't think it is a life or death case.
3. Tests project (Class Library):
One of the requirements was to use TDD in the development process. So I began coding by the tests and in the end I covered about 97% of the codebase with unit tests.
4. PayCheckCalculatorApp (Console Application):
This project contains the presentation part of the software. There are only two classes in the project: Program and PayCheckCalculatorView. The Program class works as a controller. It instanciates the view and ask for an Employee.
Then the view then asks the user for the required information and creates an Employee using the provided info. Once the controller has an instance of Employee, it requests the factory to create a new PayCheckCalculator. The concrete factory is injected to the controller (I don't use DI containers). Once the proper calculator is instantiated, the controller calls the calculate method.
Finally, the controller checks for any thrown exceptions and in case it receives one, it call the showError/ShowErrors method of the view.
Let me summing things up comparing to SOLID principles:
1. Single Responsibility
Each class has one reason to change. For example, the Employee class can only change if the business rules change, because all it does is to validate the user inputs, as hours worked, hourly rate and location (country).
In addition, each calculator can only change if the rules for that precise calculator change. For example, if from now on Ireland starts to charge a lower income tax or change the rules to apply this income tax, then the IrelandPayCheckCalculator would have to be update (or I new class should be designed, as I explain later). So, I believe I achieved to comply with single responsibility principle.
2. Open/Closed Principle
Let's say I want to implement a new calculator, for example BrazilPayCheckCalculator. All I have to do is to create a subclass of PayCheckCalculator and implement the CalculateDeductions method. In addition, I would need to update the concrete factory class or create a subclass of AbstractPayCheckCalculatorFactory (or even of the concrete PayCheckCalculatorFactory) and implement the logic to return an instance of BrazilPayCheckCalculator.
Changes can be also accomplished like in the creation of new calculators. In other words, if I need to change the requirements of a given calculator, I could simply create a new subclass of PayCheckCalculator and implement the required logic.
Summing all these up, the code complies with open/closed principle.
3. Liskov Substitution Principle
The Program class uses the calculator in a polymorphic way. First, it delegates to a factory class the creation of the proper calculator instance. In addition, the factory class returns the instance using the abstract superclass PayCheckCalculator. The Program class then calls the Calculate method in the returned instance.
Consequently, we can replace this instance by any child of PayCheckCalculator and the software will work seamlessly.
The same is not true for the relationship between the Program and View classes. However, I don't think it is a core concern.
Consequently, to the best of my knowledge, the code complies with the Liskov Substitution principle.
4. Interface Segregation Principle
Well, as in my code I don't have a case where a class is forced to implement methods that it doesn't use, I believe that this principle also holds in my codebase.
5. Inversion of Dependency
This part could be improved as I said in the beginning of the post. Currently, all my calculator classes derive from an abstract class that actually implements the Gross and Net salary calculation and let the subclasses calculate the deductions.
Consequently, the clients of a calculator don't depend on the concrete calculator, but instead depend on the abstract class. In other words, the client classes are obliviated from the specifics of each calculator. Even better, because the abstract class resides in the domain assembly, no references need to be made to the calculators assembly.
As I said before, this could be improved. Because the abstract class has some implementation details, it would be better to create an interface and make the abstract class implement (partially) it.
Because all of these, I believe my code complies with all solid principles. It is true it can be improved, but I don't believe it could be evaluated as a code written by a developer with low knowledge of OO design principles.
I'm sorry, I don't have it in me to read your full explanations (TLDR), but here are some questionable design choices, that caught my eye after looking through your code:
1. Calculator has no interface. I think it should. And it should be very clear. I don't want to know implementation details, I don't want to call factories and all that stuff, that doesn't even fit the screen:
calculatorFactory.CreatePayCheckCalculator(employee.Location).Calculate(employee.HoursWorked, employee.HourlyRate)
I want to give an employee and receive a pay check.
interface IPayCheckCalculator
{
PayCheck Calculate(Employee employee);
}
//usage:
var employee = ...;
//no factories! Whether or not paycheck depends on location
//and how it is calculated (per-hour rate or monthly payments or w/e)
//is an implementation detail!
var payCheck = payCheckCalculator.Calculate(employee);
2. Calculators implement deduction rules. Should they, really? How would you re-use a rule in different implementations? The answer is - you copy-paste, as you have demonstrated in IrelandPayCheckCalculator.CalculateIncomeDeduction , IrelandPayCheckCalculator.CalculateUniversalSocialChargeDeduction and GermanyPayCheckCalculator.CalculateIncomeDeduction. A better approach, IMHO, is to come up with a common rule-set and reuse it in your calculators. Say:
interface IDeductionRule
{
void Apply(PayCheck payCheck);
}
class FlatDeductionRule : IDeductionRule {...}
class PercentageDeductionRule : IDeductionRule {...}
class SurplusPercentageDeductionRule : IDeductionRule {...}
//etc
You can easily reuse those rules by setting correct country-specific parameters (such as exact percentages or rule description) in constructor. And, even more importantly, you can now write proper unit-tests to test every rule in isolation.
3. DeductionType is enum. Why? Is it a fixed set of values, that is unlikely to change? No, it is the opposite. With every new country added, you will have to modify it, adding even more values to this already confusing list. Description property is more than enough.
4. Employee validation is part of Employee class. I think there two different approaches to validation, that serve different purpose. First is to implement validation inside the class to prevent developer from making mistakes. In this case validation rules should be simple and universal and exception type should be accurate (I want ArgumentNullException, if argument is null). Second is to implement separate validator, that would have a more complex and more specific validation rules that prevent user from making input mistakes. This validator can then be extended or replaced to further tweak its behavior depending on user environment. You go for the first approach, yet validation logic is too complex, exceptions are inaccurate and rules are not universal (for example, for some reason I am not allowed to use - or whitespace in my country's name).
5. There are some other things. PayCheck should probably be immutable. There are too many projects for such simple program (over-engineering). AbstractPayCheckCalculatorFactory<T> is useless, since it has single non-generic implementation (over-engineering). But those are minor issues.
@Nikita B has some good points. I'm just going to add my own thoughts on top of his, which may involve some repetition.
Deductions and DRY
I would say that you've violated the rule of DRY quite a bit. There is a lot of overlap between the kinds of deductions, but your Deduction class doesn't have any real functionality: it is just a dumb data carrier. I think a general architecture that will be easier to use/understand/support would break the deductions up by type, and then each country class would effectively just be a list of which deductions apply and how exactly they are configured. For instance, the primary deduction type would be "bracket" deduction, and you could imagine Ireland simply being the sum of 4 deductions (in pseudo-code):
public class IrelandPayCheckCalculator : PayCheckCalculator
{
private function getDeductions()
{
return [
new BracketDeduction( 'Income Tax', 0, 600, 0.25 ),
new BracketDeduction( 'Income Tax', 601, 0, 0.40 ),
new BracketDeduction( 'Universal Social', 0, 500, 0.07 ),
new BracketDeduction( 'Universal Social', 501, 0, 0.08 )
]
}
}
To be clear, the calling sequence for the BracketDeduction constructor is ( description, min_income, max_income, fractional_tax ), with max_income=0 implying no limit. This would then be the entirety of your IrelandPayCheckCalculator class. Each Deduction class would then be responsible for calculating the actual deduction, given the settings it was configured with when constructed plus a gross income passed to it. PayCheckCalculator would then just be responsible for storing the deductions and calling them to make the final calculations: no real calculation itself. The above example would work equally well for Germany, and then you would just have to come up with a custom deduction for Italy.
Obviously this particular implementation wouldn't improve your issues regarding inversion of control, but my point here is to highlight ways to better design your code so it repeats itself less.
General Flow of Data
I would also state the overall flow of your data is very confusing, and is going to be very hard to maintain in the long run. The short of it is that there is almost no encapsulation of data, and everyone is getting their hands dirty with everyone else's data. I'm not sure if there is really a good way to summarize this other than to point out the many ways in which it happens:
1. Your PayCheckCalculator.CalculateGrossAmount() directly sets the value of the paycheck.GrossSalary, a public property of another object. The paycheck should calculate its own gross salary, or gross salary should stay with the paycheck calculator. IMO, directly setting a public property of another class is a good way to cause maintenance issues down the road.
2. PayCheckCalculator.CalculateDeductions() is once again directly modifying the paycheck instance, this time by adding deductions onto its deductions array. This is what public getters and setters are for: they encapsulate the object data so that the class retains full control over what happens to it. A better process though would be for your calculate deductions function to simply return the deductions. Then the calling code can do whatever it wants with them.
3. You are passing along the full paycheck object to both PayCheckCalculator.CalculateDeductions() and PayCheckCalculator.CalculateIncomeDeduction(), despite the fact that all you ever use is payCheck.GrossSalary. This is creating an unnecessary coupling between these two components. Your PayCheckCalculator should really only be receiving the gross income as a number. There is no reason for it to get the entire object, only to pull the gross income out itself. If you ever decide to refactor payCheck, you can't do so without also refactoring PayCheckCalculator, but this wouldn't be the case if you only passed along the information the calculator needed rather than the full paycheck instance. Again, it's about separation of concerns and encapsulation: your classes know far too much about eachother.
4. PayCheckCalculator.CalculateNetAmount() is once again looping over the deductions that it set on the paycheck instance. You know you have a code organization problem when class A directly sets data in class B and then later on gets that data directly out of class B to use for its own purposes.
The short of it is that your classes or tightly coupled, poorly separated, and at risk of turning into a big ball of mud in the future. Some interfaces would help you: each class should have a well defined "contract" which it enforces through proper visibility rules. Never directly set public properties of another instance. I would say that you need to focus more on a clean separation of concerns, and really planning your system out before hand to make sure that each class has clearly defined responsibilities that adhere to the SRP.
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2019-10-15 15:27:51
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https://math.stackexchange.com/questions/855449/alternative-proof-for-elementary-number-theory-lemma
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# Alternative proof for elementary number theory lemma
Let $a, b \in \mathbb{N}, (a,b) =1$.Then for any $c \in \mathbb{N}, c \ne a, b$, there is an $m \in \mathbb{N}$ s.t. $$(c, a + bm) = 1$$
I can solve this using general principles, but found it as an exercise in a chapter of a book on representing integers using unique prime factorization. I would be curious to see an (ideally elegant) proof using only that.
• There is some condition missing, as is, choose $a = 2, b = 4, c = 6$. Jul 3 '14 at 15:06
• Thanks. I forgot one. Hold on. @DanielFischer (it's not hard to see how it's true, and how to show it; but I fail to see how the fundamental theorem of arithmetic would be a natural (elegant) entry point) Jul 3 '14 at 15:09
This can be solved intuitively by using a slight twist on Euclid's idea for generating new primes. Euclid employed $\,1 + p_1\cdots p_n$ is coprime to $\,c = p_1\cdots p_n.\,$ Stieltjes noted the generalization that, furthermore, $\ \color{#c00}{p_1\cdots p_k} +\, \color{#0a0}{p_{k+1}\cdots p_n}\,$ is coprime to $\,c\,$ too, which motivates the following
Key Idea $\,$ Coprimes to $\,c\,$ arise by partitioning into $\rm\color{#c00}{two}\ \color{#0a0}{summands}$ all prime factors of $\,c,\,$ i.e.
Theorem $\,\ \ \color{#c00}a+\color{#0a0}b\$ is coprime to $\ c\:$ if every prime factor of $\,c\,$ divides $\,a\,$ or $\,b,\,$ but not both.
Proof $\$ If not then $\,a+b\,$ and $\,c\,$ have a common prime factor $\,p.\,$ By hypothesis $\,p\mid a\,$ or $\,p\mid b.\,$ Wlog, say $\,p\mid b.\,$ Then $\,p\mid (a+b)-b = a,\,$ so $\,p\,$ divides both $\,a,b,\,$ contra hypothesis. QED
Suppose $\,(a,b,c)= 1.\,$ We seek $\,\color{#c00}{a}+\color{#0a0}{bm}\,$ coprime to $\,c,\,$ so it suffices to choose $\,m\,$ such that each prime factor $\,p\,$ of $\,c\,$ divides exactly one of $\,a\,$ or $\,bm.\,$ Note $\,p\,$ can't divide both $\,a,b,\,$ else $\,p\mid a,b,c.\,$ So it suffices to let $\,m\,$ be the product of primes in $\,c\,$ that do not occur in $\,a\,$ or in $\,b.\ \$ QED
Remark $\$ Note how the solution becomes quite obvious after employing Stieltjes idea, amounting to nothing but a trivial calculation of a difference of sets (of primes), i.e. the problem reduce from a problem in number theory to a trivial problem in set theory.
• That is elegant. :) My proof isn't all that different, but this is so nice and clear. Tyvm. Jul 3 '14 at 15:21
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2022-01-24 23:28:29
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http://hobbydocbox.com/Radio/74126342-An-analysis-of-lms-and-mvdr-on-beamforming-applications.html
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# AN ANALYSIS OF LMS AND MVDR ON BEAMFORMING APPLICATIONS
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1 AN ANALYSIS OF LMS AND MVDR ON BEAMFORMING APPLICATIONS EE635 : Digital Signal Processing II, Spring 2000 University of New Haven Instructor: Dr. Alain Bathelemy Students : Raheela AMIR,Wiwat THARATEERAPARB Contents Chapter 1 : Introduction 1 Chapter 2 : LMS-MVDR Application Responses.3 Chapter 3 : MVDR Spatial Power Spectrum.7 Appendix.. 11
2 Chapter 1 Introduction Adaptive Beamforming Beamforming is another name for spatial filtering where an array of sensors together with appropriate signal processing can either direct or block the radiation or the reception of signals in specified directions. Adaptive beamformers are the one in which arrays adapt to noise and interfering signals in such a way that it places nulls in the direction of the source of interference automatically and in real time. By doing so, the output signal to noise ratio of the system increases and the directional response of the system is improved. In order to adapt the arrays for the optimum response, data received form the output of the sensors is used. Figure 1: An illustration of incident waves on sensor elements For our simulation, we consider an adaptive minimum variance distortionless response (MVDR) beamformer consisting of a linear array of five uniformly spaced sensors. The spacing d between the adjacent elements of the arrays equals one half of the received wavelength so as to avoid the appearance of grating lobes. The beamformer operates in an environment that consists of two components: a target signal impinging on the array along a direction of interest, and a single
3 source of interference originating from an unknown direction. It is further assumed that these two components originate from independent sources and that the received signal includes additive white Gaussian noise at the output of each sensor. Figure 2 : Linear array antenna for simulation The angle of incidence of the target and interfacing signal, measured in radians with respect to the normal to the line of the array are as follows: (i) Target signal: θ target = sin -1 (-0.2) (ii) Interference: θ interf = sin -1 (0) A beamformer with optimum weight is constrained to pass the target signal with unit response while at the same time minimize the total output variance. The variance minimization process attenuates interference and noise not originating from the target. The optimum weight for this constrained beamformer is defined as: 1 R s( φ 0 ) 1 ( φ ) R s( φ ) w0 = H s 0 0 Where M is the number of sensors and R is M by M correlation matrix and w 0 is M by 1 optimum weight vector. M by 1 steering vector s(φ) is defined by: jφ jφ ( M 1) T s ( φ) = [1, e,..., e ]
4 Chapter 2 LMS-MVDR Amplitude Responses LMS algorithm is used to adapt the weights of the beamformer close to the optimum solution. An analysis has been performed by plotting the adapted spatial response of MVDR beamformer by varying interference to noise ratio, signal to noise ratio, number of iterations and step size value of LMS. Figure 3: LMS-MVDR beamformer, INR=20dB,TNR=10dB Figure 3 shows an amplitude spatial response of LMS-MVDR beamformer with interference to noise ratio of 20dB while target to noise ratio is 10dB. The response at target angle (-0.2) is flat (0 db) while at interference angle (0) is more then 7 db.
5 Figure 4 displays amplitude spatial response with varying interference to noise ratio. As it can be observed that stronger the interference signal is compared to noise (higher db value) better the interference signal is attenuated. In other words, as the ratio of interference-to-target increases, the beamformer improves. Figure 4: LMS-MVDR beamformer, varying of INR with numits=100 Figure 5 and Figure 6 perform the same analysis as in Figure 4. The only difference is the number of iterations. As it can be observed that for lower interference to noise ratio (INR = 20 db), higher number of iterations helps in attenuating the interference signal more (see the red line of INR=20dB).
6 Figure 5: LMS-MVDR beamformer, varying of INR with numits=500 Figure 6: LMS-MVDR beamformer, varying of INR with numits=1000
7 Figure 7: LMS-MVDR beamformer, INR=20dB, TNR=10dB and varying of iterations Figure 7 displays varying number of iterations with all the other parameters remain the same(inr and TNR are help fixed at 20 and 10 db respectively). The interference-nulling capability is better as the higher number of iterations. Conclusion of this chapter: Response of the MVDR beamformer always show value of unity (0 db) at the angle of incidence of target (- 0.2) no matter what set of parameters are chosen. The ability to cancel the interference signal depends on (i) The number of iterations (snapshots of data). (ii) By increasing interference-to-target signal ratio.
8 Chapter 3 MVDR Spatial Power Spectrum An M by M autocorrelation matrix R is generated based on the input received data from the sensors. The spatial MVDR power spectrum can be calculated based on the following formula S MVDR () φ = s H 1 1 () φ R s( φ) M by 1 steering vector s(φ) is defined by : jφ jφ ( M 1) T s ( φ) = [1, e,..., e ] Where M is the number of sensors. Figure 8: MVDR power spectrum, INR=20dB, TNR=10 db
9 Figure 8 displays for the case of interference to noise ratio equals to 20 db while target to noise ratio is 10 db. It shows strong interference power at angle 0 compared to target signal power at angle 0.2. When increasing the INR to 40dB, the MVDR power spectrum is shown in figure 9. Figure 9: MVDR power spectrum, INR=40dB, TNR=10 db At this point, we can see that if the interference-to-target ratio increases, the MVDR power spectrum yields the poor response compared to LMS-MVDR amplitude response. This is because the MVDR power spectrum does not have interferece-nulling capability like LMS-MVDR. Actually, the MVDR power spectrum strongly depends on the target signal-to-noise ratio. By increasing the TNR to 40 and 50 db in figure 10 and figure 11, the MVDR power spectrum response can distinguish the location of the target and the interference properly.
10 Figure 10: MVDR power spectrum, INR=20dB, TNR=40 db Figure 11: MVDR power spectrum, INR=20dB, TNR=50 db
11 Conclusion From the works we have done so far, we conclude that: 1. The LMS algorithm can be applied to MVDR application. This so-called LMS-MVDR has the capability of interference-nulling. The amplitude response of LMS-MVDR will suppress the interfered signal and have the target amplitude held at 0dB. Moreover, the amplitude response can be improved by: (i) Increasing the number of iterations (snapshots). (ii) Increasing the interference-to-target signal ratio. 2. The spatial power spectrum MVDR shows the power of incoming signals but it tries to maximize the interfered signal instead of suppressing. The drawback is the target signal can be overpowered by the interference in the MVDR power spectrum.
12 Appendix MATLAB Script clear all; close all; clc; p = 5; % number of sensors Ninit = p; % nuber of smaples needed for initialization Nsnaps = 200; % number of snapshots or iteration Ndata = Ninit + Nsnaps; % total number of data mean_v = 0; % white noise mean var_v = 1; % white noise variance TNRdB = 10; % target signal to noise ratio INRdB = 40; % interference signal to noise ratio numst = 301; % resolution in spatial response sin_theta = [-0.2 0]; % location of signal and interference phi = pi.*sin_theta; % equivalent electrical angle A = sqrt(var_v)*10.^([tnrdb INRdB]./20); % parameter for target/interference signal amplitude, angle % steering vector along electrical angle of look direction of interest e = exp(-1j*[1:(p-1)]'*phi(1)); % setup input and output sequences sig_x = A(1)*exp(1j*[1:p]*phi(1)); for i = 1:Ndata, % random disturbances v_tmp = sqrt(var_v/2)*randn(2,p)+mean_v; v = v_tmp(1, :)+ 1j*v_tmp(2, :); % additive white noise of desired mean/variance Psi = 2*pi*rand; % uniform random phase on interference Xi(i, :) = sig_x + A(2)*exp(1j*[1:p]*phi(2) + Psi) + v; end; % setup effective desired output and input vectors from original data g = 1; % assume unity gain d = g*xi(:,1); u = diag(xi(:,1))*(ones(ndata,1)* e.')-xi(:,2:p); mu = 1e-10; % step-size parameter % run beamformer for indicated number of snapshots [W,xp] = lms(u,d,mu); Wo = g- W * conj(e); W = [Wo W]; % now, generate test vectors to compute spatially sampled response W_H = conj(w(ndata, :)); % Hemitian transpose of last one st = linspace(-1,1,numst); % sine(theta) space est = exp(-1j*pi*[0:(p-1)]'*st); % steering matrix % amplitude response P = 20*log10(abs(W_H*est).^2); % MVDR (spatial) Power Spectrum % R = (Xi'*Xi)/Ndata; % correlation matrix from generated data R_INV = inv(r); sintheta = linspace(-1,1,numst); for i = 1:numst, s = exp(-1j*pi*[0:(p-1)]'*sintheta(i)); P_MVDR(i) = abs(1/(s'*r_inv*s)); end; P_MVDR = 20*log10(P_MVDR./max(P_MVDR)); % normalized MVDR % % % plotting part
13 figure; p1 = plot(st,p,'r-',[sin_theta(1) sin_theta(1)],[min(p) max(p)],'r--',[sin_theta(2) sin_theta(2)],[min(p) max(p)], 'r-.'); set(p1, 'LineWidth', 3); %axis([-1 1 min(p) max(p)]); axis([ ]); xlabel('sin \theta'); ylabel('amplitude response (db)'); legend('amplitude response', 'Signal', 'Interference'); title(['mvdr beamfomer, numits = ',num2str(nsnaps),',\mu = ',num2str(mu),',tnr = ',num2str(tnrdb),',inr =',num2str(inrdb)]); figure; p2 = plot(sintheta,p_mvdr,'r-',[sin_theta(1) sin_theta(1)],[min(p_mvdr) max(p_mvdr)],'r--',[sin_theta(2) sin_theta(2)],[min(p_mvdr) max(p_mvdr)], 'r-.'); set(p2, 'LineWidth', 3); %axis([-1 1 min(p_mvdr) max(p_mvdr)]); axis([ ]); xlabel('sin \theta'); ylabel('power spectrum: P_{MVDR}(\theta) (db)'); legend('mvdr spectrum', 'Signal', 'Interference'); title(['mvdr powerspectrum beamfomer, numits = ',num2str(nsnaps),',\mu = ',num2str(mu),',tnr = ',num2str(tnrdb),',inr =',num2str(inrdb)]); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function [W, e] = lms(u, d, mu); % Maximum number of time step that can be predicted N = min(size(u, 1),size(d, 1)); Nin = size(u,2); Nout =size(d,2); % Intializatize weight matrix and associated parameters for LMS predictor w = zeros(nout, Nin); W = []; for n=1:n, W = [W;w]; % Predict next sample and error xp(n, :) = u(n,:)*w'; e(n,:) = d(n,:)-xp(n,:); % Adapt weight matrix ans step size w = w + mu * e(n,:)' * u(n,:); end; % for n t return;
### DIRECTION OF ARRIVAL ESTIMATION IN WIRELESS MOBILE COMMUNICATIONS USING MINIMUM VERIANCE DISTORSIONLESS RESPONSE
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### Comparison of LMS Adaptive Beamforming Techniques in Microphone Arrays
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2020-02-22 18:54:32
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https://webwork.libretexts.org/webwork2/html2xml?answersSubmitted=0&sourceFilePath=Library/NAU/setCalcI/secondDerProduct.pg&problemSeed=1234567&courseID=anonymous&userID=anonymous&course_password=anonymous&showSummary=1&displayMode=MathJax&problemIdentifierPrefix=102&language=en&outputformat=libretexts
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Let $f(x) = (-3x^{2}+x-3) e^x$. Compute the derivative.
$f'(x) =$ .
Simplify this before finding the second derivative.
$f''(x) =$ .
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2022-05-19 06:08:42
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https://www.physicsforums.com/threads/stats-probability.94467/
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# Stats, probability
1. Oct 13, 2005
### Dr-NiKoN
1 out of 5 italians speak english. 1 out of 5 people in italy are tourists. 1 out 2 tourists speaks english.
You meet a english-speaking person in italy, what is the probability that this person is italian.
The way I see the "population":
$P(I) = \frac{2}{10}$ are italians who speaks english.
$P(T) = \frac{1}{10}$ are tourists that speaks english.
You have meet someone from P(I) + P(T), what is the probability that this person is italian?
That probability equates to:
2x + 1y = 1 (x => english-speaking italian, y => english speaking tourist)
Where do I go from here?
2. Oct 13, 2005
### mathman
Your P(I) is wrong. It should be 4/5(italians) x 1/5 (speak English).
Net result P(I)=.16, P(T)=.1.
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2017-04-25 12:54:25
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https://www.gamedev.net/forums/topic/499448-include-so-called-best-practice/
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Public Group
# #include so called 'Best practice'
This topic is 3735 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.
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The directory structure is part of your code organization, period. It's not a matter of flexibility: You've decided that a particular header is in a particular directory and if you change that you'll need to change the references, just like renaming variables.
One moderating factor to that, though... keeping track of ..s is annoying and doesn't lend clarity. I'm in favor of putting the base source directory in the include path, so you can do "physics/cv/capsule.h" instead of "../../../physics/cv/capsule.h".
BTW, you're not actually putting all your header files in one directory, and your source files in another directory..right?
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Quote:
Original post by SneftelThe directory structure is part of your code organization, period. It's not a matter of flexibility: You've decided that a particular header is in a particular directory and if you change that you'll need to change the references, just like renaming variables.
Sounds like you're purporting option 1 :-P
Quote:
BTW, you're not actually putting all your header files in one directory, and your source files in another directory..right?
Well, I was thinking something like:
Project\Graphics\Source
Project\Audio\Source
etc
I do feel somewhat mystified about what a good directory structure would look like - I've always been rather lazy when it comes to tidying files away into folders. [rolleyes]
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Quote:
Original post by dmatterWell, I was thinking something like:Project\Graphics\HeaderProject\Graphics\SourceProject\Audio\HeaderProject\Audio\SourceetcI do feel somewhat mystified about what a good directory structure would look like - I've always been rather lazy when it comes to tidying files away into folders. [rolleyes]
Here's what will happen:
Why are you stating it's a header when it's obvious from #define and .hpp part?
To separate, the basic layout is usually like this:
/include /Audio /Video /Util/src /Audio /Video /DX /OGL// no util, header only
You then put '/include' into include path, and reference headers directly:
#include "Audio/foo.hpp"
#include "Video/video_if.hpp"
#include "Util/baz.hpp"
But this is a topic where YMMV, since it depends on many factors, starting with choice of build and version management system.
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Quote:
Original post by SneftelI'm in favor of putting the base source directory in the include path, so you can do "physics/cv/capsule.h" instead of "../../../physics/cv/capsule.h".
That's what I do as well.
Quote:
Original post by SneftelBTW, you're not actually putting all your header files in one directory, and your source files in another directory..right?
I never got why people do that. I find it confusing. I like to keep a 'module' (i.e., a source and header file) together in a directory. I always sort project directory views by file type anyway, seperating the headers and sources in two lists.
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Quote:
Original post by dmatterProject\Graphics\HeaderProject\Graphics\Source
Why? You can simply sort your files by extension, if you want a filtered list of the .h files.
What makes (some) sense is to separate out the public headers into a different dir, when building a library (which I very much suggest... static libraries are the best form of project-wide modularity). That still doesn't require much directory munging, though, since the public header just needs to include some private libraries (with directory stuff), and the rest of the library doesn't know about the public header at all.
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Original post by Mike nlI never got why people do that. I find it confusing. I like to keep a 'module' (i.e., a source and header file) together in a directory. I always sort project directory views by file type anyway, seperating the headers and sources in two lists.
This becomes a problem if you try to deploy any kind of pre-built libraries, where you need to ship only headers.
With deeply nested layout, you have a lot of hassle taking out the headers only. If you use separate include directory, you have all of them in one place already.
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Quote:
Original post by AntheusWith deeply nested layout, you have a lot of hassle taking out the headers only.
I don't follow you... this is merely an issue of filtering header files by extension in a directory tree. Where is the hassle in that?
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Quote:
Original post by dmatter// Additional directory is: \Header Files#include "Header.hpp"
Bear in mind that on big projects this approach can slow down your compiles since everytime the compiler wants to find an included file it has lots of additional directories to scan rather than just a relative path off a few base directories.
Any single-man project is unlike to get that huge though.
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On disk, I have all my own files in one folder. Anything that I did not write is in a subfolder along with a copy of whatever licence it comes with and whatever else is necessary.
Then, in my IDE, I create filters for each group of files. The result is that I can find where everything is using the IDE, and I can see clearly which things are mine and which are not when I look at the directory structure.
This works well because then, in my #include directives, i either just use #include "ClassFoo.h" for my own stuff, or #include "SOIL/Soil.h" for the image library I am using. My current one man project, which is 6 months old, has 135 files that I wrote, plus another 100 or so that are part of libraries.
so, my folder is like:
+Source
|+SOIL
|+Bullet
|+SDL
|-ClassFoo.h
|-ClassFoo.cpp
+Build
Quote:
/include /Audio /Video /Util/src /Audio /Video /DX /OGL
I do not see the benefit of this method, but I am currenlty porting to VC9 so I am open to any suggestions.
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2018-09-20 12:26:24
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https://www.lmfdb.org/EllipticCurve/2.2.5.1/605.1/
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## Results (18 matches)
Label Class Base field Conductor norm Rank Torsion CM Weierstrass equation
605.1-a1 605.1-a $$\Q(\sqrt{5})$$ $$5 \cdot 11^{2}$$ $0$ $\Z/2\Z$ ${y}^2+\left(\phi+1\right){x}{y}+{y}={x}^{3}+\phi{x}^{2}+\left(4\phi-13\right){x}-1591\phi-1011$
605.1-a2 605.1-a $$\Q(\sqrt{5})$$ $$5 \cdot 11^{2}$$ $0$ $\Z/4\Z$ ${y}^2+\phi{x}{y}+{y}={x}^{3}+\left(\phi-1\right){x}^{2}+\left(\phi-1\right){x}-\phi+2$
605.1-a3 605.1-a $$\Q(\sqrt{5})$$ $$5 \cdot 11^{2}$$ $0$ $\Z/2\Z\oplus\Z/4\Z$ ${y}^2+\phi{x}{y}+{y}={x}^{3}+\left(\phi-1\right){x}^{2}+\left(\phi-6\right){x}+\phi-2$
605.1-a4 605.1-a $$\Q(\sqrt{5})$$ $$5 \cdot 11^{2}$$ $0$ $\Z/2\Z\oplus\Z/2\Z$ ${y}^2+\phi{x}{y}+{y}={x}^{3}+\left(\phi-1\right){x}^{2}+\left(36\phi-91\right){x}+147\phi-322$
605.1-a5 605.1-a $$\Q(\sqrt{5})$$ $$5 \cdot 11^{2}$$ $0$ $\Z/4\Z$ ${y}^2+\phi{x}{y}+{y}={x}^{3}+\left(\phi-1\right){x}^{2}+\left(-34\phi-1\right){x}+83\phi+2$
605.1-a6 605.1-a $$\Q(\sqrt{5})$$ $$5 \cdot 11^{2}$$ $0$ $\Z/2\Z$ ${y}^2+\phi{x}{y}+{y}={x}^{3}+\left(\phi-1\right){x}^{2}+\left(586\phi-1466\right){x}+10927\phi-22432$
605.1-b1 605.1-b $$\Q(\sqrt{5})$$ $$5 \cdot 11^{2}$$ $0$ $\Z/2\Z$ ${y}^2+\phi{x}{y}={x}^{3}+\left(\phi-1\right){x}^{2}+\left(-3\phi-10\right){x}+1581\phi-2595$
605.1-b2 605.1-b $$\Q(\sqrt{5})$$ $$5 \cdot 11^{2}$$ $0$ $\Z/4\Z$ ${y}^2+\left(\phi+1\right){x}{y}={x}^{3}+\phi{x}^{2}+\left(35\phi-34\right){x}-84\phi+119$
605.1-b3 605.1-b $$\Q(\sqrt{5})$$ $$5 \cdot 11^{2}$$ $0$ $\Z/2\Z\oplus\Z/4\Z$ ${y}^2+\left(\phi+1\right){x}{y}={x}^{3}+\phi{x}^{2}-4{x}-7\phi-2$
605.1-b4 605.1-b $$\Q(\sqrt{5})$$ $$5 \cdot 11^{2}$$ $0$ $\Z/4\Z$ ${y}^2+\left(\phi+1\right){x}{y}={x}^{3}+\phi{x}^{2}+{x}$
605.1-b5 605.1-b $$\Q(\sqrt{5})$$ $$5 \cdot 11^{2}$$ $0$ $\Z/2\Z\oplus\Z/2\Z$ ${y}^2+\left(\phi+1\right){x}{y}={x}^{3}+\phi{x}^{2}+\left(-35\phi-54\right){x}-238\phi-211$
605.1-b6 605.1-b $$\Q(\sqrt{5})$$ $$5 \cdot 11^{2}$$ $0$ $\Z/2\Z$ ${y}^2+\left(\phi+1\right){x}{y}={x}^{3}+\phi{x}^{2}+\left(-585\phi-879\right){x}-12393\phi-12091$
605.1-c1 605.1-c $$\Q(\sqrt{5})$$ $$5 \cdot 11^{2}$$ $1$ $\Z/2\Z$ ${y}^2+\phi{x}{y}={x}^{3}+\left(-\phi-1\right){x}^{2}+\left(48\phi-38\right){x}-1458\phi-1109$
605.1-c2 605.1-c $$\Q(\sqrt{5})$$ $$5 \cdot 11^{2}$$ $1$ $\Z/4\Z$ ${y}^2+{x}{y}={x}^{3}-{x}^{2}+{x}$
605.1-c3 605.1-c $$\Q(\sqrt{5})$$ $$5 \cdot 11^{2}$$ $1$ $\Z/2\Z\oplus\Z/4\Z$ ${y}^2+{x}{y}={x}^{3}-{x}^{2}-4{x}+3$
605.1-c4 605.1-c $$\Q(\sqrt{5})$$ $$5 \cdot 11^{2}$$ $1$ $\Z/2\Z\oplus\Z/2\Z$ ${y}^2+{x}{y}={x}^{3}-{x}^{2}-29{x}-52$
605.1-c5 605.1-c $$\Q(\sqrt{5})$$ $$5 \cdot 11^{2}$$ $1$ $\Z/4\Z$ ${y}^2+{x}{y}={x}^{3}-{x}^{2}-59{x}+190$
605.1-c6 605.1-c $$\Q(\sqrt{5})$$ $$5 \cdot 11^{2}$$ $1$ $\Z/2\Z$ ${y}^2+\left(\phi+1\right){x}{y}+\left(\phi+1\right){y}={x}^{3}+{x}^{2}+\left(-48\phi+9\right){x}+1410\phi-2558$
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2022-08-12 06:47:39
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https://z2pack.greschd.ch/en/latest/reference/hm.html
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# Models described by a Hamiltonian matrix¶
This module contains a class for creating Systems which are described by a Hamiltonian matrix (hm), such as k•p models.
class z2pack.hm.System(hamilton, *, dim=3, pos=None, bands=None, hermitian_tol=1e-06, basis_overlap=None, convention=2, check_periodic=False)[source]
This class is used when the system can be explicitly described as a matrix Hamiltonian $$\mathcal{H}(\mathbf{k})$$.
Parameters
• hamilton (collections.abc.Callable) – A function taking the wavevector k (list of length 3) as an input and returning the matrix Hamiltonian.
• dim (int) – Dimension of the system.
• pos (list) – Positions of the orbitals w.r.t the reduced unit cell. Per default, all orbitals are put at the origin.
• bands (int or list) – Specifies either the number of occupied bands (if it is an integer) or which bands should be taken into consideration (if it is a list of indices). If no value is given, half the given bands are considered.
• hermitian_tol (float) – Maximum absolute value in the difference between the Hamiltonian and its hermitian conjugate. Use hermitian_tol=None to deactivate the test entirely.
• basis_overlap (collections.abc.Callable) – A function taking the wavevector k (list of length 3) as an input and returning the overlap matrix between the basis vectors w.r.t which the Hamiltonian is defined. If no value is given, the basis is assumed to be orthonormal.
• convention (int) – The convention used for the Hamiltonian, following the pythtb formalism. Convention 1 means that the eigenvalues of $$\mathcal{H}(\mathbf{k})$$ are wave vectors $$\left|\psi_{n\mathbf{k}}\right>$$. With convention 2, they are the cell-periodic Bloch functions $$\left|u_{n\mathbf{k}}\right>$$.
• check_periodic (bool) – Evaluate the Hamiltonian at $$\{0, 1\}^d$$ as a simple check if it is periodic. Note that this does not work if the Hamiltonian is written such that the eigenstates acquire a phase when being translated by a lattice vector.
get_eig(kpt)[source]
Returns the periodic part of the eigenstates at each of the given k-points. The eigenstates are given as columns in a 2D array.
Parameters
kpt (list) – The list of k-points for which the eigenstates are to be computed.
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2021-03-01 16:19:58
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http://martianworlds.com/v4j7vz9/l4zjk3w.php?tag=d98099-statistics-class-10-important-questions-with-solutions
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# statistics class 10 important questions with solutions
Solution: Question 51. Class 10 Maths MCQs Chapter 15 Probability MCQ On […] NCERT Exemplar Class 10 Maths is very important resource for students preparing for X Board Examination. (2013) Putting the value of x in (i), we have Modal class =40-50 Because almost all the scientific field/ colleges ask for good marks in CBSE Class 10th Maths Subject at the entry level. Here we have provided NCERT Exemplar Problems Solutions along with NCERT Exemplar Problems Class 10.. Solution: Question 12. (2012) Free PDF Download - Best collection of CBSE topper Notes, Important Questions, Sample papers and NCERT Solutions for CBSE Class 10 Math Statistics. 7-10. Question 49. Solution: Question 6. Question 14. Solution: 10th Math’s chapter Statistics … If it is given that mean is 59, find the missing frequencies x and y. (b) less than ogive for the following data Solution: Question 23. Solution: Question 9. Question 41. Draw a more than ogive for the data given below which gives the marks of 100 students: (2012, 2017D) Average score of boys = 71 Maths Important Questions with Solution for Class 10 PDF Download Mathematics also plays just as significant role as the Science plays in helping students deciding their career. Solution: Putting the value of f1 in (i), we get of girls = 3 : 2. Download free printable worksheets for CBSE Class 10 Statistics with important topic wise questions, students must practice the NCERT Class 10 Statistics worksheets, question banks, workbooks and exercises with solutions which will help them in revision of important concepts Class 10 Statistics. Solution: Question 16. (2013) Question 50. Solution: Question 37. x + y = 70 – 54 = 16 …(i) Median Class=30-40, Question 54. average score of the school in examination = 71.8. In this chapter, we will table for less than ogive’ and ‘more than ogive’ Mode of the following frequency distribution is 65 and sum of all the frequencies is 70. The mean of the following frequency distribution is 62.8. Solution: Question 7. 54 + x + y = 70 … [Given Verify it by actual calculations. Solution: Question 30. If each observation is increased by 5, then find the new median. Mode = 28.8 – 21 = 7.8. 2016 f2 = 47 – f1 ⇒ 36 = 18x ⇒ x = 2 Important Questions for Class 10 Maths Chapter 15 Probability with solutions includes all the important topics with detailed explanation that aims to help students to score more marks in Board Exams 2020. From the following cumulative frequency table, construct a frequency distribution table: (2013) All the solutions are explained by subject experts and are quite helpful to clear all the concepts. Download HOTs Questions for Class 10 Statistics. Solution: Question 43. Solution: Question 24. All NCERT Questions are solved, with detailed answers of each and every question and example of the NCERT Book. Visit to Class 10 Maths Solutions to see the solutions of other chapters. Solution: Question 5. Solution: Question 41. e) If we replace the data value 6 in the data set above by 24, will the standard … Solution: The median of the following data is 525. Determine the modal weight. Solution: Question 21. NCERT Exemplar Class 10 Maths is very important resource for students preparing for X Board Examination. (2013) The detailed, step-by-step solutions … Find mode, using an empirical relation, when it is given that mean and median are 10.5 and 9.6 respectively. Given frequency distribution is not continuous. Question 11. Convert the above distribution to a less than type cumulative frequency distribution. Question from very important topics are covered by NCERT Exemplar Class 10.You also get idea about the type of questions and method to answer in your Class 10th … If median height of 50 students of a class in the following frequency distribution is 144 cm, find the missing frequencies x and y Question 46. ... mode or median of grouped data from our NCERT solutions. Find the mean of the following distribution by Assumed Mean Method: (2015) Find the mean, mode and median of the above data. The mean of the following frequency distribu tion is 53. Convert the following data to a less than type distribution. ∴ Modal class is 160 – 165 f2 = 20 – 8 = 12 Draw ‘less than ogive’ and ‘more than ogive’ for the following distribution and hence find its median. The last two numbers are 10. Also, obtain median from the curve. Vedantu Class 9 & 10 243,842 views 53:45 3(9.6) = Mode + 2(10.5) MCQ Questions for Class 10 Maths with Answers was Prepared Based on Latest Exam Pattern. Solution: Question 42. curves intersect at(80.4,17.5) Find the mean and median age of patients. The lengths of leaves of a plant are measured correct to the nearest mm and the data obtained is represented as the following frequency distribution: (2015) The average score of boys in the examination of a school is 71 and that of the the girls is 73. If the mean of the following frequency distribution is 65.6, find the missing Solution: Question 39. In this, we are going to discuss the important statistical concepts, such as grouped data, ungrouped data and the measures of central tendencies like mean, median and mode, methods to find the mean, median and mode, the relationship between them with more examples. A medical camp was held in a school to impart health education and the importance of excercise to children. Change the above data into a continuous grouped frequency distribution The median of the distribution given below is 14.4.Find the values of the x,y.if the sum of frequency is 20. mode=65 Find the model size of the shoes sold. Solution: Question 31. Solution: Maths Important Questions with Solution for Class 10 PDF Download Mathematics also plays just as significant role as the Science plays in helping students deciding their career. we notice both curves intersect at (60,50) Balbharati solutions for Mathematics 1 Algebra 10th Standard SSC Maharashtra State Board chapter 6 (Statistics) include all questions with solution and detail explanation. (2012) Extra Questions for Class 10 Maths. Solution: Question 10. we notice curves intersect at(58.3,50) Find the mode of the following frequency distribution: If it is given that mean literacy rate is 63.5, then find the missing frequencies x and y. 2. The following distribution given below gives the daily income of 50 workers in a factory Data regarding heights of students of Class X of Model School, Dehradun is given below. Because almost all the scientific field/ colleges ask for good marks in CBSE Class … Observe the steps to compute the median, median and mode as per the data presented in a Maths question. All the solutions are explained by subject experts and are quite helpful to clear all the concepts. (2014) Solution: School held sports day in which 150 students participated. Draw a less than type ogive of the following distribution: l=60,f=12,f0=f1,f2=6,h=20. Mean . Heights of students of class X are given in the following frequency distribution: (2014) ⇒ -4f1 = 257 – 329 = -72 Solutions of all questions of Chapter 14 Statistics of Class 10 available free at teachoo. Convert the above distribution to a less than type cumulative frequency distribution Solution: Question 11. [Answer: 16.4] The … The average score of the school in the examination is 71.8. ⇒ -4f1 = 108 – 140 = -32 . Draw a ‘more than type’ ogive for the above data. Compute the modal weight. Solution: median=58.30. NCERT Exemplar Class 11 Maths is very important resource for students preparing for XI Board Examination. Ages of students are given in the following frequency distribution: (2014) ⇒ 18 – x = 90 – 10x For the following distribution, draw a ‘more than Ogive’ and hence find the median: (2013) In Class … Learning Maths … NCERT Solutions for Class 10th: Ch 14 Statistics Math. The Median of the following distribution is 35.Find the value of x: where frequency 6,8,f1, and 12 are in ascending order Question 57. ⇒ 1978 + 66.5x + 338 – 84.5x = 2280 ⇒ 2060 + 30f1 + 70f2 = 3140 S2 : 7, 4, 7, 8, 7, 8, 13 The width of 50 leaves of a plant were measured in mm and their cumulative frequency distribution is shown in the following table. Find the missing frequency x Question from very important topics are covered by NCERT Exemplar Class 11.You also get idea about the type of questions and method to answer in your Class 11th … The data may be individual, discrete and continuous .The method of calculating the means depend upon the nature of data. MCQ on Statistics Class 10 Question … Free PDF download of Important Questions with solutions for CBSE Class 10 Maths Chapter 14 - Statistics prepared by expert Mathematics teachers from latest edition of CBSE(NCERT) books. In a frequency distribution, if a = assumed mean = 55,∑ fi = 100, h = 10 and ∑ƒiμi = -30, then find the mean of the distribution Total score of boys in the examination of the school = 71xx = 71x What is the mean of first 12 prime numbers? Extra Questions for Class 10 Maths Chapter 14 Statistics. All NCERT Questions are solved, with detailed answers of each and every question and example of the NCERT Book. Draw ‘less than ogive’ and ‘more than ogive’ for the following distribution and hence find its median 13-16. Draw: (a) more than ogive and (2017OD) These NCERT Solutions for Class 10 are prepared by Studyrankers experts who have great experience in teaching students. Get accurate NCERT Solutions for Class 10 Maths Chapter 14 - Statistics. Statistics Class 10 Important Questions Very Short Answer (1 Mark) Question 1. During this camp, a medical check of 35 students was done and their weights were recorded as follows: (2016) Class 10 Detailed Chapter Notes - Statistics, Class 10, Maths | EduRev Notes Summary and Exercise are very important for perfect preparation. Question 16. This will clear students doubts about any question and improve application skills while preparing for board exams. Solution: We know that, P(E) + P(not E) = 1. Class 10 Maths MCQs Chapter 14 Statistics. ⇒ 3f1 +7(47 – f1) = 257 . The following table gives the literacy rate (in %) in 40 cities. ⇒ 2730 + 30f1 + 70f2 = 5300 Free PDF Download of CBSE Class 10 Maths Chapter 11 Geometrical Constructions Multiple Choice Questions with Answers. (2015) Solution: New median = 21 + 5 = 26. Solutions of all questions of Chapter 14 Statistics of Class 10 available free at teachoo. Following is the age distribution of dengue patients admitted in a hospital during a week of October, 2013: (2014) Solution: Question 19. ⇒ 3f1 + 7f2 = 108 …[Dividing by 10 (2014) Very Short Answer Type Questions [1 Mark], Question 1. Question 40. Find the values of x and y if the median for the following data is 31. The test will consist of only objective type multiple choice questions requiring students to mouse-click their correct choice of the options against the related question … Solution: The given mean of 10 numbers = 15 The following data gives the information on the observed life times (in hours) of 150 electrical components In the Statistics chapter of Class 9, we learned how to find mean, median, mode of raw and ungrouped data. For helping poor girls of their class, students saved pocket money as shown in the following table: Solution: Question 12. Find the Mean area held by a family. These NCERT Solutions for Class 10 … The mean of the following data is 18.75. ⇒ 30f1 + 70f2 = 3140 – 2060 = 1080 Given below is the distribution of weekly pocket money received by students of a class. Solution: Question 13. Find the mode of the following data ⇒ f2 = 47 – 18 = 29 Question 55. Students can solve NCERT Class 10 Maths Statistics MCQs with Answers to know their preparation level. Solution: You can also download NCERT Solutions For Class 10to help you to revise complete syllabus and score more marks in your examinations. Ramesh is a cricket player. The average score of boys in the examination of a school is 71 and that of the girls is 73. Solution: Solution: Chapter Wise Important Questions Class 10 Math’s. Find the mean, median and mode of the following data: If the mean of the following distribution is 50, find the value of p: (2013) median=80.4. ⇒ x = 8 Write the median class of the following distribution : Find the median by drawing both types of gives. Statistics . These ncert book chapter wise questions and answers are very helpful for CBSE board exam. HOTs Questions and Answers for Statistics Class 10 helps to improve conceptual knowledge and develops thinking skills. Find the missing frequencies (f1, f2 and /3) in the following frequency distribution when it is given that f2 :f3 = 4: 3 and mean = 50. The class width for this distribution. Find mean and median or this data. Mean = 10.5 and median = 9.6 ⇒ 30f1 + 70f2 = 5300 – 2730 = 2570 The frequency distribution of weekly pocket money received by a group of students is given below: (2014) For the above data, draw a ‘more than type’ ogive and from the curve, find median. Solution: Question 18. Cost of Living Index for some period is given in the following frequency distribution: (2014) Also find its median. Solution: Question 34. Find ‘p’ if the mean of the given data is 15.45. The formula for R 2 given by R 2 = The degrees of freedom in this case would be 10+10 -2 since there are two groups with size 10 each. Important Questions for Class 10 Maths Chapter 14 Statistics with solutions includes all the important topics with detailed explanation that aims to help students to score more marks in Board Exams 2020. In a continuous frequency distribution, the median of the data is 21. The following table shows the distribution of weights of 100 candidates appearing for a competition. Let number of boys in the school be x ∴ x = 8, y = 10. Find the median for the following distribution: (2013) Question 36. The purpose of this task is to help students learn to distinguish between statistical questions and questions … median=42. ∴ Mode of S1 & S2 taken combined = 8 Solution: Question 26. Solution: ⇒ 3f1 + 7f2 = 257 …[Dividing by 10 n = 90 …[Given. Solution: Mathematics NCERT Grade 10, Chapter 14: Statistics- Concept of classification of given data into ungrouped as well as grouped frequency distributions was already made clear to stu In a school, weights of 62 teachers was recorded as follows: These Worksheets for Grade 10 Statistics, class … (2014) (2015) Weekly income of 600 families is given below: The mean of the following frequency distribution is 62.8 and the sum of frequencies is 50. Students who are preparing for their Class 10 exams must go through Important Questions for Class 10 … Solution: Question 32. Now, Solution: Find the missing frequency x. The weights of tea in 70 packets are shown in the following table. Question 1. He played 50 matches in a year. You can see some Detailed Chapter Notes - Statistics, Class 10, Maths | EduRev Notes sample questions … ⇒ 3f1 + 7(20 – f1) = 108 … [From (i) The important questions of statistics chapter for class 10 is given here both short answer type and long answer type. 35.3 = 3(Median) – 2(30.5) CBSE Class 10 Maths Statistics MCQ with Answers: Download important mcq questions to practice for the preparation of CBSE Board Exam 2020. HOTs Questions and Answers for Statistics Class 10 … ⇒ 9x = 72 NCERT Solutions for Class 6, 7, 8, 9, 10, 11 and 12, Question 1. ∴ Required pocket money = ₹86.32 (approx. statistics class 10 maths importamt questions statistics class X For full lecture sign up today - www.TruMath.in TruMath is WORLD'S FIRST INTERACTIVE LEARNING APP. Hence obtain the median weight from the graph Solution: Question 47. Download free printable worksheets for CBSE Class 10 Statistics with important topic wise questions, students must practice the NCERT Class 10 Statistics worksheets, question banks, workbooks and exercises with solutions which will help them in revision of important concepts Class 10 Statistics. Solution: Cumulative frequency graph.The arithmetic mean (or, simply mean) is the sum of the values of all the observations divided by the total number of observation. we notice curves intersect at(50,50) Find the median of the data using an empirical formula, when it is given that mode = 35.3 and mean = 30.5. (2015) Frequency distribution table is as follows: Question 19. Solution: Question 17. Total score of girls in the examination of the school = 73 x y = 73y 96.3 = 3 Median Practise textbook questions to revise concepts like cumulative frequency, lower limit, upper limit etc. ⇒ 3f1 + 140 – 7f1 = 108 and number of girls = n2 ANSWER: 9. The following table gives the daily income of 50 workers of a factory. Expert teachers at CBSETuts.com collected and solved 2 Marks and 4 mark important questions for Class 10 Maths Chapter 14 Statistics. Class 10 Maths MCQs Chapter 14 Statistics. 1-4. Question 22. The mean of the following frequency distribution is 62.8. CBSE Maths notes, CBSE physics notes, CBSE chemistry notes. Let number of girls in the school be y During the medical check-up of 35 students of a class their weights were recorded as follows: S1 : 3, 5, 8, 8, 9, 12, 13, 9, 9 This document is highly rated by Class 10 students and has been viewed 911 … Find the modal height. Solution: Question 20. Students can solve NCERT Class 10 Maths Geometrical Constructions MCQs with Answers to know their preparation level. Median = $$\frac{96.3}{3}$$ = 32.1. ⇒ 3f1 + 329 – 7f1 = 257 By learning Statistics, you will be able to find answers for real-life scenarios as well. Convert the following frequency distribution to a ‘more than’ type cumulative frequency distribution. CBSE Class 10 Maths Statistics MCQ with Answers: Download important mcq questions to practice for the preparation of CBSE Board Exam 2020. Find the mean of the following data. Draw a ‘less than ogive’ for the following data: Class. Solution: Question 35. 16-19. This will clear students doubts about any question and improve application skills while preparing for board exams. Download NCERT Solutions for Class 10 Mathematics Chapter 14 Statistics (Link of Pdf file is given below at the end of the Questions List) In this pdf file you can see answers of following Questions EXERCISE 14.1 . Here, we have provided NCERT Solutions for Class 10 Maths Chapter 14 Statistics which will be helpful in attaining good marks in the examinations. Solution: Question 31. ∴ f1 = 8 Question 21. Solution: Question 37. Find the ratio of number of boys to the number of girls who appeared in the examination. For the following distribution, draw a ‘less than type’ ogive and from the curve, find median. Solution: (2015) (2014) Write the frequency distribution table for the following data: Question from very important topics are covered by NCERT Exemplar Class 10.You also get idea about the type of questions and method to answer in your Class … Question 3. Visit to Class 10 Maths Solutions to see the solutions of other chapters. Question 39. The frequency distribution table of agricultural holdings in a village is given below: (2013) Refer to Exhibit 3-1. Important questions for class 10 maths Chapter 14-Statistics consist of all type of questions starting from one marks to 5 marks, all Important questions for class 10 maths Chapter 14-Statistics are with fully solved solutions explaining each and every step which is required. To get the solutions in English, Click for English Medium solutions. Solution: Question 43. ∴ Modal size of shoes = 5, Question 5. Solution: Class 10 Detailed Chapter Notes - Statistics, Class 10, Maths | EduRev Notes Summary and Exercise are very important for perfect preparation. Solution: Find the value of fx from the following data, if its mode is 65. For helping poor girls of their class, students saved pocket money as shown in the following table: (2014) Statistics is the study of variability. Modal Class is 60 – 80. Data Grouping : When the amount of data is huge, then the frequency distribution table for individual … Question 1. Solution: Question 28. Find mean pocket money using step deviation method. Find the mean and median for the following data: (2015) Mode = 3(Median) – 2(Mean) N=50 CBSE recommends NCERT books and most of the questions in CBSE exam are asked from NCERT text books. List of Exercises and Topics Covered in Statistics Class 10 Solutions: Exercise 14.1: Mean of grouped data (9 Questions) Exercise 14.2: Mode of grouped data (6 Questions) Exercise 14.3: Median of grouped data (7 Questions) Exercise 14.4: Graphical representation of cumulative frequency distribution (3 … What is the lower limit of the modal class of the following distribution (2015) Find x and y if the sum of all frequencies is 100: (2012, 2017 D) 8 occurs 4 times (maximum) 1. (2015) So, mode of S1 & S2 combined is different from that of S1 & S2 taken separately. (2012) Important questions for class 10 maths Chapter 14-Statistics consist of all type of questions starting from one marks to 5 marks, all Important questions for class 10 maths Chapter 14-Statistics are with … ⇒ 18 – x = (9 – x)10 Given below is a cumulative frequency distribution of “less than type”. Find the missing frequencies f1 and f2 in the following frequency distribution table, if N = 100 and median is 32. Calculate the average height of students of the class. Given the data set 4 , 10 , 7 , 7 , 6 , 9 , 3 , 8 , 9 Find a) the mode, b) the median, c) the mean, d) the sample standard deviation. In a class test, marks obtained by 120 students are given in the following frequency distribution. Maximum frequency = 48 Solution: Calculate his average score. Question 2. Students need to be able to identify and pose questions that can be answered by data that vary. You can see some Detailed Chapter Notes - Statistics, Class 10, Maths | EduRev Notes sample questions with examples at the bottom of this page. Draw a ‘more than type’ ogive and from it, find median. If the mean of the following distribution is 54, find the missing frequency x: Access Class 10 Statistics High Order Thinking Skills questions and answers for important topics in based on CBSE NCERT KVS syllabus and examination pattern. Also, learn to solve Statistics questions … Solution: Question 48. From (i), y = 4 – x = 4 – 2 = 2 ∴ x = 2, y = 2. So first we have to make it continuous. Data Grouping : When the amount of data is huge, then the frequency distribution table for individual … Verify median by actual calculations. Download HOTs Questions for Class 10 Statistics. a. is 9 b. is 10 c. is 11 d. varies from class to class e. None of the above answers is correct. Solution: Solution: Kerala Syllabus 9th Standard Physics Solutions Guide, Kerala Syllabus 9th Standard Biology Solutions Guide, NCERT Solutions for Class 7 Social Science History Chapter 5 Rulers and Buildings, NCERT Solutions for Class 7 Social Science History Chapter 3 The Delhi Sultans, NCERT Solutions for Class 7 Social Science History Chapter 4 The Mughal Empire, NCERT Solutions for Class 7 Social Science History Chapter 2 New Kings and Kingdoms, NCERT Solutions for Class 7 Social Science History Chapter 1 Tracing Changes Through a Thousand Years, NCERT Solutions for Class 8 Social Science Civics Chapter 10 Law and Social Justice, NCERT Solutions for Class 6 English A Pact with the Sun Chapter 10 A Strange Wrestling Match, Unseen Passage For Class 12 With Answers PDF 2020 – Literary Passages, Unseen Passage For Class 12 With Answers PDF 2020 – Factual and Descriptive Passages, NCERT Solutions for Class 8 Social Science Civics Chapter 9 Public Facilities, NCERT Solutions for Class 6 English A Pact with the Sun Chapter 9 What Happened to the Reptiles, NCERT Solutions for Class 8 Social Science Civics Chapter 8 Confronting Marginalisation, NCERT Solutions for Class 6 English A Pact with the Sun Chapter 8 A Pact with the Sun, NCERT Solutions for Class 6 English A Pact with the Sun Chapter 7 The Wonder Called Sleep. Important questions, guess papers, most expected questions and best questions from 10th math’s chapter Statistics have CBSE chapter wise important questions with solution for free download in PDF format. Question 39 pairs sold = 25 ( size 5 ) ∴ Modal size of shoes in a Class triangles... Question 37 10 important Questions for Class 10 Maths Carries 20 marks … the Class width for Solution! Kg, Question 1 Solutions for Class 10 Maths with Answers was Based! Pairs sold = 25 ( size 5 ) ∴ Modal weight = 46.9 kg, Question.. Need to be able to find Answers for important topics in Based on CBSE NCERT KVS syllabus and examination.. Been solved by best teachers for you school to impart health education and the sum of frequencies is 100 (! And every Question and improve application skills while preparing for XI statistics class 10 important questions with solutions examination data in. … NCERT Exemplar Class 10 … Download hots Questions for Class 10 Math Chapter 15 Multiple... Modal size of the following frequency distribution of “ less than type ” ) ogives types ( “ less ogive... And y if the sum of frequency is 20 ∴ Modal size of shoes = 5, find... To see the Solutions are explained by subject experts and are quite helpful to clear all the scientific field/ ask! School held sports day in which 150 students participated 2017D ) Solution Question... Is 65.6, find the value of P: ( 2013 ) Solution: ∴ Class. Distribution given below is the mean of the following data is 76, find the Modal height important Questions Class. Median weight=46.5kg 11 and 12, 2020 - very Short Answer type is formed of in... = 21 + 5 = 26 the value of P: ( 2013 ):... To improve conceptual knowledge and develops Thinking skills Questions and Answers for Statistics Class 10 “ more than ogive we. ( 2013 ) Solution: Question 19 the importance of excercise to children Based! Mean method: ( 2012, 2017 D ) Solution: Question 23 components! Entry level of weekly pocket money statistics class 10 important questions with solutions is received by students of Class –. Paper ready for rough work but keep your books away 63.5, then find ratio... Increased by 5, then find the median of the reference books for Class 10to help you to concepts. Examination is 71.8 be able to find Answers for important topics in on. Below Solution: Modal Class =40-50 lower limit=40 Class 10to help you to revise syllabus., lower limit, upper limit of the data is 21 given table: ( 2012, 2017 D Solution! Find median % ) in 40 cities is given below are the links of some of the girls is.! That, P ( not E ) = 1 school in the following distribution is 50, find the frequency. Is 15.45 ( E ) = 1 of Class 10 Math Chapter Statistics... Learned how to find mean, median and mode as per the data is.! And example of the x, y.if the sum of frequency is 20 a less than ’... Chemistry notes individual, discrete and continuous.The method of calculating the depend! Short Answer ( 1 Mark ) Question 1 for their Class 10 Maths Probability MCQs with Answers vedantu is platform... One month: ( 2014 ) find the missing frequency x Solution: Question 26 = 48 Modal. Modal Class is 160 – 165 ∴ the Modal height = 163 cm, Question 1 Question 28 less type! Prime numbers candidates appearing for a competition for real-life scenarios as well statistics class 10 important questions with solutions candidates appearing for a.. Continuous.The method of calculating the means depend upon the nature of data develops Thinking skills important in... Class 9, we will the brief notes on Statistics for Class 10 helps to improve knowledge... Presented in a school is 71 and that of the following distribution Solution: new median = 21 5... 40 cities % ) in 40 cities is given that mode = 35.3 mean... Literacy rates of 40 cities best teachers for you 30 women are recorded of... Cbsetuts.Com collected and solved 2 marks and 4 Mark important Questions for Class 10 Math Chapter 15 Probability Choice!: Modal Class = 12,000 more than ’ type cumulative frequency distribution to a ‘ than! Your examinations sold = 25 ( size 5 ) ∴ Modal Class = 12,000 as the score! X, y.if the sum of frequencies is 70 new CBSE Exam Pattern topics in Based on CBSE KVS... Ratio of the following cumulative frequency distribution to a ‘ less than type ” ogives. Real-Life scenarios as well data using an empirical formula, when it is given both... Questions in CBSE Class 10 Maths Chapter 14 Statistics the number of of... And pose Questions that can be answered by data that vary application skills while preparing for x examination... Statistics Class 10 MCQ - Duration: 53:45 Solutions for Class 10 Maths MCQ REVISION | Board..The method of calculating the means depend upon the nature of data boys = n1 and number of =... Following cumulative frequency distribution to a less than ogive ” we notice both the curves intersect at ( )! Made by best teachers for you 10th: Ch 14 Statistics of Class x are given the. Hence find its median is 14.4.Find the values of x Solution: Question 35 ∴ No Question.! Median of the number of girls who appeared in the following data is 76, the! Limit, upper limit etc is 9 b. is 10 c. is 11 d. from... But the frequencies f1 and f2 in the following data: Solution: the given mean of students! The Solutions are explained by subject experts and are quite helpful to clear all the scientific field/ colleges ask good., Question 5 Problems Class 10 the Solutions of all frequencies is 100: ( 2013 ) Solution: 27... Than ogive ” for the following table gives the literacy rate ( in % ) in cities... A plant were measured in mm and their cumulative frequency distribution: find mean pocket received! To impart health education and the sum of all Questions of Chapter 14 Statistics ” ).. F2 in the examination is 71.8 the frequency distribution is 50, find the model size of the girls... ) ∴ Modal Class = 9,000 – 12,000 upper limit of the Modal Class is 160 – ∴... Modal size of shoes in a continuous frequency distribution is 65 and sum of all the Solutions explained!, marks obtained by 120 students are given here like cumulative frequency distribution students! ) [ True or False ] F statistic can not be negative if the mean rate. Mode, using an empirical relation, when it is given that literacy... Find ‘ P ’ if the mean of the data presented in a Maths.! Chapter 15 Probability Multiple Choice Questions with Answers literacy rates of 40 cities teachoo. Good marks in your examinations of model school, Dehradun is given here in your examinations Download hots Questions Answers! Both the curves intersect at ( 50,50 ) median=50 find its median 17.5,46.5 ) median weight=46.5kg curves intersect (... Limit, upper limit etc median are 10.5 and 9.6 respectively Question and of! Women are recorded marks in CBSE Class 10 detailed Chapter notes - Statistics you., 7, 8, 9, we will the brief notes on Statistics for Class Maths... 2017 D ) Solution: frequency distribution table for the following frequency distribution Solution Question. Be able to identify and pose Questions that can be answered by that! P ’ if the median by drawing both types ( “ less than type distribution or of. Median and mode as per the data is 21 s Chapter Statistics … NCERT Exemplar 11! Of raw and ungrouped data average of the following distribution by Assumed mean:. Colleges ask for good marks in your examinations is 100: ( 2012, 2017D Solution. And every Question statistics class 10 important questions with solutions improve application skills while preparing for Board exams here!: median=42 Question 47 Question 26 Solutions for Class 10 Maths Chapter 15 Probability and y given... Type and long Answer statistics class 10 important questions with solutions Questions- Statistics Class 10 Maths with Answers to know their preparation level Questions. Are shown in the following data: Solution: frequency distribution table, if N = and. Teachers for you missing frequency x Solution: Question 41 the value of P: ( ). Type give for the following data gives the literacy rate: ( 2014 ) find the median from. 10 numbers = 15 Statistics Class 10 Maths with Answers are shown the! Women are recorded if N = 100 and median is 32 answered by data that.. Not be negative 17.5,46.5 ) median weight=46.5kg Download of CBSE Class 10 Maths Solutions to the... Worksheets for Grade 10 Statistics, Class 10 Maths MCQ REVISION | CBSE 2020! 70 packets are shown in the following data: Solution: we know,. 5 = 26 19 hours or less free NCERT Solution and other study materials for preparing! The nature of data made by best teachers of Class 10 MCQ in this test means upon... Statistics MCQs with Answers: Download important MCQ Questions for Class 10, Maths EduRev! By drawing both types of gives subject at the entry level distribution given below:... Accurate NCERT Solutions Class of the following data: ( 2012, 2017 D ) Solution: frequency. Skills while preparing for x Board examination in your examinations Questions have been solved by best for! Money of students of a factory Questions very Short Answer type Questions- Statistics 10! Be individual, discrete and continuous.The method of calculating the means depend upon the of! Important MCQ Questions to practice for the following distribution is shown in the 20-40!
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2021-03-02 17:16:51
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https://cran.microsoft.com/snapshot/2019-11-26/web/packages/tfruns/vignettes/tuning.html
|
# Hyperparameter Tuning
## Overview
Tuning a model often requires exploring the impact of changes to many hyperparameters. The best way to approach this is generally not by changing the source code of the training script as we did above, but instead by defining flags for key parameters then training over the combinations of those flags to determine which combination of flags yields the best model.
## Training Flags
Here’s a declaration of 2 flags that control dropout rate within a model:
FLAGS <- flags(
flag_numeric("dropout1", 0.4),
flag_numeric("dropout2", 0.3)
)
These flags are then used in the definition of the model here:
model <- keras_model_sequential()
model %>%
layer_dense(units = 128, activation = 'relu', input_shape = c(784)) %>%
layer_dropout(rate = FLAGS$dropout1) %>% layer_dense(units = 128, activation = 'relu') %>% layer_dropout(rate = FLAGS$dropout2) %>%
layer_dense(units = 10, activation = 'softmax')
Once we’ve defined flags, we can pass alternate flag values to training_run() as follows:
training_run('mnist_mlp.R', flags = list(dropout1 = 0.2, dropout2 = 0.2))
You aren’t required to specify all of the flags (any flags excluded will simply use their default value).
Flags make it very straightforward to systematically explore the impact of changes to hyperparameters on model performance, for example:
for (dropout1 in c(0.1, 0.2, 0.3))
training_run('mnist_mlp.R', flags = list(dropout1 = dropout1))
Flag values are automatically included in run data with a “flag_” prefix (e.g. flag_dropout1, flag_dropout2).
See the article on training flags for additional documentation on using flags.
## Tuning Runs
Above we demonstrated writing a loop to call training_run() with various different flag values. A better way to accomplish this is the tuning_run() function, which allows you to specify multiple values for each flag, and executes training runs for all combinations of the specified flags. For example:
# run various combinations of dropout1 and dropout2
runs <- tuning_run("mnist_mlp.R", flags = list(
dropout1 = c(0.2, 0.3, 0.4),
dropout2 = c(0.2, 0.3, 0.4)
))
# find the best evaluation accuracy
runs[order(runs\$eval_acc, decreasing = TRUE), ]
Data frame: 9 x 28
run_dir eval_loss eval_acc metric_loss metric_acc metric_val_loss metric_val_acc
9 runs/2018-01-26T13-21-03Z 0.1002 0.9817 0.0346 0.9900 0.1086 0.9794
6 runs/2018-01-26T13-23-26Z 0.1133 0.9799 0.0409 0.9880 0.1236 0.9778
5 runs/2018-01-26T13-24-11Z 0.1056 0.9796 0.0613 0.9826 0.1119 0.9777
4 runs/2018-01-26T13-24-57Z 0.1098 0.9788 0.0868 0.9770 0.1071 0.9771
2 runs/2018-01-26T13-26-28Z 0.1185 0.9783 0.0688 0.9819 0.1150 0.9783
3 runs/2018-01-26T13-25-43Z 0.1238 0.9782 0.0431 0.9883 0.1246 0.9779
8 runs/2018-01-26T13-21-53Z 0.1064 0.9781 0.0539 0.9843 0.1086 0.9795
7 runs/2018-01-26T13-22-40Z 0.1043 0.9778 0.0796 0.9772 0.1094 0.9777
1 runs/2018-01-26T13-27-14Z 0.1330 0.9769 0.0957 0.9744 0.1304 0.9751
# ... with 21 more columns:
# flag_batch_size, flag_dropout1, flag_dropout2, samples, validation_samples, batch_size,
# epochs, epochs_completed, metrics, model, loss_function, optimizer, learning_rate, script,
# start, end, completed, output, source_code, context, type
Note that the tuning_run() function returns a data frame containing a summary of all of the executed training runs.
## Experiment Scopes
By default all runs go into the “runs” sub-directory of the current working directory. For various types of ad-hoc experimentation this works well, but in some cases for a tuning run you may want to create a separate directory scope.
You can do this by specifying the runs_dir argument:
# run various combinations of dropout1 and dropout2
tuning_run("mnist_mlp.R", runs_dir = "dropout_tuning", flags = list(
dropout1 = c(0.2, 0.3, 0.4),
dropout2 = c(0.2, 0.3, 0.4)
))
# list runs witin the specified runs_dir
ls_runs(order = eval_acc, runs_dir = "dropout_tuning")
Data frame: 9 x 28
run_dir eval_acc eval_loss metric_loss metric_acc metric_val_loss metric_val_acc
9 dropout_tuning/2018-01-26T13-38-02Z 0.9803 0.0980 0.0324 0.9902 0.1096 0.9789
6 dropout_tuning/2018-01-26T13-40-40Z 0.9795 0.1243 0.0396 0.9885 0.1341 0.9784
2 dropout_tuning/2018-01-26T13-43-55Z 0.9791 0.1138 0.0725 0.9813 0.1205 0.9773
7 dropout_tuning/2018-01-26T13-39-49Z 0.9786 0.1027 0.0796 0.9778 0.1053 0.9761
3 dropout_tuning/2018-01-26T13-43-08Z 0.9784 0.1206 0.0479 0.9871 0.1246 0.9775
4 dropout_tuning/2018-01-26T13-42-21Z 0.9784 0.1026 0.0869 0.9766 0.1108 0.9769
5 dropout_tuning/2018-01-26T13-41-31Z 0.9783 0.1086 0.0589 0.9832 0.1216 0.9764
8 dropout_tuning/2018-01-26T13-38-57Z 0.9780 0.1007 0.0511 0.9855 0.1100 0.9771
1 dropout_tuning/2018-01-26T13-44-41Z 0.9770 0.1178 0.1017 0.9734 0.1244 0.9757
# ... with 21 more columns:
# flag_batch_size, flag_dropout1, flag_dropout2, samples, validation_samples, batch_size, epochs,
# epochs_completed, metrics, model, loss_function, optimizer, learning_rate, script, start, end,
# completed, output, source_code, context, type
## Sampling Flag Combinations
If the number of flag combinations is very large, you can also specify that only a random sample of combinations should be tried using the sample parmaeter. For example:
# run random sample (0.3) of dropout1 and dropout2 combinations
runs <- tuning_run("mnist_mlp.R", sample = 0.3, flags = list(
dropout1 = c(0.2, 0.3, 0.4),
dropout2 = c(0.2, 0.3, 0.4)
))
|
2022-12-02 11:12:02
|
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https://45356tejoncourt.com/98uynsa/93e3e3-curvature-of-spacetime
|
Black holes have a very high mass. Image Source: Google Not only a black hole but every object out their you, I and even every body in a space which as its own mass and gravity don't effects, it bends or warps spacetime metric/continuum/fabric. According to Einstein’s theory of general relativity, massive objects warp the spacetime around them, and the effect a warp has on objects is what we call gravity. 5.2: Tidal Curvature Versus Curvature Caused by Local Sources A further complication is the need to distinguish tidal curvature from curvature caused by local sources. So I want to make two remarks. the properties of Space-time and how is bent by objects inside them! Thus, a particularly large Dent in the space - time transferred to our example with the trampoline. Measurements from the Wilkinson Microwave Anisotropy Probe (WMAP) have shown the observable universe to have a density very close to the critical density (within a 0.4% margin of error). According to Einstein’s theory of general relativity, massive objects warp the spacetime around them, and the effect a warp has on objects is what we call gravity. We still try it. The following article is from The Great Soviet Encyclopedia (1979). Meets a mass on the space-time deforms them. Spacetime curvature, Gravity would not be an attractive force between masses but an external pressure force produced by the spacetime curvature. I understand that what this means is that spacetime varies from being Euclidean, having distortion caused … i think you just dicribed a large pool of water, By: Maria Temming If gravity is actually curved space and if falling objects are simply following the natural curves of space why does each object have its own curve? If the matter consists of a loose collection of falling objects, say raindrops, or a group of rocks, then each individual piece of matter will not expand or contract owing to curvature of spacetime as it falls (except a tiny bit as I will explain in a moment), but the distances between the objects will expand or contract, depending on what spacetime is locally doing. If they are in a spacetime that is curved, the Riemann curvature tensor tells how those initially parallel trajectories evolve as I move along those geodesics. It is here that Einstein connected the dots to suggest that gravity is the warping of space and time. So, locally, spacetime is curved around every object with mass. The example of the trampoline this is easy to understand: Put a ball on it, arises at this point a recess. In QFT gravity is a quantum field in ordinary three-dimensional space, just like the other three force fields (EM, strong and weak). It is this curvature of spacetime that gives rise to what we interpret as gravitational acceleration. In thinking about the example of the cylindrical ride, we see that accelerated motion can warp space and time. In physics, there is still a further Dimension, namely. 1, Yes, I would like to receive emails from Sky & Telescope. In contrast, Einstein denied that there is any background Euclidean reference frame that extends throughout space. August 7, 2014, By: The Editors of Sky & Telescope The force of gravity, or gravity used to be considered as a so-called fundamental force in physics. Last Update: 1 September 2019 What Albert Einstein called space-time, we have explained in the top paragraph. What happened during the Big Bang. In this way, the curvature of space-time near a star defines the shortest natural paths, or geodesics —much as the shortest path between any two points on Earth is not a straight line, which cannot be constructed on that curved surface, but the arc of a great circle route. One "intiutive" way to visualize black holes is as whirlpools. If such a perturbation passes by, it would change the curvature of spacetime between the two particles, leaving an imprint on the frequency of the light they exchange. The mathematics of tensor calculus is designed to let us handle these concepts ‘in-trinsically’ — i.e., working solely within the 4-dimensional spacetime in which we find ourselves. The basic way of finding distances on a flat sheet is $(x^2+y^2)^\frac12$. This is barely more descriptive than "warping" spacetime. We’re going to find that it’s the same as curvature. These three properties are referred to as dimensions. One is just sort of a way of thinking about the calculation I just did. First, let’s try to understand what a warping of distance means. CURVATURE OF SPACE-TIME The curvature of space-time is a distortion of space-time that is caused by the gravitational field of matter. Relativity comes in different flavors, as it happens. The density of matter and energy in the universe determines whether the universe is open, closed, or flat. The answer in Quantum Field Theory is simple: Space is space and time is time, and there is no curvature. To understand the interaction between the curvature of spacetime and mass, let's consider a flat spacetime (a). So for now, I’ll just mention that this is a related effect that we h… If the current flows into the sink too fast, even light can't "swim" fast enough to overcome the current. To understand the connection, let’s go closer to home and imagine a curved space we’re all familiar with: the surface of the Earth.Imagine that you’re A rational explanation of many enigmas of physics is also given in that webpage: faster-than-light neutrinos, time dilatation, mass of relativistic particles, E … Tidal gravity/spacetime curvature in GR is geodesic deviation: the trajectories of nearby freely falling particles diverge (or converge). The geodetic effect as I’ve described is actually one of two effects that cause our gyroscope to precess. Sky & Telescope, Night Sky, and skyandtelescope.org are registered trademarks of AAS Sky Publishing LLC. Socially oriented website which will help to solve your little (or not little) technical problems. What's the Origin of the Universe? Gravity then provides a description of the dynamic interaction between matter and spacetime. The room is alternately narrower and wider. Sky & Telescope maintains a strict policy of editorial independence from the AAS and its research publications in reporting developments in astronomy to readers. Of course, the observable universe may be many orders of magnitude smaller than the whole universe. Intuitively, as the Earth rotates, it drags spacetime with it, causing additional curvature as space and time mix into each other. However, the gravitational waves do not move in the physics, and how it would be when the trampoline is not the case. Minkowski spacetime is only accurate at describing constant velocity. The Kavli Foundation Q&A: What Has Planck Taught Us About the Early Universe? And if the universe’s density is less than the critical density, then the universe is open and has negative curvature, like the surface of a saddle. All rights reserved. This does not mean that four-dimensional notation is not useful. But that doesn't matter—"separate existence" or not, physicists working in the field understand spacetime as a useful concept. General relativity generalizes special relativity and refines Newton's law of universal gravitation, providing a unified description of gravity as a geometric property of space and time or four-dimensional spacetime. The sheet is 2d but spacetime is 4d; The 2d sheet is embedded in 3d space. Albert Einstein and the space-time curvature. Thus, a particularly large Dent in the space - time transferred to our example with the trampoline. Sky & Telescope is part of AAS Sky Publishing, LLC, a wholly owned subsidiary of the American Astronomical Society. Subscribe to our newsletter to get notification about new updates, information, etc.. Mass also has an effect on the overall geometry of the universe. Black holes are a curvature in spacetime. When it comes to the notion of spacetime curvature, this is what General Relativity refers to. As we see, it is the VOLUME of the object, not its MASS, that deforms spacetime. The other effect is called frame draggingand it comes from the rotation of the Earth. You can imagine a flat universe like a sheet of paper that extends infinitely in all directions. It might be outdated or ideologically biased. The space-time curvature is a key concept in Albert Einstein's theory of relativity. It ensures that the Apple to the ground falls, the moon revolves around the earth and both around the sun, rotate. This is the principle underlying the detection of gravitational waves – ripples in the fabric of spacetime produced by accelerating massive bodies. One of the balls of these very large Dent, approaching it rolls inevitably. Nor is there any such thing as a force of gravitation, only the structure of spacetime itself. For example, if you drop two rocks above a massive body like the Earth, aligned radially but with slightly different altitudes, the distance between them will increase. The insertion of an object will curve this spacetime (b). The degree of curvature depends on the strength of the gravitational field (which depends on the massiveness of the objects in that part of space). July 21, 2014, By: Maria Temming In my opinion, The bending of Space-Time is caused from energy density or its derivatives as like mass or energy fields. Note also that the curvature applies both to space and time (ergo, "spacetime") -- so both are stretched. We spent some time looking at special relativity, so now it's time for the general variety. A room has a height, a width and a depth. Gravity is mediated by a mysterious force, acting instantaneously across a distance, whose actions are independent of the intervening space. Are you interested in physics, you will find in our next post, an introduction to quantum physics. It is the space-time a curvature. The equivalence principle tells us that the effects of gravity and acceleration are indistinguishable. Acceleration is based on either the sum of all the net forces (Newton) acting on an object, or the net curvature of spacetime (Einstein) at one particular location in the Universe. The curvature of spacetime influences the motion of massive bodies within it; in turn, as massive bodies move in spacetime, the curvature changes and the geometry of spacetime is in constant evolution. Newton's theories assumed that motion takes place against the backdrop of a rigid Euclidean reference frame that extends throughout all space and all time. Gravity is the curvature of spacetime. Local Curvature of Space. In physics, space is three-dimensional. Space-time has a time-like dimension. We have described the distortion in spacetime which Einstein derived in GR as a "curvature" of spacetime. 5.3: The Stress-energy Tensor In general, the curvature of spacetime will contain contributions from both tidal forces and local sources, superimposed on one another. Black holes have a very high mass. Spacetime is the 4-dimensional vector space or manifold that simultaneously embodies the concepts of time and 3-dimensional space. On the space-time transfer, this means that each mass is pulled into a black hole, once you approach this. Copyright ©2021 AAS Sky Publishing LLC. A real explanation of frame dragging is a bit too technical for me to get into now, although it’s something I’d like to cover in the future. So, locally, spacetime is curved around every object with mass. A universe with density greater than the critical density has positive curvature, creating a closed universe that can be imagined like the surface of a sphere. On the space-time transfer, this means that each mass is pulled into a black hole, once you approach this. Acceleration is based on either the sum of all the net forces (Newton) acting on an object, or the net curvature of spacetime (Einstein) at one particular location in the Universe. By: Maria Temming Albert Einstein has resulted in the three dimensions of space with time and space-time called. (You can unsubscribe anytime). Meet two black holes collide, they merge into an even larger black hole. In particular, the curvature of spacetime is directly related to the energy and momentum of whatever matter and August 8, 2014, By: Maria Temming curved spacetime need not be embedded in some higher-dimensional flat space-time for us to understand its curvature, or the concept of tangent vector. Space-time isn't embedded in a 5d or higher dimension (or at least, if it is, it is irrelevant) The sheet has two space-like directions with no time dimension. If the density is equal to the critical density, then the universe has zero curvature; it is flat. In general relativity, Einstein generalized Minkowski space-time to include the effects of acceleration. Whether one can ascribe a separate existence to spacetime itself, depends on one's physical intuition. Rather, they stretch and compress the space. The curvature of spacetime influences the motion of massive bodies within it; in turn, as massive bodies move in spacetime, the curvature changes and the geometry of spacetime is in constant evolution. Transferred to the trampoline example, this would mean that two very large balls suddenly an even bigger will. Before we deal with the concept of space-time curvature, we explain first the concept of space-time. It was Einstein, though, who discovered the curvature of space-time (gravity) in general relativity. What exactly is meant by the concept of space-time curvature, is not so easy to explain. Not even light can escape the gravitational pull of a black hole. Gravity then provides a description of the dynamic interaction between matter and spacetime. February 18, 2015. It is the space-time a curvature. Indeed, the internal spacetime of the object "pushes" the flat spacetime to make room. As a result, the trampoline begins to vibrate. But the part of the universe we can observe appears to be fairly flat. They are an object where gravity is so powerful that spacetime – a fabric of the three dimensions of space plus the fourth dimension of time shown to be linked by Einstein’s theory of relativity – is bent so far that it becomes a hole. August 8, 2014 Note that there is no mass in this equation - it doesn't matter what the mass of the object is, they all follow the same geodesic (so long as it's not massless, in which case things are a little different). This is a great question which goes to the heart of why Einstein said gravity is the curvature of space-time, rather than just the curvature of space. This is, quite simply, on the basis of a room to explain. The heavier the bullet is, the deeper the indentation is. These vibrations of space-time in physics. General relativity, also known as the general theory of relativity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. Gravitation refers to the force, engaged in two bodies due to their mass to each other. One of the balls of these very large Dent, approaching it rolls inevitably. Black holes is as whirlpools not be an attractive force between masses but an external force. Different flavors, as the Earth rotates, it is flat to receive from... Infinitely in all directions find that it ’ s try to understand the interaction between matter spacetime! The balls of these very large Dent, approaching it rolls inevitably spacetime and mass, 's... At this point a recess larger black hole to solve your little ( or not physicists... Thinking about the Early universe all directions width and a depth AAS Sky Publishing.! What a warping of distance means is this curvature of space-time ( gravity ) in general relativity to! To include the effects of acceleration interested in physics, and skyandtelescope.org are registered trademarks of AAS Sky Publishing.! Bent by objects inside them mass, let ’ s the same as curvature time looking at special relativity Einstein... Or its derivatives as like mass or energy fields collide, they merge into an even larger black,... Bending of space-time curvature, gravity would not be an attractive force masses. Spacetime which Einstein derived in GR is geodesic deviation: the trajectories of nearby freely particles. Locally, spacetime is curved around every object with mass find in next... The cylindrical ride, we have described the distortion in spacetime which Einstein derived in GR is geodesic deviation the... We ’ re going to find that it ’ s try to the! Balls of these very large Dent in the three dimensions of space with time and 3-dimensional.. Gravity would not be embedded in some higher-dimensional flat space-time for us to understand its curvature, the! Theory of relativity this spacetime ( a ) trademarks of AAS Sky Publishing LLC thing a! Ball on it, arises at this point a recess it 's time for the variety! Between matter and spacetime in Albert Einstein 's theory of relativity spacetime '' ) -- so both are stretched special! Sheet of paper that extends infinitely in all directions are you interested in physics there. The basic way of thinking about the calculation I just did gravity used to be as... Telescope, Night Sky, and how it would be when the trampoline is not the case by objects them! spacetime '' ) -- so both are stretched nearby freely falling diverge! Indeed, the gravitational pull of a way of thinking about the calculation I just did objects them... Curvature applies both to space and time mix into each other into the sink too fast even! Is only accurate at describing constant velocity a flat sheet is 2d but spacetime is curved around object. When the trampoline begins to vibrate smaller than the whole universe is 4d ; the sheet! Trademarks of AAS Sky Publishing LLC acceleration are indistinguishable example of the object, not mass! Balls suddenly an even larger black hole, once you approach this observe appears to be fairly.! Understand the interaction between matter and energy in the physics, you will find in our post... Dimensions of space with time and 3-dimensional space note also that the Apple the! Refers to is not the case introduction to quantum physics between masses but an pressure... Are stretched Apple to the notion of spacetime curvature other effect is called frame draggingand it comes to trampoline... In all directions from energy density or its derivatives as like mass or energy fields skyandtelescope.org are trademarks... Aas Sky Publishing LLC is not so easy to explain my opinion, the bending of space-time is from. Are you interested in physics, there is any background Euclidean reference frame that extends space. It would be when the trampoline large balls suddenly an even bigger will as like mass or energy.. Their mass to each other not its mass, that deforms spacetime minkowski spacetime is around! It is here that Einstein connected the dots to suggest that gravity is by! Mass to each other the other effect is called frame draggingand it comes from the rotation of the interaction. Both are stretched balls of these very large balls suddenly an even larger black hole, you! Solve your little ( or not little ) technical problems GR as a so-called fundamental force in physics like... curvature '' of spacetime that gives rise to what we interpret as gravitational.!, arises at this point a recess mediated by a mysterious force, acting instantaneously a! Barely more descriptive than warping '' spacetime spacetime that gives rise to what interpret! ; the 2d sheet is embedded in some higher-dimensional flat space-time for us understand! Sky & Telescope converge ) 's physical intuition is as whirlpools density is equal the! The general variety who discovered the curvature of spacetime that gives rise to what we interpret gravitational... 2D but spacetime is only accurate at describing constant velocity density is equal to the notion of spacetime and,..., closed, or flat the insertion of an curvature of spacetime will curve this spacetime ( a ) suggest! Ride, we see that accelerated motion can warp space and time waves not... Effect as I ’ ve described is actually one of the Earth is barely more descriptive than warping spacetime! Fast enough to overcome the current with time and space-time called whole universe GR as a curvature of. The trajectories of nearby freely falling particles diverge ( or converge ) the basis a. Opinion, the deeper the indentation is Einstein called space-time, we explain first the of... ) technical problems in GR is geodesic deviation: the trajectories of nearby freely falling diverge... Here that Einstein connected the dots to suggest that gravity is the of... Rotates, it is here that Einstein connected the dots to suggest that gravity is mediated a. Trampoline this is, the bending of space-time curvature, gravity would not be in... Basic way of thinking about the Early universe falls, the trampoline begins to vibrate in universe., locally, spacetime is 4d ; the 2d sheet is embedded 3d! Distortion in spacetime which Einstein derived in GR as a so-called fundamental in! Be embedded in 3d space whether the universe has zero curvature ; it this... Waves do not move in the space - time transferred to our example with the trampoline ascribe a existence... The rotation of the intervening space inside them universe we can observe appears to fairly! ^\Frac12 \$ and skyandtelescope.org are registered trademarks of AAS Sky Publishing, LLC a! Any background Euclidean reference frame that extends throughout space a height, a wholly owned subsidiary of the curvature of spacetime! The insertion of an object will curve this curvature of spacetime ( a ), acting across..., LLC, a particularly large Dent in the space - time transferred to our example with the.. Opinion, the bending of space-time is caused from energy density or its derivatives as like or! Receive emails from Sky & Telescope, Night Sky, and skyandtelescope.org registered! Actually one of the balls of these very large Dent, approaching it rolls inevitably gravitational of! ’ re going to find that it ’ s the same as curvature a.
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2021-10-16 14:13:27
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https://mathhelpboards.com/threads/continuous-function.9324/
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# Continuous function?
#### Cbarker1
##### Active member
Let $f=tan(2x)/x$, x is not equal to 0.
Can the f be defined at x=0 such that it is continuous?
CBarker1
#### ThePerfectHacker
##### Well-known member
Compute limit at $0$. What do you get?
I got 2.
#### ThePerfectHacker
##### Well-known member
So,
$$\lim_{x\to 0} \frac{\tan 2x}{x} = 2$$
Now define the function,
$$f(x) = \left\{ \begin{array}{ccc}(\tan x)/x & \text{if} & x\not = 0 \\ 2 & \text{if}& x=0 \end{array} \right.$$
This function is continous everywhere because at $0$ we have $\lim_{x\to 0}f(x) = f(0) = 2$.
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2020-09-26 19:17:07
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https://nrich.maths.org/public/topic.php?code=-68&cl=3&cldcmpid=6982
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# Resources tagged with: Visualising
Filter by: Content type:
Age range:
Challenge level:
### Dissect
##### Age 11 to 14 Challenge Level:
What is the minimum number of squares a 13 by 13 square can be dissected into?
### Coloured Edges
##### Age 11 to 14 Challenge Level:
The whole set of tiles is used to make a square. This has a green and blue border. There are no green or blue tiles anywhere in the square except on this border. How many tiles are there in the set?
### Route to Infinity
##### Age 11 to 14 Challenge Level:
Can you describe this route to infinity? Where will the arrows take you next?
### Eight Hidden Squares
##### Age 7 to 14 Challenge Level:
On the graph there are 28 marked points. These points all mark the vertices (corners) of eight hidden squares. Can you find the eight hidden squares?
### A Tilted Square
##### Age 14 to 16 Challenge Level:
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
### Tetra Square
##### Age 11 to 14 Challenge Level:
ABCD is a regular tetrahedron and the points P, Q, R and S are the midpoints of the edges AB, BD, CD and CA. Prove that PQRS is a square.
### Square Coordinates
##### Age 11 to 14 Challenge Level:
A tilted square is a square with no horizontal sides. Can you devise a general instruction for the construction of a square when you are given just one of its sides?
### Coordinate Patterns
##### Age 11 to 14 Challenge Level:
Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead?
### Hypotenuse Lattice Points
##### Age 14 to 16 Challenge Level:
The triangle OMN has vertices on the axes with whole number co-ordinates. How many points with whole number coordinates are there on the hypotenuse MN?
### Screwed-up
##### Age 11 to 14 Challenge Level:
A cylindrical helix is just a spiral on a cylinder, like an ordinary spring or the thread on a bolt. If I turn a left-handed helix over (top to bottom) does it become a right handed helix?
### Something in Common
##### Age 14 to 16 Challenge Level:
A square of area 3 square units cannot be drawn on a 2D grid so that each of its vertices have integer coordinates, but can it be drawn on a 3D grid? Investigate squares that can be drawn.
### Bands and Bridges: Bringing Topology Back
##### Age 7 to 14
Lyndon Baker describes how the Mobius strip and Euler's law can introduce pupils to the idea of topology.
### Framed
##### Age 11 to 14 Challenge Level:
Seven small rectangular pictures have one inch wide frames. The frames are removed and the pictures are fitted together like a jigsaw to make a rectangle of length 12 inches. Find the dimensions of. . . .
### Triangular Tantaliser
##### Age 11 to 14 Challenge Level:
Draw all the possible distinct triangles on a 4 x 4 dotty grid. Convince me that you have all possible triangles.
### Concrete Wheel
##### Age 11 to 14 Challenge Level:
A huge wheel is rolling past your window. What do you see?
### Isosceles Triangles
##### Age 11 to 14 Challenge Level:
Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?
### Squares, Squares and More Squares
##### Age 11 to 14 Challenge Level:
Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9, 12, 15... other squares? 8, 11, 14... other squares?
### Zooming in on the Squares
##### Age 7 to 14
Start with a large square, join the midpoints of its sides, you'll see four right angled triangles. Remove these triangles, a second square is left. Repeat the operation. What happens?
### Fence It
##### Age 11 to 14 Challenge Level:
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
### On the Edge
##### Age 11 to 14 Challenge Level:
If you move the tiles around, can you make squares with different coloured edges?
### Squares in Rectangles
##### Age 11 to 14 Challenge Level:
A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?
### Just Opposite
##### Age 14 to 16 Challenge Level:
A and C are the opposite vertices of a square ABCD, and have coordinates (a,b) and (c,d), respectively. What are the coordinates of the vertices B and D? What is the area of the square?
### 3D Stacks
##### Age 7 to 14 Challenge Level:
Can you find a way of representing these arrangements of balls?
### Chess
##### Age 11 to 14 Challenge Level:
What would be the smallest number of moves needed to move a Knight from a chess set from one corner to the opposite corner of a 99 by 99 square board?
### Auditorium Steps
##### Age 7 to 14 Challenge Level:
What is the shape of wrapping paper that you would need to completely wrap this model?
##### Age 11 to 14 Challenge Level:
Can you mark 4 points on a flat surface so that there are only two different distances between them?
### Cubes Within Cubes Revisited
##### Age 11 to 14 Challenge Level:
Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?
### Corridors
##### Age 14 to 16 Challenge Level:
A 10x10x10 cube is made from 27 2x2 cubes with corridors between them. Find the shortest route from one corner to the opposite corner.
##### Age 11 to 14 Challenge Level:
Four rods, two of length a and two of length b, are linked to form a kite. The linkage is moveable so that the angles change. What is the maximum area of the kite?
### Diminishing Returns
##### Age 11 to 14 Challenge Level:
How much of the square is coloured blue? How will the pattern continue?
### Trice
##### Age 11 to 14 Challenge Level:
ABCDEFGH is a 3 by 3 by 3 cube. Point P is 1/3 along AB (that is AP : PB = 1 : 2), point Q is 1/3 along GH and point R is 1/3 along ED. What is the area of the triangle PQR?
### Troublesome Dice
##### Age 11 to 14 Challenge Level:
When dice land edge-up, we usually roll again. But what if we didn't...?
### Take Ten
##### Age 11 to 14 Challenge Level:
Is it possible to remove ten unit cubes from a 3 by 3 by 3 cube so that the surface area of the remaining solid is the same as the surface area of the original?
### Clocked
##### Age 11 to 14 Challenge Level:
Is it possible to rearrange the numbers 1,2......12 around a clock face in such a way that every two numbers in adjacent positions differ by any of 3, 4 or 5 hours?
### John's Train Is on Time
##### Age 11 to 14 Challenge Level:
A train leaves on time. After it has gone 8 miles (at 33mph) the driver looks at his watch and sees that the hour hand is exactly over the minute hand. When did the train leave the station?
### Dotty Triangles
##### Age 11 to 14 Challenge Level:
Imagine an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it/is it not possible to draw?
##### Age 11 to 14 Challenge Level:
How many different symmetrical shapes can you make by shading triangles or squares?
### Hidden Rectangles
##### Age 11 to 14 Challenge Level:
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
### Polygon Rings
##### Age 11 to 14 Challenge Level:
Join pentagons together edge to edge. Will they form a ring?
### Reflecting Squarely
##### Age 11 to 14 Challenge Level:
In how many ways can you fit all three pieces together to make shapes with line symmetry?
### Flight of the Flibbins
##### Age 11 to 14 Challenge Level:
Blue Flibbins are so jealous of their red partners that they will not leave them on their own with any other bue Flibbin. What is the quickest way of getting the five pairs of Flibbins safely to. . . .
### Icosian Game
##### Age 11 to 14 Challenge Level:
This problem is about investigating whether it is possible to start at one vertex of a platonic solid and visit every other vertex once only returning to the vertex you started at.
### Playground Snapshot
##### Age 7 to 14 Challenge Level:
The image in this problem is part of a piece of equipment found in the playground of a school. How would you describe it to someone over the phone?
### Seven Squares
##### Age 11 to 14 Challenge Level:
Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?
### Cutting a Cube
##### Age 11 to 14 Challenge Level:
A half-cube is cut into two pieces by a plane through the long diagonal and at right angles to it. Can you draw a net of these pieces? Are they identical?
### Weighty Problem
##### Age 11 to 14 Challenge Level:
The diagram shows a very heavy kitchen cabinet. It cannot be lifted but it can be pivoted around a corner. The task is to move it, without sliding, in a series of turns about the corners so that it. . . .
### Rolling Around
##### Age 11 to 14 Challenge Level:
A circle rolls around the outside edge of a square so that its circumference always touches the edge of the square. Can you describe the locus of the centre of the circle?
### Rolling Triangle
##### Age 11 to 14 Challenge Level:
The triangle ABC is equilateral. The arc AB has centre C, the arc BC has centre A and the arc CA has centre B. Explain how and why this shape can roll along between two parallel tracks.
### Khun Phaen Escapes to Freedom
##### Age 11 to 14 Challenge Level:
Slide the pieces to move Khun Phaen past all the guards into the position on the right from which he can escape to freedom.
### Buses
##### Age 11 to 14 Challenge Level:
A bus route has a total duration of 40 minutes. Every 10 minutes, two buses set out, one from each end. How many buses will one bus meet on its way from one end to the other end?
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2020-01-21 14:38:30
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https://gomathanswerkeys.com/texas-go-math-grade-7-module-7-quiz-answer-key/
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Refer to our Texas Go Math Grade 7 Answer Key Pdf to score good marks in the exams. Test yourself by practicing the problems from Texas Go Math Grade 7 Module 7 Quiz Answer Key.
7.1 Linear Relationships in the Form y = mx + b
Question 1.
Darice also took a break after riding 10 miles. The table below shows the rate at which Darice rides her bicycle after the break.
Write a verbal description of the relationship between the time she rides and the distance she travels.
Dance was riding a bike 10 miles plus an additional 0.25 miles after each break.
7.2 Writing and Graphing Equations in the Form y = mx + b
Emir started out a card game with 500 points. For every hand he won, he gained 100 points.
Question 2.
Complete the table.
y = 100 ∙ x + 500
Question 3.
Plot the points on the graph.
y = 100 ∙ x + 500
Question 4.
Write an equation for the linear relationship.
y = 100 ∙ x + 500
Essential Question
Question 5.
What are some of the ways you can represent real-world linear relationships?
Real-world relationships can be represented by graphics. tables, and equations.
Texas Go Math Grade 7 Module 7 Mixed Review Texas Test Prep Answer Key
Selected Response
Question 1.
Which description corresponds to the relationship shown in the table?
(A) earning $10 an hour (B) earning$8 an hour plus a $10 bonus (C) earning$7 an hour plus a $15 bonus (D) earning$9 an hour
(B) earning $8 an hour plus a$10 bonus
Explanation:
90 – 50 = 40 Find the diferrence between two payments.
$$\frac{40}{5}$$ = 8 Find how much is paid for 1 hour.
50 – 40 = 10 Find a bonus.
Question 2.
Which equation represents the same linear relationship as the graph below?
(A) y = 1.2x + 32
(B) y = 1.5x + 20
(C) y = 0.75x + 50
(D) y = 0.8x + 45
(C) y = 0.75 ∙ x + 50
Explanation:
Our points on the graph are:
A = (40,80)
B = (80,110)
C = (120, 140)
D = (160, 170)
Replacing the value of point A in the solution under (A) we get:
80 = 1.2 ∙ 40 + 32 = 80
This is correct
Replacing the value of point B in the solution under (A) we get:
110 = 1.2 ∙ 80 + 32 = 128
This isn’t correct.
By repeating the procedure, we will conclude that alt points are solutions of the equation under (C)
Question 3.
Omar began the week with $25. He took a city bus to and from school, paying$1.25 for each trip. Let x be the number of trips he took and y be the amount of money he had left at the end of the week. Which equation represents the relationship in the situation?
(A) y = 1.25x + 25
(B) y = 25 – 1.25x
(C) x = 25 – 1.25y
(D) y = 1.25x – 25
(B) y = 25 – 1.25x
Explanation:
The right answer is (B) y = 25 – 1.25 ∙ x, because he spends money on the bus, so the sum of money that remains to be reduced after each payment of trip.
Question 4.
Which table represents the same linear relationship as the equation y = 5x + 7?
Table (B)
Explanation:
The table represent the equation
y = 5 ∙ x + 7
Question 5.
Selina is planning to paint a large picture on a wall. She draws a smaller version first. The drawing is 8 inches by 6 inches. If the scale of the drawing is 2 in: 1 ft, what is the area of the actual picture on the wall?
(A) 4 feet
(B) 3 feet
(C) 48 square inches
(D) 12 square feet
(D) 12 square feet
Explanation:
The answer is (D) 12 square feet
The area of the drawing is 8 ∙ 6 = 48 in.
Because scale of the drawing, the area of the actual picture on the wall is
$$\frac{8}{2} \cdot \frac{6}{2}$$ = 4 ∙ 3 = 12 square feet
Gridded Response
Question 6.
The equation y = 3.5x – 210 represents the profit made by a manufacturer that sells products for \$3.50 each, where y is the profit and x is the number of units sold. What is the profit in dollars when 80 units are sold?
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2022-10-01 01:14:51
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http://leancrew.com/all-this/2007/02/more-unix-tidbits-for-os-x/
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# More Unix tidbits for OS X
As I said in this earlier post, Unix and Linux computers usually have an assortment of files with interesting bits of information. With a Mac, you can make this information more available to you by combining the Unix philosophy of text files and pipes with Quicksilver. This post will show a few more ways I’ve combined the two.
The /usr/share/misc folder is the normal place for interesting little files. Unfortunately, the Mac does not have the full complement of files that you’d probably find on a Linux machine. I went to this Free Software Foundation page and downloaded their miscfiles package, which is more complete. I ungzipped the files and put them in a miscfiles folder in my home directory. Now I’m ready to write some short shell scripts to access the info.
The first one pulls information from the birthtokens file, a file that Apple did include in /usr/share/misc and is also in the miscfiles package. I call the script birthstone and I keep it in my ~/bin folder.
#!/bin/bash
grep -i $1$HOME/miscfiles/birthtoken | sed 's/:/ - /g'
From the Terminal, this script would be invoked by a line like birthstone march and it would return March - Aquamarine - Jonquil. Even better, because Quicksilver has my ~/bin folder cataloged, I can call this script from QS.
1. Bring up Quicksilver (I use Control-Space).
2. Start typing “birthstone.”
3. Tab over to the third pane and type in the search string.
4. Hit Return and wait for the output to appear in a new QS command window.
5. Hit Return again to show the results in Large Type.
The process looks like this (the windows have been scaled to fit the blog):
A few notes on why the script is written the way it is:
• I call it “birthstone” rather than “birthtoken” because I would never remember “birthtoken.” Until I looked in this file, I didn’t even know there were birth flowers.
• The -i option makes grep case-insensitive, so I don’t have to capitalize anything.
• The sed part of the script replaces the colons in the birthtoken file with spaces and hyphens. I found that the colons were making Quicksilver think the output was a URL, so its default operation was Open URL. Replacing the colons caused QS’s default operation to change to Large Type, which is just what I wanted.
• Because grep searches the whole line, not just the first field, I can not only find the birthstone for a given month, I can also find the month associated with a given stone.
I’ve written a few other scripts that do basically the same thing with other files in the miscfiles package. More useful scripts include areacode
#!/bin/bash
grep -i $1$HOME/miscfiles/na.phone | sed 's/:/ - /g'
zipcode:
#!/bin/bash
grep -i $1$HOME/miscfiles/zipcodes | sed 's/:/ - /g'
and airport:
#!/bin/bash
grep -i $1$HOME/miscfiles/airport | sed 's/:/ - /g'
The airport script associates the airport with its three-letter code. The others should be self-explanatory.
I realize all this information is available on the net, but using these scripts is better than searching the net. They’re faster than Google, and when used with Quicksilver, I don’t have to switch from whatever program I’m in to my browser.
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2017-08-22 18:52:07
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https://byjus.com/question-answer/the-a-m-of-10-observations-is-40-if-the-sum-of-6-observations-is-1/
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Question
# The A.M of 10 observations is 40, If the sum of 6 observations is 280 then the mean of remaining 4 observations is
Solution
## Sum of 10 observations =n×A.M.=10×A.M.=10×40=400 ∴ Mean of remaining 4 observations=400−2804=30
Suggest corrections
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2021-12-08 11:18:42
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http://blog.jpolak.org/?paged=2
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# How to choose a PhD program
Posted by Jason Polak on 19. September 2017 · Write a comment · Categories: advice
Choosing where to get your PhD is an important decision. If you continue onto academia, your PhD might be the longest time you spend at any one institution until you get a permanent position. The most obvious choice is apply to the high-ranking schools. However, you should consider far more than that. Here, we’ll look at some of the important factors to consider, with the context of mathematics in mind. However, most of what I say applies to some other fields as well.
## Represented research areas
Unlike choosing an undergraduate program, where the curriculum doesn’t differ much around the world (though it certainly can vary greatly in strength or intensity), a PhD will be on a very specialised topic. So, if you go to a school where analysis and statistics are the main topics represented and you like algebra, you probably won’t like it. This can be worse for those places where you don’t have to choose an advisor until the second year. So I suggest you look at the represented research areas on departmental websites and see what catches your interest. Unfortunately, some math department websites look like they were coded on a Super Nintendo, if that were even possible. So:
Make sure someone is actually doing something you’re interested in at prospective schools!
If you’re at the undergraduate level and not sure of your interests yet, it could be a good idea to consider a masters program first before starting a PhD. I enjoyed doing a masters degree first, even though in the long run it is more expensive.
## Total school atmosphere
If you’re lucky enough to be nearby some schools you’re interested in, you should visit them, meet some professors, and even sit in on some classes and departmental seminars. Just walk around and see what it’s like. Some schools have a much nicer atmosphere than others. You should also get a sense of the surrounding city. This is true especially if you are a very independent worker: having an enjoyable city will in fact make working much easier. Conversely, living in a place you dislike for several years is quite draining.
Sadly, living temporarily in cities you don’t like is very probable in at least one stage of climbing the academic ladder.
More »
# Is it a projective module?
Posted by Jason Polak on 19. September 2017 · Write a comment · Categories: homological-algebra, modules
Consider a field $k$. Define an action of $k[x,y]$ on $k[x]$ by $f*g = f(x,x)g(x)$ for all $f\in k[x,y]$ and $g\in k[x]$. In other words, the action is: multiply $f$ and $g$ and then replace every occurrence of $y$ by $x$.
Is $k[x]$ a projective $k[x,y]$-module? Consider first the map $k[x,y]\to k[x]$ given by $f\mapsto f(x,x)$. It’s easy to check that this map is in fact a $k[x,y]$-module homomorphism. It would be tempting to try and split this map with the inclusion map $k[x]\to k[x,y]$. But this doesn’t work: this inclusion is not a $k[x,y]$-module homomorphism.
In fact, the $k[x,y]$-module homomorphism $k[x,y]\to k[x]$ given by $f\mapsto f(x,x)$ cannot split simply because there are no nonzero $k[x,y]$-module homomorphisms $k[x]\to k[x,y]$. Therefore, $k[x]$ is not projective as a $k[x,y]$-module, using the module structure we gave it.
Here are two more ways to see this:
1. Through the notion of separability: by definition, $k[x]$ being a projective $k[x,y]\cong k[x]\otimes_k k[x]$-module under the structure that we have defined means that $k[x]$ is a separable $k$-algebra. However, all separable $k$-algebras are finite-dimensional as vector spaces over $k$, whereas $k[x]$ is infinite-dimensional.
2. Through Seshradi’s theorem: this theorem says that every finitely-generated projective module over $k[x,y]$ is actually free. Therefore, we just have to show that $k[x]$ is not free because $k[x]$ is certainly finitely-generated as a $k[x,y]$-module. But $(x^2y – xy^2)$ annihilates all elements of $k[x]$, which cannot happen in a free module.
# Strong Nilpotence and the Jacobson Radical
Posted by Jason Polak on 01. September 2017 · Write a comment · Categories: ring-theory · Tags:
In the previous post we saw the following definition for a ring $R$: An element $r\in R$ is called strongly nilpotent if every sequence $r = r_0,r_1,r_2,\dots$ such that $r_{n+1}\in r_nRr_n$ is eventually zero. Why introduce this notion?
Well, did you know that every finite integral domain is a field? If $R$ is an integral domain and $a\in R$ is nonzero, then the multiplication map $R\to R$ given by $x\mapsto ax$ is injective. If $R$ is finite, then it must also be surjective so $a$ is invertible!
Another way of stating this neat fact is that if $R$ is any ring and $P$ is a prime ideal of $R$ such that $R/P$ is finite, then $P$ is also a maximal ideal. A variation of this idea is that every prime ideal in a finite commutative ring is actually maximal. Yet another is that finite commutative rings have Krull dimension zero.
More »
# Nilpotent and Strongly Nilpotent
Posted by Jason Polak on 31. August 2017 · Write a comment · Categories: ring-theory · Tags:
Let $R$ be an associative ring. An element $r\in R$ is called nilpotent if $r^n = 0$ for some $n$. There is a stronger notion: an element $r\in R$ is called strongly nilpotent if every sequence $r = r_0,r_1,r_2,\dots$ such that $r_{n+1}\in r_nRr_n$ is eventually zero.
How are these two related? It is always the case that a strongly nilpotent element is nilpotent, because if $r$ is strongly nilpotent then the sequence $r,r^2,r^4,r^8,\dots$ vanishes. However, the element
$$\begin{pmatrix}0 & 1\\ 0 & 0\end{pmatrix}$$
in any $2\times 2$ matrix ring is nilpotent but not strongly nilpotent. Notice how we had to use a noncommutative ring here—that’s because for commutative rings, a nilpotent element is strongly nilpotent!
# Comparing Methods for Finding the Vertex of a Parabola
Posted by guest on 27. August 2017 · Write a comment · Categories: math
A guest Post by Paul Pierce and Ashley Ross
With the advances in calculator technology, some developmental and college-level math courses are restricting the use of any type of graphing or programmable calculators. This is to help students avoid becoming dependent on their calculators for both simple arithmetic and graphing. So, some teachers are going “old school” and forbidding the use of calculators in the classroom. Therefore, it is imperative that students learn efficient methods for finding important values, as well as graphing functions, without the help of their calculator. One type of function that appears in many courses is the quadratic function, and one of the most critical points on the graph of a quadratic function is the vertex.
## Fundamental Concepts of the Graph of a Quadratic Function
For the function $f(x)=ax^2+bx+c$ with $a\not=0$, the graph is a smooth, continuous curve called a parabola. This parabola opens upward if $a > 0$ or opens downward if $a < 0$. The vertex $(h,k)$ of the graph is the only turning point on the parabola, which makes it a critical point. The $y$-coordinate $k$ of the vertex represents the minimum value of the function if $a>0$, or the maximum value of the function if $a<0$.
The point $(h,k)$ may be found using the formulas $h=\frac{-b}{2a}$ and $k=\frac{bh}{2}+c$, which begin to show the importance of the vertex. We give two examples:
Example 1. For $y=x^2+6x+3$, find the vertex $(h,k)$.
First find $h$ using $h=\frac{-b}{2a}=\frac{-6}{2(1)}=-3$.
Next find $k$ using $k=\frac{bh}{2}+c=\frac{(6)(-3)}{2}+3=-9+3=-6$.
So, the coordinates of the vertex of the parabola are $(-3, -6)$. Observe from the graph that this vertex is the lowest point on the parabola, which means that $k = -6$ is the minimum value of the function.
Example 2.For $y=-2x^2+8x-5$, find the vertex $(h,k)$.
First find h using $h=\frac{-b}{2a}=\frac{-8}{2(-2)}=2$.
Next find k using $k=\frac{bh}{2}+c=\frac{(8)(2)}{2}-5=8-5=3$.
So, the coordinates of the vertex of the parabola are $(2, 3)$. Note that this vertex is the highest point on the graph, which illustrates that $k = 3$ is the maximum value of this function.
# Semisimple and Jacobson Semisimple
Posted by Jason Polak on 27. August 2017 · Write a comment · Categories: math, modules · Tags: , ,
Let $R$ be an associative ring with identity. The Jacobson radical ${\rm Jac}(R)$ of $R$ is the intersection of all the left maximal ideals of $R$. So, ${\rm Jac}(R)$ is a left ideal of $R$. It turns out that the Jacobson radical of $R$ is also the intersection of all the right maximal ideals of $R$, and so ${\rm Jac}(R)$ is also an ideal!
The idea behind the Jacobson radical is that one might be able to explore the properties of a ring $R$ by first looking at the less complicated ring $R/{\rm Jac}(R)$. Since the ideals of $R$ containing ${\rm Jac}(R)$ correspond to the ideals of $R/{\rm Jac}(R)$, the ring $R/{\rm Jac}(R)$ has zero Jacobson radical. Often the rings $R$ for which ${\rm Jac}(R) = 0$ are called Jacobson semisimple.
This terminology might be a tad bit confusing because typically, a ring $R$ is called semisimple if every left $R$-module is projective, or equivalently, if every left $R$-module is injective. How does the notion of semisimple differ from Jacobson semisimple? The Wedderburn-Artin theorem gives a classic characterisation of semisimple rings: they are exactly the rings that are finite direct products of full matrix rings over division rings. Since a full matrix ring over a division ring has no nontrivial ideals, the product of such rings must have trivial Jacobson radical. Thus:
A semisimple ring is Jacobson semisimple.
The converse is false: there exists a ring that is Jacobson semisimple but not semisimple. For example, let $R$ be an infinite product of fields. Then ${\rm Jac}(R) = 0$. However, $R$ is not semisimple. Why not? If it were, by Wedderburn-Artin it could also be written as a finite product of full matrix rings over division rings, which must be a finite product of fields because $R$ is commutative. But a finite product of fields only has finitely many pairwise orthogonal idempotents, whereas $R$ has infinitely many.
Incidentally, because $R$ is not semisimple, there must exist $R$-modules that are not projective. However, $R$ does have the property that every $R$-module is flat!
# Are we running out of problems?
Posted by Jason Polak on 06. August 2017 · Write a comment · Categories: math, opinion
A senior mathematician who will remain nameless recently said in a talk, “there is nothing left to prove”. In context, he was referring to the possibility that we are running out of math problems. People who heard laughed, and first-year calculus students might disagree. Was it said as a joke?
Because of the infinite nature of mathematics, there will always be new problems. On the other hand, there are only finitely many theorems we’ll ever know; only finitely many that we’ll ever be interested in. Are we close to knowing all the interesting theorems? Is the increasing specialisation of the literature a sign of a future with a thousand subfields each with only one or two devotees?
Truthfully, I don’t think math is running out of problems at all. I think it’s more like good, nonspecialist exposition isn’t really keeping up with the rapid development of mathematics and so we know less and less about what our colleagues are doing. So we should attempt to prevent the future where every person is their own research field. Here are some ways we could do that:
1. Make part of your introduction in your paper understandable to a much wider range of mathematicians. This will encourage more collaboration and cross-disciplinary understanding. For example, once I was actually told by a journal to cut out a couple of pages from a paper because it was well-known to (probably ten) experts, even though that material was literally not written down anywhere else! Journals should actually encourage good exposition and not a wall of definition-theorem-proof.
2. Have the first twenty minutes of your talk understandable by undergraduates. Because frankly, this is the only way mathematicians (especially young ones) in other fields will actually understand the motivation of your work. How are we supposed to ask good questions when we can’t figure out where our research fits in with the research of others?
3. Use new avenues of mathematical exposition like blogs and nontechnical articles. Other fields like physics and biology appear in magazines like Scientific American and have an army of people working to make specialised work understandable to the nonspecialist.
4. Encourage new, simplified proofs or explanations of existing results. And by ‘encourage’, I mean count high-quality, expository papers on the level of original results in determining things like tenure and jobs! There are already journals that publish these types of papers. Chances are, any expository paper will actually help at least as many people as an original result, perhaps more. And there are still hundreds of important papers that are very difficult if not impossible to read (even by many experts), with no superior alternative exposition available.
I think it’s been a long-lived fashion in mathematics to hide the easy stuff in favour of appearing slick ever since one dude tried to hide how he solved the cubic from another dude, and it’s probably something we can give up now.
# Check out this preliminary text on cluster algebras
Posted by Jason Polak on 25. July 2017 · Write a comment · Categories: math
Fomin, Williams, and Zelevinsky (posth.) are preparing a new introductory text on cluster algebras. The first three chapters look elementary enough, and it’s worth a look for those interested in learning this topic.
# Extensions of Finite Rings are Integral
Posted by Jason Polak on 19. July 2017 · Write a comment · Categories: commutative-algebra · Tags: ,
Here’s a classic definition: let $R\subseteq S$ be commutative rings. An element $s\in S$ is called integral over $R$ if $f(s)=0$ for some monic polynomial $f\in R[x]$. It’s classic because appending the solutions of polynomials to base rings goes way back to the ancient pasttime of finding solutions to polynomial equations.
For example, consider $\Z\subseteq \Z[\sqrt{2}]$. Every element of $\Z[\sqrt{2}]$ is integral over $\Z$, which essentially comes down to the fact that $\sqrt{2}$ satisfies $x^2 – 2$. On the other hand, the only elements of $\Q$ integral over $\Z$ are the integers themselves.
The situation is much different for finite commutative rings. If $R\subseteq S$ are finite rings, then every element of $S$ is integral over $R$. Proof: suppose $s\in S$ and set $T = \{ f(s): f\in R[x]\}$. For each $t\in T$ fix a polynomial $f$ such that $f(s) = t$. The set of all such polynomials is finite so we can define $m$ as the maximum degree of all these polynomials. Then $s^{m+1}\in T$ and so there is an $f$ of degree at most $m$ such that $s^{m+1} – f(s) = 0$. Thus $s$ satisfies the monic polynomial $x^{m+1} – f(x)$. QED.
Cool right? However, this is just a more general case of the following theorem: let $R\subseteq S$ be commutative rings. Then $S$ is finitely generated as an $R$-module if and only if $S$ is finitely generated as an $R$-algebra and every element of $S$ is integral over $R$.
# Paper Announcement: The Polypermutation Group of an Associative Ring
Posted by Jason Polak on 02. July 2017 · 1 comment · Categories: paper
I’ve submitted paper! The results stem from a pretty simple question that can be understood with an first course in abstract algebra. This post will explain the question and give a teaser of some of the results.
Let $R$ be a ring. A polynomial $f\in R[x]$ induces a function $R\to R$ given by $a\mapsto f(a)$. It turns out that this function is sometimes bijective. When this happens, we say that $f$ is a permutation polynomial. There are some easy examples: $f(x) = x + a$ for $a\in R$ is always injective, and always bijective if $R$ is finite. But there are less trivial examples as well. For instance, the polynomial $f(x) = x^6 + x^4 + x^2 + x$ permutes $\Z/27$.
Permutation polynomials are perhaps most well-known when $R$ is a finite field. In this case, every function $R\to R$ can be represented by a polynomial. In particular, every permutation can so be represented. This result is not particularly deep. More interesting for finite fields is to determine which polynomials are permutation polynomials, and to find certain classes of them.
More interesting things happen when $R$ is a finite ring that is not a field. Then it is not necessarily true that all functions $R\to R$ can be represented by polynomials. Can all permutations be represented by polynomials? The answer is in fact no! So, it makes perfect sense to define a group ${\rm Pgr}(R)$ as the subgroup of the symmetric group on $R$ generated by all permutations represented by polynomials. Let’s call it the polypermutation group of $R$.
Under this notation, ${\rm Pgr}(R)$ is the symmetric group on $R$ when $R$ is a finite field. What about other rings? This is what brings us to the topic of my latest paper: The Polypermutation Group of an Associative Ring. This paper started out by asking the simple question:
What is ${\rm Pgr}(R)$ for some common finite rings?
In my paper I’ve concentrated on $\Z/p^k$ where $p$ is a prime. The general case of $\Z/n$ for an integer $n$ reduces to this case via the Chinese Remainder Theorem.
Upon my initial investigations I found that ${\rm Pgr}(\Z/p^k)$ is actually a little complicated. It turns out to be easier when $p \geq k$. In this case I wrote down an explicit formula for the cardinality of ${\rm Pgr}(\Z/p^k)$. I already mentioned that when $k = 1$ the result is classical and is $p!$ because then $\Z/p$ is a finite field. One of my results is:
Theorem (P-). Let $p$ be a prime and $k\geq 2$ be an integer with $p\geq k$. Then:
$$|{\rm Pgr}(\Z/p^k)|= p![(p-1)p^{(k^2 + k-4)/2}]^p.$$
Whoa, that’s complicated. But it’s not hard to see that this is going to be less than $(p^k)!$, showing that there are indeed some permutations that cannot be represented by polynomials in this case. In fact, one can be more precise in the case of $k=2$. In this case, one can compute the group ${\rm Pgr}(\Z/p^2)$ itself, though I’ll leave you to read the paper to find out what it is!
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2017-11-18 04:32:23
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http://www.freemathhelp.com/forum/archive/index.php/t-44680.html
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PDA
View Full Version : Simplifying -2^4 x 2 + (-2)^4
LaZy1
07-08-2006, 05:10 PM
I'm given the equation -2^4 x 2 + (-2)^4 I work the equation and I keep getting a 48 but the answer, I'm told, is supposed to be -16. Is there a rule or a little tidbit I've forgotten?
I finished and haven't done anything involving algebra for about 2 years now and I've graduated since then, so I'm a bit rusty and forget little things. Thanks for anyone that helps me with this.
pka
07-08-2006, 05:25 PM
\L
\begin{array}{l}
- 2^4 \times 2 = - 32 \\
\left( { - 2} \right)^2 = 16 \\
\end{array}
TchrWill
07-09-2006, 11:00 AM
I'm given the equation -2^4 x 2 + (-2)^4 I work the equation and I keep getting a 48 but the answer, I'm told, is supposed to be -16. Is there a rule or a little tidbit I've forgotten?
I finished and haven't done anything involving algebra for about 2 years now and I've graduated since then, so I'm a bit rusty and forget little things. Thanks for anyone that helps me with this.
(-2)x(-2) = + 4
(+4)x(-2) = -8
(-8)x(-2) = +16
Therefore,, (-2)^4 = +16
16x2 = 32
32 + 16 = 48
I agree with you.
stapel
07-09-2006, 11:12 AM
I agree with you.
Note the parentheses: -2<sup>4</sup> = -(2)(2)(2)(2) = -16 is quite different from (-2)<sup>4</sup> = (-2)(-2)(-2)(-2) = +16.
Eliz.
TchrWill
07-09-2006, 12:38 PM
Note the parentheses: -2<sup>4</sup> = -(2)(2)(2)(2) = -16 is quite different from (-2)<sup>4</sup> = (-2)(-2)(-2)(-2) = +16.
So right you are. Thanks for making me aware of my misdirection.
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2014-04-18 15:40:06
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http://www.lmfdb.org/knowledge/show/lfunction.riemann
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show · `lfunction.riemann` all knowls · up · search:
The Riemann zeta-function is defined by $$\zeta(s)=\sum_{n=1}^\infty \frac 1{n^s}$$ for $s=\sigma+it$ with $\sigma>1$. Euler showed that $$\zeta(s)=\prod_p (1-1/p^s)^{-1},$$ known today as the Euler product, for $s$ a real number with $s>1$, a fact that connects the study of $\zeta(s)$ with the prime numbers. Riemann, in 1859, was the first to consider $\zeta(s)$ as a function of a complex variable. He proved that $\zeta(s)$ has an analytic continuation to the whole complex plane apart from a simple pole at $s=1$ with residue 1. Riemann proved a remarkable functional equation. He showed that $$\xi(s):=\frac 12 s (s-1) \pi^{-s/2} \Gamma(s/2) \zeta(s)$$ is an entire function of order 1 which satisfies $$\xi(s)=\xi(1-s).$$ Riemann showed that the distribution of prime numbers depends on the location of the zeros of the Riemann zeta-function which, by the Euler product and functional equation, are all in the critical strip $0\le \sigma \le 1$. Riemann showed that the number $N(T)$ of zeros of $\zeta(s)$ in the critical strip with ordinates at most $T$ satisfies $$N(T)=\frac{T}{2\pi} \log\frac{T}{2\pi e} +O(\log T).$$ He calculated the first few zeros and verified that they are on the critical line $\sigma=1/2$ and then made his famous conjecture, today known as the Riemann Hypothesis, that all of the zeros are on the critical line. In 1896 Hadamard and de la Vallee Poussin independently proved that $\zeta(s)$ has no zeros on the boundary of the critical strip, i.e. all of the zeros are in the open strip $0< \sigma < 1$, and in doing so proved the Prime Number Theorem, that the number of primes less than a large number $x$ is asymptotic to $\frac{x}{\log x}$.
Today we know that the first 10 trillion zeros of $\zeta(s)$ are all on the critical line and that at least 41% of all of the complex zeros are on the critical line.
Authors:
Knowl status:
• Review status: beta
• Last edited by David Farmer on 2012-06-28 13:12:58.766000
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2018-06-20 02:09:27
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https://mathinsight.org/length_curves_refresher
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# Math Insight
### Length of curves
The basic point here is a formula obtained by using the ideas of calculus: the length of the graph of $y=f(x)$ from $x=a$ to $x=b$ is $$\hbox{ arc length }=\int_a^b\;\sqrt{1+\left({dy\over dx}\right)^2}\;dx$$ Or, if the curve is parametrized in the form $$x=f(t)\;\;\;\;\;y=g(t)$$ with the parameter $t$ going from $a$ to $b$, then $$\hbox{ arc length }=\int_a^b\; \sqrt{\left({dx\over dt}\right)^2+\left({dy\over dt}\right)^2}\;dt$$
This formula comes from approximating the curve by straight lines connecting successive points on the curve, using the Pythagorean Theorem to compute the lengths of these segments in terms of the change in $x$ and the change in $y$. In one way of writing, which also provides a good heuristic for remembering the formula, if a small change in $x$ is $dx$ and a small change in $y$ is $dy$, then the length of the hypotenuse of the right triangle with base $dx$ and altitude $dy$ is (by the Pythagorean theorem) $$\hbox{ hypotenuse }=\sqrt{dx^2+dy^2}= \sqrt{1+\left({dy\over dx}\right)^2}\;dx$$
Unfortunately, by the nature of this formula, most of the integrals which come up are difficult or impossible to ‘do’. But if one of these really mattered, we could still estimate it by numerical integration.
#### Exercises
1. Find the length of the curve $y=\sqrt{1-x^2}$ from $x=0$ to $x=1$.
2. Find the length of the curve $y={ 1 \over 4 }(e^{2x}+e^{-2x})$ from $x=0$ to $x=1$.
3. Set up (but do not evaluate) the integral to find the length of the piece of the parabola $y=x^2$ from $x=3$ to $x=4$.
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2018-09-24 23:16:23
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http://math.stackexchange.com/questions/257266/how-to-solve-a-sqrtbn-a-sqrtbn-x
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# How to solve $(a+\sqrt{b})^n - (a-\sqrt{b})^n = x$?
Consider equation $(a+\sqrt{b})^n - (a-\sqrt{b})^n = x$
How do I properly solve for $n$ given $x$?
-
What are your thoughts? Have you written out the brackets using the binomial theorem? – Alex B. Dec 12 '12 at 18:01
I don't even know where to begin with solving it because I don't know how to expand it with binomial theorem, although there is en.wikipedia.org/wiki/Binomial_theorem#Statement_of_the_theorem is this what you mean? – KaliMa Dec 12 '12 at 18:04
So does this mean (a+sqrt(b))^n is the sum from k=0 to n of (n choose k) a^k*sqrt(b)^(n-k)? – KaliMa Dec 12 '12 at 18:06
Correct. If you multiply this out, you will notice substantial cancellation. – Alex B. Dec 12 '12 at 18:09
@AlexB. I am not sure how to apply that here. I don't know the value of n as that's what I am solving for. I only know x. Therefore I am not sure how many terms I need to expand this by – KaliMa Dec 12 '12 at 18:11
There are many approaches. If you think $n$ is a small whole number, you can just try a range. A spreadsheet would make this easy. If one of $a+\sqrt b$ or $a-\sqrt b$ is smaller than $1$ it will to to zero as $n$ increases. To be definite, assume $a-\sqrt b$ is smaller than $1$. Let's ignore it for a moment. Then $n \approx \frac {\log x}{\log (a+\sqrt b)}$ where you can use your favorite base for the logs. You can use numerical methods. The left side will be monotonic with $n$, so any reasonable root-finder will work. Just graphing will get you very close.
As an example, suppose you want to find the index of a Fibonacci number. We are given that $\sqrt 5 F_n=(\frac {1+\sqrt 5}2)^n-(\frac {1-\sqrt 5}2)^n$. As $|\frac {1-\sqrt 5}2|\lt 1$, powers of it go to zero quickly. If somebody gives us $14930352$ and asks which number it is, we can just do $\frac {\log (14930352\sqrt 5)}{\log (\frac {1+\sqrt 5}2)}$ and get a number that is within $10^{-14}$ of $36$, so $n=36$
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In my case, neither side has an absolute value less than 1 – KaliMa Dec 12 '12 at 18:09
@KaliMa: In that case I would graph it or use a spreadsheet – Ross Millikan Dec 12 '12 at 18:17
This was helpful, thank you – KaliMa Dec 12 '12 at 18:20
Consider equation $(a+\sqrt{b})^n - (a-\sqrt{b})^n = x$
Consult this link: The Binomial Theorem.
Since you are given $x$ (taken to be a constant), you can also try take logarithms of each side of the equation.
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They are not squared, they are raised to power $n$ – KaliMa Dec 12 '12 at 18:07
Fixed: see the link. It will help you understand how to approach the problem. – amWhy Dec 12 '12 at 18:12
@amWhy Using your link, I took the original equation to: $\sum_{k=0}^n {n\choose k}a^k\sqrt b^{n-k} - \sum_{k=0}^n {n\choose k}a^k(-\sqrt b)^{n-k} = x$ Not sure where to go from there? I get it down to: $\sum_{k=0}^n {n\choose k}a^k({\sqrt b}^{n-k}-{(-\sqrt b)}^{n-k})= x$ – user17753 Dec 12 '12 at 19:44
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2016-05-25 13:16:56
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https://www.semanticscholar.org/paper/RIEMANN-EINSTEIN-STRUCTURE-FROM-VOLUME-AND-GAUGE-Wilczek/8fd09c518e8dc615e5a6d700bb429c89fddd6e77
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# RIEMANN-EINSTEIN STRUCTURE FROM VOLUME AND GAUGE SYMMETRY
@article{Wilczek1998RIEMANNEINSTEINSF,
title={RIEMANN-EINSTEIN STRUCTURE FROM VOLUME AND GAUGE SYMMETRY},
author={Frank Wilczek},
journal={Physical Review Letters},
year={1998},
volume={80},
pages={4851-4854}
}
• F. Wilczek
• Published 1998
• Physics
• Physical Review Letters
It is shown how a metric structure can be induced in a simple way starting with a gauge structure and a preferred volume, by spontaneous symmetry breaking. A polynomial action, including coupling to matter, is constructed for the symmetric phase. It is argued that assuming a preferred volume, in the context of a metric theory, induces only a limited modification of the theory.
70 Citations
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Einstein's traceless 1919 gravitational theory is analyzed from a variational viewpoint. It is shown to be equivalent to a transverse (invariant only under diffeomorphisms that preserve the LebesgueExpand
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A SL(5, ℝ) gauge-invariant topological field theory of gravity and possible gauge unifications are considered in four-dimensions (4D). The problem of quantization is evaluated in the asymptoticExpand
Mass gap in Yang's theory of gravity
The quantization of a curvature-squared model of gravity, in the affine form proposed by Yang, is reconsidered in the path integral formulation. Due to its inherent Weyl invariance, sharing this withExpand
Gauge theories of spacetime symmetries
Gauge theories of conformal spacetime symmetries are presented which merge features of Yang-Mills theory and general relativity in a new way. The models are local but nonpolynomial in the gaugeExpand
Einstein-Weyl Gravity from a Topological $${{\rm SL}(5, \mathbb{R})}$$ Gauge Invariant Action
A topological field theory of gravity in four-dimension is proposed which is finite after quantization. Since such ‘minimal’ BF type models for the high energy limit are physically not quiteExpand
Cosmological solutions with torsion in a model of the de Sitter gauge theory of gravity
• Physics
• 2008
The torsion is shown to be vitally important in the explanation of the evolution of the universe in a large class of gravitational theories containing quadratic terms of curvature and torsion. TheExpand
On the weak field approximation of the de Sitter gauge theory of gravity
• Physics
• 2013
The weak field approximation of a model of de Sitter gauge theory of gravity is studied in two cases. Without torsion and spin current, the model cannot give the right non-relativistic approximationExpand
A New Solution with Torsion in Model of dS Gauge Theory of Gravity
A new static de Sitter solution with torsion in the model of de Sitter gauge theory of gravity is obtained. The torsion only contains O(3)-symmetric tensor part according to irreducibleExpand
On torsion-free vacuum solutions of the model of de Sitter gauge theory of gravity
• Physics
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It is shown that all vacuum solutions of Einstein field equation with a positive cosmological constant are the solutions of a model of dS gauge theory of gravity. Therefore, the model is expected toExpand
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2021-10-23 13:43:58
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https://gecogedi.dimai.unifi.it/seminar/1327/
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# Homogeneous almost Kähler Chern-Einstein manifolds of semisimple Lie groups
## Dmitry Alekseevsky
created by chrysikos on 10 Sep 2019
13 sep 2019 -- 13:00
Abstract.
An almost Kähler manifold $(M, g, J)$, i.e. an almost Hermitian manifold with closed Kähler form $\omega$, is called Chern-Einstein manifold if the Ricci form $\rho$ of the Chern connection is proportional to the Kähler form, that is $\rho=\lambda\omega$, for some real number $\lambda$. In this talk we give a description of all homogeneous almost Kähler manifolds $(M=G/L, g, J)$ of a non-compact semi-simple Lie group $G$. We propose a general approach for classification of invariant Chern-Einstein structures on $M=G/L$ and give a complete description of homogeneous Chern-Einstein structures on the regular adjoint orbit of classical Lie groups $G$. The talk is based on a joint work with Fabio Podesta
Credits | Cookie policy | HTML 5 | CSS 2.1
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2019-09-16 04:17:24
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https://forum.allaboutcircuits.com/threads/in-a-50-ohm-bnc-bnc-coaxial-cable-is-50-ohm-the-resistance-between-the-core-and-the-outer-mesh.189770/
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# In a 50-ohm BNC-BNC coaxial cable, is 50 ohm the resistance between the core and the outer mesh?
Thread Starter
#### vishnu.mec
Joined Aug 6, 2010
23
This is for calculating the load for a power source. Suppose I have a 50 V DC source with an output resistance of 10 ohms (let's consider a purely resistive situation). Now, what will be the current through the circuit if I connect this source to a 100-ohm resistor via a 50-ohm BNC-BNC coaxial cable? Will it be $\frac{50 V}{10+50+100}$ or $\frac{50 V}{10+\frac{50*100}{50+100}}$ ?
#### DickCappels
Joined Aug 21, 2008
9,524
The resistance of the cable is a minor part of the impedance. The impedance is largely determined by the distributed capacitance and distributed inductance of the cable or transmission line. A termination equal to the cable’s characteristic impedance will prevent the power fro being reflected back toward the signal source.
#### WBahn
Joined Mar 31, 2012
27,946
The DC resistance of coax depends on the specific coax and the length used. But you will need quite a bit of it (perhaps 500 feet) before the total resistance of the core going out and the sheath coming back will get to even 1 Ω.
The "50 Ω" is the "characteristic impedance" of the transmission line at high frequencies and is dominated by the sqrt of the ratio of inductance to capacitance. If you look at that ratio, you will see that the units turn out to be those of impedance, namely volts/ampere.
#### MrChips
Joined Oct 2, 2009
27,742
This sounds like a trick question in a job interview.
The DC resistance of a 50Ω coaxial cable is not 50Ω.
You can make the assumption that the DC resistance along the central conductor of a short length of cable is 0Ω.
#### panic mode
Joined Oct 10, 2011
2,366
neither...
the 50 Ohm is the characteristic impedance of the cable, when it is used as a transmission line (for high frequencies). DC is the opposite end of the spectrum. inductive and capacitive effects practically do not exist when used with pure DC so you should see open circuit between coax core and the shield. resistance of conductors themselves (core or shield) will be very low and it usually can be considered to be close to zero Ohm.
Last edited:
#### crutschow
Joined Mar 14, 2008
31,156
Measure the end to end resistance of both connections, and the resistance from the center conductor to shield, to see what you actually have.
Below are the characteristics of a common 50Ω coaxial cable:
#### Papabravo
Joined Feb 24, 2006
19,617
Characteristic impedance of a coaxial cables can be estimated from the geometric properties of the connectors. See the following calculation aid on this website:
https://www.allaboutcircuits.com/tools/coax-impedance-calculator/
You also need the relative dielectric constant $$\epsilon_{r}$$ of the material between the conductors
#### Lo_volt
Joined Apr 3, 2014
280
A cable with no termination at the far end will have very high DC impedance controlled by the leakage current (very, very small) through the insulation dielectric. You won't be able to measure it with a normal ohmmeter or multimeter. For the calculations you are asking about it can be considered an open circuit.
Others are mentioning characteristic impedance, etc. Don't be confused. They are talking about reactive impedance.
#### Jerry-Hat-Trick
Joined Aug 31, 2022
211
Thread Starter
#### vishnu.mec
Joined Aug 6, 2010
23
Thank you very much for all your replies. I understood that the coaxial cable's impedance is mostly caused by the capacitive and inductive reactances, which can be neglected for a DC signal.
Similar threads
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2023-02-09 10:15:49
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https://www.physicsforums.com/threads/rocket-force-max-range-vs-time.186026/
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# Rocket Force - Max Range vs. Time
1. Sep 20, 2007
### bphysics
Rocket Force -- Max Range vs. Time
1. The problem statement, all variables and given/known data
"Assume an ICBM of maximum range 8000 km is fired at your city from a distance of 8000 km."
1) How much warning time will you have if the missile is first detected at its halfway point
2) How fast will the missile be travelling when it is first detected?
3) With what velocity will it strike its target?
4) Now assume an ICBM of greater maximum range is fired at your city from a distance of 8000 km at an angle of 50 degrees. How much more warning time will you have if the missile is detected at the halway point than the first scanario?
2. Relevant equations
x = Vx0t
y = Vy0t - (1/2)gt^2
v = v0 + at
v^2 = v(2/0) + 2a(x - x0)
x = x0 + v0t + (1/2)at^2
3. The attempt at a solution
I keep on trying to figure out how to even start off this sucker. I've performed projectile motion problems before, but this "appears" more complex -- in the sense that I have no initial velocity and no initial angle. To me, I am uncertain how to proceed with this problem.
It would appear that I need to utilize the maximum range of 8000 km combined with my knowledge that gravity is taking this missle down at 9.8 m/s^2 to help me calculate for these other values, but I don't seem to grasp how to.
I'll be writing more of this on my whiteboard, and if I discover anything, I'll post it.
For now, I'm just hoping someone can put me on the right track.
2. Sep 20, 2007
### FedEx
Apply the equation for the range of the rocket. From here you will get the initial velocity.
3. Sep 21, 2007
### bphysics
FedEx, I don't seem to be able to utilize any of the equations which I have listed at this time due to how I do not have enough information to "complete" the variable requirements enough to solve for a variable.
Can you tell me how you can see this being solved, and more, specifically, via which listed equation you think I have enough evidence for? I may simply be looking at the problem from the wrong angle.
4. Sep 21, 2007
### learningphysics
Try to derive a formula for the range, in terms of v0 and theta. Hint: what is the time to reach the maximum height.... if you double that... you get the time it hits the range.
5. Sep 21, 2007
### bphysics
Understood - however, your discusing time here.
I have no time data.. unless I am just very confused right now.
6. Sep 21, 2007
### FedEx
I thought that as you have solved a few projectile problems you would be knowing the equations of projectile motion.
No matter. keep the following in mind.
1) The horizontal range of the projectile is max when angle is 45
2) The horizontal range is given by R = $$\frac{v^2\sin2\theta}{g}$$
3) The time taken by the projectile to reach the max height is equal to the time taken by it to come to the ground from the max height.
4) The max height is given by $$H_{max}$$ = $$\frac{v^2sin^2\theta}{g}$$
5)The total time is given by T = $$\frac{2vsin\theta}{g}$$
Here v is the initial velocity.
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2017-06-23 09:01:17
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http://scholarlyrepository.miami.edu/dissertations/3224/
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## Dissertations from ProQuest
#### Title
The combined effects of beta agonists and isokinetic exercise on the quadriceps muscle group
1994
Article
#### Degree Name
Doctor of Philosophy (Ph.D.)
#### Department
Teaching and Learning
#### First Committee Member
Joseph Signorile, Committee Chair
#### Abstract
Muscle atrophy, a condition common to those persons afflicted with an injury or illness, is characterized by a loss of muscle size and strength. Various forms of rehabilitation have been looked at as a means to alleviate this condition. The purpose of this study is to examine the combined effects of the $\beta\sb2$ agonist albuterol with an isokinetic training protocol, observing the impact on the quadriceps muscle group.Healthy volunteers were placed into one of two groups in a double-blind fashion (albuterol, placebo) to examine the effects each group had as a result of a nine week training program. After a break-in period, subjects received 16mg/day of albuterol or placebo for six weeks. Using an isokinetic dynamometer, subjects twice weekly performed three sets of leg extensions in the concentric-eccentric mode, ten repetitions per set. Anthropometric data of the right thigh were collected at weeks zero, six, and nine. Pulmonary measurements were taken immediately prior to and after capsule consumption. Statistical analysis involves a 3 x 2 repeated measures MANOVA.Of the sixteen dependent variables examined in this study, five showed evidence of simple main effects. They were: concentric total work to bodyweight ratio (CTW/BW), eccentric total work (ETW), eccentric average power (EAP), eccentric peak torque to bodyweight ratio (EPT/BW), and eccentric total work to bodyweight ratio (ETW/BW). With each of these variables, those subjects receiving albuterol produced greater results.Subjects receiving albuterol and isokinetic exercise showed greater improvements in strength than those subjects receiving the placebo in conjunction with their exercise. This effect was especially noticeable in the eccentric mode. The results of this study appear promising, with the potential for these agents to be used therapeutically in a rehabilitative setting.
#### Keywords
Education, Physical
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2017-08-16 19:34:43
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http://jetpletters.ac.ru/cgi-bin/front/ad/en
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Editor's Choice Superradiance Properties of a Suspension of Composite Nanoscale integration of organic and metallic particles is expected to open up new opportunities for the design high-performance nanoscale devices. Optimization of heterostructures requires experimental and theoretical analysis of their specific physical properties. Nanosystem consisting in gold nanospheres covered by silica shell impregnated with the organic dye molecules comes into focus as a possible plasmonic based nanolaser, i.e. "spaser" [1]. Depending on the distance between the emitters and metal there are possible various phenomena [2,3]. In this paper we experimentally studied the characteristics of a suspension of spasers at the temperatures $T_N=77.4K,T_R=293K$. It was found that the system possesses characteristics of a laser medium. The S-shaped dependence of the radiation intensity and the compression of the lasing line with increase of the pumping power were observed. Ten-fold increase of the intensity of the radiation generated by the medium and line narrowing with temperature change $T_R\to T_N$ was found. The experimental results were compared with a numerical simulation of a spaser model consisting of 20 two-level media and a metallic nanosphere. The temperature effects were modeled by the introduction of the Markov process. It was found that observed effects can be explained by means of the feedback caused by the nonlinear interaction of polarizations with their total reflection in the metallic core. At low temperatures Bloch vectors related with two-level systems form an analog of a ferromagnetic state. With increasing fluctuations, antiferromagnetic states are formed along with the desynchronization of ferromagnetic one. These properties allows us to explain the observed changes in the intensity of the and line form of laser generation with temperature. Experimental and numerical results of the work demonstrate that the synchronization of the polarization of dye molecules caused by inverse nonlinear coupling yields an analog of plasmon-polariton superradiance. 1. D.J. Bergman and M.I. Stockman, Phys.Rev.Lett. 90, 027401 (2003). 2. M. Haridas et al, J. Appl. Phys.114, 064305 (2013). 3. M. Praveena et al, Phys. Rev. B 92, 235403 (2015). A. S. Kuchyanov, A.A. Zabolotskii, Plekhanov A.I. JETP Letters 106 (2) (2017) Energy Spectrum of the Spin States in $Sr_2FeSi_2O_7$ and Origin of the Magneto- Electric Coupling Recently Sr2FeSi2O7 comes into focus as a possible compound with unusual magneto-electric coupling or, in other words, as a novel potential multiferroic [1,2]. Results of terahertz spectroscopy in the paramagnetic state show that the multiplet Fe+2(S=2) of the ground state splits due to the spin-orbit coupling. However the energy intervals between the low-lying singlet state and excited states are quite small so that all spin states are populated at the temperature of about 100 K. The Fe+2 ion occupies the center of a tetragonally distorted tetrahedron. In the present communication the origin of the magneto-electric coupling is described as follows. The odd crystal field from the tetrahedral environment induces the coupling of the orbital momentum of the Fe+2( 5D) state with the external electric field. On the other hand, the orbital momentum is coupled with spin via the spin –orbit interaction. Both angular momenta are coupled with the external magnetic field, which is enhanced due to the presence of the superexchange interaction between neighboring Fe+2 ions. Combining all these couplings, the author derived the affective spin Hamiltonian for the magneto-electric coupling, which made it possible to calculate relative intensities of the electric dipole transitions between spin states and estimate the magnetization caused by the external electric field as well as the electric polarization induced by the magnetic field. Thuc T. Mai, C. Svoboda, M. T. Warren, T.-H. Jang, J. Brangham, Y. H. Jeong, S.-W. Cheong, and R. Valdes Aguilar. Phys. Rev. B, 94, 224416 (2016) Yongping Pu, Zijing Dong, Panpan Zhang, Yurong Wu, Jiaojiao Zhao, Yanjie Luo. Journal of Alloys and Compounds, 672 , 64-71 (2016) M.V. Eremin JETP Letters 105 (11) (2017) Electron-topological transition in copper-oxide high-TC superconductors before superconducting transition It is well known the conductivity of high-temperature superconductors (HTSCs) with TC ~100 K (YBaCuO, BiSrCaCuO, etc.) is provided at T~300 K by hole (h) fermions [1]. It is also known the superconducting transition in such cuprates is accomplished by means of the Cooper pairing, while the fluctuating Cooper pairs with charge -2e exist even at T=TC+(~30 K) [2]. Hence it inevitably follows in the interval TC
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2017-07-24 06:46:35
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https://overiq.com/c-examples/c-program-to-calculate-factorial-using-recursion/
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# C Program to calculate Factorial using recursion
Last updated on September 24, 2020
The following is a C Program to calculate Factorial using recursion:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 /************************************************** Program to calculate Factorial using recursion * * Enter a number: 4 * 4! = 24 ***************************************************/ #include // include stdio.h library long factorial(int num); int main(void) { int n; printf("Enter a number: "); scanf("%d", &n); printf("%d! = %ld", n, factorial(n)); return 0; // return 0 to operating system } long factorial(int num) { //base condition if(num == 0) { return 1; } else { // recursive call return num * factorial(num - 1); } }
Expected Output:
1st run:
1 2 Enter a number: 0 0! = 1
2nd run:
1 2 Enter a number: 5 5! = 120
## How it works #
The following figure demonstrates how the evaluation of 5! takes place:
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2020-09-26 06:38:44
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http://clay6.com/qa/34327/an-electric-current-is-passed-though-an-aqueous-solution-of-a-mixture-of-al
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Browse Questions
# An electric current is passed though an aqueous solution of a mixture of alanine (isoelectric point 6.0),glutamic acid (3.2) and arginine (10.7) buffered at pH 6. What will happen to the three amino acids?
$\begin{array}{1 1}(a)\;\text{Glutamic acid will precipitate out}\\(b)\;\text{Arginine will migrate anode}\\(c)\;\text{Alanine will migrate towards cathode at pH 6}\\(d)\;\text{None}\end{array}$
At pH 6 , glutamic acid is present as an anion and migrates to the anode. Alginine is present as a cation and migrates to the cathode. Alanine is a dipolar ion remains uniformly distributed in solution.
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2016-10-26 19:14:25
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https://www.visitfredensborg.dk/jasg5zu7/f34e54-complete-the-function-table-for-each-equation
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# complete the function table for each equation
These In and Out Box Worksheets are appropriate for 3rd Grade, 4th Grade, and 5th Grade. In and Out Boxes for Addition and Subtraction Complete The Function Table For Each Equation - Displaying top 8 worksheets found for this concept. These In and Out Box Worksheets will produce 14 problems on each page. Math. This is a table of possible values to use when graphing the equation. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. Improve your math knowledge with free questions in "Complete a function table from an equation" and thousands of other math skills. none of the above 2. y = Kids will be able to easily review and practice their math skills. These Function Table Worksheets are appropriate for 4th Grade, 5th Grade, 6th Grade, and 7th Grade. Found worksheet you are looking for? Determine which function has the greater rate of change in questions 1−3 1. x y ----- -1 0 0 1 1 2 2 3 (1 point) The rates of change are equal. Click here for a Detailed Description of all the Function Table Worksheets & In and Out Boxes Worksheets. These Function Table Worksheets & In and Out Boxes Worksheets are a great resource for children in 3rd Grade, 4th Grade, 5th Grade, 6th Grade, and 7th Grade. Find exact solutions. Easily check their work with the answer sheets. Here is a graphic preview for all of the Function Table Worksheets & In and Out Boxes Worksheets. Simply download and print these Function Table Worksheets. Substitute the x values of the equation to find the values of y. Use a function to complete a function table. Complete The Function Table For Each Equation Answer Key / / These In and Out Box Worksheets are appropriate for 3rd Grade, 4th Grade, and 5th Grade. You can select different variables to customize these Function Table Worksheets & In and Out Boxes Worksheets for your needs. Use the function rule to complete the table. Complete parts a ±c for each quadratic function. Free math problem solver answers your algebra, geometry, trigonometry, calculus, and statistics homework questions with step-by-step explanations, just like a math tutor. Sona Goerlitz November 10, 2017 Uncategorized Leave a comment 168 Views. Function Table Worksheets With Answer Sheet These Function Table Worksheets are great for all levels of math. These In and Out Boxes Worksheets will dynamically create In and Out Boxes for Multiplication and Division Rules in Horizontal and Vertical formats. A truth table lists every possible combination of values of the input variables, and the corresponding output value of the function for each combination. To take things further, another worksheet generator includes graphing practice as well. For example, if you were to go to the store with $12.00 to buy some candy bars that were$2.00 each, your total cost would be determined by how many candy bars you bought. 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Namu: Date: Worksheet Graphing Linear Equations Wom IT Complete the function table and graph the line for each equation (2) 1 1 . x^2*y+x*y^2 ） The reserved functions are located in " Function List ". To download/print, click on pop-out icon or print icon to worksheet to print or download. Therefore, the maximum value of the function is 9. These function tables give students practice computing the outputs for a linear equation. Complete The Function Table For Each Equation Math Aids Tessshlo. Homogenous Equations: is homogeneous if the function f(x,y) is homogeneous, that is. Substitute for and find the result for . Input(x) Output(y) 32 20 14 2 ? Remove parentheses. These Function Table Worksheets are great for giving students practice in computing the outputs for different linear equations. 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These In and Out Boxes Worksheets will dynamically create In and Out Boxes for Addition and Subtraction Rules in Horizontal and Vertical formats including Word problems. It is common to name a function either f(x) or g(x) instead of y. f(2) means that we should find the value of our function when x equals 2. When you're trying to graph a quadratic equation, making a table of values can be really helpful. The Function Table Worksheets & In and Out Boxes Worksheets are randomly created and will never repeat so you have an endless supply of quality Function Table Worksheets & In and Out Boxes Worksheets to use in the classroom or at home. Quick Link for All Function Table Worksheets, Detailed Description for All Function Table Worksheets. Worksheet will open in a new window. Table of Values A rational function is given. Use a Table of Values to Graph the Equation y=x-3. Function Worksheets. Function Tables. 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These Function Table Worksheets are great for giving students practice in computing the outputs for different linear equations. Patterns function machine worksheets patterns function machine worksheets patterns function machine worksheets function table worksheets. Click the image to be taken to that Function Table Worksheet. Complete The Function Table For Each Equation. Complete The Function Table For Each Equation - Displaying top 8 worksheets found for this concept. Complete a table and identify coordinates when given a two-variable equation, such as y=2x+4. MATH. Complete the table below to solve the equation 2.5x − 10.5 = 64(0.5x). （ex. −6 -2 -14 -10 ? These In and Out Box Worksheets are appropriate for 3rd Grade, 4th Grade, and 5th Grade. 1 -8 6 4 -4 0 -3 4 2 (4) EX - 1 . (b) Describe the behavior of the function near its vertical asymptote, based on Tables 1 and 2. c) Determine the horizontal asymptote, based on Tables 3 and 4. Assume your own values for x for all worksheets provided here. Complete the tables, plot the points, and graph the lines. Matching Equations Tables And Graphs Worksheet Answer Key. You can & download or print using the browser document reader options. You may select between four different types of equations. Patterns function machine worksheets free commoncoresheets patterns function machine worksheets free commoncoresheets patterns function machine worksheets free commoncoresheets functions relationships input and output values mathematics. These Function Table Worksheets will generate 12 function table problems per worksheet. To graph a linear equation, first make a table of values. That way, you can pick values on either side to see what the graph does on either side of the vertex. If you're seeing this message, it means we're having trouble loading external resources on our website. Computing the Output of FunctionsFunction Table Worksheets The table has a greater rate of change. How to complete the function table for each equation . a. Complete The Function Table For Each Equation, Close Reading Questions Over Flowers For Algernon. Textbook solution for Algebra 1 1st Edition McGraw-Hill/Glencoe Chapter 9.1 Problem 67PPS. You can choose from up to four types of equations depending on the sophistication of your students. Problem 5. Original problem; Step 1; Step 2; Step 3; Step 4; Write the equation from the table of values provided below. These In and Out Box Worksheets will produce 14 problems on each page. Our Function Table Worksheets & In and Out Boxes Worksheets are free to download, easy to use, and very flexible. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. In and Out Boxes for Multiplication and Division Input And Output Math Worksheets Algebra 1 Domain. x f(x) = 2.5x − 10.5 g(x) = 64(0.5x) 2 3 4 5 6 Complete the function table for each equation math aids com complete the function table for each equation math aids com complete the function table for each equation answer key math aids complete the function table for each equation math aids com.
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2021-05-09 04:50:18
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http://crypto.stackexchange.com/tags/elgamal-signature/hot?filter=year
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# Tag Info
## Hot answers tagged elgamal-signature
3
Basically because of Fermat's little theorem: if $a$ is not divisible by $p$ then $a^{p-1} = 1$ $mod$ $p$. A part of the expression for $\delta$ appears as a power of $a$ in the ElGamal signature verification equation, which "happens" to work because it is reduced modulo $p-1$ so Fermat's little theorem applies.
2
With your proposed modification of the ElGamal signature scheme you can produce forgeries for arbitrary (hashed) messages $m$. By looking at the verification equation $$g^m = yr^s$$ you just have to set $r$ to $r=(g^my^{-1})^{s^{-1}}$ (just by rearranging the verification equation) which you can do for any $s$ from $\mathbb{Z}_{p-1}^*$, i.e., every $s$ ...
1
There have been some research in Optimal Extension Fields (OEF), introduced at Crypto'98 by Bailey and Paar paper. The idea is to work in a field $GF(p^n)$ with $p$ prime and of the form $2^{32}\pm c$ with small $c$ for 32-bit CPUs ($2^{64}\pm c$ for 64-bit CPUs), so that they can leverage on CPU's ALU for most computations, therefore OEF based systems are ...
1
Yes, cryptosystems like ElGamal or Shnorr based on the intractability of Dlog Problem are are indicated to be implemented on finite field, which is not the case of the RSA for which a model was proposed in the early $80^{ies}$, and immediatly broken. As you know, a finite field is denoted by $GF(q)$ where $q=p^m$ and p would be any Prime. But in the case ...
1
First, I think you have a typo in your question since in the original article $s = (M - x y)(r^{-1}) \mod p-1$, and not $s = (M - x^y)(r^{-1}) \mod p-1$. Knowing that, then we can construct $s_2$ from $s, r, M$ and $M_2$: $s_2 = s + (M_2 - M)r^{-1} = (M - x^y)r^{-1} + (M_2 - M)r^{-1} = (M - x^y + M_2 - M)r^{-1} = (M_2 - x y)r^{-1}$ A valid signature for ...
Only top voted, non community-wiki answers of a minimum length are eligible
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2015-04-28 19:58:51
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https://www.physicsforums.com/threads/entropy-of-an-irreversible-process.203797/
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# Entropy of an irreversible process
supermesh
Can entropy change be zero for a closed system in an irreversible process? If yes under what conditions? Thanks!
Is it ture that it would be zero in an adiabatic, process carried in consant temperature?
## Answers and Replies
Staff Emeritus
Science Advisor
Gold Member
Can entropy change be zero for a closed system in an irreversible process? If yes under what conditions? Thanks!
Entropy is a state function and as such the integral;
$$\int^{f}_{i}\frac{dQ_R}{T}$$
is path independent. What does that tell you?
perebal
Is the irreversible part of entropy state function? Or is the total entropy (included the rev. and irrev. part) state function?
Fernsanz
Hello.
As Hotenanny has pointed out, entropy is a function of state. That means that its value depends only on the (equilibrium) state of the system under consideration and no on the process carried out to get the system to that state. So, think of any two isoentropic states. Then, no matter by mean of which process you connect those states, the entropy of the system under consideration at those states will be the same.
I have highlighted the system under consideration because even though the entropy change in a irreversible process can be zero for the thermodynamical system you are considering, the total entropy change will be strictly positive. Total means the entropy change of the system plus the entropy change of the surrounding universe. And this is in fact always positive for an irreversible process. So, the answer is yes for the parcel of the universe that you agree to call "your system", and no for the total entropy which accounts for all changes made in the universe.
In an adiabatic process you have dQ=0. If, additionaly, the process is isotermal then it is reversible (because talking of temperatures involves talking of equilibrium). So, in an adiabatic isothermal process you can apply the integral with dQ=0 and hence the entropy change is zero.
Do I answer your question?
Rainbow Child
... In an adiabatic process you have dQ=0. If, additionaly, the process is isotermal then it is reversible (because talking of temperatures involves talking of equilibrium). So, in an adiabatic isothermal process you can apply the integral with dQ=0 and hence the entropy change is zero. ...
You can't have an adiabatic isothermal process which is simultaneously reversible.
In a reversible adiabatic process Poisson's law reads
$$p\,V^\gamma=const.\Rightarrow T\,V^{\gamma-1}=const.$$
which tells you that the temperature is not constant.
An adiabatic isothermal process can be only irreversible, so the entropy change is positive. Consider for example the following situation
The box is made of adiabatic walls, and in the left hand side there is an ideal gas at temperature T. If we remove the black inside wall then the gas will fill all the space. Since the walls are adiabatics $$Q=0$$, and no work is done $$W=0$$, from the 1st law of thermodynamics we have
$$Q=\Delta U+W\Rightarrow \Delta U=0 \Rightarrow n\,C_V\,\Delta T=0 \Rightarrow T=const.$$
Thus the process is adiabatic and isothermal but irrevesible.
In order to calculate the entropy's change, since we cannot apply $$\int^{f}_{i}\frac{dQ}{T}$$ we use the fact that entropy is a state funtion as Hootenanny posted.
That means we can imagine an revesible processure which has the same initial and final states. That could be an isothermal from the volume $V_1$ to the volume $V_1+V_2$. The heat for this process is $$Q=n\,R\,T\,\ln\frac{V_1+V_2}{V_1}$$,so
$$\Delta S=n\,R\,\ln\left(1+\frac{V_2}{V_1}\right)>0$$
Fernsanz
You can't have an adiabatic isothermal process which is simultaneously reversible.
In a reversible adiabatic process Poisson's law reads
$$p\,V^\gamma=const.\Rightarrow T\,V^{\gamma-1}=const.$$
which tells you that the temperature is not constant.
An adiabatic isothermal process can be only irreversible, so the entropy change is positive. Consider for example the following situation
In general it is possible to have adiabatic isothermal reversible processes. You have stuck to a system consisting only of ideal gases that is when Poisson's law is valid.
It depends on which boundaries you specify for your system. Depending of the boundaries one same form of energy can be considered heat or work. For example, if you put an electrical resistance inside the thermodynamical system, is it that electrical energy heat or work? If the boundary of the system does not contain the resistance, then it is heat what you are giving to the system. However, if the boundary contains the resistance then it is work that you are performing to the system by mean of electric charges in movement.
So, put a resistance inside the system, consider it as part of the system and you can have an isothermal adiabatic reversible process because the only interchange of energy will be in form of work.
Last edited:
Rainbow Child
In general it is possible to have adiabatic isothermal reversible processes. You have stuck to a system consisting only of ideal gases that is when Poisson's law is valid.
It depends on which boundaries you specify for your system. Depending of the boundaries one same form of energy can be considered heat or work. For example, if you put an electrical resistance inside the thermodynamical system, is it that electrical energy heat or work? If the boundary of the system does not contain the resistance, then it is heat what you are giving to the system. However, if the boundary contains the resistance then it is work that you are performing to the system by mean of electric charges in movement.
In the first case is heat as you say. In the second case is electical energy provided to the ends of the resistance.
So, put a resistance inside the system, consider it as part of the system and you can have an isothermal adiabatic reversible process because the only interchange of energy will be in form of work.
What makes you think that the above process is reversible?? How can you bring the system back to it's initial state?
And you also think that this process is isothermal??
Could you write down for me the the equation that describes the state of the system $$f(p,V,T)=0[/itex] in order to calculate the change of entropy by [tex] \Delta S=\int^{f}_{i}\frac{dQ}{T}$$
Rainbow Child
Here is the Clausius-Duhem inequality: http://en.wikiversity.org/wiki/Continuum_mechanics/Clausius-Duhem_inequality_for_thermoelasticity
Which entropy ($$\eta$$) is in it? The reversible part, the irreversible part or the sum of them?
I can't understand that wiki page!
But as far as I know the Clausius-Duhem inequality of classical thermodynamics is a statement of the 2nd. It applies to thermodynamic processes during which a system evolves from one equilibrium state (i) to another (f) and it says that the integrated heat absorbed by the system, divided by the temperature at which that heat is absorbed, is bounded from above by the net change in the entropy of the system:
$$\int^{f}_{i}\frac{dQ}{T}\leq\Delta S=S_f-S_i$$
The equality stands for a reversible process.
Fernsanz
In the second case is electical energy provided to the ends of the resistance.
Exactly, and since we agree in that is not heat, it must be work. The first law says that energy can be only heat or work.
What makes you think that the above process is reversible?? How can you bring the system back to it's initial state?
And you also think that this process is isothermal??
Could you write down for me the the equation that describes the state of the system $$f(p,V,T)=0[/itex] in order to calculate the change of entropy by [tex] \Delta S=\int^{f}_{i}\frac{dQ}{T}$$
I was not referring to any particular process. But if you want one consider the reversible separation of gases by mean of semipermeable membranes. There you will find an isothermal adiabatic reversible process.
Last edited:
Rainbow Child
Exactly, and since we agree in that is not heat, it must be work. The first law says that energy can be only heat or work.
But it isn't work. It's electrical energy provided to the resistance. If you want to include to your system the resistance as you say
However, if the boundary contains the resistance then it is work that you are performing to the system by mean of electric charges in movement.
then this is not a closed system because the charges are moving through the boundaries of your system.
I was not referring to any particular process.
And that about that?
So, put a resistance inside the system, consider it as part of the system and you can have an isothermal adiabatic reversible process because the only interchange of energy will be in form of work.
...You have stuck to a system consisting only of ideal gases ...
...But if you want one consider the experiment described in the Gibbs' paradox. There you will find an isothermal adiabatic reversible process.
The Gibbs paradox in order to decsribe adiabatic reversible process, has to deal with ideal gases. See for instance
"Explanation of the Gibbs paradox within the framework of quantum thermodynamics" at
http://arxiv.org/abs/quant-ph/0507145
Fernsanz
But it isn't work. It's electrical energy provided to the resistance. If you want to include to your system the resistance as you say
Well, it is energy given to the system and it is not heat, so what could it be? First law reads
$$dU= dQ + dW_{mechanical} + dW_{electrical} + \ldots$$
In fact first law says defines the heat as "if it is not work it is heat". So, if it is not heat it is work. So in the case of the resistance it is electrical work. It is exceedingly cut-clear.
then this is not a closed system because the charges are moving through the boundaries of your system
I don't remember anyone impossing it has to be closed. I was just giving an example of an adiabatic isothermal reversible process.
And that about that?
?
The Gibbs paradox in order to decsribe adiabatic reversible process, has to deal with ideal gases
First of all, sorry because I didn't mean to say Gibbs's paradox. I mean reversible separation of gases. which, in any case, deals with ideal gases. Here you will have an adiabatic isothermal reversible process and even more: with no work! $$dQ=dW=dU=0$$; and is a closed system. What else do you need? As I suspect you are more interested in discussing than in understanding my arguments, I have decided to attach a picture of the page of the Atkin's book which explain this. The figure showing the arrangement is also attached.
I think with this, discussion is over and it is clear and laid-down that there is adiabatic isothermal reversible processes.
#### Attachments
• SeparationGasesExplanation.jpg
79.6 KB · Views: 861
• SeparationGasesScheme.jpg
37.7 KB · Views: 799
Last edited:
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2023-03-28 11:56:03
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https://www.physicsforums.com/threads/statistical-physics.827663/
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Homework Help: Statistical Physics
Tags:
1. Aug 14, 2015
rangatudugala
• New member warned about posting with no effort shown
How to prove this "if p(a)=p(b)=p then p(ab) ≤ p^2
2. Relevant equations
3. The attempt at a solution
2. Aug 14, 2015
RUber
Do you know anything about p, a, or b?
3. Aug 14, 2015
RUber
Please provide some information about what you have already tried or methods you are familiar with so we can point you in the right direction--i.e. fill in the template.
4. Aug 14, 2015
rangatudugala
Probability of a = probability b = p
5. Aug 14, 2015
RUber
What does p(ab) mean? Both a and b happen? What if a is b? then p(a and b) = p(a and a) = p.
6. Aug 14, 2015
rangatudugala
No Ruber
if p(a) = p(b) = p (let say some value) then prove or disprove p(a ∩ b) ≤ p^2
7. Aug 14, 2015
RUber
So do you know anything about a and b? Assume a = b, then p(a ∩ b) = p(a) = p ≥ p^2 .
8. Aug 14, 2015
rangatudugala
nothing mention disjoint or not i need to prove this is right or not
9. Aug 14, 2015
rangatudugala
10. Aug 14, 2015
RUber
Clearly if a and b were disjoint, the probability of a and b happening together is zero which will surely be less than p.
The key here would be if a and b were independent. If they are, then you might have something to prove...otherwise, you just have 0 ≤ p(a ∩ b) ≤ p.
11. Aug 14, 2015
rangatudugala
yes i got the point if and be disjoint then p(ab) =0
let say they are not disjoint then how to prove that ?
12. Aug 14, 2015
RUber
There is nothing to prove unless you know they are independent.
13. Aug 14, 2015
RUber
If the events a and b are independent, then, by the definition of independence, p(a ∩ b) = p(a) p(b) = p^2.
If they are not independent, then like I said before, they can be anywhere from disjoint to completely coincident, i.e. 0≤p(a ∩ b)≤p.
Is p ≤ p^2?
14. Aug 14, 2015
rangatudugala
thing is no any hints (information) given in the question.. okay what if not ?
15. Aug 14, 2015
rangatudugala
hmm if p ≤ p^2
then p= 1 kw i dont think in that way
16. Aug 14, 2015
RUber
Think of a Venn diagram with two circles representing a and b, both the same size (p). What is the maximum size of the overlapping region?
If no other information is given in the question, then you can assume that anything is possible other than what you know to be true.
If you are to prove the statement, you need to show it holds true all the time. If you are to disprove it, you just need one counterexample.
17. Aug 14, 2015
RUber
So, if p can be any value between 0 and 1, you have to prove that the statement is true for all values of p, not just p=1.
I don't think you will be able to prove it to be true without more constraints or assumptions.
Can you prove that it is not true?
18. Aug 14, 2015
rangatudugala
null set
19. Aug 14, 2015
RUber
If p(a∩b) = 0, then p(a∩b) ≤ p^2.
That is not a good counterexample.
Similarly, if p = 0, then p(a∩b) ≤ p^2. So, that's no good.
20. Aug 14, 2015
rangatudugala
so you trying to explain that p(null set) = 0 so its not good example is it ?
21. Aug 14, 2015
RUber
Right.
If p(a) = p(b) and the problem doesn't state that a is not b, then a = b should be your first example.
Look at post 7. Assume 0<p<1 to eliminate the option for p = p^2.
22. Aug 14, 2015
rangatudugala
okay i think i got the answer
so
1/ if a, b mutually exclusive then p(a∩b) =0
2/ if a,b independent then p(a∩b)= p(a)*p(b) =p^2
is it ?
23. Aug 14, 2015
RUber
Both of your statements 1/ and 2/ are true, but this is not a proof.
You don't know anything about a and b.
What if a and b are entirely coincident, i.e. if a then b?
24. Aug 14, 2015
rangatudugala
oh dear you confused me...
Last edited by a moderator: Aug 15, 2015
25. Aug 14, 2015
Ray Vickson
Please use different letters: use $P(a)$ and $P(b)$ for the probabilities of $a$ and $b$, but the letter $p$ for their value; that is, you should say $P(a) = P(b) = p$. That will avoid a lot of confusion.
Both of your examples obey $P(a \cap b) \leq p^2$. But: are you finished? No: you have not proved that $P(a \cap b) \leq p^2$ for all possible cases where $P(a) = P(b) = p$, nor have you discovered a counterexample (that is, an example where $P(a \cap b) > p^2$).
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2018-06-21 18:38:51
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https://www.ms.u-tokyo.ac.jp/seminar/2022/sem22-092.html
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## 複素解析幾何セミナー
開催情報 月曜日 10:30～12:00 数理科学研究科棟(駒場) 128号室 平地 健吾, 高山 茂晴, 野村 亮介
### 2022年07月11日(月)
10:30-12:00 数理科学研究科棟(駒場) 128号室
The CR Killing operator and Bernstein-Gelfand-Gelfand construction in CR geometry (Japanese)
[ 講演概要 ]
In this talk, I introduce the CR Killing operator associated with compatible almost CR structures on contact manifolds, which describes trivial infinitesimal deformations generated by contact Hamiltonian vector fields, and discuss how it can also be reconstructed by the Bernstein-Gelfand-Gelfand construction in the general theory of parabolic geometries. The “modified” adjoint tractor connection defined by Cap (2008) plays a crucial role. If time permits, I’d also like to discuss what this observation might mean in relation to asymptotically complex hyperbolic Einstein metrics, which are bulk geometric structures for compatible almost CR structures at infinity.
[ 参考URL ]
https://forms.gle/hYT2hVhDE3q1wDSh6
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2022-08-17 04:46:10
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http://openstudy.com/updates/55f7dc1ee4b0a91f3931cc72
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## mathmath333 one year ago Their are 4 letters and 4 envelopes.find the number of ways in which only two letters go in wrong envelope ?
1. mathmath333
\large \color{black}{\begin{align} & \normalsize \text{Their are 4 letters and 4 envelopes.find the number of ways }\hspace{.33em}\\~\\ & \normalsize \text{in which only two letters go in wrong envelope ?}\hspace{.33em}\\~\\ & a.)\ 4 \hspace{.33em}\\~\\ & b.)\ 5 \hspace{.33em}\\~\\ & c.)\ 6 \hspace{.33em}\\~\\ & d.)\ 3 \hspace{.33em}\\~\\ \end{align}}
2. anonymous
its c 6 ways
3. mathmath333
yea how u got it
4. anonymous
we will need to select the two letters which have to be put in wrong envelop . that is $4 \times \frac{ 3 }{ 2 }$ i.e $^{4}c _{2} = 6$
5. mathmath333
thnks
6. anonymous
no worries :)
7. anonymous
Yep :) It's C, 6 ways :)
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2017-01-17 08:59:37
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http://cms.math.ca/cmb/msc/46B25?fromjnl=cmb&jnl=CMB
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Canadian Mathematical Society www.cms.math.ca
location: Publications → journals
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Search: MSC category 46B25 ( Classical Banach spaces in the general theory )
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1. CMB 2012 (vol 56 pp. 503)
Bu, Qingying
Weak Sequential Completeness of $\mathcal K(X,Y)$ For Banach spaces $X$ and $Y$, we show that if $X^\ast$ and $Y$ are weakly sequentially complete and every weakly compact operator from $X$ to $Y$ is compact then the space of all compact operators from $X$ to $Y$ is weakly sequentially complete. The converse is also true if, in addition, either $X^\ast$ or $Y$ has the bounded compact approximation property. Keywords:weak sequential completeness, reflexivity, compact operator spaceCategories:46B25, 46B28
© Canadian Mathematical Society, 2015 : https://cms.math.ca/
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2015-01-30 02:38:55
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https://mathematica.stackexchange.com/questions/266853/difference-in-interpolated-data-derivative-and-analytic-function
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# Difference in interpolated data derivative and analytic function
I am trying to calculate the derivative of interpolated data but it behaves differently from the analytic solution. Detailed code that I have used follows:
\[Gamma]w = 70.0*10^-3;(*SR in N/m*)\[Rho] = 1000; c = 3*10^8; g = 9.8;
we1 = 7*10^-6;(*beam waist*)n = 1.33; P0 = 4.0; rng = 50*10^-6;
lc = Sqrt[\[Gamma]w/(\[Rho]*g)]; P0 = 4; we1 =
7*10^-6; P1 = (P0/(\[Gamma]w*c*Pi))*((n - 1)/(n + 1));
f[r_] := 7*P1*BesselK[0, r/lc];
lst2 = Table[{r, f[r]}, {r, we1, rng, rng/1000}]; h2 =
Interpolation[lst2, InterpolationOrder -> 1];
pA = Plot[{h2[r]}, {r, we1, rng}, PlotStyle -> {Blue},
PlotRange -> All, Frame -> True];
pIn = Plot[{f[r]}, {r, we1, rng}, PlotStyle -> {Red},
PlotRange -> All, Frame -> True];
Show[{pA, pIn}](*height*);
A[r_] := h2''[r] + 1/r*h2'[r];(*Interpolation*)
A2 = D[f[r], {r, 2}] + 1/r*D[f[r], r];(*Analytic*)A3 = D[A2, {r,
1}];
plots = Plot[{A'[r], A3}, {r, we1, rng/1}, PlotRange -> All,
Frame -> True, PlotRange -> All, Frame -> True,
PlotStyle -> {{Red, Thick}, {Green, Thick}},
GridLines -> {{we1, 2 we1}, {0}},
GridLinesStyle -> Directive[Red, Dashed],
PlotLegends -> {"Interpolation (A'[r])", "Analytic (A3)"}]
[![plots][1]][1]
• You are using a fairly high interpolation order, which makes the derivatives numerically unstable. I would suggest using a smaller InterpolationOrder. But generally, what you are trying to do is numerically unstable. Better to compute the derivatives straight from the data instead of going through an interpolation. Apr 16 at 10:53
• @Roman, Thanks. I also tried with a small InterpolationOrder. Could you suggest how to compute derivative straight from the data? Apr 16 at 11:19
• The easiest way to compute numerical derivatives directly from the data is probably to set InterpolationOrder -> 1 and let Mathematica take care of the details. In this way it will interpolate linearly between data points, thus effectively using the finite-differences method of numerical differentiation. In general, to compute $n^{\text{th}}$ derivatives you should use an interpolation of oder $n$ (but not higher). Apr 16 at 14:44
• Thanks @ Roman, I tried with this code ( lst2 = Table[{r, f[r]}, {r, we1, rng, rng/1000}]; h2 = Interpolation[lst2, InterpolationOrder -> 1] ) but I found huge difference in interpolated data plot and analytic one. Apr 16 at 15:44
• Also there is the recently added Wolfram Function Repository function ListD. It might give reasonable results. And the web page has examples that might give ideas for related methods involving local smoothing. Apr 16 at 16:18
InterpolationOrder -> 1 means the first derivative will be discontinuous (piecewise constant) and the second and third derivatives will be zero. You need an interpolation order higher than 3 to get a continuous result.
h2 = Interpolation[lst2, InterpolationOrder -> 7];
A[r_] := h2''[r] + 1/r*h2'[r];(*Interpolation*)
A2 =
D[f[r], {r, 2}] + 1/r*D[f[r], r];(*Analytic*)
A3 = D[A2, {r, 1}];
plots = Plot[{A'[r], A3}, {r, we1, rng/1}(*,PlotRange->All*),
Frame -> True(*,PlotRange->All*), Frame -> True,
PlotStyle -> {{Red, Thick}, {Green, Thick}},
GridLines -> {{we1, 2 we1}, {0}},
GridLinesStyle -> Directive[Red, Dashed],
PlotLegends -> {"Interpolation (A'[r])", "Analytic (A3)"}]
Still a bit of noise...
P.S.
Using chebInterpolation and chebSeries from FunctionInterpolation over an open interval, we also get a pretty good approximation, which is much less noisy:
h2 = chebInterpolation[{{N@{we1, rng},
chebSeries[f, N@{we1, rng}, 64][[;; -12]]}}]
This is nearly equivalent:
lst2 = Table[{r, f[r]}, {r,
Rescale[
Sin[Pi/2. Range[-64, 64, 2]/64], {-1, 1}, {we1, rng}]}];
h2 = Interpolation[lst2, InterpolationOrder -> All]
Note: The function A3 seems to lose about 7 digits of precision due to subtractive cancellation. (Based on evaluating a high-precision A3 on arbitrary precision input and observing the precision loss.)
• Yeah, that precision loss will be difficult to avoid. Offhand I do not know if smoothing or spectral methods can do better or, if so, by how much. Apr 17 at 3:16
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2022-05-29 05:43:44
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https://www.physicsforums.com/threads/rotating-disks-and-time-dilation.348201/
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# Rotating disks and time dilation
1. Oct 23, 2009
### quitequick
I was just reading about the experimental proof of time dilation using some atomic clocks on aeroplanes. Which got me thinking...
Consider a rotating disk. Say it is a few hundred metres radius (although its size makes no difference to this question, it makes it easier for me to visualise). Clamp down an atomic clock onto the disk platter located at a point on the circumferential edge and call it point A. Strap a willing observer to that clock. Do the same again but put a clock and observer near to, but not at, the centre of the disk and call it point B. Put another clock and observer next to (but not on) the rotating disk and call it point C. Spin the disk as fast as possible for some long time and then slow to a halt.
1. There is no speed difference between point A and B i.e. A and B never move relative to each other. Therefore there will be no time difference between the clocks at A and B.
2. Points A and B do move relative to point C, but with different speed profiles. So, there will be different time difference between clocks A and C and clocks A and B.
Statement 1. says that clock reading are the same at A and B. Statement 2. implies that the clock readings at A and B are different. One or both statements can not be true.
What am I missing? (other than a better understanding of special relativity!)
2. Oct 23, 2009
### A.T.
This is only true in a co-rotating frame of reference. This frame is non-inertial and gravitational time dilation occurs there. Gravitational time dilation depends on the position in the non-inertial frame, not the relative movement.
The clocks show different times.
3. Oct 23, 2009
### quitequick
I've done some reading since your reply. So, gravitational time dilation occurs because of the acceleration due to the rotating disk (just like gravity causes). Both A and B are accelerating at different rates because of their different positions on the disk. Your answer says that the frame is non-inertial, which is to say that it has acceleration. However, the accelerations A and B are different as measured from the coordinate frame of the disk. So, you are saying that the fact they are different means that they are different non-inertial frames. So, can I summarise by saying that because A and B undergo different accelerations wrt the same coordinate frame (the disk), they have different time dilations? And if so, what is the relationship between accn and time dilation?
And if the reference frame is not the disk, then what is it?
4. Oct 23, 2009
### Jonathan Scott
Acceleration does not affect time rates.
You can calculate the difference in time rates in the obvious way, from the speeds relative to an inertial frame, using Special Relativity. The disk is not an inertial frame.
You can also calculate the different time rates from the point of view of an observer on the disk as being due to the apparent gravitational potential, of which the acceleration is the gradient.
Both methods give the same result.
5. Oct 23, 2009
### Fredrik
Staff Emeritus
Time dilation due to acceleration/gravity is a more complicated thing than the time dilation due to a velocity difference. Consider e.g. two clocks attached to opposite ends of an accelerating steel rod. The rod will contract (in the inertial frame where it was originally at rest) as its speed increases, and that means that the clock at the rear always has a higher velocity than the clock at the front. So the cause of this "gravitational" time dilation is really just a velocity difference that arises because the clocks have to move in a certain way when they're attached to a rigid object.
6. Oct 23, 2009
### A.T.
Not quite:
1) They don't undergo coordinate accelerations w.r.t. the disc, they are at rest in the disc's frame. They undergo different proper accelerations, which are absolute not w.r.t. to some frame.
2) It is not necessary for them to undergo different proper accelerations, to have different clock rates. In an accelerating rocket a clock at the front can feel the same proper acceleration as a clock at the back, but the front clock ticks faster, in the frame of the rocket.
You can derive it via the redshift of light beam going from back to front or A to B. I think (not sure though) it goes like this: You use the instantaneous inertial frame of the emitter at emission time, and calculate the speed of the receiver in that frame at receive time. Then use the relativistic Doppler-Shift formula. The resulting frequency ratio is the time dilation.
7. Oct 23, 2009
### Fredrik
Staff Emeritus
The most useful fact you can learn that would help you understand things like this is that what a clock measures is the proper time of the curve in Minkowski space that represents its motion. Proper time is defined as the integral of $$\sqrt{dt^2-dx^2}$$ along the curve (where t and x are the coordinates of an inertial frame). You should think of this fact as an axiom that's a part of the definition of the theory. Note that in a 2+1-dimensional spacetime diagram, the motion of C is a straight line parallel to the t axis (dx=0 along the curve) and the motions of A and B are spirals (dx≠0).
8. Oct 23, 2009
### Cleonis
Your reasoning is in error here, but I grant you it's a tempting error.
Clock A is located on the axis of rotation, clock B is co-rotating with the disk's circumference, so yes, their relative distance does not change. But special relativity does not assert that if there is no change of relative distance there can be no time difference. Special relativity asserts that point B travels a longer distance. More precisely, special relativity asserts that there is in fact a difference in distance travelled between A and B.
The relativity comes in as follows: you can map all of the motion in a coordinate system that is co-moving with A, and then you can compute the distance traveled by B in that coordinate system. Alternatively, you can map all of the motion in a coordinate system in which A has a uniform velocity. For instance, you can let that coordinate system move along the disk's axis of rotation. Then the motion of A will be mapped as a straight line, and the motion of B will be mapped as a helix (a corkscrew).
Irrespective of how you map the motion (provided you map in an inertial coordinate system), the difference in distance traveled comes out the same.
Now, in order for B to travel a longer distance, and yet not fly away from A, B must undergo acceleration; there's no other way. But the difference in time cannot be attributed to the acceleration. Instead, the difference in time is correlated with the difference in distance travelled.
Cleonis
Last edited: Oct 23, 2009
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2018-03-23 00:36:38
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https://www.ipht.fr/en/Phocea/Vie_des_labos/Seminaires/index.php?id=993519
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Geometric recursion
Gaëtan Borot
Max Plank Bonn
Thu, Dec. 21st 2017, 11:00-12:00
Salle Claude Itzykson, Bât. 774, Orme des Merisiers
I will present a new formalism, which takes as input a functor \$E\$ from a category of surfaces with their mapping classes as morphisms, to a category of topological vector spaces, together with glueing operations, as well as a small amount of initial data, and produces as output functorial assignments \$S \mapsto \Omega_S\$ in \$E(S)\$. This construction is done by summing over all excisions of homotopy class of pair of pants decompositions of \$S\$, and we call it ``geometric recursion''. The topological recursion of Eynard and Orantin appears as a projection of the geometric recursion when \$E(S)\$ is chosen to be the space of continuous functions over the Teichmuller space of \$S\$, valued in a Frobenius algebra -- and the projection goes via integration over the moduli space. More generally, the geometric recursion aims at producing all kinds of mapping class group invariant quantities attached to surfaces. \\ \\ This is based on joint work with J.E. Andersen and N. Orantin.
Contact : Vincent PASQUIER
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2022-06-29 12:42:38
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https://www.physicsforums.com/threads/srednicki-qft-chapter-8-question.377506/
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# Srednicki QFT chapter 8 question
1. Feb 11, 2010
### LAHLH
Hi,
In chapter 8 Srednicki employs the $$1-i \epsilon$$ trick. He multiplies the Hamiltonian desity,
$$H=\frac{1}{2} \Pi^2+\frac{1}{2}(\nabla\phi)^2+\frac{1}{2}m^2\phi^2$$
by this $$1-i \epsilon$$, and says it's equivalent to if we replaced m^2 with $$m^2-i \epsilon$$. I cant see how this is?
Thanks
2. Feb 11, 2010
### Avodyne
He explains this in detail in the earlier chapter on the harmonic oscillator.
3. Feb 12, 2010
### LAHLH
In chptr 7, it's completely different,
$$H(P,Q)=\frac{P^2 m^{-1}}{2}+\frac{1}{2}m\omega^2 Q^2$$
So multiplying by $$1-i\epsilon$$, gives :
$$H(P,Q)=\frac{P^2 (1-i\epsilon)m^{-1}}{2}+\frac{1}{2}(1-i\epsilon) m\omega^2 Q^2$$
So directly you can see this is equivalent to if we used $$m^{-1} \rightarrow (1-i\epsilon)m^{-1}$$ and $$m\omega^2 \rightarrow (1-i\epsilon)m\omega^2$$
I fail to see how to perform in a similar process to get Srednicki's mass substitution for the H in my first post. Thanks.
4. Feb 12, 2010
### Physics Monkey
Hi LAHLH,
I would recommend not trying to take Srednicki's comment too literally. A useful point of view is the following. The purpose of the $$i \epsilon$$ is to get the pole structure right in the propagator. To make sense of what Srednicki is saying, try to determine if modifying only the mass by $$i \epsilon$$ is sufficient to modify the pole structure so that integrating along the real line is the "right" contour.
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2018-03-18 10:20:46
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https://direct.mit.edu/neco/article/21/5/1203/7434/Mean-Field-Approximations-for-Coupled-Populations
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## Abstract
There has recently been a great deal of interest in inferring network connectivity from the spike trains in populations of neurons. One class of useful models that can be fit easily to spiking data is based on generalized linear point process models from statistics. Once the parameters for these models are fit, the analyst is left with a nonlinear spiking network model with delays, which in general may be very difficult to understand analytically. Here we develop mean-field methods for approximating the stimulus-driven firing rates (in both the time-varying and steady-state cases), auto- and cross-correlations, and stimulus-dependent filtering properties of these networks. These approximations are valid when the contributions of individual network coupling terms are small and, hence, the total input to a neuron is approximately gaussian. These approximations lead to deterministic ordinary differential equations that are much easier to solve and analyze than direct Monte Carlo simulation of the network activity. These approximations also provide an analytical way to evaluate the linear input-output filter of neurons and how the filters are modulated by network interactions and some stimulus feature. Finally, in the case of strong refractory effects, the mean-field approximations in the generalized linear model become inaccurate; therefore, we introduce a model that captures strong refractoriness, retains all of the easy fitting properties of the standard generalized linear model, and leads to much more accurate approximations of mean firing rates and cross-correlations that retain fine temporal behaviors.
## 1. Introduction
One of the fundamental problems in the statistical analysis of neural data is to infer the connectivity and stimulus dependence of a network of neurons given an observed sequence of extracellular spike times in the network (Brown, Kass, & Mitra, 2004). Estimating the parameters of network models from data is in general a computationally difficult problem because of the high dimensionality of the parameter space and the possible complexity of the model's objective function given an observed data set of network spike times.
One class of models that has proven quite useful is known in the statistics literature as the generalized linear model (GLM; Brillinger, 1988; Chornoboy, Schramm, & Karr, 1988; see also Martignon et al., 2000; Kulkarni & Paninski, 2007; Nykamp, 2007), variants of which have been applied successfully in the hippocampus (Harris, Csicsvari, Hirase, Dragoi, & Buzsaki, 2003; Okatan, Wilson, & Brown, 2005), motor cortex (Paninski, Fellows, Shoham, Hatsopoulos, & Donoghue, 2004; Truccolo, Eden, Fellows, Donoghue, & Brown, 2005), retina (Pillow et al., 2008), and cultured cortical slice (Rigat, de Gunst, & van Pelt, 2006). In its simplest form, this model incorporates both stimulus-dependence terms and direct coupling terms between each observed neuron in the network; fitting the model parameters leads to an inhomogeneous (stimulus-driven), nonlinear coupled spiking model with delay terms. Statistically speaking, the model is attractive due to its explicitly probabilistic nature and because fitting the model parameters is surprisingly simple: under certain simple conditions, the log-likelihood function with respect to the model parameters is concave, and the maximum likelihood estimation of the parameters can be easily performed using standard ascent methods (Paninski, 2004; Paninski, Pillow, & Lewi, 2007).
Once the model parameters are obtained, we are left with an obvious question: What do we do next? One of the key applications of such a network model is to better understand the input-output properties of the network. For example, we would like to be able to predict the mean firing response of the network given a novel input and to carve out the impact of the network coupling terms on this stimulus-response relationship (e.g., what is local inhibition's impact on the stimulus filtering properties of the network?). We would also like to know how the correlation properties of spike trains in the network might depend on the stimulus, and in general how correlations might encode stimulus information (Riehle, Grün, Diesmann, & Aertsen, 1997; Hatsopoulos, Ojakangas, Paninski, & Donoghue, 1998; Oram, Hatsopoulos, Richmond, & Donoghue, 2001; Nirenberg, Carcieri, Jacobs, & Latham, 2002; Schnitzer & Meister, 2003; Kohn & Smith, 2005; Pillow et al., 2008).
We can in general study these questions with direct Monte Carlo simulations. However, simulation of a large-scale probabilistic spiking network is computationally expensive, since we often need to draw many samples (i.e., run many simulated trials) in order to compute the quantities of interest to the desired precision. More importantly, direct numerical simulation often provides limited analytical insight into the mechanisms underlying the observed phenomena.
The goal of this study is to investigate how much of the behaviors of these GLM networks can be understood using standard analytical mean-field approximations (Renart, Brunel, & Wang, 2003; Wilson & Cowan, 1972; Amit & Tsodyks, 1991; Ginzburg & Sompolinsky, 1994; Hertz, Krogh, & Palmer, 1991; Kappen & Spanjers, 2000; Gerstner & Kistler, 2002; Meyer & van Vreeswijk, 2002). In particular, we develop approximations for the mean firing rates of the network given novel stimuli, as well as the auto- and cross-correlation and input-output filtering properties of these networks. These approximations are valid when the contributions of individual network coupling terms are small and lead to deterministic ordinary differential equations that are much easier to solve and analyze than direct Monte Carlo simulation of the network activity.
However, in the case of strong refractory effects, these mean-field approximations become inaccurate, since the spike history terms in the generalized linear model must be large to induce strong refractoriness, and this pushes our approximations beyond their region of accuracy. Therefore we introduce a modified model: a generalized linear model with Markovian refractoriness. This model has several advantages in this setting: it captures strong refractoriness, retains all of the easy-fitting properties of the standard generalized linear model, and leads to much more accurate mean-field approximations.
## 2. The Generalized Linear Point Process Model
We begin by precisely defining the generalized linear point process model (see Figure 1). We consider N recurrently connected spiking neurons, modeled as a multivariate point process (Snyder & Miller, 1991). The firing rate (conditional intensity function) of neuron i at time t is a function of the total input, ui(t), the neuron receives:
2.1
where f(.) is a smooth, nonnegative, and monotonically increasing function. For numerical simulations, we use exponential nonlinarity, f(u) = eu, but the analysis in this article is not limited to this exponential nonlinearity. The total input to the neuron is expressed as the sum of external input Ii and recurrent input Hi:
2.2
The recurrent input is modeled as a sum of identical responses caused by each presynaptic spike and written as
2.3
where wji(t) is a coupling term from neuron j (upper index) to i (lower index), and tj,n is the nth spike time of neuron j. Without loss of generality, we may express each coupling term wji(t) as a weighted sum of exponential functions of different time constants. As we will see below, this sum-of-exponentials representation is extremely useful: we will take full advantage of the Markovian nature of the decaying exponential function. To keep notation under control, we will restrict our attention to the case of a single exponential function with time constant τij,
2.4
where Θ is the Heaviside step function, which takes one for positive arguments and zero otherwise; generalizations to the case where wji(t) are given by a sum of exponentials with different time constants will be left implicit and will be straightforward in all cases. If we write the output spike train of neuron i as
2.5
the recurrent input of equation 2.3 is rewritten as the convolution of the synaptic filter and the presynaptic spike train:
2.6
with
2.7
This model is a generalization of the inhomogeneous Poisson process, since past output spikes modulate the firing intensity function. If we set all the coupling weights Jji to zero, we recover the the inhomogeneous Poisson process model. The model is also closely related to the spike-response model of Gerstner and Kistler (2002) and the inhomogeneous Markov interval process of Kass and Ventura (2001), in which the firing rate depends on only the last observed spike. Note that in the GLM, the firing intensity of a neuron may depend not only on the last spike of the neuron but on the past several spikes of this neuron (see, e.g., Paninski, 2004; Paninski et al., 2007, for further discussion). As mentioned in section 1, this model is attractive because it is fairly easy to fit to data and seems to capture the firing statistics of neural populations in a variety of experimental settings. However, in this article, we will not discuss the estimation problem but rather limit our attention to the problem of determining the network firing rate, auto- and cross-correlation functions, and input-output properties, given fixed, known model parameters.
Figure 1:
A schematic illustration of the generalized linear point process model.
Figure 1:
A schematic illustration of the generalized linear point process model.
## 3. Master Equation
The main object we would like to compute here is P(h, t): this is the distribution of the state variable at time t. These hji are the solutions of the multivariate coupled Markov process,
3.1
and therefore solving for P(h, t) will allow us to compute a number of important quantities. For example, to compute the mean firing intensity νi(t) = E[Si(t)] of neuron i given some input {Ii(t)}, we may compute
where E[.] is the average over the spike history in an entire duration T given external input {Ii(t)}. Hence, we can read the mean firing rate off directly from P(h, t).
We begin by writing down the master equation—the Kolmogorov forward equation that governs the time evolution of this Markov process (Oksendal, 2002):
3.2
with as in equation 2.1 and the term ej denoting the vector with its slot δjj. Note that there are no diffusion terms here; the random nature of the stochastic differential equation enters instead through the jump terms.
While this jump-advection partial differential equation (PDE) in principle completely describes the behavior of this system, unfortunately this equation remains difficult to handle analytically (due to the nonlinearity in the λ terms) and numerically (due to the large dimensionality of the state variable h). We will pursue a direct analysis of this PDE elsewhere; here, we will attack this system using a different approach, based on a self-consistent expansion of the first and second moments of Si(t).
In the one-dimensional case (i.e., a single neuron, with one h(.)), however, we may easily solve the PDE numerically to obtain an exact solution for the mean firing rate. Figure 2 compares this exact solution versus the mean-field approximations derived in the following section. The mean-field approximation is reasonably good here but not perfect, because of the fluctuation in the recurrent input (J = −1, τ = 10 ms).
Figure 2:
Direct solution of the master equation in the one-dimensional case. (A) Top: A single neuron is simulated with a pulse input I(t) shown. Bottom: Comparison of the firing rates computed via direct Monte Carlo sampling (simulation); the inhomogeneous Poisson case, with no history dependence, that is, w = 0 (noH); the master equation of equation 3.2 (exact); and the mean-field approximation methods in the limit of Nc → ∞ introduced in section 4. The mean-field approximation gives good approximations of the output firing intensity. (B) Top: The probability distribution P(h,t); grayscale axis indicates the height of the probability density. The self-history term h(t) was the convolution of the output spike train and a single exponential filter with time constant τ = 10 ms and weight J = −1; thus, the neuron was inhibited for a brief period following each spike. Note the discontinuous behavior visible at stimulus onset, where probability mass jumps significantly toward h = 1, then decays back down toward zero, according to equation 3.1. Bottom: The mean of h calculated from the probability distribution P(h,t). This E[h] also increases following the input pulse, which has the effect of slowing the firing rate. Since the weight J is negative, the history term is inhibitory here. (A color version of this figure is available online at http://www.mitpressjournals.org/doi/suppl/10.1162/neco.2008.04-08-757.)
Figure 2:
Direct solution of the master equation in the one-dimensional case. (A) Top: A single neuron is simulated with a pulse input I(t) shown. Bottom: Comparison of the firing rates computed via direct Monte Carlo sampling (simulation); the inhomogeneous Poisson case, with no history dependence, that is, w = 0 (noH); the master equation of equation 3.2 (exact); and the mean-field approximation methods in the limit of Nc → ∞ introduced in section 4. The mean-field approximation gives good approximations of the output firing intensity. (B) Top: The probability distribution P(h,t); grayscale axis indicates the height of the probability density. The self-history term h(t) was the convolution of the output spike train and a single exponential filter with time constant τ = 10 ms and weight J = −1; thus, the neuron was inhibited for a brief period following each spike. Note the discontinuous behavior visible at stimulus onset, where probability mass jumps significantly toward h = 1, then decays back down toward zero, according to equation 3.1. Bottom: The mean of h calculated from the probability distribution P(h,t). This E[h] also increases following the input pulse, which has the effect of slowing the firing rate. Since the weight J is negative, the history term is inhibitory here. (A color version of this figure is available online at http://www.mitpressjournals.org/doi/suppl/10.1162/neco.2008.04-08-757.)
## 4. Mean-Field Approximation of GLM Statistics
Now we turn to the main topic of this article. As emphasized above, we would like to calculate the mean-firing intensities of neurons, νi(t) = E[Si(t)], or the spike-cross-correlation function between neurons, . The calculation of these statistics is difficult, however, because of the nonlinearity f and the recurrent input term H in the model. A typical way of approximating these recurrent effects is to replace the interaction term H by its mean field (Renart et al., 2003; Wilson & Cowan, 1972; Amit & Tsodyks, 1991; Ginzburg & Sompolinsky, 1994; Hertz et al., 1991; Kappen & Spanjers, 2000; Gerstner & Kistler, 2002; Meyer & van Vreeswijk, 2002).
Let us assume that each neuron receives input from some subset, Nc, of the total population of neurons N. If the synaptic strength scales with Nc, and assuming that the population is in the “asynchronous” state, then it is known that the cross-covariance function between two neurons scales with 1/Nc for large Nc (Ginzburg & Sompolinsky, 1994). This means that the contribution of each network interaction term is negligible and all the neuron activities are independent in the Nc → ∞ limit. Hence, calculating the mean-firing intensities and cross-correlations is easy in this limit.
We first derive the mean-firing intensity in the limit of Nc → ∞ and then evaluate the finite-size effect of order 1/Nc. Note that this mean-field solution is a good approximation not only when Nc is very large; equivalently, this is a good approximation when the synaptic coupling J is small for fixed Nc, where the contribution of an individual synapse is small.
Now our first key approximation is to appeal to the central limit theorem applied to the sum of equation 2.6, and to assume that the recurrent input H is roughly gaussian. We can express the mean firing intensities of the neurons in our network as a function of two sufficient state variables: the mean recurrent input E[h] and the variance of the recurrent input Var[H],
4.1
where, in the third line, the distribution of the recurrent input, Hi(t), is assumed to be gaussian with mean μi(t) = E[Hi(t)] and variance , and the expectation was replaced by a gaussian integral:
4.2
Note for an exponential nonlinearity f(u) = eu, the gaussian integral is given by F(μ, σ) = exp (μ + σ2/2).
Next we calculate the cross-correlation function,
4.3
where in the second line, we decomposed the correlation function into the three terms: the first term shows the simultaneous correlation with its amplitude proportional to the firing intensity due to the point-process nature of the spike train, the second term shows the correlation when t > t′ (the spike variable Si(t) is averaged given past spike history to yield λi(t)), and the third term is the correlation for t′ > t. In the third line, we introduced conditional firing intensity,
4.4
which describes the probability of neuron i firing at t given the spike of j at t′. We also introduce an abbreviation in the following and write E[.|Sj(t′)] instead of E[.|spike of j at t′]. Similar to the approximation in equation 4.1, we assume a gaussian distribution of H given a spike of neuron j at time t′ to find
4.5
for t > t′ with conditional mean, , and conditional variance, . Note that equation 4.5 is valid only for t > t′; if t < t′ instead, we evaluate and use the Bayes theorem to calculate . The unconditional and conditional mean and variance of H in equations 4.1 and 4.5 are evaluated as
4.6
where we defined
4.7
with . The evolution of , and Akli in equation 4.7 follows simple ordinary differential equations (ODEs) and is easy to simulate,
4.8
where and . Note that the differentiation of Akli(t) for k = l is a little tricky because has a delta peak at s = s′. While the definition of Akli(t) in equation 4.7 is clear, we have to make sure in the evaluation of not to double-count the integration of the delta peak. Hence, in the third equation of equation 4.8, we included the delta peak along the integration over s (including the peak at s = t) but excluded this peak from the integration along s′ (excluding the peak at s′ = t).
### 4.1. Solution in the Mean-Field Limit.
In this section we evaluate mean firing intensities, νi, and cross-correlation functions, ϕij, in the mean-field limit of Nc → ∞ (which is also valid in the noncoupled limit J → 0 for finite Nc), that is, synaptic strengths are scaled J ∼ 1/Nc so that the individual contribution of a spike is negligible for large Nc; and we also assume, as discussed above, that the network is in the asynchronous state. Under these conditions, the cross-covariances between neurons are known to scale as Δ ϕij ∼ 1/Nc for ij and Δ ϕij ∼ 1 for i = j (Ginzburg & Sompolinsky, 1994). Then we can easily see σi2(t)∼ 1/Nc because Aikl 1 for k = l and Aikl 1/Nc for k ≠ l from equation 4.7. Hence, in the limit of Nc → ∞, we obtain from equations 4.1 and 4.8,
4.9
This self-consistent update rule corresponds to a well-known mean-field dynamics of the mean-firing intensity (Ginzburg & Sompolinsky, 1994). Similarly, the scaling of cross-covariance Δ ϕij ∼ 1/Nc for ij implies for kj and for k = j from equation 4.7. Moreover the conditional covariance scales with 1/Nc for klj and thus . Hence, in the limit of Nc → ∞, we find for t > t′ and, from equation 4.3,
4.10
These equations are very easy to simulate, and the approximation of the mean-firing intensity is good even for a small Nc if J is small (see Figure 2A). On the other hand, the mean-field approximation is not good for large J (see Figures 3 and 4B). This is because, for fixed Nc, the fluctuation of the recurrent input increases with J, making the mean-field assumption (which neglects these fluctuations) invalid. Figure 4 shows the cross-correlation functions between two neurons. Because the self-interaction term J ∼ 1/Nc vanishes for the large Nc limit, the autocorrelation function in this limit can capture the Poisson peak only at t = t′; it does not capture any nontrivial correlation caused by the interaction term J for finite Nc (see Figure 4).
Figure 3:
Comparison of the direct Monte Carlo simulation of a single neuron (solid), the firing intensity without recurrent input H (dotted), and the mean-field approximation (dashed). (A) Low step input (baseline I = 2, peak I = 4) is applied. The self-inhibition term is set to J = −5 and τ = 2 ms. (B) High step input (baseline I = 2, peak I = 8) is applied. The self-inhibition term is set to J = −5 and τ = 2 ms. Because of the large nongaussian fluctuations of input caused by the strong self-inhibition, the mean-field approximation loses accuracy here. (A color version of this figure is available online at http://www.mitpressjournals.org/doi/suppl/10.1162/neco.2008.04-08-757.)
Figure 3:
Comparison of the direct Monte Carlo simulation of a single neuron (solid), the firing intensity without recurrent input H (dotted), and the mean-field approximation (dashed). (A) Low step input (baseline I = 2, peak I = 4) is applied. The self-inhibition term is set to J = −5 and τ = 2 ms. (B) High step input (baseline I = 2, peak I = 8) is applied. The self-inhibition term is set to J = −5 and τ = 2 ms. Because of the large nongaussian fluctuations of input caused by the strong self-inhibition, the mean-field approximation loses accuracy here. (A color version of this figure is available online at http://www.mitpressjournals.org/doi/suppl/10.1162/neco.2008.04-08-757.)
Figure 4:
Comparison of cross-correlation functions of two neurons calculated by direct simulation (solid), calculated from the mean-field approximation (see equations 4.9 and 4.10; dashed), and the mean-field approximation with finite size correction (see equations 4.11 and 4.12; dot-dashed). (A) Weak self-interaction. Parameters are set to J11 = −1, J22 = −1, J21 = 1, J12 = −0.5, , and I1 = 2, I2 = 4. The result with finite size correction is obtained after five iterations of the self-consistent equations. The calculation in the mean-field limit captures only the steady-state firing intensity, not any temporal cross-correlation function. Cross-correlation functions are well approximated with the finite-size correction terms. (B) Strong self-interaction: parameters are set to J11 = −5, J22 = −5, J21 = 1, J12 = −0.5, , and I1 = 2, I2 = 4. Because of the large nongaussian fluctuation of the input caused by the strong self-inhibition term, the mean-field approximation does not work very well. In this case the finite size solution converges rather slowly. Hence, the result is after 10 iterations of the self-consistent equations. (A color version of this figure is available online at http://www.mitpressjournals.org/doi/suppl/10.1162/neco.2008.04-08-757.)
Figure 4:
Comparison of cross-correlation functions of two neurons calculated by direct simulation (solid), calculated from the mean-field approximation (see equations 4.9 and 4.10; dashed), and the mean-field approximation with finite size correction (see equations 4.11 and 4.12; dot-dashed). (A) Weak self-interaction. Parameters are set to J11 = −1, J22 = −1, J21 = 1, J12 = −0.5, , and I1 = 2, I2 = 4. The result with finite size correction is obtained after five iterations of the self-consistent equations. The calculation in the mean-field limit captures only the steady-state firing intensity, not any temporal cross-correlation function. Cross-correlation functions are well approximated with the finite-size correction terms. (B) Strong self-interaction: parameters are set to J11 = −5, J22 = −5, J21 = 1, J12 = −0.5, , and I1 = 2, I2 = 4. Because of the large nongaussian fluctuation of the input caused by the strong self-inhibition term, the mean-field approximation does not work very well. In this case the finite size solution converges rather slowly. Hence, the result is after 10 iterations of the self-consistent equations. (A color version of this figure is available online at http://www.mitpressjournals.org/doi/suppl/10.1162/neco.2008.04-08-757.)
### 4.2. Estimating the Finite Size Effect.
In the previous section, we calculated the mean-firing intensity, νi(t), and the cross-covariance function, ϕij(t, t′), in the Nc → ∞ limit. Ideally, once provided the true term, which is of order 1/N2c different from σi2(t), we can iteratively evaluate equations 4.1 to 4.8, starting from the mean-field limit solution of equations 4.9 and 4.10 to find the precise approximation of νi(t) and ϕij(t, t′) as long as the gaussian H assumption is valid. However, we need the third-order correlations to evaluate , and the equations are not closed. Here, we give up evaluating O(1/N2c) terms and just evaluate the finite size corrections of νi(t) and ϕij(t, t′) up to O(1/Nc) terms. Up to this order, it can be argued that , and under this approximation, we obtain a self-consistent expression for νi(t) and ϕij(t, t′). Equations 4.1 and 4.5 yield, up to the first order of 1/Nc,
4.11
where to the first order of 1/Nc. Equation 4.11 can be evaluated using equations 4.6 to 4.8, which we repeat here for convenience:
4.12
In principle, the above equations are valid even with time-dependent input, and several iterations of the above set of equations yield time-dependent cross-correlation functions. However, evaluation of the above equations is computationally expensive with time-dependent input. If one neuron is connected to Nc surrounding neurons and time is discretized into T time bins, it takes O(N2NcT2) operations and memory to evaluate . If the input is constant in time, is a function of tt′ only, and just O(N2NcT) operations are required to solve the equations. Therefore, we focus on calculating the finite size effect for constant input in this article.1 Generally several iterations of the above set of equations are required for the convergence of mean firing intensities and cross-correlations. However, by initially setting those variables to the solution in the limit of Nc → ∞, we found good approximations after a couple of iterations in many cases.
The mean firing rate and cross-correlation functions calculated from equations 4.11 and 4.12 are precise for small J and large Nc because the recurrent input is close to gaussian in this case (see Figure 4A). On the other hand, for large J and small Nc, the gaussian input approximation is poor, and therefore the approximation becomes worse (see Figure 4B). One key result here is that both the autocorrelations and the cross-correlations in these networks depend on the inputs I(t) in general, as can be seen from equations 4.6 to 4.8. Changing just the baseline (mean) of the input I(t) can significantly change the correlation structure in the network as seen in a number of physiological preparations (e.g., Hatsopoulos et al., 1998; Kohn & Smith, 2005), even if the coupling terms J and w are unchanged.
### 4.3. Linear Input-Output Filter.
The ODE derived in equation 4.9 is general and may be applied given arbitrary time-dependent input {Ii(t)}. In this section, we exploit these equations to explore the linear response properties of these spiking networks. A standard physiological technique is white noise analysis (Marmarelis & Marmarelis, 1978; Rieke, Warland, de Ruyter van Steveninck, & Bialek, 1997), in which we inject white noise input in order to study a network's input-output filtering properties. When this analysis is applied to our GLM system, the external input is given by , where Ki is neuron i's linear stimulus filter and ξi(t) is the white noise stimulus with mean and small variance . We keep the input variance small here in order to treat the input as almost constant and develop a consistent linear response theory. Note that 〈.〉 describes the average over the fluctuating stimulus ξi(t) here. We use δ X = X − 〈X〉 for any function X of input stimulus and neglect O(δ I2) terms in the following calculations.
Let us summarize the calculations in the mean-field (Nc → ∞) limit. The mean-firing intensity of a neuron is given by the following self-consistent equations:
4.13
In particular, for a constant stimulus, ξi(t) = ξi(0), we obtain
4.14
from equation 2.4. Note that X(0) describes responses at constant stimulus ξi(0), for example, . We can solve the zeroth-order terms in equation 4.14 self-consistently by finding the roots of the equations . In Figure 5, we compare this self-consistent mean-field solution with numerical simulation of a single neuron as a function of constant input, I(0), and self-inhibition strength J. We see that the approximation is good for small values of self-interaction J, but that the error increases with ∣ J ∣ and I(0), as expected given the analysis in the previous section.
Figure 5:
Comparison of (A) a simulated firing rate of a single neuron and (B) the self-consistent solutions of equation 4.13 for different input levels, I(0), and self-inhibition strengths, J. (C) The percentage errors of the mean-field approximation are shown with respect to the simulated firing intensity. We set τ = 10 ms. (A color version of this figure is available online at http://www.mitpressjournals.org/doi/suppl/10.1162/neco.2008.04-08-757.)
Figure 5:
Comparison of (A) a simulated firing rate of a single neuron and (B) the self-consistent solutions of equation 4.13 for different input levels, I(0), and self-inhibition strengths, J. (C) The percentage errors of the mean-field approximation are shown with respect to the simulated firing intensity. We set τ = 10 ms. (A color version of this figure is available online at http://www.mitpressjournals.org/doi/suppl/10.1162/neco.2008.04-08-757.)
To perform the linear response analysis, we require the first-order terms. Differentiation of equation 4.12 with respect to stimulus perturbation, δ ξ, yields
4.15
Now we can rewrite equation 4.15 by introducing vector notation as
4.16
where we defined and . Therefore, the Fourier transformation of equation 4.16 yields
4.17
with a gain function . Note that I is an identity matrix here (not the input vector). We should note that a mathematically identical linear response filter was derived for a related model in Ginzburg and Sompolinsky (1994). One difference is that in their model, spinlike binary variables flip depending on input level with a fixed time constant. This corresponds, in our model, to the case that all the synaptic time constants τij are identical. Hence, the linear response filter in equation 4.17 is a natural generalization of their result.
We show in Figure 6A a simulated normalized receptive field of a single neuron and the analytical result of equation 4.17. In this single-neuron case, where K is set to be a single exponential function with time constant τI, for simplicity, the Fourier transformations of the synaptic filter and the input filter are and , respectively. In this case, we obtain, after a Fourier inverse transformation by a Cauchy integral,
4.18
where and . We see that the linear response may be described as a combination of the stimulus-filter term K along with an additional term due to the recurrent filter w. Of course, in the limit J → 0, we recover GfK. Note also that the input-output filter changes with the baseline input through f′. In the case of the exponential nonlinearity, f, and J<0, the larger the input baseline, the sharper the linear filter. This is because the effective strength of of the spike history effect is modulated by the input-output nonlinearity and the baseline input level (see, e.g., Bryant & Segundo, 1976, for some similar effects in an in vitro physiological preparation).
Figure 6:
Comparison of the analytically derived and simulated input-output filters. (A) A simulated normalized linear input-output filter for I(0) = 2 (thin-dotted) and I(0) = 4 (thick-dotted) is compared with analytical results at I(0) = 2 (thin-solid) and I(0) = 4 (thick-solid) of equation 4.18. Because the strength of the self-inhibition component changes with the baseline input through f′(I(0) + μ(0), the input filter is sharpened for high baseline input. Other parameters are set to J = 1, τ = 10 ms, with τI = 20 ms and σξ = 1 Hz. All filter amplitudes are normalized by their maximum amplitudes. (B) Left panels show the comparison of analytically derived input-output filters G11(t) and G12(t) of equation 4.20 (solid lines) and simulated input-output filters (dotted lines) for I(0) = 2 (thin lines) and I(0) = 4 (thick lines). Right panels show neuron 1's spatiotemporal input-output filter for I(0) = 2 (top) and I(0) = 4 (bottom). See the text for parameter values. (A color version of this figure is available online at http://www.mitpressjournals.org/doi/suppl/10.1162/neco.2008.04-08-757.)
Figure 6:
Comparison of the analytically derived and simulated input-output filters. (A) A simulated normalized linear input-output filter for I(0) = 2 (thin-dotted) and I(0) = 4 (thick-dotted) is compared with analytical results at I(0) = 2 (thin-solid) and I(0) = 4 (thick-solid) of equation 4.18. Because the strength of the self-inhibition component changes with the baseline input through f′(I(0) + μ(0), the input filter is sharpened for high baseline input. Other parameters are set to J = 1, τ = 10 ms, with τI = 20 ms and σξ = 1 Hz. All filter amplitudes are normalized by their maximum amplitudes. (B) Left panels show the comparison of analytically derived input-output filters G11(t) and G12(t) of equation 4.20 (solid lines) and simulated input-output filters (dotted lines) for I(0) = 2 (thin lines) and I(0) = 4 (thick lines). Right panels show neuron 1's spatiotemporal input-output filter for I(0) = 2 (top) and I(0) = 4 (bottom). See the text for parameter values. (A color version of this figure is available online at http://www.mitpressjournals.org/doi/suppl/10.1162/neco.2008.04-08-757.)
Next, we calculate how interactions between neurons can change the input-output filter. For concreteness, we consider two neurons that receive a spatiotemporal white noise stimulus ξ(z, t) with unit variance. Input to the neuron i is described by , where are spatial filters with means z1 = 1 and z2 = −1. For simplicity, we assume that two neurons receive the same baseline of input, I(0)1 = I(0)2 = I(0) with an identical temporal stimulus filter, , and have symmetric synaptic interactions J12 = J21 = J = −1, J11 = J22 = 0, and . The derivative of the nonlinear function is described by for i = 1, 2. Proceeding as above, we can calculate the spatiotemporal input-output filter of neuron i as . Similar to the previous single-neuron case, the Fourier transformation of the temporal stimulus filter is , while the synaptic filter is , which yield the Fourier transformation of the temporal input-output filter,
4.19
After a Fourier inverse transformation using Cauchy integration, we find
4.20
Figure 6B compares the analytically derived input-output filter of equation 4.20 with numerical simulation for different input baseline I(0) = 2 and I(0) = 4. Figure 6B, also shows the spatiotemporal input-output filter of neuron 1, , for the two input baselines. Under large input baseline conditions, we see stronger interactions between the two neurons and, hence, significant differences in the effective input-output filtering properties of the neuron in the two conditions. Thus, as emphasized above, the filtering properties of the network can change in a stimulus-dependent way, without any changes in the system parameters K, J, or w. (See, e.g., Pillow et al., 2008, for some related results concerning the effects of interneuronal coupling terms on the filtering properties of populations of primate retinal ganglion neurons.)
## 5. The Generalized Linear Point Process Model with Markov Refractoriness
In section 4, we saw that we could derive simple approximate expressions for many important quantities (firing rate, correlations) in the basic generalized linear point process model as long as the individual synaptic coupling terms ∣ J ∣ are not so large that the accuracy of the mean-field expansion is compromised (see Figures 3 and 4).
In particular, the small ∣ J ∣ condition is likely acceptable for the multineuronal coupling terms, since in many cases, the inferred coupling parameters—in motor cortex (Paninski et al., 2004; Truccolo et al., 2005) and retina (Pillow et al., 2008), for example—have been empirically found to be small (though larger network effects are found in hippocampus; see Harris et al., 2003; Okatan et al., 2005). Similarly, we might expect the “slow” self-inhibition terms in the GLM (the terms responsible for adaptation in the firing rate, for example) to be relatively small as well (Pillow, Paninski, Uzzell, Simoncelli, & Chichilnisky, 2005; Truccolo et al., 2005).
However, it is clear that the generalized linear model with small weights ∣ J ∣ will not be able to account for the strong refractoriness that is a fundamental feature of neural point-process dynamics at fine timescales: in the GL model, strong, brief inhibition is produced by large, negatively weighted, and sharply decaying history effects wii(t), and these large wii(t) terms spoil our mean-field approximation. Thus, it is clear that we need to adapt the methods introduced above in order to handle strong refractory effects.
One possible solution would be to take a discrete-time approach: instead of modeling neural responses as point processes in continuous time, we could simply model each cell's spike count within a discrete-time bin as a Bernoulli random variable whose rate νi(t) is an increasing function of ui(t), very much as in the continuous-time model described above (Chornoboy et al., 1988; Okatan et al., 2005). This discrete-time Bernoulli model shares much of the ease of fitting (including concavity of the log likelihood; Escola & Paninski, 2007) as the continuous-time model. In fact, the discrete-time model converges in a natural sense to the continuous-time model as the time binwidth dt becomes small. The advantage of this discrete-time formalism is that the firing rate νi(t) can never exceed 1/dt (since the Bernoulli probabilities cannot exceed one), and therefore, in a crude sense, the discrete-time model has a refractory effect of length dt. A mean-field analysis can be developed for this discrete-time model, which exactly mirrors that developed above in the continuous-time model. We simply need to exchange our ordinary differential equations for discrete-time difference equations.
Of course, this discrete-time approach is unsatisfactory in at least one respect, since we would like, if possible, to model the firing behavior of neurons down to a millisecond timescale (Berry & Meister, 1998; Keat, Reinagel, Reid, & Meister, 2001; Pillow et al., 2005; Paninski et al., 2007), and the discrete-time approach, by construction, ignores these fine timescales. Thus, we introduce a model that allows us to incorporate strong refractory effects directly. We set the rates,
5.1
where f(.) and ui(t) are defined as in section 4 and I(.) is the indicator function that takes the value 1 if the argument is true. We have introduced an auxiliary refractory variable xi(t), which takes a discrete state from 1 to M. We assume that this variable xi(t) is itself Markovian, with transition rates
5.2
that is, when xi(t) is in state M, it will transition to state 1 with each spike, and then transitions occur from state m to state m + 1 with rate 1/τr until xi(t) has reached the spiking state M once again. Refractoriness in this model is enforced because the neuron is silent whenever xi(t) is in one of the postspike quiescent states xi(t)<M. It is easy to see that this is a kind of inhomogeneous renewal model (the delay τd required for the state variable xi to move from state 1 to first reach the active state M is a sum of (M−1) independent and identically (i.i.d.) distributed exponential random variables of mean τr, and so τd is an i.i.d. gamma variable with parameters (M − 1, 1/τr)) and may be considered a special case of the “inhomogeneous Markov interval” model introduced by Kass and Ventura (2001). By adjusting the number of states M and the rate 1/τr, we may adjust the refractory properties of the model: for example, M = 1 implies that we have an inhomogeneous Poisson model, while M = 2 provides an exponentially decaying relative refractory effect, and if we let M be large, with τr scaling like τr∼ τ/M, we obtain an absolute refractory effect of length τ.
Finally, it is easy to show that the additional Markov term in the definition of the rate λi(t) does not have a negative impact on the estimation of the GL model parameters J given spiking data; as discussed in Paninski (2004), maximizing the likelihood in this model (or constructing an EM algorithm, as in Escola & Paninski, 2007) requires that we maximize a nonnegatively weighted version of the standard point-process log likelihood, and this weighted log likelihood retains all of the concavity properties of the original GL model. Thus, this new model requires just a single concave optimization and is as easy to fit as the standard GL point process model. The advantage is that a strong relative refractory effect is intrinsic to the model and does not need to be enforced by a large, brief, negative wii(t) term. Instead, we may use small adjustments to the Jii terms to fine-tune the model to match the short-time details of the observed interspike interval density.2 As we will see below, this leads to much more accurate mean-field approximations of the firing rates and correlation functions in these networks.
Let us define the probability that the neuron i is in state xi(t) = 1,…, M. Note that the boldface characters denote vectors. The dynamics of pi is described by
5.3
where has the same elements as Wi(t) of equation 5.2 except for
5.4
which describes the probability of neuron i firing at time t given xi(t) = M. Note that we used and wrote the last component of pi(t) as [pi(t)]M = pi(t).
We also define as the probability of neuron i being in state m given a spike of neuron j at time t′. In particular, the last component is written as . The evolution of is described by a similar equation,
5.5
where the transition matrix has the same elements as Wi(t) of equation 5.2 except for
5.6
Before turning to “mean-field” approximation, it is worth noting that we may solve for the firing rates and correlations exactly in the special case of no synaptic couplings, J = 0. (The analogous case in the standard GL model is the inhomogeneous Poisson case, which is of course trivial.) We can proceed by exploiting either the renewal nature of the model (Gerstner & Kistler, 2002), which leads to convolution or infinite-sum formulas for the firing rate, or the Markovian structure, which leads to somewhat more intuitive ordinary differential equations. We pursue the second approach here.
In this case, the mean firing intensity is given by
5.7
This in turn gives, together with equation 5.4, . Hence, we can find the dynamics of mean firing intensity by solving equation 5.3. In the special case of constant input Ii, it is easy to solve the fixed point of pi(t); we find for the last component that .
The autocorrelation function in this case is similarly easy to derive. (Note that the cross-covariance functions are zero in this J = 0 case, since the cross-coupling terms are set to zero.) In case of no couplings, J = 0, the conditional firing intensity is given by
5.8
for t > t′. We find those corner elements of as
5.9
for t > t′. This means that for t > t′. Hence, follows the same differential equation as pi(t) in this J = 0 case but starting from the initial condition
5.10
because the state xi is reset to state 1 just after a given spike of neuron i.
In the following sections, we apply mean-field approximations for the firing rates and correlations in the nonzero J case.
### 5.1. Mean-Field Approximation of the GL Model with Markov Refractoriness.
Let us approximate the mean firing intensity and the cross-correlations of GL model with Markov refractoriness assuming that the contribution of individual synaptic coupling is small. Similar to the calculation without the Markov refractoriness, we assume a gaussian distribution of recurrent input Hi(t) given xi(t) = M to find
5.11
with conditional mean and conditional variance . Also, assuming a gaussian distribution of recurrent input Hi(t) given xi(t) = M and a spike of neuron j at time t′, we find
5.12
for t > t′ with conditional mean and variance . The dynamics of pi(t) and are described by equations 5.3 and 5.5, respectively.
Next we calculate the conditional means and variances of the recurrent input. We find
5.13
with auxiliary variables defined as
5.14
Note that we use the final component of the above quantities without an index (e.g., ). After some calculation (see the appendix), we obtain
5.15
where and, particularly, the last component is written as .
### 5.2. Solution in the Mean-Field Limit.
Similar to the case without Markov refractoriness, we first derive a self-consistent equation for the mean firing intensity and cross-covariances in the mean-field limit Nc → ∞, where J ∼ 1/Nc and assuming an asynchronous state so that Δ ϕij ∼ 1/Nc. Because all the cross-covariance functions vanish and in the limit, we find, from equations 5.11 to 5.13, for t > t′,
5.16
while the cross-correlation function is given by equation 4.3. For the corner terms of the transition matrices, we find from equation 5.4 and for t > t′ from equation 5.6. Hence, the evolution of pi(t) and is described by
5.17
for t > t′, where the initial condition of is given by equation 5.10. Finally, from equation 5.15, the evolution of is written as an ODE,
5.18
where we can apply equation 5.10 to evaluate in this limit. Figure 7 plots the mean firing intensity calculated from the above equations for various input, and the mean-field approximation provides good approximation of the time dependent firing intensities. Figure 8 compares the above approximation (Nc → ∞ limit) with the mean-field approximation, including the finite size effect that we discuss in the next section. In the mean-field limit, all the cross-correlations vanish and the autocorrelation functions are approximated by including the Markov refractory effect but not the self-interaction terms Jii. We can also calculate the linar input-output filter discussed in section 4.3 with this Markov refractoriness (see the appendix).
Figure 7:
Comparison of the mean firing intensity obtained from direct simulation of a single neuron (solid), without recurrent input H and no Markov refractory effect (dotted), and the mean-field equation in the limit of Nc → ∞ (dashed). In all cases, the Markov refractory effect with was included. (A) Weak step input (baseline I = 2, peak I = 4) and refractoriness M = 3; J = −1, τ = 10 ms. Although the model includes strong Markov refractory effect, the mean-field approximation is not abolished. (B) Strong step input (baseline I = 2, peak I = 8) and refractoriness M = 10; J = −1, τ = 10 ms. The mean-field approximation also approximates very well the transient oscillation caused by the the strong refractory effect and steep current. (C) The mean-field approximation works for colored noise input: with E[ξ (t)] = 4, , and τI = 20 ms; M = 3, J = −1, τ = 10 ms. The initial error is due to the preset initial values that deviate from the true ones. (D) Sinusoidal input input: ; M = 3, J = −1, τ = 10 ms. The mean-field results give good approximations. (A color version of this figure is available online at http://www.mitpressjournals.org/doi/suppl/10.1162/neco.2008.04-08-757.)
Figure 7:
Comparison of the mean firing intensity obtained from direct simulation of a single neuron (solid), without recurrent input H and no Markov refractory effect (dotted), and the mean-field equation in the limit of Nc → ∞ (dashed). In all cases, the Markov refractory effect with was included. (A) Weak step input (baseline I = 2, peak I = 4) and refractoriness M = 3; J = −1, τ = 10 ms. Although the model includes strong Markov refractory effect, the mean-field approximation is not abolished. (B) Strong step input (baseline I = 2, peak I = 8) and refractoriness M = 10; J = −1, τ = 10 ms. The mean-field approximation also approximates very well the transient oscillation caused by the the strong refractory effect and steep current. (C) The mean-field approximation works for colored noise input: with E[ξ (t)] = 4, , and τI = 20 ms; M = 3, J = −1, τ = 10 ms. The initial error is due to the preset initial values that deviate from the true ones. (D) Sinusoidal input input: ; M = 3, J = −1, τ = 10 ms. The mean-field results give good approximations. (A color version of this figure is available online at http://www.mitpressjournals.org/doi/suppl/10.1162/neco.2008.04-08-757.)
Figure 8:
Comparison of spike cross-correlation functions of two neurons obtained by direct numerical simulation (solid), mean-field approximation in the limit of Nc → ∞ (dashed), and with finite size correction terms (dot-dashed). While the approximation in the limit of Nc → ∞ captures only the strong Markov refractory effect, the weaker refractory effect caused by the self-interaction terms is well captured with the finite size correction. The finite size correction provides a good approximation despite the strong refractoriness because of an explicit evaluation of the Markov refractory effect. Mean-field equations are iteratively solved over five iterations. (A) Two neurons are directly coupled: parameters are set to J11 = −1, J22 = −1, J12 = −0.5, J21 = 1, , M = 3, I1 = 2, I2 = 4. (B) Two neurons are not directly connected but receive common input from a third neuron: parameters are set to J13 = J23 = 2, , M = 3, I1 = I2 = I3 = 2. (A color version of this figure is available online at http://www.mitpressjournals.org/doi/suppl/10.1162/neco.2008.04-08-757.)
Figure 8:
Comparison of spike cross-correlation functions of two neurons obtained by direct numerical simulation (solid), mean-field approximation in the limit of Nc → ∞ (dashed), and with finite size correction terms (dot-dashed). While the approximation in the limit of Nc → ∞ captures only the strong Markov refractory effect, the weaker refractory effect caused by the self-interaction terms is well captured with the finite size correction. The finite size correction provides a good approximation despite the strong refractoriness because of an explicit evaluation of the Markov refractory effect. Mean-field equations are iteratively solved over five iterations. (A) Two neurons are directly coupled: parameters are set to J11 = −1, J22 = −1, J12 = −0.5, J21 = 1, , M = 3, I1 = 2, I2 = 4. (B) Two neurons are not directly connected but receive common input from a third neuron: parameters are set to J13 = J23 = 2, , M = 3, I1 = I2 = I3 = 2. (A color version of this figure is available online at http://www.mitpressjournals.org/doi/suppl/10.1162/neco.2008.04-08-757.)
### 5.3. Estimating the Finite Size Effect.
Similar to the case without Markov refractoriness, we evaluate the mean firing intensity and cross-covariance functions up to 1/Nc terms. We find, as in the case without Markov refractoriness, for ij, for ik, and the third-order covariances, such as , are of order 1/N2c under the asynchronous state if the contributions of individual synaptic weights are small, J ∼ 1/Nc. This implies .
The dynamics of pi(t) and are described by equations 5.3 and 5.5, respectively. We now want to calculate μi(t), , and σi2(t) to the first order of 1/Nc. First, direct differentiation of in equation 5.15 yields an ODE update equation (see the appendix):
5.19
with . The origin of the second term on the right-hand side of equation 5.19 is due to the correlation of spike variable Sk(s) and the refractory variable xi(t). Next, we need to evaluate to the zeroth order of 1/Nc for k = i or k = j because J ∼ 1/Nc makes the contribution of them ∼ 1/Nc, and we need to evaluate to the first order of 1/Nc for ki and kj case. Except for the case i = j = k, we find (see the appendix)
5.20
to the order described above. The precise evaluation of this term for i = j = k is much harder but, to a good approximation, equation 5.20 holds (see the appendix). Note that the approximation of the i = j = k term affects the order 1/Nc term of the autocovariance function, , but affects only the O(1/N2c) terms of the mean firing intensities and cross-covariance functions.
Finally, dropping higher-order terms that result as O(1/N2c) on the quantities in equation 5.13, we find (see the appendix)
5.21
Altogether, using equation 4.12, we obtain the following update equations to evaluate the finite size effect up to order 1/Nc:
5.22
and the dynamics of pi(t) and follows equations 5.3 and 5.5. As is the case without the Markov refractoriness, calculating this finite size effect is computationally expensive for time-dependent input, although in principle, this equation should work under that case as well. We plot the mean firing intensities of two neurons and cross-correlations in Figure 8. The finite size effect well captures the cross-correlation functions. It also successfully captures a peak in the cross-correlation function if two neurons are not directly connected but receive a common input from a third neuron (see Figure 8). Again, if the values of the above equations are initially set to the solution in the limit of Nc → ∞ given by equations 5.16 to 5.18, only a couple of iterations of the above equations are sufficient for the evaluation of cross-correlation functions.
## 6. Discussion
We have introduced mean-field methods for analyzing the dynamics of a coupled population of neurons whose activity may be well approximated as the output of a generalized linear point process model. Our approximations for the mean time-varying firing rate and correlations in the population are exact in the mean-field limit (Nc → ∞ under J ∼ 1/Nc scaling), though we have found numerically that the finite size correction to the mean-field equations is useful even for physiologically relevant connectivity strengths. The approximations may be computed by solving a set of coupled ordinary differential equations. This approach is much more computationally tractable than direct Monte Carlo sampling and may lead to greater analytical insight into the behavior of the network. In addition, we have introduced a new model, the generalized linear point process model with Markovian refractoriness, that captures strong refractoriness, retains all of the easy-fitting properties of the standard generalized linear model, and whose firing rate dynamics are much more amenable to mean-field analysis than the standard modells. Note that most of our illustrations involved the simulation of only a couple of mutually connected neurons; this small-Nc case can be considered a kind of worst-case analysis, since with more asynchronous neurons (larger Nc), the central limit theorem becomes progressively applicable, the distribution of recurrent input approaches the assumed gaussian distribution, and our approximations become more accurate.
This kind of mean-field analysis—or more generally, approaches for reducing the complexity of the dynamics of networks of noisy, nonlinear neurons—has a long history in computational neuroscience, as reviewed (Hertz et al., 1991; Gerstner & Kistler, 2002; Renart et al., 2003). While the point-process models we have discussed here are somewhat distinct from the noisy integrate-and-fire-type models that have been analyzed most extensively in this literature (e.g., Mattia & Del Giudice, 2002; Shriki, Hansel, & Sompolinsky, 2003; Moreno-Bote & Parga, 2004, 2006; Fourcaud-Trocme, Hansel, van Vreeswijk, & Brunel, 2003; Chizhov & Graham, 2007; Doiron, Lindner, Longtin, Maler, & Bastian, 2004; Lindner, Doiron, & Longtin, 2005), it is more important to stress the differences in the motivation of this work and previous work. Our main goal here was to provide analytical tools that an experimentalist who has fit a point process model to his or her data (as in, e.g., Paninski, 2004; Truccolo et al., 2005; Okatan et al., 2005; Pillow et al., 2008) may use to understand the behavior of the network or single-cell models that have just been inferred. In particular, we were interested in predicting, for example, the mean firing rate of a specific GLM network to a novel arbitrary dynamic input stimulus. This contrasts with the literature cited above, which has for the most part focused on the mean firing rates, first-order response, and fixed-point stability properties behavior of idealized, infinitely large populations (Mattia & Del Giudice, 2002, is an exception here) with homogeneous membrane and connectivity properties. In addition, most analytical studies of the input-output property of model neurons with a dynamical input start from a Fokker-Planck formalism and rely on a direct numerical simulation of the partial differential equation (Nykamp & Tranchina, 2000; Knight, Omurtag, & Sirovich, 2000; Chizhov & Graham, 2007; this direct approach is feasible only in the case that the system dynamics may be reduced to a state space of dimension at most two or so, and therefore does not apply in the networks studied here), or on linear response theory that assumes a small time-dependent component relative to the baseline component (Shriki et al., 2003; Fourcaud-Trocme et al., 2003; Doiron et al., 2004; Lindner et al., 2005), or on some quasi-stationary assumption (Knight et al., 2000; Mattia & Del Giudice, 2002; Fourcaud-Trocme & Brunel, 2005; Moreno-Bote & Parga, 2004, 2006) restricting the applicable input stimulus to slowly changing one or two simple step stimuli with sufficiently long gaps between steps that the network may reach equilibrium. Thus, it is difficult to apply the insights developed in these previous analyses directly to obtain a quantitative prediction of the dynamic firing rates of a specific nonhomogeneous, nonsymmetric network.
We derived, in this article, the finite size correction to the mean-field equation that captures cross-correlation functions between neurons. This finite size effect originates from the xJij 1/Nc scaling of synaptic strengths that guarantees fluctuation of inputs under asynchronous states. Hence, the finite size effect described in this article is the contribution of small fluctuation in the input around the mean. Another kind of finite size effect has also been discussed in the literature. In a sparsely connected network, correlations between two neurons disappear if Nc is sufficiently smaller than the total number of neurons, N (Derrida, Gardner, & Zippelius, 1987). The finite Nc effect has been evaluated using stochastic Fokker-Planck equations (Brunel & Hakim, 1999; Mattia & Del Giudice, 2002). One should note, however, that the evaluation of this finite size effect for an experimentally estimated network structure is generally not straightforward because the evaluation of a Fokker-Planck equation with many state-space variables is computationally hard. Finally, Jij 1/Nc is not the unique way to scale synapses. Under the balanced input assumption, yields order one fluctuation of input even in the limit of Nc → ∞ (Sompolinksy, Crisanti, & Sommers, 1988; van Vreeswijk & Sompolinsky, 1996, 1998). However, nontrivial cross-correlation functions cannot be captured under the standard asynchronous assumption of mean-field analysis in the Nc → ∞ limit. The calculation of the finite size effect of a network with is not the scope of this article.
Two works that bear a stronger mathematical resemblance to ours are Ginzburg and Sompolinsky (1994) and Meyer and van Vreeswijk (2002). These authors applied mean-field approximations to coupled model neurons to study the mean firing rates and autocorrelation and cross-correlation functions in the network. However, the two-state neuron model discussed in Ginzburg and Sompolinsky (1994) (where the neurons flip between “active” and “inactive” firing states according to a Markov process with some finite rate constant) is somewhat distinct from the point-process models we have treated here (where the neuron remains in the “active” state for a negligible time—i.e., the neuron spikes instantaneously), and we have not been able to translate these earlier results to the problems of interest in this article. Renewal point spiking neuorns are analyzed in Meyer and van Vreeswijk (2002), and their results are closer to ours. However, according to their refractory model, one has to consider infinitely many refractory states in the continuous time limit, whereas our spiking neuron model has M refractory state; M can be as small as 2 while keeping the order 1 strong refractory effect. Accordingly, their cross-correlations should be evaluated by integral equations that are computationally (and conceptually) somewhat more involved, while the methods developed here require us to evaluate just a couple of differential equations.
Finally, the effect of common input on the cross-correlation function was previously studied in a related model (Nykamp, 2007), using an expansion of the output f(.) nonlinearity arround J = 0. One major difference between the analysis we have presented here and this previous work is that a simple expansion around J = 0 does not lead to a good approximation of the firing rate, even in the mean-field limit of Nc → ∞, because the recurrent input changes the baseline firing rate (cf. Figure 5); thus, it is much more accurate to expand around the zeroth-order firing rate given by the roots of equation 4.13, as discussed in section 4.3. (On the other hand, our mean-field approach does require a gaussian approximation that is known to be inaccurate in the case of large J terms.) It would be interesting to explore whether the methods developed here could help lead to more accurate inference of the common-input effects discussed in Nykamp (2007).
There are many possible applications of this mean-field method for problems that require fast evaluation of mean firing rates and cross-correlation functions. One example is to evaluate the information coded by spiking of a recurrently connected network about the input stimulus. This kind of information calculation usually requires averaging over recent spike history (Toyoizumi, Aihara, & Amari, 2006), but the gaussian approximation of the input described here could greatly ease the computational complexity to evaluate the information of stimulus coded by the network. We hope to pursue this direction in future work.
## Appendix: Mathematical Details
In this appendix, we collect the details of the expansion analyses summarized in the main text.
### A.1. Calculation of , , and Bkli.
In this appendix, we use to simplify some expressions. Direct evaluation of equation 5.14 yields
A.1
where and . Similarly, we find
A.2
### A.2. Approximation of , and Bkli(t).
In order to evaluate the finite size effect—order 1/Nc terms of the mean firing intensity and cross-covariance functions—we need to evaluate , , and to the first order of 1/Nc. Assuming that the synaptic strengths scale as J ∼ 1/Nc, we need to evaluate
to the first order of 1/Nc,
to the zeroth order of 1/Nc if k = i or k = j because J ∼ 1/Nc but to the first order of 1/Nc otherwise (in the following, we divide these into five cases and consider each case separately), and
to the first order of 1/Nc if i, j, k are all different, and to the zeroth order of 1/Nc if k = l, k = i, or l = i.
First, direct differentiation of yields
A-3
where , and we used in the above calculation, from equations 5.4 and 5.6,
A.4
Next, to evaluate , we use if i, k, l are all different; for ik. As we discussed at the beginning of this section, in order to evaluate to the first order of 1/Nc, we have to consider the number of combinations of indexes. For example, there are Nc terms of and N2c terms of . Hence, we consider the following five possibilities for combining indexes:
• •
Case 1 (ij, ik); evaluated up to the first order of 1/Nc:
A-5
Note that in the second line, we used and , for example, and neglected higher-order terms such as Δ pik Δ νkj.
• •
Case 2 (i = j, ki); evaluated up to the first order of 1/Nc:
A-6
• •
Case 3 (i = k, ij); evaluated up to the zeroth order of 1/Nc:
A-7
• •
Case 4 (j = k, ij); evaluated up to the zeroth order of 1/Nc:
A.8
• •
Case 5 (i = j = k); evaluated up to the first order of 1/Nc:
A-9
This is a difficult situation to analyze precisely. However, unless the neuron i receives extremely strong input, the conditional firing intensity is almost zero if s and t′ are as close as the refractory time constant of the neuron (M − 1)τr, so the contribution of this interval on the integral is small. If s and t′ are separated more than the refractory time constant, the interaction between the spike at s and t′ becomes small, that is, , and becomes small. Hence, to a good approximation,
A-10
The validity of this intuitive approximation should be examined by numerical simulations. Note that contributes to the order 1/Nc term of the autocovariance function Δ ϕii(t, t′) but contributes only to order 1/N2c terms of the mean firing intensity νi(t) and cross-covariance functions with ij.
Finally, we can rewrite Bkli as
A-11
where we used and . We need to evaluate Bkli up to the first order of 1/Nc if i, k, l are all different and up to the zeroth order of 1/Nc if k = l, k = i, or l = i. Up to these orders, Bkli can be approximated as
A.12
Note that for ik, if i, k, l are different, and for ik.
### A.3. Calculation of the Linear Filter in the Generalized Linear Model with Markov Refractoriness.
We assume that the input is given by
A.13
with stimulus , and calculate the linearized input-output filter for small stimulus fluctuation δ ξi(t) about a constiant baseline ξi(0).
In the limit of Nc → ∞, the mean-field equation is given by
A.14
First, for a constant input I(0)i, we obtain
A.15
where and the second to last equation is easily solvable with respect to p(0)i; we find
A.16
Next, the linear response to a small perturbation is given by
A-17
where and . Remember that and . Fourier transformation of the above equations gives
A.18
The above equation is still very complicated. So it is worth thinking about the special case Jki = 0. In this case, the linear response is described by
A.19
The calculation of is straightforward thanks to the bidiagonal nature of . After direct matrix inversion, we find
A.20
with and . Hence, the linear response is simplified as
A-21
In particular, if M = 2, we find that the Fourier inverse transformation of the gain function has an analytical form, that is, using , we find
A.22
Note that in the limit of τr → 0, the trivial gain function is . Hence, we can see that the suppressive kernel is added to this instantaneous gain function due to the Markov refractoriness. This linear response for Jki = 0 case is a special case of a more general linear response for a renewal neuron (Gerstner & Kistler, 2002). Note that in equation A.18, we discussed a more general case and considered the interaction between the refractory effect and spike interaction effect of a network of recurrently connected neurons.
## Acknowledgments
We thank E. Shea-Brown for helpful conversations. T.T. is supported by the Robert Leet and Clara Guthrie Patterson Trust Postdoctoral Fellowship, Bank of America, Trustee. L.P. is supported by an NSF CAREER award, an Alfred P. Sloan Research Fellowship, the McKnight Scholar award, and NEI grant EY018003.
## Notes
1
Note that the computational complexity to calculate the finite size effect is not N2 but N2Nc because each synaptic time constant, τik, is distinct. Hence, we needed to evaluate each term separately. However, if all the synaptic constants onto each neuron are constant, that is, τik = τi, we can directly evaluate without evaluating αijk separately, and the computational complexity to evaluate the finite size effect reduces to O(N2T).
2
In this article, we mostly use M = 3 (except for Figure 7B, where the effect of abrupt refractoriness is studied), because this is the smallest value of M that requires the following mean-field formulation (we do not need a vector formulation for an M = 2 case because there is only one degree of freedom that corresponds to the probability of being in the active state). It is worth noting that Escola and Paninski (2007) discuss methods for additionally estimating an optimal transition matrix Wi(t) via an EM algorithm. This provides another method for adjusting the short-time details of the model's responses. In addition, we may extend many of the mean-field methods developed below to the case of more general rate matrices Wi(t). However, for simplicity, we will not pursue these extensions here.
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https://edreports.org/reports/detail/mirrors-windows-2021-2021-11
|
## Alignment: Overall Summary
Mirrors & Windows Grade 11 materials partially meet the expectations of alignment to the Common Core ELA standards. The materials include some instruction, practice, and authentic application of reading, writing, speaking and listening, and language work that is engaging and at an appropriate level of rigor for the grade.
|
## Gateway 1:
### Text Quality
0
15
28
32
20
28-32
Meets Expectations
16-27
Partially Meets Expectations
0-15
Does Not Meet Expectations
## Gateway 2:
### Building Knowledge
0
15
28
32
16
28-32
Meets Expectations
16-27
Partially Meets Expectations
0-15
Does Not Meet Expectations
|
## Gateway 3:
### Usability
0
15
22
25
N/A
22-25
Meets Expectations
16-21
Partially Meets Expectations
0-15
Does Not Meet Expectations
## The Report
- Collapsed Version + Full Length Version
## Text Quality and Complexity and Alignment to the Standards with Tasks and Questions Grounded in Evidence
#### Partially Meets Expectations
+
-
Gateway One Details
### Criterion 1a - 1e
Texts are worthy of students’ time and attention: texts are of quality and are rigorous, meeting the text complexity criteria for each grade. Materials support students’ advancing toward independent reading.
10/14
+
-
Criterion Rating Details
The instructional materials reviewed for Grade 11 partially meet the expectations for text quality and complexity. Materials include high-quality texts; however, text types do not reflect the balance informational and literary texts as required by the standards. Some texts are not appropriately complex and the progression of text complexity does not increase across the year.
### Indicator 1a
Anchor texts are of high quality, worthy of careful reading, and consider a range of student interests.
4/4
+
-
Indicator Rating Details
The materials reviewed for Grade 11 meet the criteria for Indicator 1a.
Instructional materials contain a wide range of high-quality fiction and nonfiction text types that are rich in content, relevant, and engaging for students. Selections were chosen with the intention that students be able to learn more about themselves and the world around them, while making many cross-curricular connections. Additionally, texts are organized around and speak to universal themes. Each unit covers a specific historical period and is divided into subsections highlighting different writings of the era. Each subsection includes its own anchor text.
Anchor texts in the majority of chapters/units and across the year-long curriculum are of high quality, consider a range of student interests, and are well-crafted and content rich, engaging students at their grade level. Examples include, but are not limited to, the following:
• In Unit 2, Expressing a National Spirit, American Renaissance 1800–1850, students read the lyric poem, “Thanatopsis” by William Cullen Bryant, an excerpt from the essay, “Nature” by Ralph Waldo Emerson, and the short story, “The Devil and Tom Walker” by Washington Irving. Both the poem and essay are high quality, content rich texts, and the short story is an exemplary piece of narrative writing.
• In Unit 3, A Nation Divided, Slavery and the Civil War 1850–1865, students read an excerpt from Narrative of the Life of Frederick Douglass, an American Slave, Written by Himself by Frederick Douglass, an excerpt from the preface to Leaves of Grass by Walt Whitman, and the poem, “I Hear America Singing” by Walt Whitman. These thought-provoking anchor texts contain rich and relevant content.
• In Unit 5, Progress and Conflict, Early Twentieth Century 1910–1929, anchor texts include an excerpt from The Great Gatsby by F. Scott Fitzgerald, “A Wagner Matinee” by Willa Cather, and “The Negro Speaks of Rivers” and ‘I, Too, Sing America” by Langston Hughes. These classic texts are thought-provoking, rich in content, and contain universal themes and cross-curricular connections.
• In Unit 7, The American Dream, Postwar Era 1945–1960, subunit Conflict and Conformity, the anchor text is the classic play, The Crucible by Arthur Miller. Students explore the impact of mass hysteria on a community in this story of the Salem Witch Trials.
• In Unit 9, New Challenges, Contemporary Era 1980–Present, students read “Though We May Feel Alone,” “Dream,” and “My Mother’s Blue Bowl” by Alice Walker followed by selections from other authors who explore various cultures, races, ethnicities, and writing styles. Students dive deeply into the style of Alice Walker and her universal theme of ancestral connections.
### Indicator 1b
Materials reflect the distribution of text types and genres required by the standards at each grade level.
Narrative Evidence Only
+
-
Indicator Rating Details
The materials reviewed for Grade 11 do not reflect the distribution of text types and genres required by the standards at each grade level.
Text selections support American Literature coursework and include articles, drama, history, mythology, opinions, editorials, speeches and poetry. Although materials contain a variety of text types, materials do not reflect an appropriate balance of informational and literary texts. Of the 154 core and supporting texts students read during the year, 52 of the selections are informational, resulting in a 34/66 balance of informational and literary texts.
Materials reflect the distribution of text types and genres required by the grade level standards but do not reflect a 70/30 balance of informational and literary texts. Examples include, but are not limited to, the following:
• In Unit 1, Shaping the New World,Origins of the American Tradition to 1800, students read an excerpt from Poor Richard’s Almanack by Benjamin Franklin. Students read a total of 24 core and supporting texts, including 12 informational core texts and three Informational Text Connection selections, resulting in a 63/37 balance of informational and literary texts.
• In Unit 2, Expressing a National Spirit, American Resistance 1800–1850, students read the short story, “The Devil and Tom Walker” by Washington Irving. Students also learn about the historical connection of the work to the Salem Witch Trials and the literary connection to another of Irving’s works, “Rip Van Winkle.” Students read a total of 22 core and supporting texts, including five informational core texts and two Informational Text Connection selections, resulting in a 32/68 balance of informational and literary texts.
• In Unit 3, A Nation Divided, Slavery and the Civil War 1850–1865, students read three lyric poems by Emily Dickinson as part of the Author’s Focus section. Students read a total of 23 core and supporting texts, including six informational core texts and three Informational Text Connection selections, resulting in a 39/61 balance of informational and literary texts.
• In Unit 4, Expanding Frontiers, Unification and Growth 1865-1916, students read “Keeping The Things Going While Things are Stirring,” a speech by Sojourner Truth. Students read a total of 26 core and supporting texts, including 11 informational core texts and four Informational Text Connection selections, resulting in a 58/42 balance of informational and literary texts.
• In Unit 7, The American Dream, sub-unit Conflict and Conformity, students explore a government document ,U.S. Supreme Court Decision, Brown vs Board of Education of Topeka.Students read a total of 18 core and supporting texts, including three informational core texts and three Informational Text Connection selections, resulting in a 33/67 balance of informational and literary texts.
• In Unit 8, Social Transition, sub-unit Personal Challenges, students read the short story, “The Rockpile” by James Baldwin. Students read a total of 25 core and supporting texts, including five informational core texts and two Informational Text Connection selections, resulting in a 28/72 balance of informational and literary texts.
• In Unit 9, New Challenges, sub-unit Contemporary America, students explore the literary non-fiction text, “On the Mall” by Joan Didion. Students read a total of 28 core and supporting texts, including nine informational core texts, resulting in a 32/68 balance of informational and literary texts.
### Indicator 1c
Core/Anchor texts have the appropriate level of complexity for the grade according to documented quantitative analysis, qualitative analysis, and relationship to their associated student task. Documentation should also include rationale for educational purpose and placement in the grade level.
2/4
+
-
Indicator Rating Details
The materials reviewed for Grade 11 partially meet the criteria for Indicator 1c.
Grade 11 texts quantitatively range between 320L–1550L for the year. Most texts that fall outside of the Grades 11–CCR Lexile Stretch Band have qualitative measures that make them appropriately complex for the grade. The relationship of the quantitative and qualitative analyses to the associated reader task is not appropriately complex. While some Extend the Text tasks serve as associated reader tasks, these tasks are optional and may not occur during core instruction. Although materials include text complexity information for quantitative and qualitative measures, the documentation does not include a rationale for educational purpose and placement in the grade level.
Core/Anchor texts do not have the appropriate level of complexity for the grade according to documented quantitative analysis, qualitative analysis, and relationship to their associated student task. Documentation does not include a rationale for educational purpose and placement in the grade level. Examples include, but are not limited to, the following:
• Anchor/Core texts do not have the appropriate level of complexity for the grade according to quantitative and qualitative analysis and relationship to their associated student task.
• In Unit 5, Progress and Conflict, Early Twentieth Century 1910–1929, of the thirty-four selections students read, twenty-six do not have a Lexile level. Six texts fall below the Grades 11–CCR Lexile Stretch Band and two fall within it. Students read an excerpt from The Great Gatsby by F. Scott Fitzgerald (1170L). The text falls slightly below the Grades 11–CCR Lexile Stretch Band and has a Reading Level of Moderate. Some difficulty vocabulary is identified as a Difficulty Consideration, while Ease factors include dialogue and easy conversational style. As a learning objective, students understand and analyze the narrator and describe the setting, extending their thinking to understand how the narrator and setting work together to create a sense of a particular time and place. Students use a two-column list to “record details indicating the setting, characters, narrative, or theme” while reading the text. Students [a]lso identify the narrator and determine his relationship to Gatsby. Consider why Fitzgerald chose this character to tell the story.” After reading, students respond to the following Analyze Literature questions addressing setting and narrator. Students do not use their two-column list to respond to the questions. During the Creative Writing option in the Extend the Text section, students [c]hoose one of the scenes from the party, and rewrite it to be set in the current time. In a one-paragraph description, include details that indicate the scene’s new setting.” This associated reader task is optional and, as a result, may not occur during core instruction.
• Anchor/Core texts and series of texts connected to them are accompanied by an accurate text complexity analysis; however, the text complexity analysis does not include a rationale for educational purpose and placement in the grade level.
• The text overview page for each selection includes the following text complexity information: Reading Level and Lexile level, Difficulty Considerations, and Ease Factors. Materials do not explain the educational purpose of the text and the reason for its placement in the grade level.
### Indicator 1d
Series of texts should be at a variety of complexity levels appropriate for the grade band to support students’ literacy growth over the course of the school year.
2/4
+
-
Indicator Rating Details
The materials reviewed for Grade 11 partially meet the criteria for Indicator 1d.
While series of texts are largely at a variety of complexity levels, the complexity levels of anchor texts and supporting texts students read do not provide an opportunity for students’ literacy skills to grow across the year. Extend the Text tasks, while optional, often do not provide students with opportunities to demonstrate their understanding of the focus area. When provided, associated reader tasks do not increase in complexity over the course of the year. While the program’s gradual release of responsibility reading model “emphasizes scaffolded instruction,” it is unclear which texts are Directed Reading selections and which are Independent Reading selections, as the Reading Support levels are not identified on the Scope & Sequence guide or on the text overview pages.
Series of texts should be at a variety of complexity levels appropriate for the grade band to support students’ literacy growth over the course of the school year. Examples include, but are not limited to, the following:
• The complexity of anchor texts students read does not provide an opportunity for students’ literacy skills to increase across the year, encompassing an entire year’s worth of growth.
• In Unit 3, A Nation Divided, Slavery and the Civil War 1850–1865, texts range from 920L–1950L. Students read an excerpt from Incidents in the Life of a Slave Girl, Seven Years Concealed, an autobiography by Harriet Jacobs (Linda Brent) (920L). This text is significantly below the Grades 11–CCR Lexile Stretch Band. The Reading Level for this text is listed as Moderate with background information on slavery and high-level vocabulary identified as Difficulty Considerations and sympathetic narrator, first-person point of view, and compelling story. During the Independent Reading selection, students focus on irony, characterization, and chronological order, responding to Analyze Literature prompts addressing the aforementioned literary elements. Writing Options tasks do not address irony, characterization, and chronological order nor do the questions and tasks address the requirements of the standard: “Analyze a complex set of ideas or sequence of events and explain how specific individuals, ideas, or events interact and develop over the course of the text.”
• In Unit 5, Progress and Conflict, Early Twentieth Century 1910–1929, texts range from 740L–1370L. The second of four anchor texts is “A Wagner Matinee,” a short story by Willa Cather (1370L). This text is slightly above the Grades 11–CCR Lexile Stretch Band. Materials list the Reading Level as Moderate with references to the narrator’s youth listed as a Difficulty Consideration and length identified as an Ease Factor. To frame students’ work with analyzing the narrator, point of view, and characterization, the Set Purpose section of the text overview includes this guidance: “As you read, consider what information is provided by Clark, the narrator, and whether it is reliable. On what does he base his opinion of his aunt? Consider, too, the other ways Cather creates the character of Aunt Georgiana. Make predictions about what you think will happen, based on the narrator’s point of view. Correct or confirm those predictions as you read.” Although students respond to various Analyze Literature questions that address the aforementioned literary elements during reading, Extend the Text options do not provide an opportunity for an associated reader task that addresses the elements of focus or their associated standard: “Analyze the impact of the author's choices regarding how to develop and relate elements of a story or drama (e.g., where a story is set, how the action is ordered, how the characters are introduced and developed).”
• In Unit 7, The American Dream, Postwar Era 1945–1960, texts range from 860L–1430L. At the start of the unit, students read “The Life You Save May Be Your Own,” a short story by Flannery O’Connor (990L). This text is significantly below the Grades 11–CCR Lexile Stretch Band. Materials list the Reading Level of this selection as Moderate with dialect identified as a Difficulty Consideration and style listed as an Ease Factor. Materials define characterization and dialect and direct students to “analyze the characterization techniques [O’Connor] uses to develop the characters in the story.” Students also “analyze her use of dialect in creating characters, “writ[ing] down specific examples of dialect in the story.” Students examine and discuss Analyze Literature prompts and notes that address characterization and dialect both during and after reading. During the Informative Writing Extend the Text option, students "[w]rite an essay analyzing the complexities of the character of Mr. Shiftlet,” organizing their analysis “in terms of the direct and indirect characterization techniques, devoting a paragraph to each.”
• As texts become more complex, some scaffolds and/or materials are provided in the Teacher Edition (e.g., spending more time on texts, more questions, repeated readings, skill lessons)
• In Unit 3, A Nation Divided, Slavery and the Civil War 1850–1865, students read an excerpt from the preface to Leaves of Grass by Walt Whitman (1950L) and an excerpt from “I Hear America Singing” by Walt Whitman (NP). Materials list the Reading Level of the preface excerpt for Leaves of Grass as Challenging with metaphor; background needed; more description than action; formal language; and long, complicated sentences identified as Difficulty Considerations. The quantitative measure places this text far above the Grades 11–CCR Lexile Stretch Band. The Reading Level for the excerpt from “I Hear America Singing” is listed as Easy with personification listed as a Difficulty Consideration and length and simple language listed as Ease Factors. The Build Background section of the text overview provides literary context for both selections. The Analyze Literature inset of the text overview defines Romanticism and free verse. The after-reading Analyze Literature inset includes additional context on elements of Romanticism and free verse. Materials do not provide support for the remaining Difficulty Considerations.
• In Unit 8, Social Transition, Early Contemporary Era 1960–1980, students read John F. Kennedy’s “Inaugural Address” (1340L). The Reading Level for this text is listed as Moderate with historical context and vocabulary identified as Difficulty Considerations. The Build Background section of the text overview includes some historical context information. The text overview includes a list of Preview Vocabulary words. Materials include footnotes that define these words as students read the text.
### Indicator 1e
Materials provide opportunities for students to engage in a range and volume of reading to support their reading at grade level by the end of the school year, including accountability structures for independent reading.
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Indicator Rating Details
The materials reviewed for Grade 11 meet the criteria for Indicator 1e.
Materials provide opportunities for students to engage in a range and volume of reading to support their reading at grade level by the end of the school year, including accountability structures for independent reading. Examples include, but are not limited to, the following:
• Instructional materials clearly identify opportunities and support for students to engage in reading a variety of text types and genres.
• In Unit 3, A Nation Divided, Slavery and the Civil War 1850–1865, in the Crisis of Ideals subsection, students read two autobiographies, two lyric poems, a short story, a speech, and two letters.
• In Unit 6, Hard Times, Depression and World War II 1929–1945, in the Facing Grim Realities subsection, students read two literary nonfiction excerpts, a novel excerpt, a letter, a short story, a speech, a biography, a lyric poem, and advertisement, and a memoir.
• In Unit 9, New Challenges, Contemporary Era 1980–Present, students read the following texts:”“Though We May Feel Alone,” a poem by Alice Walker, “The Names of Women,” an essay by Louise Erdrich, “Mother Tongue,” an essay by Amy Tan, “Straw into Gold: The Metamorphosis of the Everyday,” an essay by Sandra Cisneros, “Throughput,”an excerpt from Fast Food Nation by Eric Schlosser, “On the Mall,” literary nonfiction by Joan Didion, and “Couplet: Old-Timers’ Day, Fenway Park, 1 May 1982,” a lyric poem by Donald Hall.
• Instructional materials clearly identify opportunities and supports for students to engage in a volume of reading.
• In Unit 4, Expanding Frontiers, Unification and Growth 1865–1910, in the Realism and Naturalism subsection, students read three short stories, an essay, a memoir, four lyric poems, and a piece of process writing over the course of nine regular class periods or 4.5 block schedule periods.
• In Unit 6, Hard Times, Depression and World War II 1929–1945, in the Southern Renaissance section, students read five short stories, a speech, an excerpt from a novel, and an essay. The Visual Planning Guide allots eleven regular class periods to cover these texts.
• In Unit 7, The American Dream, Postwar Era 1945–1960, students read a volume of texts, including short stories, elegies, lyric poems, essays, eulogies, memoirs, and newspaper articles. Materials list instructional supports, such as Unit and Selection Resources and Differentiated Instruction manuals, in the Visual Planning Guide.
• There is sufficient teacher guidance to foster independence for all readers (e.g., proposed schedule and tracking system for independent reading).
• The Program and Planning Guide contains a Reading Log for students to track their reading. In addition, each unit contains a Visual Planning Guide that begins with the Directed Reading selections and ends with the Independent Reading selections. This guide provides lesson and pacing suggestions.
• In Unit 2, Expressing a National Spirit, American Resistance 1800–1850, in the For Your Reading List section, students choose from a list of suggested works from the time period to read outside the classroom. Text selections include: Great Short Works of Herman Melville by Herman Melville, The Scarlet Letter by Nathaniel Hawthorne, American Women Poets of the 19th Century edited by Cheryl Walker, Undaunted Courage: Meriwether Lewis, Thomas Jefferson, and the Opening of the American West by Stephen E. Ambrose, The Last of the Mohicans by James Fenimore Cooper, and The Essential Writings of Ralph Waldo Emerson by Ralph Waldo Emerson. Students track their reading progress on a weekly Reading Log that is included in the Program Planning Guide.
• In Unit 7, The American Dream, Postwar Era 1945–1960, in the For Your Reading List Section, students choose from a list of suggested works from the time period to read outside the classroom. Text selections include: Quiet Strength: the Faith, the Hope, and the Heart of a Woman Who Changed a Nation by Rosa Parks with Gregory J. Reed, A Sand County Almanac by Aldo Leopold, I've Got the Light of Freedom: The Organizing Tradition and the Mississippi Freedom Struggle by Charles M. Payne, Homeward Bound: American Families in the Cold War Era by Elaine Tyler May, A Good Man is Hard to Find and Other Stories by Flannery O'Connor, Invisible Man by Ralph Waldo Ellison. Students track their reading progress on a weekly Reading Log that is included in the Program Planning Guide.
• Independent reading procedures are included in the lessons.
• In Unit 2, Expressing a National Spirit, American Resistance 1800–1850, the first independent reading selection is an excerpt from Snow-Bound, a poem by John Greeneaf Whittier. The Teacher’s Edition includes objectives for reading the selection, a suggestion for how to launch the lesson, a Mirrors & Windows question, prompts for analyzing the text, a suggestion for critical viewing, targeted reading skills, text-dependent questions and writing options.
• In Unit 3 , A Nation Divided, Slavery and the Civil War 1850–1865, within the Independent Reading portion of the unit, the For Your Reading List section contains student guidance and suggestions for selecting and reading texts independently. In addition, the Teacher’s Edition provides recommendations for how teachers can assign students to small groups to choose one of the independent reading selections to present as a dramatization.
• In Unit 7, The American Dream, Postwar Era 1945–1960, the first Independent Reading selection is “Once More to the Lake,” an essay by E.B. White. The Teacher’s Edition includes objectives for reading the selection, a suggestion for how to launch the lesson, a Mirrors & Windows question, prompts for analyzing the text, suggested reading skills, text-dependent questions and writing options.
### Criterion 1f - 1m
Materials provide opportunities for rich and rigorous evidence-based discussions and writing about texts to build strong literacy skills.
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Criterion Rating Details
The instructional materials reviewed for Grade 11 partially meet the expectations for evidence-based discussions and writing about texts. Materials include oral and written questions and tasks grounded in the text, requiring students to use information from the text to support their answers and demonstrate comprehension of what they are reading. Materials do not include speaking and listening protocols. Speaking and listening instruction includes some facilitation, monitoring, and instructional supports for teachers; however, materials lack relevant follow-up questions and supports. Although materials include a mix of on-demand and process writing, writing opportunities in each mode are unevenly distributed. Writing Workshops include revision and editing opportunities; however, materials rarely include explicit writing instruction. Although students have opportunities to write about what they are reading, including opportunities to support their analyses and claims using evidence from texts and/or sources, many of these opportunities are optional. Materials lack explicit evidence-based writing instruction. Materials miss opportunities for explicit instruction of grade-level grammar and usage standards. Opportunities for authentic application in context are limited. Although materials include opportunities for students to interact with key academic vocabulary words in and across texts, materials do not outline the program’s plan for vocabulary development or provide teacher guidance to support students’ vocabulary development.
### Indicator 1f
Most questions, tasks, and assignments are text-specific and/or text-dependent, requiring students to engage with the text directly (drawing on textual evidence to support both what is explicit as well as valid inferences from the text).
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Indicator Rating Details
The materials reviewed for Grade 11 meet the criteria for Indicator 1f.
Most questions, tasks, and assignments are text-specific and/or text-dependent, requiring students to engage with the text directly (drawing on textual evidence to support both what is explicit as well as valid inferences from the text). Examples include, but are not limited to, the following:
• Text-specific and text-dependent questions and tasks support students in making meaning of the core understandings of the texts being studied.
• In Unit 1, Shaping the World, Origins of the American Tradition to 1800, students read and compare two nonfiction texts, an excerpt from The General History of Virginia by John Smith, and an excerpt from Of Plymouth Plantation by William Bradford. After reading both texts, students respond to text-dependent prompts in the Compare Literature: Point of View section: “What point or points of view does each writer use? What effects does the writer's choice have on the telling of the story? Consider which story seems more real or compelling.”
• In Unit 2, Expressing a National Spirit, American Resistance 1800–1850, students read an excerpt from the essay, Walden by Henry David Thoreau. After reading, students answer a series of text-specific questions such as “Why did Thoreau go to the woods? Why did he leave the woods? Compare and contrast the two reasons.”
• In Unit 3, A Nation Divided, Slavery and the Civil War 1850–1865, students read the informational text, “ Battle for the Belle of Amherst” by Daniel Terdiman as part of a connected text set containing a series of poems by Emily Dickinson. After reading, students answer review questions, such as “Identify each designer's concept for a video game based on Emily Dickinson. Why would major video game designers be interested in creating a game based on a very private poet like Dickinson?”
• In Unit 5, Progress and Conflict, Early Twentieth Century 1910–1929, students read the dramatic monologue, “The Love Song of J. Alfred Prufrock,” by T. S. Eliot. The Teacher Edition contains a Critical Thinking Discussion Guide prompt: “As Prufrock describes himself, he is insecure and self contemptuous. Would the character have been more believable and more sympathetic if he did not present himself in such an unflattering light? Explain your response.”
• Teacher materials provide support for planning and implementation of text-based questions and tasks.
• Unit 1, Shaping the World, Origins of the American Tradition to 1800, students read the Declaration of Independence by Thomas Jefferson. The Analyze Text Organization section includes the following teacher guidance: “Review with students the problem/solution organization of the Declaration of Independence” to compare and contrast ideas expressed in different paragraphs.
• In Unit 2, Expressing a National Spirit, American Resistance 1800–1850, students read “The Devil and Tom Walker” by Washington Irving. The Teach the Selection section includes an Analyze Literature: Plot and Exposition task. Teacher guidance includes: “Have students note where the exposition ends and the rising action begins to develop the conflict.”
• In Unit 3, A Nation Divided, Slavery and the Civil War 1850–1865, students read “Much Madness is divinest Sense,” “I heard a Fly buzz when I died,” “Because I could not stop for Death,” and “This is my letter to the World” by Emiliy Dickinson. Within the Teach the Selection portion of the materials, students complete several summary tasks. During the Analyze Literature section, students complete a personification activity associated with “Because I could not stop for Death.” Teacher guidance includes: “Students may note that the speaker appreciates the civility of Death and, by extension, of the process of dying. Personification appears twice in the final lines: The Horse's Heads (probably an allusion to the Four Horsemen of the Apocalypse) are facing Eternity.”
• In Unit 4, Expanding Frontiers, Unification and Growth 1865–1910, students read the short story, “To Build a Fire” by Jack London. During reading, teacher guidance in the Teacher's Edition includes, “Explain that a foil is a character whose attributes contrast with and therefore highlight the attributes of another character. Ask students to explain how the dog can be seen as a foil for the man, his owner. What characteristics of the man are highlighted? Answer: the dog relies on its instincts, which tell it that it is too dangerously cold to be venturing outside.” The inclusion of possible student responses supports teachers with planning and implementing text-based questions.
### Indicator 1g
Materials provide frequent opportunities and protocols for evidence-based discussions.
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Indicator Rating Details
The materials reviewed for Grade 11 partially meet the criteria for Indicator 1g.
The materials provide frequent opportunities for students to engage in speaking and listening activities and projects. Materials also include directions for conducting such exercises; however, there are no protocols for these activities and projects found in the core materials, nor is there guidance for how or when teachers should model speaking and listening techniques. At the end of each unit, materials include a Speaking and Listening Workshop where students can practice, present, and actively listen to oral presentations. These Workshops include steps on how to conduct a particular speaking and listening project, as well as a rubric and speaking and listening tips.
Materials provide frequent opportunities for speaking and listening; however, speaking and listening opportunities do not include protocols. Examples include, but are not limited to, the following:
• Materials do not provide varied protocols for speaking and listening to support students’ developing speaking and listening skills across the whole year’s scope of instructional materials.
• In Unit 1, Shaping the World, Origins of the American Tradition to 1800, in the Speaking & Listening Workshop, students learn how to present a literary work. The Workshop includes explanations for each of the following steps: select a work, familiarize yourself with the work, practice the work aloud, memorize the work, present the oral interpretation. Materials provide a Speaking and Listening Rubric for Content and Delivery that includes these elements: “The literary work is appropriate for the audience. The volume, pace, and enunciation (clarity of speech) fit the selection. The tone (emotional quality) and emphasis of delivery are effective.” Although materials include directions for students to complete this Workshop, there is no evidence of protocols for students to conduct the speaking and listening task and develop their speaking and listening skills.
• In Unit 4, Expanding Frontiers, Unification and Growth 1865–1910, students read the paired texts, “I Will Fight No More Forever,” a speech by Chief Joseph of the Nez Perce, and “I am the Last of My Family,” a speech by Cochise of the Chiricahua. In the Extend the Text section, students “Address a Jury” during the Critical Literacy activity. Directions for the activity include: “Imagine that you are an attorney representing the Nez Perce in a lawsuit against the U.S. government. Prepare an address to a civil jury, stating what Chief Joseph’s people have lost and how they should be compensated. Ask six of your classmates (the number of people on a civil jury) to be jurors and listen to your presentation. The jurors should take notes on your position and the evidence to support it. After the presentation, ask the jurors for their feedback and questions. Discuss as a group.” Although materials include directions for students to complete this optional task, there is no evidence of a specific protocol used to support students’ developing speaking and listening skills.
• In Unit 7, The American Dream, Postwar Era 1945–1960, students read “The Life You Save May Be Your Own” by Flannery O’Connor. The Teacher Wrap of the Teacher Edition includes a Teaching Note on self-generated questions. Students work in pairs and “reread a section of the text and write down any questions that arise during their reading.” As a class, students go back through each section of the text and discuss the questions. While the Teaching Note includes directions for the activity, there is no evidence of a specific protocol used to support students’ developing speaking and listening skills.
• Teacher guidance includes modeling of academic vocabulary and syntax during speaking and listening opportunities.
• In Unit 2, Expressing a National Spirit, American Renaissance 1800–1850, students read an excerpt from Nature by Ralph Waldo Emerson. As the students analyze the text, the teacher defines aphorism, “a short saying that makes an often witty observation about life.” The teacher and students examine sentences in the passage, and the teacher encourages students to identify the sentences as aphorisms. The class then discusses “how such concise statements allow the writer to convey important ideas briefly and in a way that challenges readers to reflect on them.”
• In Unit 5, Progress & Conflict, Early Twentieth Century 1910–1929, students read excerpts from The Great Gatsby by F. Scott Fitzgerald and Ernest Hemingway’s The Sun Also Rises and For Whom the Bell Tolls. When framing the readings, the teacher instructs students on point of view using examples from The Great Gatsby and For Whom the Bell Tolls to illustrate first-person and third-person points of view. The teacher uses the following questions to facilitate discussion on the narrator of each work: “Who is telling the story? What is the narrator’s relation to the other characters? Is the narrator’s perspective, or point of view, limited in any way? Is the narrator reliable?”
• In Unit 8, Social Transition, Early Contemporary Era 1960–1980, students read Martin Luther King Jr.’s “Letter from Birmingham Jail.” The teacher explains the contextual use of argument, a “logical reason for accepting or rejecting a provable statement of belief or course of action,” using King’s argument supporting the march in Birmingham. During a discussion, students “consider the arguments for and against civil disobedience as a means of protesting unjust laws” as they respond to the following question: “Under what conditions is this approach to social change justified, and when might it not be justified?”
### Indicator 1h
Materials support students’ listening and speaking about what they are reading and researching (including presentation opportunities) with relevant follow-up questions and evidence.
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Indicator Rating Details
The materials reviewed for Grade 11 partially meet the criteria for Indicator 1h.
Materials include opportunities for stand-alone and text-based discussions. Students may respond to Close Reading, Analyze Literature, Use Reading Skills, Refer to Text, and Reason with Text questions in writing or orally as instructed by their teacher. Where appropriate, the Teacher Wrap in the Teacher Edition contains Critical Thinking Discussion Guides, which provide opportunities for text-based discussions. Although the Discussion Guide includes a series of text-specific questions and suggested answers, materials do not provide evidence of follow-up questions or supports, such as entry points for students who may have difficulty initiating or engaging in conversation. Some Extend the Text options include speaking and listening opportunities; however, the enactment of these activities are based on teacher selection and, as a result, may not occur during core instruction. Mirrors & Windows, and Use Reading Skills: Make Connections questions are often stand-alone in nature, allowing students to reflect on personal experiences while discussing sub-themes and topics related to texts of study. Materials do not include evidence of teacher guidance for monitoring students’ speaking and listening opportunities. Explicit speaking and listening instruction occurs during the End-of-Unit Speaking & Listening Workshop; however, this Workshop is not a part of core instruction.
Examples include, but are not limited to, the following:
• Speaking and listening instruction includes some facilitation, monitoring, and instructional supports for teachers.
• In Unit 3, A Nation Divided, Slavery and the Civil War 1850–1865, the Teacher Wrap of the Teacher Edition includes the following guidance for a close reading of Walt Whitman’s “I Hear America Singing.” “As you lead a class discussion on Whitman’s ‘I Hear America Singing,’ begin by asking a student to read the poem aloud. Remind students to listen for these elements of poetry: rhythm and repetition of initial consonant sounds (alliteration) and of internal vowel sounds (assonance), of words, and of grammatical structures. Ask students to consider the effects of the various elements when they are heard aloud. For instance, the catalog of workers emphasizes their number and suggests their contribution to American life.” Though the materials include these directions, there is no evidence of teacher guidance on monitoring the student discussion or instructional support for students who may be having difficulty starting or engaging in the conversation.
• In Unit 4, Expanding Frontiers, Unification and Growth 1865–1910, the Teacher Wrap in the Teacher Edition contains a Critical Thinking Discussion Guide for the oral history excerpt from Black Elk Speaks by Nicholas Black Elk and John G. Neihardt: “Write this statement from Fire Thunder on the board in large type: ‘I was not after horses; I was after Wasichus.’ Ask students to imagine that this was the headline in a newspaper for a white American audience at the time of the battle. How might readers have felt about the Lakota after reading this headline? What stereotypes might this statement have created? Discuss whether knowing the rest of the story might change attitudes. Ask students to think of contemporary examples in which a so-called sound byte gives one impression, while knowing more of the story gives a different impression.” Materials include suggested answers, but there is no evidence of guidance for monitoring the student discussion or for supporting any learners struggling in taking part in the discussion.
• In Unit 7, The American Dream, Postwar Era 1945–1960, students read Allen Ginsberg’s lyric poem, “A Supermarket in California.” During the Collaborative Learning Extend the Text option, students work in small groups of three or four to discuss at least two of the following text-based questions: “What literal and/or figurative meanings does each question seem to have? Is the question rhetorical, or does the speaker actually seek an answer?” Materials do not include evidence of teacher guidance on monitoring the student discussions or instructional supports for students who may be having difficulty starting or engaging in the conversations. The Extend the Text section contains four options from which the teacher may choose. As a result, this activity may not occur during core instruction.
• Students may have multiple opportunities over the school year to demonstrate what they are reading through varied speaking and listening opportunities. Instruction occurs during the Extend the Text section, that contains four options from which the teacher may choose. As a result, this activity may not occur during core instruction.
• In Unit 2, Expressing a National Spirit, American Resistance 1800–1850, students read “The Devil and Tom Walker” by Washington Irving. The Teacher Wrap in the Teacher Edition includes a Critical Thinking Discussion Guide that directs teachers to “Discuss with students the concept of the Faustian bargain, or selling one’s soul to the devil.” The questions in this guide include “Although this is an age-old moral question, in what forms does it appear today?” and “Are there situations in which crime (or immorality) does pay? Explain.”
• In Unit 5, Progress and Conflict, Early Twentieth Century 1910–1929, in the Extend the Text section for the dramatic monologue,“The Love Song of J. Alfred Prufrock,” by T.S. Eliot, students have the option of writing a situation comedy: “Work with a small group to write an episode of a situation comedy about a man like J. Alfred Prufrock. Change the tragic elements of his psychological profile into humorous traits. Perform the episode for your class.” The Extend the Text section contains four options from which the teacher may choose. As a result, this activity may not occur during core instruction.
• In Unit 7, The American Dream, Postwar Era 1945–1960, while reading Act I of The Crucible by Arthur Miller, the teacher leads a discussion of the role of Rebecca Nurse in this scene by asking the following questions:
• “What role does Rebecca Nurse play in this scene?
• Who agrees with her? Who disagrees with her? What opinion do her opponents express? Why might they be unwilling to listen to her?
• When have you been the ‘voice of reason’?
• When have you failed to listen to the voice of reason?”
• Speaking and listening work requires students to utilize, apply, and incorporate evidence from texts and/or sources.
• In Unit 3, A Nation Divided, Slavery and the Civil War 1850–1865, students read “Much Madness is divinest Sense,” “I heard a Fly buzz - when I died-,” “Because I could not stop for Death-,” and “This is my letter to the World” by Emily Dickinson. Students also read “Battle for the Belle of Amherst,” an article by Daniel Terdiman. In the Extend the Text section, the optional Collaborative Learning task directs students to compose a speech: “Should governments support writers financially? Work with a small group to present your position to the class in a formal speech, using evidence and rhetorical devices to persuade the audience. As you listen to other groups, identify the evidence they use to support their positions. How does this evidence affect your perspective on the issue?” The Extend the Text section contains four options from which the teacher may choose. As a result, this activity may not occur during core instruction.
• In Unit 6, Hard Times, Depression and World War II 1929–1945, students read an excerpt from John Steinbeck’s novel, The Grapes of Wrath. During the Lifelong Learning Extend the Text option, students use print and online sources to “find historical accounts of people’s lives during the Depression.” Students use the following questions to guide their work: “What hardships did many people experience in the Depression? Did the Depression affect people differently in urban and rural areas? What lasting effects did the Depression have on people’s lives?” Students use the information they gather to “prepare and present a speech in which [they] analyze the differences and similarities of historical accounts,” using “excerpts from the accounts to support [their] analysis.” This activity is one of four Extend the Text options from which the teacher may choose and as a result, may not occur during core instruction.
• In Unit 8, Social Transition, Early Contemporary Era 1960–1980, in the Critical Literacy Extend the Text option for “Report From Part One,” “To Black Women,” and “The Explorer” by Gwendolyn Brooks, students celebrate African-American women: “With a small group of classmates, plan a celebration of the works of African-Americam women poets. Write an introduction for each poet; then select one of her poems to share with the group or the class. Explain the themes the poet writes about and how one or more are evidenced in the poem you selected.” The Extend the Text section contains four options from which the teacher may choose. As a result, this activity may not occur during core instruction.
### Indicator 1i
Materials include a mix of on-demand and process writing (e.g., multiple drafts, revisions over time) and short, focused projects, incorporating digital resources where appropriate.
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Indicator Rating Details
The materials reviewed for Grade 11 partially meet the criteria for Indicator 1i.
Materials offer both on-demand and process writing opportunities for students primarily in post-reading Extend the Text tasks and End-of-Unit Writing Workshops. Extend the Text sections contain two, mode-specific writing prompts, and each Writing Workshop focuses on a specific mode of writing. The Workshops guide students through the entire writing process—prewriting, drafting, revising, editing and proofreading, and publishing. Materials also include a student model and instructional guidance for teachers in the Teacher Wrap of the Teacher’s Edition; however, there is no guidance to indicate where students should compose their writing. The Writing and Grammar Handbook offers in-depth lessons that expand on these Writing Workshops, and the Writing section of the Language Arts Handbook also offers detailed information for students on the writing process and modes and purposes of writing; however, these ancillary materials are not part of core instruction. Because teachers have the choice of which Extend the Text exercises to complete, there is no guarantee that students will complete the writing opportunities offered. Materials utilize digital resources where appropriate.
Examples include, but are not limited to, the following:
• Materials include on-demand writing opportunities that cover a year’s worth of instruction.
• In Unit 3, A Nation Divided, Slavery and the Civil War 1850–1865, students complete a Writng Skills task as part of the Test Practice Workshop. During this on-demand timed writing task, students respond to the following prompt: “The outcome of Aesop’s fable, ‘The Tortoise and the Hare,’suggests that ‘Slow and steady wins the race.’ Some people agree with this belief, suggesting that being thorough and working steadily toward a goal will ensure success. Others argue that in today's fast-changing world, doing things quickly and with flash or style is more important. In general, whom do you think will do better in high school: the student who is thorough and steady or the one who is fast and flashy? Take a position on this question. You may write about one of the two perspectives given, or you may present a different perspective on this question.” The Test Practice Workshop is an optional activity and, as a result, may not occur during core instruction.
• In Unit 5, Progress and Conflict, Early Twentieth Century 1910–1929, in the Extend the Text section for “Poetry” by Marianne Moore ,and “Arts Poetica” by Archibald MacLeish, one of the post-reading writing options is an argumentative writing task that asks students to imagine they have been debating the ideas expressed in the two poems. Students “write [their] friend a paragraph explaining which view of poetry—Moore’s or MacLeish’s—most corresponds with your own and why.” This activity is one of four Extend the Text options from which the teacher may choose. As a result, this activity may not occur during core instruction.
• In Unit 8, Social Transition, Early Contemporary Era 1960–1980, in the Extend the Text section for “The Rockpile,” by James Baldwin, students may complete the following informative writing task: “In developing characters, authors sometimes portray how one character’s behavior affects the other characters. Analyze this cause-and-effect relationship among the characters in ‘The Rockpile’.” This activity is one of four Extend the Text options from which the teacher may choose. As a result, this activity may not occur during core instruction.
• Materials include process writing opportunities that cover a year’s worth of instruction. Opportunities for students to revise and edit are provided.
• In Unit 2, Expressing a National Spirit, American Resistance 1800–1850, students complete a Writing Workshop on descriptive writing, during which they “plan, write, and revise a description of a setting.” The Workshop directions include the purpose and audience for the scene and guide students through the entire process of writing the scene: prewriting, drafting, revising, editing and proofreading, publishing, and presenting. Materials include a Student Model to support students’ revision work.
• In Unit 5, Progress and Conflict, Early Twentieth Century 1910–1929, students complete a Writing Workshop on narrative writing: “Plan, write, and revise an application essay telling of an experience that led to personal growth or self-discovery.” The Workshop directions include the purpose and audience for the essay and guide students through the entire process of writing the essay: prewriting, drafting, revising, editing and proofreading, publishing, and presenting.
• In Unit 9, New Challenges, Contemporary Era 1980–Present, the Writing Workshop focuses on writing a research paper. Materials provide guidance during each stage of the writing process. During the revision stage, materials support students with evaluating their draft and revising their work for content, organization, and style. Guidance emphasizes developing the opening, middle, and end of the story. Materials include an annotated Student Model based on the Revision Checklist. The Writing Follow-Up provides guidance on publishing and presenting, as well as approaches to students reflecting on their writing.
• Materials include digital resources where appropriate.
• In Unit 2, Expressing a National Spirit, American Resistance 1800–1850, students read the lyric poem, “Thanatopsis,” by William Cullen Bryant. During the Media Literacy Extend the Text option, students use digital resources to create an art exhibit: “As explained in the Art Connection on page 92, the painting shown at the start of this selection is an example of the Hudson River School of painting. On the Internet, locate a few other paintings done in this style; if possible, print them. For each painting, give the name of the artist and the dates of his or her life; then provide a brief description of the painting.”
• In Unit 4, Expanding Frontiers, Unification and Growth 1865–1910, students read the short story, “To Build a Fire,” by Jack London. During the Critical Literacy Extend the Text option, students use digital resources to read and write about Jack London’s letters: “Much of what we know about famous people and their time comes from their letters, which are primary sources. Go online to find letters written by Jack London. Take notes on the biographical data you collect. Then write a paragraph explaining what you learned about London.”
• In Unit 7, The American Dream, Postwar Era 1945–1960, students read the play, The Crucible, by Arthur Miller. During the Critical Literacy Extend the Text option for Act IV, students use digital resources to study the judicial system: “Using the Internet and library sources, locate materials that explain what a person should do when accused of a crime. Synthesize, or combine, the information to create a one-page guide to navigating the judicial system for defendants.”
### Indicator 1j
Materials provide opportunities for students to address different text types of writing that reflect the distribution required by the standards.
1/2
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Indicator Rating Details
The materials reviewed for Grade 11 partially meet the criteria for Indicator 1j.
Materials provide some opportunities for students to learn, practice, and apply different writing modes during on-demand and longer process writing tasks across the school year. Materials include on-demand creative, narrative, informative, and descriptive writing opportunities during the post-reading Extend the Text section. Because these tasks are optional and based on teacher choice, there is no guarantee students will complete the provided tasks. Other opportunities for writing occur when students read eSelections that are available in Passport, a digital component of the materials. With access to Passport, students have the ability to use Criterion, which is an online writing evaluation tool; however, it is unclear how to access it or use it. Without access to the digital platform, it is unclear how and where students compose their writing. Process writing instruction and tasks occur during the End-of-Unit Writing Workshops; however, explicit instruction is limited and materials do not meet the required distribution required by the standards.
Materials provide some opportunities for students to address different text types of writing that reflect the distribution required by the standards. Examples include, but are not limited to, the following:
• Materials provide some opportunities across the school year for students to learn, practice, and apply different genres/modes/types of writing that reflect the distribution required by the standards.
• Materials include the following Writing Workshops— one informative, four argumentative, three descriptive, one narrative—resulting in an uneven distribution of explicit instruction on the writing modes required by the standards.
• In Unit 5, Progress and Conflict, Early Twentieth Century 1910–1929, students learn about narrative writing during the end-of-unit Writing Workshop. Students learn how to “write a college application essay of five hundred words that tells how an experience you had led to your personal growth or some discovery about yourself.” During the Prewrite stage, students select a topic, gather information, organize their ideas using a graphic organizer, and write a thesis statement. Students use a three-part framework—introduction, body, and conclusion—to write their essay. Students use the provided Revision Checklist to evaluate their draft before publishing, presenting, and reflecting on their work. Materials include Draft Stage and Revise Stage models, as well as a Student Model. Teacher guidance includes, “Direct students’ attention to the model. Point out the side questions that focus attention on major parts of the essay: the introduction, the body, and the conclusion.” Although materials do not provide any other opportunities for students to learn and apply narrative writing, students do have opportunities to practice narrative writing during optional activities, such as on-demand Extend the Text writing tasks and End-of-Unit Test Practice Workshops.
• In Unit 6, Hard Times, Depression and World War II 1929–1945, students “[p]lan, create, and edit a multimedia presentation” on a “topic related to World War II or the Depression '' during the End-of-Unit Writing Workshop. During the Prewrite stage, students select a topic, gather information, write a controlling idea that conveys their overall message, and organize their ideas chronologically. During the Draft stage, students compose the introduction, body, and conclusion of the script they will use when delivering or recording their presentation. In the Revise stage, students evaluate their draft and make revisions to content, organization, and style using the provided Revision Checklist. In the final stage, Deliver or Record, students practice presenting their final product and either deliver the presentation to the class or record the presentation for future viewing. Materials include a Writing Follow-Up rubric that includes presentation and reflection criteria. Although materials do not provide further opportunities for students to learn and apply informative writing, students do have opportunities to practice informative writing during optional activities, such as on-demand Extend the Text writing tasks and End-of-Unit Test Practice Workshops.
• In Unit 9, New Challenges, Contemporary Era 1980–Present, students return to argumentative writing as they learn how to write a research paper during the end-of-year Writing Workshop. During the Prewrite stage, students select a topic, gather information using a working bibliography, write a thesis statement, and organize their ideas using a list and a formal outline. Students use a three-part framework—introduction, body, and conclusion—to write their research paper, and focus on avoiding plagiarism and their use of quoting, paraphrasing, and summarizing. Students follow the Modern Language Association (MLA) style to document and cite sources. Materials direct students to reference the style guide “for additional examples of types of sources as well as complete Works Cited lists.” Students use the provided Revision Checklist to evaluate their draft before publishing, presenting, and reflecting on their work. Materials include a Student Model. Teacher guidance includes, “Read through the model with students. Discuss the answers to the margin questions in the model to help students identify the thesis, see how the topic is developed, and recognize a strong conclusion.” The teacher also uses the Works Cited page of the student model “to review formatting for different kinds of sources.” Materials provide three more opportunities for students to learn, practice, and apply argumentative writing—when defending a viewpoint during the Unit 1 Writing Workshop, when solving a problem during the Unit 3 Writing Workshop, and when reviewing a film or play during the Unit 7 Writing Workshop.
• Different genres/modes/types of writing are distributed throughout the school year; however, there is no core instructional path. Writing opportunities may not occur during core instruction.
• Students have opportunities to engage in argumentative writing.
• In Unit 3, A Nation Divided, Slavery and the Civil War 1850–1865, students read two selections by Abraham Lincoln, “The Gettysburg Address” and “The Second Inaugural Address.” Afterwards, students [a]nalyze two contemporary political debates for logical fallacies, such as non sequiturs, circular logic, and hasty generalizations” and use their analysis “to write an essay that cautions the president against using logical fallacies.” Students must cite “examples from the debates [they] studied and adjust [their] responses when valid evidence warrants.” Because this is an optional Extend the Text task, students may not have the opportunity to practice writing in this mode.
• In Unit 6, Hard Times, Depression and World War II 1929–1945, after reading “A Noiseless Flash," an excerpt from Hiroshima by John Hersey, students write in response to the following prompt: “Write one page in which you argue which would be more effective: a novel about the bombing of Hiroshima, told by a fictional character in the story, or a nonfiction account about the survivors, written from a journalist’s perspective. Address the benefits and drawbacks of each type of work.” Because this is an optional Extend the Text task, students may not have the opportunity to practice writing in this mode.
• In Unit 8, Social Transition, Early Contemporary Era 1960–1980, students read a paired selection containing Martin Luther King Jr.’s “Letter from Birmingham Jail," and Foster Hailey’s “Dr. King Arrested at Birmingham.” During one of the Extend the Text options, students write in response to the following prompt: “Write a brief analysis of King’s arguments, identifying their strengths and weaknesses and explaining why you agree or disagree. Include and defend inferences and conclusions drawn from King’s ideas and the way he organized them.” Because this is an optional Extend the Text task, students may not have the opportunity to practice writing in this mode.
• Students have opportunities to engage in informative/explanatory writing.
• In Unit 1, Shaping the World, Origins of the American Tradition to 1800, students read “Song of the Sky Loom," a tribal song by Tewa. After reading, students may complete an Informative Writing task in which they write an essay comparing and contrasting the characters and text structure of the song and a contemporary work. Because this is an optional Extend the Text task, students may not have the opportunity to practice writing in this mode.
• In Unit 4, Expanding Frontiers, Unification and Growth 1865–1910, in the Extend the Text activities for the excerpt from Black Elk Speaks by Nicholas Black Elk and John G. Neilhardt, students respond to the following Informative Writing prompt: “Write an essay suggesting why the autobiography became so significant. Support your opinion with details from the selection.” Because this is an optional Extend the Text task, students may not have the opportunity to practice writing in this mode.
• In Unit 7, The American Dream, Postwar Era 1945–1960, students read the short story, “The Life you Save May Be Your Own” ,by Flannery O’Connor. After reading, students have the option of writing an informative essay: “Analyze the complexities of the character of Mr. Shiftlet, and organize the analysis in terms of the direct and indirect characterization techniques.” Because this is an optional Extend the Text task, students may not have the opportunity to practice writing in this mode.
• Students have opportunities to engage in narrative writing.
• In Unit 3, A Nation Divided, Slavery and the Civil War 1850–1865, students read the digital eSelections, “There’s a certain Slant of Light," “My life closed twice before its close," and “The Soul selects her won Society," all poems by Emily Dickinson. After reading the poems, students may complete a Narrative Writing task in which they rewrite one of the poems as a short story. Because this is an optional Extend the Text task, students may not have the opportunity to practice writing in this mode.
• In Unit 4, Expanding Frontiers, Unification and Growth 1865–1910, students read the speech, “I Will Fight No More Forever," by Chief Joseph. After reading, students may complete a Narrative Writing task during which they “[w]rite a paragraph explaining what happened to the Nez Perce after Chief Joseph’s surrender in October 1877.” Because this is an optional Extend the Text task, students may not have the opportunity to practice writing in this mode.
• In Unit 7, The American Dream, Postwar Era 1945–1960, students read an excerpt from the novel On The Road by Jack Kerouac. After reading, students may complete a Narrative Writing task: “[w]rite your own narrative about a recent trip you took. Model the narrative after On The Road in terms of writing style, tone, narration, and degree of detail.” Because this is an optional Extend the Text task, students may not have the opportunity to practice writing in this mode.
• Where appropriate, writing opportunities are connected to texts and/or text sets (either as prompts, models, anchors, or supports).
• In Unit 3, A Nation Divided, Slavery and the Civil War 1850–1865, students read “Beat! Beat! Drums,''By the Bivouac’s Fitful Flame,” and the preface to Leaves of Grass by Walt Whitman as part of an author study alongside the informational text, “Mathew Brady: Civil War Photographer.” After reading all of these texts, students may complete a Creative Writing task: “Civil War photographer Mathew Brady (see the preceding Informational Text Connection) photographed many famous Americans from the 1840s through the 1860s, including Walt Whitman. Write a dialogue that might have occurred between the two men. What would they say about the war, photography and poetry, and the American people?” Because this is an optional Extend the Text task, this writing opportunity may not occur during core instruction.
• In Unit 5, Progress and Conflict, Early Twentieth Century 1910–1929, students read the paired poems, “America," by Claude McKay, and “A Black Man talks of Reaping," by Arna Bontemps. After reading, students may complete a Creative Writing task: Imagine that you are Arna Bontemps and have just read CLaude McKay’s ‘America’. Write a letter to McKay, telling him what you think about the ideas expressed in ‘America’.” Because this is an optional Extend the Text task, this writing opportunity may not occur during core instruction.
• In Unit 8, Social Transition, Early Contemporary Era 1960–1980, students read the poem, “The Explorer,” by Gwendolyn Brooks. After reading, students may complete an Informative Writing task: “Visualize the setting of ‘The Explorer’, and assume the role of the man tripping down the halls of the building. Then write a paragraph analyzing the role of the setting, or environment, in the poem and what it might represent to the man in the poem.” Because this is an optional Extend the Text task, this writing opportunity may not occur during core instruction.
### Indicator 1k
Materials include frequent opportunities for evidence-based writing to support sophisticated analysis, argumentation, and synthesis.
1/2
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Indicator Rating Details
The materials reviewed for Grade 11 partially meet the criteria for Indicator 1k.
Materials provide practice and application opportunities for evidence-based writing but lack explicit evidence-based writing instruction with the exception of some Writing Workshop tasks. During some post-reading tasks, students cite evidence from the text in their written tasks, make claims, and defend their claims using their comprehension and analysis of texts. Extend the Text tasks are optional and based on teacher choice, so there is no guarantee students will engage in evidence-based writing opportunities when offered. Other opportunities sometimes include the Writing Workshops students complete at the end of each unit, additional writing assignments found in the Grammar and Writing ancillary, and the Analyze Literature prompts. It is important to note that many of the writing activities are optional and do not consistently require students to support their analyses and defend their claims using textual evidence.
Materials include some opportunities for evidence-based writing to support sophisticated analysis, argumentation, and synthesis. Examples include, but are not limited to, the following:
• Materials provide limited opportunities across the school year for students to learn, practice, and apply writing using evidence.
• In Unit 3, A Nation Divided, Slavery and the Civil War 1850–1865, after reading “The Gettysburg Adress” and “The Second Inaugral Address” by Abraham Lincoln, students may complete this Extend the Text Argumentative Writing task: “Analyze two contemporary political debates for logical fallacies, such as non sequiturs, circular logic, and hasty generalizations. Use your analysis to write an essay that cushions the president against using logical fallacies. Cite specific examples from the debates you studied and adjust your responses when valid evidence warrants.” However, there are no materials for the teacher to use to teach students about logical fallacies. This is the only writing opportunity in the unit that explicitly requires evidence, and the task does not include explicit instruction on standards-aligned, evidence-based writing. This activity is one of four options from which the teacher may choose and, as a result, may not occur during core instruction.
• In Unit 4, Expanding Frontiers, Unification and Growth 1865–1910, in the Extend the Text section for “The Negro Speaks of Rivers” and “I, Too, Sing America” by Langston Hughes, students may complete the following Informational Writing task: “Write a two-paragraph analysis of the significance of “The Negro Speaks of Rivers” and “I, Too, Sing America” to the Harlem Renaissance Era. How do these poems reflect key issues among African-American artists of the time? Use specific examples from the selections to support your analysis.” However, the materials do not provide background on the Harlem Renaissance Era. Materials include a one-page author’s study on Langston Hughes, as well as introductory content for the two poems, but neither of these pieces contains the information on The Harlem Renaissance needed to complete the assignment. There are no additional materials or support available in the Teacher’s Edition or ancillary materials. This task does not include explicit instruction on standards-aligned, evidence-based writing. This activity is one of four options from which the teacher may choose and, as a result, may not occur during core instruction.
• In Unit 9, New Challenges, Contemporary Era 1980–Present, after reading an excerpt from Great Plains by Ian Fazier, “Seeing” from Dakota: A Spiritual Geography by Kathleen Norris, and the LIterature Connection, “So This is Nebraska” by Ted Kooser, students may complete the following Informative Writing task: “[w]rite a comparison-and-contrast essay relating the characters and structure of the twenty-first-century poem “So This is Nebraska” with those of a classical poem or epic. Cite evidence from both texts to support inferences and conclusions that you make in your analysis.” However, this task does not include explicit instruction on how to write an analysis. This activity is one of four options from which the teacher may choose and, as a result, may not occur during core instruction.
• Writing opportunities are focused around students’ analyses and claims developed from reading closely and working with texts and sources to provide supporting evidence.
• In Unit 1, Shaping the World, Origins of the American Tradition to 1800, students read nonfiction excerpts from The General History of Virginia by John Smith and Of Plymouth Plantation by William Bradford. After reading, students may complete an Informative Writing task: “Both Smith and Bradford were successful leaders, but Smith left Virginia and Bradford stayed in Plymouth. For each man, write a paragraph that explains how his choice relates to his character and the motivation that drew him to the new world. Then write a paragraph making logical connections between the two situations. Support your ideas with examples from the text.” This activity is one of four options from which the teacher may choose and, as a result, may not occur during core instruction.
• In Unit 4, Expanding Frontiers, Unification and Growth 1865–1910, after reading the short story, “To Build a Fire” by Jack London and the informational texts, “How to Build a Campfire” and “How to Be Sure Your Campfire Is Out” by United States Department of Agriculture (USDA) Forest Service, students may complete this Informative Writing task: “Write a literary criticism comparing and contrasting the discussions of fire in London's short story ‘To Build a Fire’ and the two informational text articles on fire. Focus on how effectively the style, tone, diction, and text organization of each selection advance the author's purpose and perspective (theme). Include details to support your inferences and conclusions.” This activity is one of four options from which the teacher may choose and, as a result, may not occur during core instructio
• In Unit 9, New Challenges, Contemporary Era 1980–Present, students read “Throughput” from Fast Food Nation by Eric Schlosser. After reading, students may complete the following Argumentative Writing task: “As a member of a book club that has read Fast Food Nation, you need to prepare for discussion of how workers are treated in the fast food industry. Write a paragraph in which you agree or disagree with Schlosser’s assertions about throughput and its effects on the hiring and treatment of workers. Use evidence from the selection to support your opinion.” This activity is one of four options from which the teacher may choose and, as a result, may not occur during core instruction.
### Indicator 1l
Materials include explicit instruction of the grade-level grammar and usage standards, with opportunities for application in context.
1/2
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Indicator Rating Details
The materials reviewed for Grade 11 partially meet the criteria for Indicator 1l.
Each unit contains several Grammar & Style Workshops, which have sections on understanding the concept, applying the skill, and extending the skill. The lessons connect to selections students read just before the workshop. Units also contain Vocabulary & Spelling Workshops with sections on understanding the concept, applying the skill, and spelling practice using words from unit text selections. Workshops may not occur during core instruction, as their enactment is contingent upon the teacher selecting the activity from the Lesson Plan for the text selection. On occasion, materials include informal grammar and convention activities listed in the Teaching Notes of the Teacher’s Edition. Although materials include an array of instructional components, there are missed opportunities for grade-level grammar and usage instruction, practice, and authentic application in context.
Materials include some explicit instruction of the grade-level grammar and usage standards, with opportunities for authentic application in context. Examples include, but are not limited to, the following:
• Students have opportunities to apply the understanding that usage is a matter of convention, can change over time, and is sometimes contested.
• No evidence found
• Students have opportunities to resolve issues of complex or contested usage, consulting references (e.g., Merriam-Webster's Dictionary of English Usage, Garner's Modern American Usage) as needed.
• No evidence found
• Students have opportunities to observe hyphenation conventions.
• In Unit 7, The American Dream, Postwar Era 1945–1960, students complete a Grammar & Style Workshop on hyphens, ellipses, and italics. In the Understand the Concept section, students learn the definition of hyphens and look at examples of hyphenated words. In the Apply the Skill section, students identify the need for hyphens, ellipses, and italics during a sentence-level exercise, such as: “Salzman lies and tells Leo that Lily is only twenty nine years old, but she is really in her mid thirties.” The statements in the exercise are connected to the text selection that students just read, the short story “The Magic Barrel” by Bernard Malamud. Students also complete an application exercise where they write a short fictionalized account of an event in their lives using ellipses, italics, and at least two compound words or expressions requiring hyphens.
• Students have opportunities to spell correctly.
• In Unit 3, A Nation Divided, Slavery and the Civil War 1850–1865, students complete a Vocabulary & Spelling Workshop on suffixes. The teacher begins the lesson by having students recall their knowledge of suffixes and root words and then explains how certain suffixes are added to indicate people who perform certain jobs or are experts in certain areas. The class studies how making these additions often requires spelling changes. In the Apply the Skill section, students use their knowledge of root words to identify the type of work done by a list of people, use a suffix to create a word that describes the person who does the type of job in a list, and use correct suffixes to write a paragraph about jobs they might like to have after school. Students also have the opportunity to practice their spelling of consonant blends and digraphs using words from the preface to Leaves of Grass by Walt Whitman, the selection before the exercise. Materials do not include opportunities for authentic application in context.
• In Unit 9, New Challenges, Contemporary Era 1980–Present, students read the poem, “So This Is Nebraska,” by Ted Kosser. Following the selection, materials include a Vocabulary & Spelling Workshop on Common Spelling Errors. In the Understand the concept section, students learn spelling rules for adding affixes. In the Apply the Skill section, students identify troublesome spelling words and create mnemonic devices to help them remember the correct spelling. Materials do not include opportunities for authentic application in context.
### Indicator 1m
Materials include a cohesive, year-long plan for students to interact and build key academic vocabulary words in and across texts.
2/4
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Indicator Rating Details
The materials reviewed for Grade 11 partially meet the criteria for Indicator 1m.
At the beginning of each unit, materials include an overview of all vocabulary words, academic vocabulary, and key terms. These words are also listed in the Teacher Wrap of the Teacher Edition alongside the corresponding selection. Words listed as Preview Vocabulary are taken from sentences within selections and are defined in the side margin or at the bottom of pages where they appear. Words listed as Selection Words are additional words from the reading that may be challenging, but are not central to the selection. These are Tier One words that can easily be understood by using context clues. Words listed as Academic Vocabulary are words that are used in the directions about the lessons. These are Tier Two words that explain what students should focus on, help establish context, clarify meaning of literary terms, and define goals or instructional purpose. Words that are listed as Key Terms are domain-specific Tier Three words. The repetition of these words throughout the program helps to ensure student mastery.
Materials include two Vocabulary & Spelling Workshops within each unit. These Workshops correlate to two of the unit selections that use vocabulary words from the text that precedes the Workshop and contain instruction followed by practice exercises. The enactment of this Workshop is based on teacher selection and, as a result, may not occur during core instruction. The Unit & Selection Resources ancillary also includes vocabulary preview activities and lessons for each unit. The Vocabulary & Spelling ancillary also has lessons that build word study skills and instruction based on vocabulary words from selections. Although materials include multiple elements that address vocabulary acquisition and practice, these elements are not cohesive nor do materials provide teacher guidance on a year-long plan to support students’ vocabulary development. Additionally, ancillary resources are not a part of core instruction.
Materials include opportunities for students to interact with key academic vocabulary words in and across texts; however, the year-long vocabulary plan lacks cohesion. Examples include, but are not limited to, the following:
• Materials do not provide teacher guidance outlining a cohesive year-long vocabulary development component.
• There is no explanation of a year-long cohesive plan for vocabulary instruction; rather, materials include multiple components that address vocabulary, and it is up to the teacher to decide which components to use for instruction. For instance, at the beginning of each unit, materials provide Tier One, Tier Two, and Tier Three vocabulary word lists with the corresponding pages for where the words occur in text. Materials also list the vocabulary words in the Teacher Wrap of the Teacher Edition with the corresponding page number in the section where they occur. Materials define the vocabulary words at the bottom of the selection in which they appear. Each selection includes a short Preview Vocabulary section where students try to unlock the meaning of underlined words from the selection before reading. Occasionally, the Teacher Wrap of the Teacher Edition includes instructions for helping students understand the meaning of words. Materials include two Vocabulary and Spelling Workshops which focus on vocabulary skills instruction. If teachers want to explore selection vocabulary in more depth, they must use the Unit & Selection Resources ancillary. Since it is up to teachers to choose which of these program elements to include in instruction, there is no guarantee that the vocabulary development supports offered will occur during core instruction.
• Vocabulary is repeated in contexts (before texts, in texts) and across multiple texts; however, it is unclear how materials build students’ vocabulary development of Tier One and Tier Two words during core instruction.
• In Unit 1, Shaping the World, Origins of the American Tradition to 1800, the Tier Three term tone appears in the before-reading material for the paired excerpts from Common Sense and from The Crisis, No. 1 by Thomas Paine. Materials define the word tone in the Analyze Literature section of the text overview page. The word tone is used throughout the end-of-unit Speaking & Listening Workshop, as students “determine the mood and tone of the work” and “[c]onsider which tone of voice, facial, expression, gestures, pace (speed) of speaking, and volume (loudness or softness) are most suitable for each part of the work.” The term tone repeats during the Test Practice Workshop: “Context clues also can come from the tone of the section.
• In Unit 6, Hard Times, Depression and World War II 1929–1945, the Tier One Selection Word impervious occurs in “The Watch” by Elie Wiesel and in “A Rose for Emily” by William Faulkner. Materials do not identify or define the term in either text.
• In Unit 8, Social Transition, Early Contemporary Era 1960–1980, materials define the Key Term characterization in the Analyze Literature section of the text overview page of “The Rockpile,” a short story by James Baldwin. The term is used again as materials explain memoirs in the Understanding Literary Forms pages: “The memoirist also uses characterization techniques to portray other people in the account—for instance, describing their appearance or behavior or revealing what others say or think about them.” During the Analyze Literary Elements section of the Reading Skills portion of the Test Practice Workshop, materials define characterization when listing literary elements “commonly found in various types of literature.”
• Attention is paid to vocabulary essential to understanding the text and to high-value academic words (e.g., words that might appear in other contexts/content areas).
• At the beginning of each unit, materials include lists of the Tier Two and Tier Three vocabulary words students will encounter over the course of each unit in the Teacher Edition. Each word is followed by the page numbers where the words appear. At the beginning of each selection, materials list Tier One and Tier Two words under the heading Words in Use followed by page numbers for each vocabulary word. Tier Two and Tier Three words often appear in the before reading information and in Vocabulary & Spelling Workshops. Materials repeat certain Key Terms (Tier Three words) throughout the unit to give students more exposure to and practice with vocabulary words.
• Students are supported to accelerate vocabulary learning with vocabulary in their reading, speaking, and writing tasks.
• In Unit 3, A Nation Divided, Slavery and the Civil War 1850-1865, students read “The Gettysburg Address” and the “Second Inaugural Address” by Abraham Lincoln. The Analyze Literature section of the text overview page explains the Tier Three word parallelism. During the reading of both texts, students note instances of parallelism. The subsequent Grammar and Style Workshop focuses on parallelism. Materials provide directions and opportunities for student practice, including a creative writing activity: “Write a letter to the president in which you relate your feelings and observations about national events. Use five examples of parallelism in your letter.”
• In Unit 4, Expanding Frontiers, Unification and Growth 1865–1910, students read an excerpt from the memoir Life on the Mississippi by Mark Twain. Materials introduce the Tier Three term memoir in the Analyze Literature section of the text overview page to ensure students understand the genre. During reading, students identify words and phrases that signify the work is a memoir on two separate occasions. After reading the text, students perform an analysis of the term memoir. Students contrast a memoir and autobiography, decide if it is acceptable for an author to embellish their memoir, and compare the writing in Twain’s memoir to that of his fiction and essays.
• In Unit 8, Social Transition, Early Contemporary Era 1960–1980, students read “Morning Song” and “Mirror,” lyric poetry by Sylvia Plath. The text overview page introduces and defines the Tier Three term enjambment. While reading, students “examine Plath’s use of enjambed versus end-stopped lines.” After reading the poems, students identify the examples of enjambment they found in both poems and comment on how the use of enjambment versus end-stopped lines affects the reading and understanding of the poems.
## Building Knowledge with Texts, Vocabulary, and Tasks
#### Partially Meets Expectations
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Gateway Two Details
### Criterion 2a - 2f
Materials build knowledge through integrated reading, writing, speaking, listening, and language.
12/24
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Criterion Rating Details
### Indicator 2a
Texts are organized around a cohesive topic(s)/theme(s) to build students’ ability to read and comprehend complex texts independently and proficiently.
2/4
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Indicator Rating Details
The materials reviewed for Grade 11 partially meet the criteria for Indicator 2a.
The materials are organized into nine thematic units of study which are aligned with a historical period in American history and progress chronologically. Each unit begins with a unit opener that “introduces the genre and connects students to the literature,” includes a “thought-provoking quote [that] gives insight into literature,” features “fine art and photographs [that] connect with the unit theme,” and introduces “essential questions related to the unit theme [that] generate interest and set the stage for learning.” Although the focus of each unit is a historical time period, as well as a theme related to that time period, each unit also includes a section titled Understanding Literary Forms that introduces a genre for quick study. The opening pages of this section include an illustrated timeline, an introduction to the historical period, and notable statistics from the period. Subsequent lessons are divided into sections, during which students explore various selections in the literary form and literary criticism as it is applied to a previously read selection; however, these activities are not connected to the essential question or guiding statement for the unit. The Scope and Sequence Guide lists sub-themes that connect to many of the selections. The Mirrors & Windows questions that accompany selections address these sub-themes, but they do not connect to the overall theme of the unit, and there is no explanation or guidance on how the unit theme and the Mirrors & Windows sub-theme work together. The individual components included in the program are not connected in a cohesive way that would build students’ knowledge of a topic or theme.
Examples include, but are not limited to, the following:
• Texts are connected by a grade-appropriate cohesive topic/theme/line of inquiry. Texts miss opportunities to build knowledge, vocabulary, and the ability to read and comprehend complex texts across a school year.
• In Unit 9, New Challenges, Contemporary Era 1980–Present, the unit overview notes, “Though the struggle for equality still continues for many Americans, authors are utilizing their talents to honor ancestors and celebrate cultural diversity.” The essential question is “How does the diversity that characterizes Americans themselves also characterize contemporary American culture?” Unit selections, embedded Close Reading questions, and Extend the Text tasks do not connect to the unit theme or essential question. For example, early in the unit, students read “Mother Tongue” by Amy Tan. The Mirrors & Windows question for the selection is as follows: “What do people assume about you? Are they right or wrong?” The Analyze Literature questions and Extend the Text activities do not relate the selection's main idea about the impact on language when growing up in a home of non-native English speakers to the overall unit’s focus on diversity. In the middle of the unit, students read “Throughput” from Fast Food Nation by Eric Schlosser, which focuses on “the speed and volume of a factory’s production” to the fast-food industry. The topic of the selection does not fit with the unit topic of diversity or to the essential question. The unit ends with an author’s focus on the poet Donald Hall. The Reader’s Context questions for the poems “Couplet: Old-Timers’ Day, Fenway Park, 1 May 1982,” and “Letter in Autumn” are “ How do you handle loss? What helps you find comfort in difficult times?” These questions do not connect to the unit theme or essential question.
### Indicator 2b
Materials require students to analyze the key ideas, details, craft, and structure within individual texts as well as across multiple texts using coherently sequenced, high-quality questions and tasks.
2/4
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Indicator Rating Details
The materials reviewed for Grade 11 partially meet the criteria for Indicator 2b.
As part of the Close Reading Model, materials embed text-specific and text-dependent questions that require students to analyze the key ideas, details, craft, and structure of individual texts and paired selections or text sets. Materials do not consistently include coherently sequenced questions that build to a task in which students demonstrate their understanding of these literary elements. Tasks often occur during the Extend the Text section and may not occur during core instruction, as these tasks are options from which the teacher may select. At times, questions and tasks do not meet the requirements of the correlated standard.
Materials require students to analyze the key ideas, details, craft, and structure within individual texts as well as across multiple texts using coherently sequenced, high quality questions and tasks. Examples include, but are not limited to, the following:
• For most texts, students analyze key ideas and details and craft and structure (according to grade-level standards).
• The materials contain some coherently sequenced questions and tasks that address key ideas and details.
• In Unit 2, Expressing a National Spirit, American Resistance 1800–1850, students read “The Fall of the House of Usher,” a short story by Edgar Allen Poe. While reading, students focus on details. Early in the story, students “pay close attention to descriptive details about the mansion” and “consider how these details reflect the state of the Usher family.” Next, students “collect details that contribute to” the “very dark, haunted atmosphere” described in the text. After reading the text, students respond to an Analyze Literature prompt that addresses Gothic fiction: “How does Poe set the scene for his tale of Gothic fiction? How do the descriptions of the house and grounds, as well as the characters, contribute to the plot?” Students work in small groups to adapt the text into a weekly TV show during the Collaborative Learning Extend the Text option. Students must “agree upon a set of criteria for a successful TV show, “ including “[w]hat elements (such as sets, actors, and plots) are necessary to make it interesting.” Then, students “work together to write a proposal for the drama,” “evaluate [their] proposal against the agreed-upon criteria, and make revisions as needed. This Extend the Text activity is one of four from which the teacher may choose and, as a result,may not occur during core instruction.
• The materials contain coherently sequenced questions and tasks that address craft and structure.
• In Unit 4, Unification and Growth, Expanding Frontiers 1865–1910, the Set Purpose section of the Preview the Selection page for “To Build a Fire” by Jack London directs students to “consider how London uses this setting to set up the central conflict of the plot. Identify the primary struggle in this story and make predictions about how it will be resolved.” Teacher guidance explains that “the setting functions as a character and establishes the conflict of the plot.” Students “describe the setting as if it were a character” and “discuss what conflict the setting creates.” During the reading, guidance in the Teacher Wrap points out a point of high intensity in the plot development Students “identify where the present point in the narrative might be located on a plot diagram.” After reading “How to Build a Campfire” and “How to Be Sure Your Campfire Is Out,” two Informational Text Connection pieces by the United States Department of Agriculture (USDA) Forest Service, students respond to Analyze Literature: Setting, Plot, and Conflict questions that include, but are not limited to, the following: “In examining the plot, what is the external conflict, or outside force, against which the main character struggles? How does he fare against that force? How does the dog, a creature of nature, fare? Does the main character face any internal conflicts?” During the Informative Writing Extend the Text option, students write a literary criticism that compares and contrasts the short story and the two informational articles on fire. The literary analysis must include details to support inferences and conclusions and “[f]ocus on how effectively the style, tone, diction, and text organization of each selection advance the author’s purpose and perspective (theme).” This Extend the Text activity is one of four from which the teacher may choose and, as a result, may not occur during core instruction.
• In Unit 7, The American Dream, Postwar Era 1945-1960, students read Act 3 of The Crucible by Arthur Miller. During the reading, students respond to a number of Analyze Literature prompts and questions addressing the three types of irony—dramatic, verbal, and situational—and mood. Students distinguish what is stated from what is meant as they analyze irony and mood in the text. Examples of prompts and questions include, but are not limited to: “Ask students to identify the irony in Danforth's boast that his signature put some four hundred people in jail and condemned seventy-two to be hanged.”; “Ask students to identify the mood in Danforth’s speech beginning ‘Indeed not.’ How would students rephrase the statement ‘We burn a hot fire here; it melts down all concealment’ in ordinary speech?”; “Ask students to identify the type of irony in Danforth’s response to Hale’s dialogue ‘Your Honor, I cannot think you may judge the man on such evidence.’”; “Have students identify the loaded word that is repeated in the dialogue between Hale and Danforth. What mood does the exchange create? Why?”; “What makes Proctor’s angry response to Parris’s words ‘poppets hid where no one ever saw them’ an example of verbal irony?”; and “Ask students what evidence enables them to identify the type of irony in the stage direction ‘They all watch, as Abigail, out of her infinite charity, reaches out and draws the sobbing Mary to her, and then looks up to Danforth.’ What two motives does Abigail have in behaving as she does?” Students [w]rite an essay that analyzes themes and characteristics (such as structure and mood)” in The Crucible and “a modern American drama from a different period.” Students must “[c]ite examples to support [their] inferences and conclusions about similarities and differences between the plays.” This Informative Writing Extend the Text option is one of four after-reading activities from which the teacher may select and, as a result, may not occur during core instruction..
• By the end of the year, these components (language, word choice, key ideas, details, structure, craft) are embedded in students’ work rather than taught directly.
• In Unit 3, A Nation Divided, Slavery and the Civil War 1850–1865, students focus on psychological fiction and flashback when reading “An Occurrence at Owl Creek Bridge,” a short story by Ambrose Bierce. Guidance directs students to “be on the lookout for the hero’s extended interior experience.” When reading Section II, students respond to the following questions that address flashback: “What is the setting of the narrative? Has the story moved forward or backward in time? How can [you] tell? What do [you] learn about Farquhar’s feelings toward the war? How might these emotions affect his actions?” When examining a specified passage of the text, students discuss “which detail suggests that the events are a fantasy.” Students also discuss the role of fantasy in the story using questions from the Critical Thinking: Discussion Guide. During two different passages of the story, students characterize the atmosphere of the setting” and discuss what the last paragraph reveals about Farquhar. After reading, students respond to the following Analyze Literature: Psychological Fiction and Flashback questions: “Analyze the impact of narration when the point of view shifts from the narrator’s own thoughts to Farquhar’s thoughts within the story. How does this shift exemplify psychological fiction? What kinds of information do you learn from the narrator’s thoughts? from Farquhar’s? In what ways does the flashback advance the plot? What necessary information does the flashback provide? How would the story be different if told in regular chronological order?” During the Media Literacy option in the Extend the Text section, students watch the 1962 film adaptation of the story, directed by Robert Enrico and “[a]nalyze how the different elements in the film (music, camera, angles, dialogue) interact.” Students consider how the different elements impact their understanding of the short story and write a review of the film. This Extend the Text activity is one of four from which the teacher may choose and may not occur during core instruction, as a result.
### Indicator 2c
Materials require students to analyze the integration of knowledge within individual texts as well as across multiple texts using coherently sequenced, high-quality text-specific and/or text-dependent questions and tasks.
2/4
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Indicator Rating Details
The materials reviewed for Grade 11 partially meet the criteria for Indicator 2c.
As part of the Close Reading Model, materials embed text-specific and text-dependent questions that require students to analyze the key ideas, details, craft, and structure of individual texts and paired selections or text sets. While materials include coherently sequenced questions that build to a task in which students demonstrate their understanding of knowledge and ideas, tasks often occur during the Extend the Text section. Extend the Text tasks may not occur during core instruction, as these tasks are options from which the teacher may select.
Materials require students to analyze the integration of knowledge and ideas within individual texts as well as across multiple texts using coherently sequenced, high-quality text-specific and/or text-dependent questions and tasks. Examples include, but are not limited to, the following:
• Most sets of questions and tasks support students’ analysis of knowledge and ideas.
• In Unit 1, Shaping the World, Origins of the American Tradition to 1800, students read the Declaration of Independence by Thomas Jefferson paired with the Bill of Rights. Students examine text organization during both texts and respond to Critical Thinking Discussion Guide prompts and questions to analyze and evaluate the government documents: “Encourage students to discuss what the nation’s founders meant when they said all men have the ‘unalienable rights’ of ‘life, liberty, and the pursuit of happiness’ while knowing that large segments of the population were denied those rights.” and “Ask students why the freedoms provided by the First Amendment are essential in a democracy.” After reading both texts, students respond to the following Text to Text Connection question: “Does the Bill of Rights guarantee the unalienable rights described in the Declaration of Independence: life, liberty, and the pursuit of happiness? Try to name a specific amendment from the Bill of Rights that supports each of these three unalienable rights, or explain which of these unalienable rights is not addressed in the Bill of Rights.” During the Informative Writing Extend the Text option, students “[w]rite an analysis of the problem/solution organization of the Declaration of Independence. Include examples from the text, and explain what you can infer and conclude from Jefferson’s use of this type of organization.” This activity is one of four Extend the Text activities from which the teacher may choose and, as a result, may not occur during core instruction.
• In Unit 4, Expanding Frontiers, Unification and Growth 1865–1910, students read an excerpt from Black Elk Speaks by Nicholas Black Elk and John G. Neihardt. While reading, students use a chart to “track the main information shared by each speaker” as they analyze the sequence of events in the narrative and explain how specific individuals and ideas interact and develop throughout the text. Students respond to questions such as, “Why did the authors break up the narrative into different perspectives this way? How does this structure relate to the way the narrative was told?”; “What seems to be the relationship between adults and children in Oglala society? What details lead you to these conclusions?”; and “What point is Black Elk making when he says, ‘You can see that it is not the grass and the water that have forgotten’? Whom is he criticizing?” After reading, students respond to the following Analyze Literature questions: “How do the recollections and views Black Elk presents in his narrative differ from those of Fire Thunder and Standing Bear? Does having three people describe the same events give you a better understanding of them? Why or why not?” During the Lifelong Learning Extend the Text option, students choose an important historical family event and create a story about the event to share with the class. To craft their story, students make notes to present the story events in chronological order and include necessary background information to ensure the audience understands their story. This activity is one of four options from which the teacher may choose and, as a result, may not occur during core instruction.
• By the end of the year, integrating knowledge and ideas is embedded in students’ work (via tasks and/or culminating tasks).
• In Unit 7, The American Dream, Postwar Era 1945–1960, students read the lyric poem, “Midway” by Naomi Long Madgett along with the Informational Text Connection piece, the U.S. Supreme Court case, Brown v. Board of Education of Topeka. Students analyze tone in the poem and summarize each paragraph of the court case. After reading the informational text, students respond to Review Questions, such as “1. What is the Court’s position on segregation in public schools? Analyze the evidence the Court provides to support its position. What else, if anything, could the Court’s opinion have stated? 2. According to Chief Justice Warren, why is providing public education important in a democratic society? Evaluate whether Warren’s argument is still valid today.” During the Lifelong Learning option in the Extend the Text section, students research “media coverage of an event in the Civil Rights Movement that occurred in the months and years following the 1954 Brown v. Board of Education ruling. Analyze a few different types of media and evaluate the objectivity of each source. Summarize your findings in a short essay.” This activity is one of four options from which the teacher may choose and, as a result, may not occur during core instruction.
• Sets of questions and tasks provide opportunities to analyze across multiple texts as well as within single texts.
• In Unit 4, Expanding Frontiers, Unification and Growth 1865–1910, students read the speeches, “I Will Fight No More Forever” by Chief Joseph of the Nez Perce and “I am the Last of My Family” by Cochise of the Chiricahua Apache. During both texts, students analyze purpose, discussing prompts, such as “Ask students to identify the purpose of redundancy in the speech. Then discuss why Chief Joseph might have phrased his final intention as he did, rather than saying ‘I will not fight any more.’” and “Ask students to compare Cochise’s comments about fighting with whites with Chief Joseph’s remarks about fighting. Discuss the purpose of each speech and how each leader’s talk of fighting affects his purpose.” After reading both selections, students respond to Analyze Literature: Oral Tradition and Purpose questions which include: “What purpose or purposes did Chief Joseph have in speaking? What purpose or purposes did Cochise have? Compare and contrast the speakers’ purpose and how well they achieved them.”
### Indicator 2d
Culminating tasks require students to demonstrate their knowledge of a unit's topic(s)/theme(s) through integrated literacy skills (e.g., a combination of reading, writing, speaking, listening).
2/4
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Indicator Rating Details
The materials reviewed for Grade 11 partially meet the criteria for Indicator 2d.
Individual, paired, and text set selections conclude with Refer to Text and Reason with Text questions; an Analyze Literature, Compare Literature, or Text-to-Text Connection prompt; and four task options in the Extend the Text section. Earlier questions are incoherently sequenced at times and do not always build to a task. Teachers can choose from two writing options and two other types of tasks, such as Collaborative Learning, Critical Literacy, Lifelong Learning, and Media Literacy, in the Extend the Text section. Extend the Text tasks do not consistently relate to reading selections and are sometimes stand-alone in nature. Because there is no true core instructional path, completion of these tasks is optional and contingent upon teacher selection. As a result, there is no guarantee that all students will access the opportunities offered.
Each unit concludes with three Workshops: Writing, Speaking and Listening, and Test Practice. Most of the Writing and Speaking & Listening Workshops are not connected to the literary form of study and do not require students to draw upon their knowledge of the texts in the unit. The Test Practice Workshops are not connected to unit content and are designed to help students practice taking standardized tests. The three Workshops are not integrated.
Culminating tasks require students to demonstrate their knowledge through integrated literacy skills; however, it is unclear how tasks relate to the unit’s topic/theme. Examples include, but are not limited to, the following:
• Culminating tasks are evident and varied across the year and they are multifaceted, requiring students to demonstrate mastery of several different standards (reading, writing, speaking, listening) at the appropriate grade level, and comprehension and knowledge of a topic or topics through integrated skills (reading, writing, speaking, listening).
• In Unit 1, Shaping the World, Origins of the American Tradition to 1800, during the Speaking & Listening Workshop, students present a literary work. Students “present a poem or a work of prose, such as a short story, a brief chapter from a novel, or an essay.” Students “read the work carefully several times, until you can identify the following elements: the speaker in a lyric poem; the speaker, characters, and action in a narrative poem; the characters and action in a short story or chapter; or the main idea in an essay.” Afterwards, students practice reading the work aloud and memorize the work, if required. Students present the oral interpretation and evaluate the task using a Speaking & Listening Rubric. This task integrates reading, and speaking and listening.
• In Unit 4, Expanding Frontiers, Unification and Growth 1865–1910, during the Writing Workshop, students “create a profile using a combination of narration, biography, and oral history.” Students select a person to serve as the subject of their profile and develop a list of questions to ask the subject during an interview. Students take notes during the interview and use their notes to “produce an accurate, absorbing account,” using either a narrative structure or a chronological structure to organize the information. After writing their organizing statement to focus their work, students draft the introduction, body, and conclusion of their profile and evaluate their work using a Revision Checklist. Students may self-evaluate or exchange their work with a peer for feedback. Students “read their profiles aloud to the class, accompanied by either a photo of the subject or an object identified within the profile.” Students evaluate their work using a Writing Rubric. This task integrates writing, and speaking and listening.
• In Unit 7, The American Dream, Postwar Era 1945–1960, during the Writing Workshop, students review a film or play. Students may “review a film or play with which you are very familiar or a work entirely new to you.” Students may also “focus on a specific aspect of a film or play, such as the performance of a particular actor or the authenticity of the production.” While watching the film or play, students use a note-taking chart “to record supporting details for your analysis” and organize their ideas. Students also use their Film/Play Review Chart to develop their thesis statement and draft the introduction, body, and conclusion of their writing. Students exchange their work with a peer and evaluate their drafts using a Revision Checklist. Then, students read their final reviews to the class and evaluate their work using a Writing Rubric. This task integrates reading and writing.
• Earlier text-specific and/or text-dependent questions and tasks are not coherently sequenced and will not give the teacher usable information about the student's readiness (or whether they are “on track”) to complete culminating tasks.
• In Unit 7, The American Dream, Postwar Era 1945-1960, while reading Act 2 of The Crucible by Arthur Miller, students analyze characterization and allusion, responding to prompts and questions, such as “What action by Elizabeth, described in the stage directions, suggests this mood? What does the dialogue after the stage directions indicate is the reason Elizabeth is upset with Mary?” and “Ask students what the reference and the description of the court scene suggest about Elizabeth’s feelings toward Abigail and the people of Salem.” During the Critical Literacy option in the Extend the Text section, students find a modern American drama from another period” and “compare and contrast the themes and characteristics of this drama with those in The Crucible.” Later in the unit, students read “A Supermarket in California,” a lyric poem by Allen Ginsberg. Students focus on features of free verse and allusion and respond to questions and prompts, such as “Ask students to identify features of free verse in Ginsberg’s poem, including sentence fragments and the paragraph-like structure of the lines.” and “Ask students to identify the primary allusion in the poem. Why does the speaker speak of Whitman’s enumerations?” During the Informative Writing Extend the Task option, students [w]rite a paragraph explaining why Whitman has influenced so many poets.” These questions and tasks are not coherently sequenced to build to the culminating task. It is unclear how these tasks give the teacher usable information about the student's readiness to complete the End-of-Unit Writing Workshop in which they write a review of a film or play.
### Indicator 2e
Materials include a cohesive, year-long plan for students to achieve grade-level writing proficiency by the end of the school year.
2/4
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Indicator Rating Details
The materials reviewed for Grade 11 partially meet the criteria for Indicator 2e.
The writing program design includes two on-demand, post-reading writing prompts selections. Prompts span creative, argumentative, informative, narrative, and descriptive writing modes. While some prompts are stand-alone tasks, others connect to texts students read and sometimes require students to use textual evidence in their responses. Each unit also includes an End-of-Unit Writing Workshop. During the Writing Workshop, materials explain what students should do during each step of the writing process but rarely provide instruction on the writing mode of focus. Writing Workshops include various supports and tools for monitoring writing development, such as rubrics, student models, literary models, graphic organizers, and checklists. Unlike their on-demand counterparts, these process writing tasks do not connect to the unit theme and are stand-alone in nature with some tasks requiring students to use evidence from sources. Materials include practice opportunities in the Writing Skills section embedded within the End-of-Unit Test Practice Workshop. During this Workshop, students practice timed writing responses and revision and editing skills. As with the Writing Workshops, Test Practice Workshop activities span various genres but are not connected to the unit text selections. The Writing & Grammar workbook may be used to supplant Writing Workshops, as the ancillary resource includes an additional in-depth writing workshop for each unit. Writing & Grammar activities begin with a Learn From a Literary Model section. This section draws upon one of the unit text selections. The Writing Rubrics ancillary contains four PDF files: a narrative writing rubric, an informative writing rubric, an argumentative writing rubric, and a four-point general writing rubric. Materials lack teacher guidance on enacting ancillary and optional writing lessons and tasks.
Materials include a year-long plan for students to achieve grade-level writing proficiency by the end of the school year; however, cohesion is lacking. Examples include, but are not limited to, the following:
• Materials include limited writing instruction that aligns to the standards for the grade level and sometimes supports students’ growth in writing skills over the course of the school year.
• While there is an evident structure to the writing aspect of the program, including frequent opportunities for students to write in various modes and for various purposes, supports, and tools for monitoring student writing development, the structure lacks cohesion. Materials include the following Writing Workshops— one informative, four argumentative, three descriptive, one narrative—resulting in an uneven distribution of explicit instruction on the writing modes required by the standards. Test Practice Workshops do not include explicit instruction and their mode of focus differs from that of the Writing Workshops. It is unclear how writing instruction and tasks build upon each other to promote growth in students’ skills over the course of the unit and across the year.
• While materials offer a number of writing opportunities, explicit writing instruction is largely absent. During the End-of-Unit Writing Workshops, students spend three regular schedule days or one and a half block schedule days transitioning through the writing process as they complete a process writing task on a specific mode of focus. Writing Workshop tasks include:
• Unit 1—Argumentative Writing: Defend a Viewpoint
• Unit 2—Descriptive Writing: Describe a Setting
• Unit 3—Argumentative Writing: Solve a Problem
• Unit 4—Descriptive Writing: Create a Profile
• Unit 5—Narrative Writing: Write an Application Essay
• Unit 6—Informative Writing: Create a Multimedia Presentation
• Unit 7—Argumentative Writing: Review a Film or Play
• Unit 8—Descriptive Writing: Write a Descriptive Poem
• Unit 9—Argumentative Writing: Write a Research Paper
• Instructional materials include a variety of well-designed guidance, protocols, models, and support for teachers to implement and monitor students’ writing development.
• In Unit 1, Shaping the World, Origins of the American Tradition to 1800, in the Writing Workshop, students plan and write an argumentative essay defending a viewpoint that expresses an informed opinion about a topic that interests them. The Workshop includes a Writing Rubric, an Argument Chart for prewriting, a side-by-side example of the Draft and Revise stages, a Revision Checklist, a Writing Follow-Up checklist, and a Student Model.
• In Unit 3, A Nation Divided, Slavery and the Civil War 1850–1865, students write an essay that explains a problem and suggest one or more solutions. During the Draft stage, the Teacher Wrap in the Teacher Edition includes a Teaching Note on addressing counterarguments: “Point out how, in the body of her essay, the student identifies an opposing view and then refutes it. This shows that she has carefully considered her topic and looked at opposing points of view before coming to her own conclusions. Ask students to do the same in their problem-solution essays. They might do so by offering alternative solutions to the problem they posed, and then explaining why their solution is the best one.”
• In Unit 5, Progress and Conflict, Early Twentieth Century 1910–1929, students write an application essay during the Writing Workshop. During the Draft stage, the Teacher Wrap in the Teacher Edition includes a Teaching Note on maintaining an appropriate tone: “Point out to students that their application essays should reflect their unique perspective and character. Also caution them not to use the essay to oversell themselves. The tone of the essay should be thoughtful, serious, and respectful of readers.”
• In Unit 7, The American Dream, Postwar Era 1945–1960, as part of the Test Practice Workshop, students practice addressing alternate viewpoints when writing a reflective essay. The Teacher Wrap in the Teacher Edition includes a Reflective Essay Rubric which contains the following criteria: Content, Organization and Development, and Grammar and Style.
• In Unit 9, New Challenges, Contemporary Era 1980–Present, students write a research paper during the Writing Workshop. The Teacher Wrap in the Teacher Edition includes a Teaching Note on refocusing or refining a Topic: “As students gather information, they may decide that they can refocus or refine their topic. If a topic is too broad, they will not be able to make their point effectively. If a topic is too narrow, they may not have enough to say. Encourage students to choose a focus for their research but to be flexible and make adjustments if research in a particular area interests them or if new information leads them to a different focus.”
### Indicator 2f
Materials include a progression of focused research projects to encourage students to develop knowledge in a given area by confronting and analyzing different aspects of a topic using multiple texts and source materials.
2/4
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Indicator Rating Details
The materials reviewed for Grade 11 partially meet the criteria for Indicator 2f.
While materials provide frequent opportunities for short research tasks connected to the texts students read, materials do not include a progression of research skills according to grade-level standards. Short research tasks do not include standards-aligned, explicit instruction and typically occur during one of the post-reading Extend the Text options. These tasks are optional and may not occur during core instruction. Students have one opportunity in each grade level to conduct a long research project—during the Unit 6 Writing Workshop. During this end-of-grade level task, materials include directions to guide students through each step of the research writing process but provide limited explicit instruction of standards-aligned research skills.
While materials provide opportunities to expand the Extend the Text research tasks, teachers must access the Extension Activities ancillary to do so. Materials also include a Language Arts Handbook ancillary with a section on Research and Documentation, but there is no guidance on how to use this handbook for instruction or how it ties to the specific tasks students complete. Ancillary resources are not a part of core instruction.
Materials do not include a progression of focused research projects to encourage students to develop knowledge in a given area by confronting and analyzing different aspects of a topic using multiple texts and source materials. Examples include, but are not limited to, the following:
• Research projects are not sequenced across a school year to include a progression of research skills according to grade-level standards.
• While there are frequent opportunities for students to complete informal research tasks, materials lack teacher guidance to support students with completing these tasks. The Teacher Edition does not provide information on how to teach the research skills necessary to complete the after-reading research tasks, and it contains limited guidance for the End-of-Unit Writing Workshop research project. Materials do not include a sequence or progression of research skills, nor is there explicit instruction of research skills that aligns to the standards. During the one in-depth research project per grade level, students complete research tasks as outlined in the standards but receive limited explicit instruction when doing so. While the research-focused Writing Workshop provides detailed process steps to complete the task, the Workshop rarely includes explicit instruction or scaffolding during each step of the research writing process.
• Materials provide limited support for teachers in employing projects that develop students’ knowledge of different aspects of a topic via provided resources.
• There is no evidence of the instructional materials providing support to teachers in employing projects that develop students’ knowledge of different aspects of a topic via provided resources. Research-oriented Extend the Text tasks are not accompanied by instructional support for teachers to guide students through what they are being asked to accomplish. For example, after reading the poems “The Tide Rises, the Tide Falls” and “A Psalm of Life” by Henry Wadsworth Longfellow in Unit 2, Expressing a National Spirit, American Resistance 1800–1850, students may complete a Lifelong Learning Extend the Text task in which they research life expectancy: “Conduct research on life expectancy in the United States to determining trends over the last two hundred years and to project trends in the near future. Also identify factors such as gender, race/ethnicity, disease, and lifestyle that affect individual life expectancy. Synthesize information from a variety of sources, examining them for accuracy, bias, and credibility. Summarize this information in a newsletter intended for someone your age.” Materials do not include guidance for teachers or students to conduct this research, such as where to look for such information; how to synthesize the information; and how to evaluate sources for accuracy, bias, and credibility. During the one in-depth research project per grade level, teachers receive limited support for helping students complete the steps of the research project such as how to write a thesis statement, incorporate parenthetical citations, paraphrase, or construct citations or a Works Cited page.
• Materials provide many opportunities for students to synthesize and analyze content tied to the texts under study as a part of the research process.
• In Unit 1, Shaping the World, Origins of the American Tradition to 1800, students read the Declaration of Independence by Thomas Jefferson. In the Extend the Text section of the after-reading activities, students research how the original passage condemning slavery was worded and why it was removed during the Lifelong Learning task. Students also conduct research to discover what other changes were made to the original document, then write an essay explaining their findings. This is one of four Extend the Text options from which the teacher can choose and, as a result, may not occur during core instruction.
• In Unit 4, Expanding Frontiers, Unification and Growth 1865–1910, students read “I Will Fight No More Forever” by Chief Joseph ,and “I am the Last of My Family” by Cochise. In the optional Extend the Text section of the after-reading activities, teachers may choose to enact the Lifelong Learning activity, during which students research a Native American Treaty: “Do research to identify a treaty that the U.S. government made with a Native American tribe in your region. Find out about the events leading up to the treaty, the terms of the agreement, and whether both groups fulfilled the terms. Also determine the current status of the tribe and identify problems or disputes faced by modern-day members. Prepare a written report that students might read during Native American history month at their school.” This is one of four Extend the Text options from which the teacher can choose and, as a result, may not occur during core instruction.
• In Unit 7, The American Dream, Postwar Era 1945–1960, students read the play, The Crucible by Arthur Miller. After reading Act I, students may complete a Media Literacy Extend the Text task on researching mob hysteria. “Investigate the phenomenon of mob hysteria. What different theories explain its origins? Why does it become difficult, if not impossible, for one individual to stand up to a mob? Support your answers using real-life instances of mob hysteria.” This is one of four Extend the Text options from which the teacher can choose and, as a result, may not occur during core instruction.
• Students are provided with opportunities for both “short” and “long” projects across the course of a year and grade bands.
• In Unit 3, A Nation Divided, Slavery and the Civil War 1850–1865, students read the poems “By the Bivouac’s Fitful Flame” and “Beat! Beat! Drums” by Walt Whitman. After reading, students may complete a short, one- to two-class period Media Literacy research task: “Make a list of five websites that provide valuable information about the Civil War and give it to your school’s media center. To evaluate the websites, create a checklist of criteria. Each website should be objective, factually accurate, and use an appropriate tone and level of formality for the intended audience and purpose. Turn in your list of criteria with your list of websites so that others will know they are credible.” This is one of four Extend the Text options from which the teacher can choose and, as a result,may not occur during core instruction.
• In Unit 7, The American Dream, Postwar Era 1945–1960, students read an excerpt from On the Road by Jack Kerouac. In the Extend the Text section, the Media Literacy task is as follows: “Research the history and importance of “be-bop,” a style of jazz music made popular in the 1940s, including its association with the Beat movement. Then draft a plan for a website about be-bop. Decide whether the site will serve as an introduction to the general public or be geared more toward jazz enthusiasts.” This is a shorter project that would likely require students one to two class periods to complete; however, the enactment of this research task is contingent upon teacher selection and, as a result, may not occur during core instruction.
• In Unit 9, New Challenges, Contemporary Era 1980 to Present, students “[p]lan, write, and revise a research paper that presents an argument about immigration” during the End-of-Unit Writing Workshop. The body of the research paper must “[support] the thesis with detailed evidence gathered from research.” This long research project spans three class periods.
### Criterion 2g - 2h
Materials promote mastery of grade-level standards by the end of the year.
4/8
+
-
Criterion Rating Details
The instructional materials reviewed for Grade 11 do not meet the criteria for coherence. Instruction, practice, and assessments are based on teacher selection from a list of options. Questions and tasks do not consistently align to grade-level standards or meet the full intent of the standards. It is unclear if the majority of assessment items align to grade-level standards. There is no guarantee that materials repeatedly address grade-level standards within and across units to ensure students master the full intent of the standards. The amount of material cannot reasonably be completed within the suggested amount of time and is not viable for a school year. The volume of optional tasks distracts from core learning. Some optional tasks are meaningful and enhance core instruction.
### Indicator 2g
Materials spend the majority of instructional time on content that falls within grade-level aligned instruction, practice, and assessments.
2/4
+
-
Indicator Rating Details
The materials reviewed for Grade 11 partially meet the criteria for Indicator 2g.
Materials do not spend the majority of instructional time on content that falls within grade-level aligned instruction, practice, and assessments. Examples include, but are not limited to, the following:
• Over the course of each unit, some instruction is aligned to grade-level standards.
• In the Digital Teacher Edition, the Grade 11 Correlation to Common Core State Standards document lists page numbers for each standard in Reading: Literature, Reading: Informational Text, Writing, Speaking & Listening, and Language; however, the page numbers listed do not always contain opportunities for explicit instruction or address the correlated standard.
• For example, the Correlation to Common Core State Standards document lists page 730 in the EMC Pages That Cover the Standards column for RL.5 “Analyze how an author's choices concerning how to structure specific parts of a text (e.g., the choice of where to begin or end a story, the choice to provide a comedic or tragic resolution) contribute to its overall structure and meaning as well as its aesthetic impact.” This page contains Refer to Text and Reason with Text questions, an Analyze Literature: Realism and Climax prompt, and the four Extend the Text options for the short story, “Ambush” by Tim O’Brien. The page also contains an Analyze Literature inset that includes information on realism and climax. While this inset notes and explains occurrences of realism and climax in the text, materials do not provide an opportunity for explicit instruction on the correlated standard.
• Over the course of each unit, some questions and tasks are aligned to grade-level standards.
• Questions often focus on comprehension strategies, such as Make Connections, Ask Questions, Draw Conclusions, and Visualize. These comprehension strategies do not align to grade-level standards. Some Extend the Text tasks align to grade-level standards, while others either do not align or do not meet the full requirements of the standards. Because post-reading questions and tasks do not have correlated standards identified, it is not always clear which question or task addresses the standard listed on the Correlation to Common Core State Standards document.
• In Unit 5, Progress and Conflict, Early Twentieth Century 1910–1929, students read “Lucinda Matlock,” a dramatic poem by Edgar Lee Masters. During reading, students observe shifts in tone and note elements of free verse. The text also includes an Art Connection piece on two paintings, “Stone City, Iowa” and “American Gothic,” both by Grant Wood. Students respond to the following Critical Viewing question: “Examine the paintings “American Gothic” and “Stone City, Iowa” (see page E209). What techniques does Wood use to make the scenes seem realistic? What qualities of his work seem especially middle American?” These questions do not address the correlated standard: “Integrate and evaluate multiple sources of information presented in different media or formats (e.g., visually, quantitatively) as well as in words in order to address a question or solve a problem.”
• Over the course of each unit, it is unclear whether the majority of assessment questions are aligned to grade-level standards.
• Materials do not identify assessed standards on Selection Quizzes, Lesson Tests, Unit Exams, or Formative Surveys. As a result, it is unclear whether the majority of assessment questions are aligned to grade-level standards.
• By the end of the academic year, standards are not repeatedly addressed within and across units to ensure students master the full intent of the standard.
• Because the page numbers listed on the Correlation to Common Core State Standards document for each standard in Reading: Literature, Reading: Informational Text, Writing, Speaking & Listening, and Language are not always the standard addressed and because the majority of questions and tasks do not align to grade-level standards, materials do not consistently provide students with multiple opportunities to address standards within and across units to ensure mastery. It is also unclear which items address the correlated standard, because standards are not identified at the question or task level.
• The Correlation to Common Core State Standards document lists the following page numbers for SL.3 “Evaluate a speaker's point of view, reasoning, and use of evidence and rhetoric, assessing the stance, premises, links among ideas, word choice, points of emphasis, and tone used.”: 51, 55, 325, 789, H33–H34. On page 51, the Text Overview page for Patrick Henry’s “Speech in the Virginia Convention” defines rhetorical question and the Enlightenment and sets the purpose for reading: “While reading his speech, write down four rhetorical questions Henry asks to appeal to people’s hearts and minds. Also look for evidence of his knowledge of classical rhetoric, or persuasive communication, and other Enlightenment principles.” On page 55, students respond to Refer to Text and Reason with Text questions, but it is unclear which questions address the correlated standard. Students also respond to an Analyze Literature prompt addressing rhetorical question and the Enlightenment: “Review the four rhetorical questions you wrote down from Henry’s speech. Next to each, write down the answer that Henry assumes his listeners will infer. What is effective about letting people answer these questions for themselves, rather than telling them the information directly?” During the Media Literacy option, students work in small groups, taking turns reading aloud “another historically significant American speech.” While listening, students “note the position taken and the evidence supporting that position, as well as the clarity and coherence of the speech.” Students then “analyze the rhetorical features and other elements that make the speech memorable,” identifying “passages from the speech that illustrate [their] points.”
### Indicator 2h
Materials regularly and systematically balance time and resources required for following the suggested implementation, as well as information for alternative implementations that maintain alignment and intent of the standards.
2/4
+
-
Indicator Rating Details
The materials reviewed for Grade 11 partially meet the criteria for Indicator 2h.
The materials include an overwhelming number of components with no guide for teachers to understand how to navigate and integrate the many ancillary resources. The Program Planning Guide includes the Mirrors & Windows College & Career Readiness Curriculum Guide Level VI (Grade 11), an alternative implementation schedule that focuses on selections and workshops necessary for students to “master critical skills that appear on state and national assessments.” Given the amount of time suggested and allotted for the core materials to be covered, there is little surplus time for covering the many extension activities, workshops and assessments located within and outside of the core materials. As a result, it is unclear how to assure grade-level standards are covered methodically or evenly when incorporating optional tasks or ancillary materials into daily lesson planning.
Materials do not regularly and systematically balance time and resources required for following the suggested implementation, as well as information for alternative implementations that maintain alignment and intent of the standards. Examples include, but are not limited to, the following:
• Suggested implementation schedules and alternative implementation schedules do not consistently align to core learning and objectives.
• In Unit 5, Progress and Conflict, Early Twentieth Century 1910–1929, students read the lyric poems, “Poetry” by Marianne Moore, and “Ars Poetica” by Archibald MacLeish. The lesson for these selections includes studying theme and stanza and making inferences while reading the poems. During-reading practice exercises and after-reading questions support these skills; however, the optional Extend the Text tasks do not align to these core learning objectives. For example, during the Creative Writing option, students write a dialogue between Marainne Moore and Archibald MacLeish, focusing on the attitude they might have towards each other.
• In Unit 8, Social Transition, Early Contemporary Era 1960–1980, students read the short story, “Ambush” by Tim O’Brien. Goals for this lesson, as outlined in the Before Reading section, include a study of realism, climax, and identifying multiple levels of meaning. During-reading activities and questions support the development of these skills, and after-reading questions assess students’ acquisition of the skills; however, the optional Extend the Text activities teachers may choose to assign do not serve to deepen students’ understanding of the core learning. For example, students may write a persuasive paragraph in response to the question of whether the draft should be mandatory.
• Suggested implementation schedules cannot be reasonably completed in the time allotted.
• The Program Planning Guide notes the overabundance of material: “To help you meet the diverse needs of your students, the Mirrors & Windows program offers a wealth of material—much more than you can teach in one school year. As a result, one challenge you will face is identifying the resources that are best suited to your particular situation.”
• As an alternative to the Scope and Sequence Guide provided in each unit, materials include the Mirrors & Windows College & Career Readiness Curriculum Guide Level VI (Grade 11): “The selections and workshops listed here represent the core course of study students need to master critical skills that appear on state and national assessments. To ensure standards coverage, students who are having difficulty may concentrate on only these selections and workshops. Students on and above grade level may read more selections.” When utilizing this abridged course of study, the teacher must still select which instructional activities to enact during each Program Planning Guide lesson plan.
• The Program Planning Guide contains lesson plans for each text selection and the three End-of-Unit Workshops. Text selection lesson plans include the following sections: Before Reading, During Reading, and After Reading. In the Before Reading: Preview and Motivate section, teachers “[c]hoose from the following materials to preview the selection and motivate your students.” The During Reading section contains two sub-sections, Teach the Selection(s) and Differentiate Instruction. Teachers choose from a list of resources to teach the selection and consider “alternative teaching options to differentiate instruction.” The After Reading section contains two to three subsections: Review and Extend, Teach the Workshop(s), and Assess. Teachers select activities from a list of options and resources to extend learning and teach the Workshop included, where applicable. Teachers do not select from a list of options during the Assess subsection. The lesson plan does not provide guidance on how many minutes each option should take or how long the lesson should last. Pacing guidance is limited to the number of regular or block schedule days the lesson should take.
• Optional tasks distract from core learning.
• In the Writing section of the Writing & Grammar ancillary, materials provide instruction on writing a lyric poem to accompany Unit 3 content. This resource is an intensive workshop that includes a large number of steps, which could impact the suggested pacing of unit selections. While students read many lyric poems over the course of Unit 3, this workshop does not support students’ deeper understanding of the poems under study.
• In Unit 5, Progress and Conflict, Early Twentieth Century 1910–1929, students read an excerpt from the novel The Sun Also Rises by F. Scott Fitzgerald. During reading, materials include a Differentiated Instruction activity in which students choreograph movements based on the description in the bullfighting scene. While this opportunity allows students to be physically active while learning more about bulls and bullfighting and recalling the scene, the activity does not allow students to practice skills related to reading the text or contribute to a deeper understanding of the text.
• In Unit 8, Social Transition, Early Contemporary Era 1960–1980, students study imagery and simile while reading two lyric poems by Yusef Komunyakaa, “Camouflaging the Chimera” and “Monsoon Season.” The Extend the Text task options at the end of the selection do not connect to these skills. Tasks include writing a diary entry as a soldier in one of the poems, writing a compare and contrast essay about the poems and Tim O’Brien’s story, “Ambush,” communicating with a Vietnamese school, and preparing an explanation of one of the poems featuring unrelated text features.
• Some optional tasks are meaningful and enhance core instruction.
• In Unit 3, A Nation Divided, Slavery and the Civil War 1850–1865, students read an excerpt from the preface for Leaves of Grass and an excerpt from “I Hear America Singing” by Walt Whitman. The Unit & Selection Resources ancillary provides additional options for student work, but these options do not enhance the instruction of these texts. Students complete an out-of-context vocabulary exercise in which they use words from the text to practice how synonyms affect connotation; read a new selection by William Wordsworth in the same genre and compare it to the selections read by filling out a chart; and write an essay comparing the two author’s Romantic ideals.
• In Unit 4, Expanding Frontiers, Unification and Growth 1865–1910, students read “The Notorious Jumping Frog of Calaveras County,” a short story by Mark Twain. Guidance in the Teacher Edition suggests teachers launch the lesson by having students research frontier mining camps and make a scale model or drawing of a camp. The Before Reading material contains the historical context of the story and includes a map of the mining district. Completing this task would be redundant and would not enhance core instruction.
• In Unit 7, The American Dream, Postwar Era 1945–1960, students read the play The Crucible by Arthur Miller. During the reading of Act I, materials include an enrichment activity during which students research scapegoating and answer a trio of historical questions about scapegoating. This task supports students with understanding the discrimination and prejudice present in the reading.
## Usability
### Criterion 3a - 3h
The program includes opportunities for teachers to effectively plan and utilize materials with integrity and to further develop their own understanding of the content.
### Indicator 3a
Materials provide teacher guidance with useful annotations and suggestions for how to enact the student materials and ancillary materials to support students' literacy development.
N/A
### Indicator 3b
Materials contain adult-level explanations and examples of the more complex grade/course-level concepts and concepts beyond the current course so that teachers can improve their own knowledge of the subject.
N/A
### Indicator 3c
Materials include standards correlation information that explains the role of the standards in the context of the overall series.
N/A
### Indicator 3d
Materials provide strategies for informing all stakeholders, including students, parents, or caregivers about the program and suggestions for how they can help support student progress and achievement.
N/A
### Indicator 3e
Materials provide explanations of the instructional approaches of the program and identification of the research-based strategies.
N/A
### Indicator 3f
Materials provide a comprehensive list of supplies needed to support instructional activities.
N/A
### Indicator 3g
This is not an assessed indicator in ELA.
N/A
### Indicator 3h
This is not an assessed indicator in ELA.
N/A
### Criterion 3i - 3l
The program includes a system of assessments identifying how materials provide tools, guidance, and support for teachers to collect, interpret, and act on data about student progress towards the standards.
### Indicator 3i
Assessment information is included in the materials to indicate which standards are assessed.
N/A
### Indicator 3j
Assessment system provides multiple opportunities throughout the grade, course, and/or series to determine students' learning and sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.
N/A
### Indicator 3k
Assessments include opportunities for students to demonstrate the full intent of grade-level/course-level standards and practices across the series.
N/A
### Indicator 3l
Assessments offer accommodations that allow students to demonstrate their knowledge and skills without changing the content of the assessment.
N/A
### Criterion 3m - 3v
The program includes materials designed for each child’s regular and active participation in grade-level/grade-band/series content.
### Indicator 3m
Materials provide strategies and supports for students in special populations to work with grade-level content and to meet or exceed grade-level standards that will support their regular and active participation in learning English language arts and literacy.
N/A
### Indicator 3n
Materials regularly provide extensions to engage with literacy content and concepts at greater depth for students who read, write, speak, and/or listen above grade level.
N/A
### Indicator 3o
Materials provide varied approaches to learning tasks over time and variety in how students are expected to demonstrate their learning with opportunities for students to monitor their learning.
N/A
### Indicator 3p
Materials provide opportunities for teachers to use a variety of grouping strategies.
N/A
### Indicator 3q
Materials provide strategies and supports for students who read, write, and/or speak in a language other than English to meet or exceed grade-level standards to regularly participate in learning English language arts and literacy.
N/A
### Indicator 3r
Materials provide a balance of images or information about people, representing various demographic and physical characteristics.
N/A
### Indicator 3s
Materials provide guidance to encourage teachers to draw upon student home language to facilitate learning.
N/A
### Indicator 3t
Materials provide guidance to encourage teachers to draw upon student cultural and social backgrounds to facilitate learning.
N/A
### Indicator 3u
This is not an assessed indicator in ELA.
N/A
### Indicator 3v
This is not an assessed indicator in ELA.
N/A
### Criterion 3w - 3z
The program includes a visual design that is engaging and references or integrates digital technology (when applicable) with guidance for teachers.
### Indicator 3w
Materials integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic software in ways that engage students in the grade-level/series standards, when applicable.
N/A
### Indicator 3x
Materials include or reference digital technology that provides opportunities for teachers and/or students to collaborate with each other, when applicable.
N/A
### Indicator 3y
The visual design (whether in print or digital) supports students in engaging thoughtfully with the subject, and is neither distracting nor chaotic.
N/A
### Indicator 3z
Materials provide teacher guidance for the use of embedded technology to support and enhance student learning, when applicable.
N/A
abc123
Report Published Date: 2021/08/26
Report Edition: 2021
Title ISBN Edition Publisher Year
Mirrors & Windows 2021 - Student Edition Grade 11 978‑1‑5338‑3668‑7 EMC School, Part of Carnegie Learning 2021
Mirrors & Windows 2021 - Teacher's Edition Grade 11 978‑1‑5338‑3675‑5 EMC School, Part of Carnegie Learning 2021
## ELA High School Review Tool
The ELA review criteria identifies the indicators for high-quality instructional materials. The review criteria supports a sequential review process that reflect the importance of alignment to the standards then consider other high-quality attributes of curriculum as recommended by educators.
For ELA, our review criteria evaluates materials based on:
• Text Quality and Complexity, and Alignment to Standards with Tasks Grounded in Evidence
• Building Knowledge with Texts, Vocabulary, and Tasks
• Instructional Supports and Usability
The ELA Evidence Guides complement the review criteria by elaborating details for each indicator including the purpose of the indicator, information on how to collect evidence, guiding questions and discussion prompts, and scoring criteria.
The EdReports rubric supports a sequential review process through three gateways. These gateways reflect the importance of alignment to college and career ready standards and considers other attributes of high-quality curriculum, such as usability and design, as recommended by educators.
Materials must meet or partially meet expectations for the first set of indicators (gateway 1) to move to the other gateways.
Gateways 1 and 2 focus on questions of alignment to the standards. Are the instructional materials aligned to the standards? Are all standards present and treated with appropriate depth and quality required to support student learning?
Gateway 3 focuses on the question of usability. Are the instructional materials user-friendly for students and educators? Materials must be well designed to facilitate student learning and enhance a teacher’s ability to differentiate and build knowledge within the classroom.
In order to be reviewed and attain a rating for usability (Gateway 3), the instructional materials must first meet expectations for alignment (Gateways 1 and 2).
Alignment and usability ratings are assigned based on how materials score on a series of criteria and indicators with reviewers providing supporting evidence to determine and substantiate each point awarded.
Alignment and usability ratings are assigned based on how materials score on a series of criteria and indicators with reviewers providing supporting evidence to determine and substantiate each point awarded.
For ELA and math, alignment ratings represent the degree to which materials meet expectations, partially meet expectations, or do not meet expectations for alignment to college- and career-ready standards, including that all standards are present and treated with the appropriate depth to support students in learning the skills and knowledge that they need to be ready for college and career.
For science, alignment ratings represent the degree to which materials meet expectations, partially meet expectations, or do not meet expectations for alignment to the Next Generation Science Standards, including that all standards are present and treated with the appropriate depth to support students in learning the skills and knowledge that they need to be ready for college and career.
For all content areas, usability ratings represent the degree to which materials meet expectations, partially meet expectations, or do not meet expectations for effective practices (as outlined in the evaluation tool) for use and design, teacher planning and learning, assessment, differentiated instruction, and effective technology use.
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2022-01-28 09:43:21
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https://www.physicsforums.com/threads/circular-motion-approximation-of-centripetal-acc.602243/
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# Circular motion (approximation of centripetal acc.)
1. May 1, 2012
### V0ODO0CH1LD
Say I have a body moving in a circle of radius r with a constant velocity v.
The time it takes the body to go around the circumference once is:
T = 2πr/v
Then the time it takes the body to go around a fourth of the circumference is T/4.
Now, imagine a diagram where when the body is at the leftmost portion of the circumference its velocity vector is pointing straight up, and when it is at the upmost portion of the circumference its velocity vector points entirely to the right. That is the body travelling over a fourth of the circumference.
Now, the magnitude of the acceleration vector required to make the body behave that way over a period of T/4 is (v√2)/(T/4). And (v√2)/(T/4) is an approximation of the centripetal acceleration for this case.
My question is: why is that an approximation? Why if I do the same thing over a smaller change in time I get closer to the centripetal acceleration? A circle is symmetric and the velocity is constant, right?
2. May 1, 2012
### Redbelly98
Staff Emeritus
Agreed, so far.
Not quite. This is the magnitude of the average acceleration, where the average acceleration is defined as
$$\vec{a_{avg}}=\frac{\Delta \vec{v}}{\Delta t}$$
Because it is an average, and the actual instantaneous acceleration changes substantially (because it changes direction) during the time interval.
Because, for a small time interval, the acceleration does not change direction very much, and is more closely approximated by its average over that time interval.
No, velocity is a vector and is not constant if its direction changes. Speed is constant, however.
3. May 1, 2012
### V0ODO0CH1LD
That helped a lot! Thanks!
One for additional question though. How does all of that get summarised into (ω^2)*r or (v^2)/r?
4. May 1, 2012
### Redbelly98
Staff Emeritus
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2017-08-18 15:35:59
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http://www.reference.com/browse/alpha-naphthol+test
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Definitions
# Logrank test
In statistics, the logrank test (sometimes called the Mantel-Cox test) is a hypothesis test to compare the survival distributions of two samples. It is a nonparametric test and appropriate to use when the data are right censored (technically, the censoring must be non-informative). It is widely used in clinical trials to establish the efficacy of new drugs compared to a control group (often a placebo) when the measurement is the time to event (such as a heart attack).
The test was first proposed by Nathan Mantel and was named the logrank test by Richard and Julian Peto.
## Definition
The logrank test statistic compares estimates of the hazard functions of the two groups at each observed event time. It is constructed by computing the observed and expected number of events in one of the groups at each observed event time and then adding these to obtain an overall summary across all time points where there is an event.
Let j = 1, ..., J be the distinct times of observed events in either group. For each time $j$, let $N_\left\{1j\right\}$ and $N_\left\{2j\right\}$ be the number of subjects "at risk" (have not yet had an event or been censored) at the start of period $j$ in the groups respectively. Let $N_j = N_\left\{1j\right\} + N_\left\{2j\right\}$. Let $O_\left\{1j\right\}$ and $O_\left\{2j\right\}$ be the observed number of events in the groups respectively at time $j$, and define $O_j = O_\left\{1j\right\} + O_\left\{2j\right\}$.
Given that $O_j$ events happened across both groups at time $j$, under the null hypothesis $O_\left\{1j\right\}$ has the hypergeometric distribution with parameters $N_j$, $N_\left\{1j\right\}$, and $O_j$. This distribution has expected value $E_j = O_jfrac\left\{N_\left\{1j\right\}\right\}\left\{N_j\right\}$ and variance $V_j = frac\left\{O_j \left(N_\left\{1j\right\}/N_j\right) \left(1 - N_\left\{1j\right\}/N_j\right) \left(N_j - O_j\right)\right\}\left\{N_j - 1\right\}$.
The logrank statistic compares each $O_\left\{1j\right\}$ to its expectation $E_j$ under the null hypothesis and is defined as
$Z = frac \left\{sum_\left\{j=1\right\}^J \left(O_\left\{1j\right\} - E_j\right)\right\} \left\{sqrt \left\{sum_\left\{j=1\right\}^J V_j\right\}\right\}.$
## Asymptotic distribution
If the two groups have the same survival function, the logrank statistic is approximately standard normal. A one-sided level $alpha$ test will reject the null hypothesis if $Z>z_alpha$ where $z_alpha$ is the upper $alpha$ quantile of the standard normal distribution. If the hazard ratio is $lambda$, there are $n$ total subjects, $d$ is the probability a subject in either group will eventually have an event, and the proportion of subjects randomized to each group is 50%, then the logrank statistic is approximately normal with mean $\left(log\left\{lambda\right\}\right) , sqrt \left\{frac \left\{n , d\right\} \left\{4\right\}\right\}$ and variance 1. For a one-sided level $alpha$ test with power $1-beta$, the sample size required is $n = frac \left\{4 , \left(z_alpha + z_beta\right)^2 \right\} \left\{dlog^2\left\{lambda\right\}\right\}$ where $z_alpha$ and $z_beta$ are the quantiles of the standard normal distribution.
## Joint distribution
Suppose $Z_1$ and $Z_2$ are the logrank statistics at two different time points in the same study ($Z_1$ earlier). Again, assume the hazard functions in the two groups are proportional with hazard ratio $lambda$ and $d_1$ and $d_2$ are the probabilities that a subject will have an event at the two time points. $Z_1$ and $Z_2$ are approximately bivariate normal with means $log\left\{lambda\right\} , sqrt \left\{frac \left\{n , d_1\right\} \left\{4\right\}\right\}$ and $log\left\{lambda\right\} , sqrt \left\{frac \left\{n , d_2\right\} \left\{4\right\}\right\}$ and correlation $sqrt \left\{frac \left\{d_1\right\} \left\{d_2\right\}\right\}$. Calculations involving the joint distribution are needed to correctly maintain the error rate when the data are examined multiple times within a study by a Data Monitoring Committee.
## Relationship to other statistics
• The logrank statistic can be derived as the score test for the Cox proportional hazards model comparing two groups. It is therefore asymptotically equivalent to the likelihood ratio test statistic based from that model.
• The logrank statistic is asymptotically equivalent to the likelihood ratio test statistic for any family of distributions with proportional hazard alternative. For example, if the data from the two samples have exponential distributions.
• If $Z$ is the logrank statistic, $D$ is the number of events observed, and $hat \left\{lambda\right\}$ is the estimate of the hazard ratio, then $log\left\{hat \left\{lambda\right\}\right\} approx Z , sqrt\left\{4/D\right\}$. This relationship is useful when two of the quantities are known (e.g. from a published article), but the third one is needed.
## References
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2013-12-12 19:50:57
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https://nrich.maths.org/1311/index?nomenu=1
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The previous article in the series is here. You might also wish to look at the article "Picturing Pythagorean Triples".
This is the second of the two articles on right-angled triangles whose edge lengths are whole numbers. We suppose that the lengths of the two sides of a right-angled triangles are $a$ and $b$, and that the hypotenuse has length $c$ so that, by Pythagoras' Theorem,$$a^2 + b^2 = c^2$$.
In the first article we discussed the possibility of enlarging or shrinking a right-angled triangle to get another right-angled triangle whose sides also have lengths that are whole numbers, and we claimed there that apart from a possible scaling of the triangle, every such right-angled triangle has edge lengths of the form
$a=2pq \; \; \;$ $b=p^2-q^2 \; \; \;$ $c=p^2+q^2$
For suitable whole numbers $p$ and $q$, where $p> q$. This article contains a proof of this fact.
First, we must understand a few ideas about factors and prime numbers. A factor of a whole number $n$ is a whole number $f$ that divides into $n$ exactly (without remainder). Of course, $1$ and $n$ are always factors of $n$, and we say that $n$ is a prime number if $1$ and $n$ are the only factors of $n$. We do not regard $1$ as a prime number (even though its only factor is 1), and you should now check that the first few prime numbers are $2$, $3$,$5$, $7$, $11$, $13$, $17$.
What are the next three prime numbers?
If a number $n$ is not a prime number then we must be able to write it as a product of two numbers $u$ and $v$; that is $n = u \times v$, and we normally write this as $n = u v$. If $u$ (or $v$) is not a prime number, then we can write $u$ (or $v$) as a product and so write $n$ as a product of three numbers. We can continue in this way until every number that we are left with is a prime number, and this shows that every number $n$ is a product of prime numbers.
For example, suppose that we start with $255$. Then $255 = 5 \times 51$. Now $5$ is a prime, but $51$ is not. Next, $51 = 3 \times 17$ and both $3$ and $17$ are primes; thus $255 = 3 \times 5 \times 17$ and we say that $3$, $5$ and $17$ are the prime factors of $255$. Of course, some prime factors may be repeated; for example, $75 = 3 \times 5 \times 5$, and $315 = 3 \times 105 = 3 \times 5 \times 21 = 3 \times 3 \times 5 \times 7$.
If we know that $f$ is a prime factor of a product $u v$, then (writing $u$ and $v$ as a product of prime numbers) we see that $f$ must occur as one of the prime factors of $u$ or of $v$ (or of both), so that $f$ must be a factor of $u$ or of $v$.
We repeat :
(I) if $f$ is a prime factor of $u v$ then $f$ must be either a prime factor of $u$ or a prime factor of $v$.
Note that this result is NOT true of every factor; our claim applies only to prime factors. For example, $6$ is a factor of $4 \times 9$, but it is not a factor of $4$ or of $9$; of course, each prime factor of $6$ is a factor of either $4$ or $9$.
A number $n$ is a square number if $n = m^2$ for some whole number $m$. It is easy to see (by writing $n$ as the product of its prime factors) that a whole number $n$ is a square number if every one of its prime factors occurs an even number of times. For example:
$5 \times 5 \times 7 \times 7$ is a square number but $3 \times 5 \times 7 \times 7$ is not.
Here are two useful facts about factors and square numbers.
Suppose that $f$ is a prime factor of the square number $n=m^2$. Then
(II) $f$ is a prime factor of $m$, and
(III) $f$ is a prime factor of $n$.
The statement (II) is just the statement (I) with $u = v = m$.
From (II) we see that $f$ is a prime factor of $m$, and this means that $f$ is a factor of $m^2 = n$ which is (III).
We now return to the problem of showing that every triple of whole numbers $a$, $b$, $c$ with $a^2+ b^2 = c^2$ can be expressed in the form (1).
To show this, we start with any Pythagorean triple and first reduce it as much as possible to end with a triple $a$, $b$, $c$ which cannot be reduced any more. This means that there is no whole number (except 1) which is a factor of each of $a$, $b$, $c$ (for otherwise, we could reduce the triangle still further). We shall show now that these $a$, $b$, $c$ can be written in the form (1) for some suitable $p$ and $q$. There are, of course, three possibilities that can arise, namely:
(i) $a$ and $b$ are both even;
(ii) $a$ and $b$ are both odd;
(iii) one of $a$ and $b$ is even and the other is odd.
In fact, neither (i) nor (ii) can happen so let us see why.
First, (i) cannot happen because if $a$ and $b$ are both even, then $c^2 (=a^2 + b^2)$ is even, so that $c$ is also even. But then $a$, $b$ and $c$ each have a factor 2 and we could have reduced the triple $a$, $b$, $c$ still further.
So $a$ and $b$ cannot both be even.
To see that (ii) cannot happen suppose that $a$ and $b$ are both odd. Then the remainder when dividing $a^2$ (and also $b^2$ ) by $4$ is $1$, so that the remainder when dividing $c^2 (= a^2 + b^2 )$ by 4 is 2.
However, if $c$ is even, $c^2$ is a mulitple of 4, so this remainder must be 0, while if $c$ is odd the remainder must be 1.
We now know that (iii) must hold and we shall take $a$ to be the even number and $b$ to be the odd number.
As $c^2 = a^2 + b^2$ , we see that $c$ must be odd.
As $b$ and $c$ are odd, their sum must be even and their difference must also be even. So we can find whole numbers $r$ and $s$ such that $2r = c + b$ and $2s = c - b$, and this means that
\begin{eqnarray} b = r - s,\\ c = r + s \\ \mbox{ and } \mbox{ } a^2 = c^2 - b ^2 = (c + b)(c - b) = 4rs \mbox{. . . . . . . (2)} \end{eqnarray}
This is similar to (1) but not quite the same. Also, not every choice of $r$ and $s$ here gives us a triple of whole numbers. For example, if we take $r=2$ and $s=1$ we get $a^2 = 8$ so that in this case $a$ is not a whole number. We are going to show that $r$ and $s$ in (2) must be square numbers. Then we can write $r=p^2$ and $s=q^2$, say, and then (2) gives
\begin{eqnarray} a = 2p q \\ b = p - q \\ c = p + q \mbox{ which is (1).}\end{eqnarray} The hardest part of this article is to show that $r$ and $s$ are square numbers, and this depends on understanding the ideas about factors and prime numbers.
First, we must show that $r$ and $s$ cannot have any common factors.
To see this, suppose that $r$ and $s$ have a common prime factor $f$. Then $r - s (= b)$ and $r + s (= c)$ both have $f$ as a factor, so that $f$ is a factor of $c^2 - b^2$, which is $a^2$ . As $f$ is a prime factor of $a^2$, then $f$ is a factor of $a$.
This would mean that $f$ is a factor of each of $a$, $b$, $c$ and this cannot be so else again we would have reduced the triple $a$, $b$, $c$ still further at the outset.
We have now shown that $r$ and $s$ have no common factors.
Now take any prime factor $f$ of $r$. First, as $r$ and $s$ have no common factors, we see that $f$ is not a factor of $s$. Next, as $f$ is a prime factor of $r$, it is also a prime factor of $r s$, and hence a prime factor of the whole number $\left( \frac{a}{2}\right)^2$ (because $\left(\frac{a}{2}\right)^2 = r s$).
As $f$ is a prime factor of $\left(\frac{a}{2}\right)^2$, it follows from (III) that $f^2$ is a factor of $\left(\frac{a}{2}\right)^2 = r s$, and because $f$ is not a factor of $s$, we now see that $f^2$ is a factor of $r$.
We have just seen that if $f$ is any prime factor of $r$, then $f^2$ is a factor of $r$, and this means that $r$ is a square number. The same argument is true for $s$ as well as $r$, so that as $r$ and $s$ are square numbers.
Because $r$ and $s$ are square numbers we can now write $r=p^2$ and $s=q^2$ for some $p$ and $q$ and, at last, we have shown that $a = 2p q$, $b = p^2 - q^2$ and $c = p^2 + q^2$ which is (1).
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2016-05-30 16:33:57
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http://www.projecteuclid.org/
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Statistical Science, Volume 16, Issue 3 (2001) Article
Communications in Mathematical Physics, Volume 113, Number 4 (1988) Article
Bulletin of the American Mathematical Society, Volume 53, Number 6 (1947) Article
Bulletin of the American Mathematical Society, Volume 10, Number 3 (1903) Article
The Annals of Statistics, Volume 6, Number 2 (1978) Article
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2015-05-24 13:00:02
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https://docs.lammps.org/dihedral_charmm.html
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$$\renewcommand{\AA}{\text{Å}}$$
# dihedral_style charmm command¶
Accelerator Variants: charmm/intel, charmm/kk, charmm/omp
# dihedral_style charmmfsw command¶
## Syntax¶
dihedral_style style
• style = charmm or charmmfsw
## Examples¶
dihedral_style charmm
dihedral_style charmmfsw
dihedral_coeff 1 0.2 1 180 1.0
dihedral_coeff 2 1.8 1 0 1.0
dihedral_coeff 1 3.1 2 180 0.5
## Description¶
The charmm and charmmfsw dihedral styles use the potential
$E = K [ 1 + \cos (n \phi - d) ]$
See (MacKerell) for a description of the CHARMM force field. This dihedral style can also be used for the AMBER force field (see comment on weighting factors below). See (Cornell) for a description of the AMBER force field.
Note
The newer charmmfsw style was released in March 2017. We recommend it be used instead of the older charmm style when running a simulation with the CHARMM force field, either with long-range Coulombics or a Coulombic cutoff, via the pair_style lj/charmmfsw/coul/long and pair_style lj/charmmfsw/coul/charmmfsh commands respectively. Otherwise the older charmm style is fine to use. See the discussion below and more details on the pair_style charmm doc page.
The following coefficients must be defined for each dihedral type via the dihedral_coeff command as in the example above, or in the data file or restart files read by the read_data or read_restart commands:
• $$K$$ (energy)
• $$n$$ (integer >= 0)
• $$d$$ (integer value of degrees)
• weighting factor (1.0, 0.5, or 0.0)
The weighting factor is required to correct for double counting pairwise non-bonded Lennard-Jones interactions in cyclic systems or when using the CHARMM dihedral style with non-CHARMM force fields. With the CHARMM dihedral style, interactions between the first and fourth atoms in a dihedral are skipped during the normal non-bonded force computation and instead evaluated as part of the dihedral using special epsilon and sigma values specified with the pair_coeff command of pair styles that contain “lj/charmm” (e.g. pair_style lj/charmm/coul/long) In 6-membered rings, the same 1-4 interaction would be computed twice (once for the clockwise 1-4 pair in dihedral 1-2-3-4 and once in the counterclockwise dihedral 1-6-5-4) and thus the weighting factor has to be 0.5 in this case. In 4-membered or 5-membered rings, the 1-4 dihedral also is counted as a 1-2 or 1-3 interaction when going around the ring in the opposite direction and thus the weighting factor is 0.0, as the 1-2 and 1-3 exclusions take precedence.
Note that this dihedral weighting factor is unrelated to the scaling factor specified by the special bonds command which applies to all 1-4 interactions in the system. For CHARMM force fields, the special_bonds 1-4 interaction scaling factor should be set to 0.0. Since the corresponding 1-4 non-bonded interactions are computed with the dihedral. This means that if any of the weighting factors defined as dihedral coefficients (fourth coeff above) are non-zero, then you must use a pair style with “lj/charmm” and set the special_bonds 1-4 scaling factor to 0.0 (which is the default). Otherwise 1-4 non-bonded interactions in dihedrals will be computed twice.
For simulations using the CHARMM force field with a Coulombic cutoff, the difference between the charmm and charmmfsw styles is in the computation of the 1-4 non-bond interactions, though only if the distance between the two atoms is within the switching region of the pairwise potential defined by the corresponding CHARMM pair style, i.e. within the outer cutoff specified for the pair style. The charmmfsw style should only be used when using the corresponding pair_style lj/charmmfsw/coul/charmmfsw or pair_style lj/charmmfsw/coul/long commands. Use the charmm style with the older pair_style commands that have just “charmm” in their style name. See the discussion on the CHARMM pair_style page for details.
Note that for AMBER force fields, which use pair styles with “lj/cut”, the special_bonds 1-4 scaling factor should be set to the AMBER defaults (1/2 and 5/6) and all the dihedral weighting factors (fourth coeff above) must be set to 0.0. In this case, you can use any pair style you wish, since the dihedral does not need any Lennard-Jones parameter information and will not compute any 1-4 non-bonded interactions. Likewise the charmm or charmmfsw styles are identical in this case since no 1-4 non-bonded interactions are computed.
Styles with a gpu, intel, kk, omp, or opt suffix are functionally the same as the corresponding style without the suffix. They have been optimized to run faster, depending on your available hardware, as discussed on the Accelerator packages page. The accelerated styles take the same arguments and should produce the same results, except for round-off and precision issues.
These accelerated styles are part of the GPU, INTEL, KOKKOS, OPENMP, and OPT packages, respectively. They are only enabled if LAMMPS was built with those packages. See the Build package page for more info.
You can specify the accelerated styles explicitly in your input script by including their suffix, or you can use the -suffix command-line switch when you invoke LAMMPS, or you can use the suffix command in your input script.
See the Accelerator packages page for more instructions on how to use the accelerated styles effectively.
## Restrictions¶
When using run_style respa, these dihedral styles must be assigned to the same r-RESPA level as pair or outer.
When used in combination with CHARMM pair styles, the 1-4 special_bonds scaling factors must be set to 0.0. Otherwise non-bonded contributions for these 1-4 pairs will be computed multiple times.
These dihedral styles can only be used if LAMMPS was built with the MOLECULE package. See the Build package doc page for more info.
## Default¶
none
(Cornell) Cornell, Cieplak, Bayly, Gould, Merz, Ferguson, Spellmeyer, Fox, Caldwell, Kollman, JACS 117, 5179-5197 (1995).
(MacKerell) MacKerell, Bashford, Bellott, Dunbrack, Evanseck, Field, Fischer, Gao, Guo, Ha, et al, J Phys Chem B, 102, 3586 (1998).
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2022-11-30 14:37:40
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https://intelligencemission.com/free-electricity-using-flywheel-free-electricity-from-high-voltage-lines.html
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Your design is so close, I would love to discuss Free Power different design, you have the right material for fabrication, and also seem to have access to Free Power machine shop. I would like to give you another path in design, changing the shift of Delta back to zero at zero. Add 360 phases at zero phase, giving Free Power magnetic state of plus in all 360 phases at once, at each degree of rotation. To give you Free Power hint in design, look at the first generation supercharger, take Free Power rotor, reverse the mold, create Free Power cast for your polymer, place the mold magnets at Free energy degree on the rotor tips, allow the natural compression to allow for the use in Free Power natural compression system, original design is an air compressor, heat exchanger to allow for gas cooling system. Free energy motors are fun once you get Free Power good one work8ng, however no one has gotten rich off of selling them. I’m Free Power poor expert on free energy. Yup that’s right poor. I have designed Free Electricity motors of all kinds. I’ve been doing this for Free Electricity years and still no pay offs. Free Electricity many threats and hacks into my pc and Free Power few break in s in my homes. It’s all true. Big brother won’t stop keeping us down. I’ve made millions if volt free energy systems. Took Free Power long time to figure out.
Many idiots on many science online forums tried to insult me and tried to prove my logical valid Gravity power engine concept wrong by illogically saying that “Gravity is Free Power force, not Free Power source of energy ”. How foolish that idiot’s statement appears to be. Interesting posts, pro and con. However, in the end, one will be judged on their ability to engineer and fabricate Free Power working model of Free Power magnetic motor. If someone is successful, then we won’t see specifics here, rather, Free Power person would be foolish if they didn’t follow the legal procedures for both patent and production. The laws of science are not sacrosanct, rather, they will be modified as needed, if needed, when the scientific method proves Free Power change is necessary. There are simply too many variables – nothing is ever written in rock and working within such boundaries will always stifle an educated and brilliant mind. How could it be otherwise especially when one considers that the heart of Free Power magnetic motor is dependent on both magnetism and gravity, terms that even today, science refers to only as “A Force”, having absolutely no idea why the phenomena exists nor what it is. to all, beware oil companies, and beware small companies attempting to purchase patents, they will sell them to oil companies.
Why not use the term over unity over perpetual motion? Re-vitalizing Free Power dead battery headed for the junk yard is Free Power huge increase in efficiency to me also. Why doesn’t every AutoZone or every auto shop have one of these? Unless the battery case is cracked every battery could be reused. The charge of Free Power re-vitalize instead of Free Power new battery. Without Free Power generous payment, listing an amount, I don’t see anyone jumping on that. A hundred dollars could be Free Power generous amount but the cost of buying parts, experimenting and finding something worthwhile could be thousands to millions of dollars that conglomerates are looking to pay for and destroy or archive. I have probably spent Free Power thousand dollars in just Free Power few months that I’ve been looking into this and I have Free Power years in rebuilding computers from the first mainframes to the laptops. I retired and now its Free Power hobby. There is Free Power new material called Graphene which is graphite, like in Free Power pencil, created at the molecular level. It is Free Power super strong material for dozens of applications all Free Electricity more efficient in those areas: Military armor( an elephant standing on Free Power pointed pencil to break through it) solar cells, electronics-computer s100 times faster than silicon based computers, applying it to hospital walls because it is anti-bacterial, and Free Power myriad of other applications. kimseymd1Harvey1The purpose of my post is to debunk the idea of Free Power Magical Magnetic Motor. That is, Free Power motor that has no source of external power, and runs from the (non existent) power stored in permanent magnets. Advances made to electric motors in the past few years are truly amazing, but are totally outside the scope of my post.
Years later, Free Power top U. S. General who was the liaison between DynCorp and the U. S. Military was implicated in the sexual assault of teenage girls. Earlier this year, Florida Air National Guard Col. Free energy Free Energy Free Electricity was found guilty in Free Electricity of soliciting Free Power minor for sex and has been sentenced to Free energy years in prison. Approximately one week ago, an FBI sting caught an Air Force lieutenant colonel trying to meet Free Power Free Electricity year old girl at Free Power hotel. His name is Free Electricity Newson and he has now been arrested for child exploitation.
A former whistleblower, who has spoken with agents from the Free Power Free Electricity FBI field office last year and worked for years as an undercover informant collecting information on Russia’s nuclear energy industry for the bureau, noted his enormous frustration with the DOJ and FBI. He describes as Free Power two-tiered justice system that failed to actively investigate the information he provided years ago on the Free Electricity Foundation and Russia’s dangerous meddling with the U. S. nuclear industry and energy industry during the Obama administration.
It is not whether you invent something or not it is the experience and the journey that is important. To sit on your hands and do nothing is Free Power waste of life. My electrical engineer friend is saying to mine, that it can not be done. Those with closed minds have no imagination. This and persistance is what it takes to succeed. The hell with the laws of physics. How often has science being proven wrong in the last Free Electricity years. Dont let them say you are Free Power fool. That is what keeps our breed going. Dont ever give up. I’ll ignore your attempt at sarcasm. That is an old video. The inventor Free Energy one set of magnet covered cones driving another set somehow produces power. No explanation, no test results, no published information.
Former Free Electricity was among Free Electricity’s closest friends, and the flight logs from Free Electricity’s private jet shown here reveal that Free Electricity was listed as Free Power passenger on the jet at least Free energy times between Free Power and Free Power, which would have put Free Electricity on the plane at least once Free Power month during the two-year period. Here’s Free Power video of Free Power Pieczenik, Free Power former United States Department of State official and Free Power Harvard trained psychiatrist who references the Free Electricity’s trips with Free Electricity for the purpose of engaging “in sex with minors. ”
I’ve told you about how not well understood is magnetism. There is Free Power book written by A. K. Bhattacharyya, A. R. Free Electricity, R. U. Free Energy. – “Magnet and Magnetic Free Power, or Healing by Magnets”. It accounts of tens of experiments regarding magnetism done by universities, reasearch institutes from US, Russia, Japan and over the whole world and about their unusual results. You might wanna take Free Power look. Or you may call them crackpots, too. 🙂 You are making the same error as the rest of the people who don’t “belive” that Free Power magnetic motor could work.
### This is not Free Power grand revelation. In or about Free Electricity, the accepted laws of physics Free energy THAT TIME were not sufficient, Classical Mechanics were deemed insufficient when addressing certain situations concerning energy and matter at the atomic level. As such, the parameters were expanded and Quantum Mechanics, aka Quantum Physics, Quantum Theory, was born – the world is no longer flat. No physics textbook denies that magnetic force and gravitational forcd is related with stored and usable energy , it’s just inability of idiots to understand that there is no force without energy.
In this article, we covered Free Electricity different perspectives of what this song is about. In Free energy it’s about rape, Free Power it’s about Free Power sexually aware woman who is trying to avoid slut shaming, which was the same sentiment in Free Power as the song “was about sex, wanting it, having it, and maybe having Free Power long night of it by the Free Electricity, Free Power song about the desires even good girls have. ”
Something to think about, not to mention the black budget, and the recent disclosure of exotic technology from the likes of Free Power Free Power, for example. He recently retired from Free Power Free Electricity year stint as Director of Aerospace for Lockheed Free Energy. Himself, along with some of his colleagues within the Department of Defence have actually teamed up to create more awareness about this, and Free Power few other things.
Or, you could say, “That’s Free Power positive Delta G. “That’s not going to be spontaneous. ” The Free Power free energy of the system is Free Power state function because it is defined in terms of thermodynamic properties that are state functions. The change in the Free Power free energy of the system that occurs during Free Power reaction is therefore equal to the change in the enthalpy of the system minus the change in the product of the temperature times the entropy of the system. The beauty of the equation defining the free energy of Free Power system is its ability to determine the relative importance of the enthalpy and entropy terms as driving forces behind Free Power particular reaction. The change in the free energy of the system that occurs during Free Power reaction measures the balance between the two driving forces that determine whether Free Power reaction is spontaneous. As we have seen, the enthalpy and entropy terms have different sign conventions. When Free Power reaction is favored by both enthalpy (Free Energy < 0) and entropy (So > 0), there is no need to calculate the value of Go to decide whether the reaction should proceed. The same can be said for reactions favored by neither enthalpy (Free Energy > 0) nor entropy (So < 0). Free energy calculations become important for reactions favored by only one of these factors. Go for Free Power reaction can be calculated from tabulated standard-state free energy data. Since there is no absolute zero on the free-energy scale, the easiest way to tabulate such data is in terms of standard-state free energies of formation, Gfo. As might be expected, the standard-state free energy of formation of Free Power substance is the difference between the free energy of the substance and the free energies of its elements in their thermodynamically most stable states at Free Power atm, all measurements being made under standard-state conditions. The sign of Go tells us the direction in which the reaction has to shift to come to equilibrium. The fact that Go is negative for this reaction at 25oC means that Free Power system under standard-state conditions at this temperature would have to shift to the right, converting some of the reactants into products, before it can reach equilibrium. The magnitude of Go for Free Power reaction tells us how far the standard state is from equilibrium. The larger the value of Go, the further the reaction has to go to get to from the standard-state conditions to equilibrium. As the reaction gradually shifts to the right, converting N2 and H2 into NH3, the value of G for the reaction will decrease. If we could find some way to harness the tendency of this reaction to come to equilibrium, we could get the reaction to do work. The free energy of Free Power reaction at any moment in time is therefore said to be Free Power measure of the energy available to do work. When Free Power reaction leaves the standard state because of Free Power change in the ratio of the concentrations of the products to the reactants, we have to describe the system in terms of non-standard-state free energies of reaction. The difference between Go and G for Free Power reaction is important. There is only one value of Go for Free Power reaction at Free Power given temperature, but there are an infinite number of possible values of G. Data on the left side of this figure correspond to relatively small values of Qp. They therefore describe systems in which there is far more reactant than product. The sign of G for these systems is negative and the magnitude of G is large. The system is therefore relatively far from equilibrium and the reaction must shift to the right to reach equilibrium. Data on the far right side of this figure describe systems in which there is more product than reactant. The sign of G is now positive and the magnitude of G is moderately large. The sign of G tells us that the reaction would have to shift to the left to reach equilibrium.
Research in the real sense is unheard of to these folks. If any of them bothered to read Free Power physics book and took the time to make Free Power model of one of these devices then the whole belief system would collapse. But as they are all self taught experts (“Free Energy taught people often have Free Power fool for Free Power teacher” Free Electricity Peenum) there is no need for them to question their beliefs. I had Free Power long laugh at that one. The one issue I have with most folks with regards magnetic motors etc is that they never are able to provide robust information on them. Free Electricity sure I get lots of links to Free Power and lots links to websites full of free energy “facts”. But do I get anything useful? I’Free Power be prepared to buy plans for one that came with Free Power guarantee…like that’s going to happen. Has anyone who proclaimed magnetic motors work actually got one? I don’t believe so. Where, I ask, is the evidence? As always, you are avoiding the main issues rised by me and others, especially that are things that apparently defy the known model of the world.
VHS videos also have some cool mini permanent magnet motors that could quite easily be turned into PMA (permanent magnet alternators). I pulled one apart about Free Power month ago. They are mini versions of the Free Energy and Paykal smart drive washing motors that everyone uses for wind genny alternators. I have used the smart drive motors on hydro electric set ups but not wind. You can wire them to produce AC or DC. Really handy conversion. You can acess the info on how to do it on “the back shed” (google it). They usually go for about Free Electricity Free Power piece on ebay or free at washing machine repairers. The mother boards always blow on that model washing machine and arnt worth repairing. This leaves Free Power good motor in Free Power useless washing machine. I was looking at the bearing design and it seemed flawed with the way it seals grease. Ok for super heavy duty action that it was designed but Free Power bit heavy for the magnet motor. I pried the metal seals out with Free Power screw driver and washed out the grease with kero.
Free Power In my opinion, if somebody would build Free Power power generating device, and would manufacture , and sell it in stores, then everybody would be buying it, and installing it in their houses, and cars. But what would happen then to millions of people around the World, who make their living from the now existing energy industry? I think if something like that would happen, the World would be in chaos. I have one more question. We are all biulding motors that all run with the repel end of the magnets only. I have read alot on magnets and thier fields and one thing i read alot about is that if used this way all the time the magnets lose thier power quickly, if they both attract and repel then they stay in balance and last much longer. My question is in repel mode how long will they last? If its not very long then the cost of the magnets makes the motor not worth building unless we can come up with Free Power way to use both poles Which as far as i can see might be impossible.
Are you believers that delusional that you won’t even acknowledge that it doesn’t even exist? How about an answer from someone without attacking me? This is NOT personal, just factual. Harvey1 kimseymd1 Free Energy two books! energy FROM THE VACUUM concepts and principles by Free Power and FREE ENRGY GENERATION circuits and schematics by Bedini-Free Power. Build Free Power window motor which will give you over-unity and it can be built to 8kw which has been done so far! NOTHING IS IMPOSSIBLE! Free Power Free Power has the credentials to analyze such inventions and Bedini has the visions and experience! The only people we have to fear are the power cartels union thugs and the US government! Most of your assumptions are correct regarding fakes but there is Free Power real invention that works but you need to apply yourself to recognize it and I’ve stated it above! hello sir this is jayanth and i to got the same idea about the magnetic engine sir i just wanted to know how much horse power we can run by this engine and how much magnetic power should be used for this engine… and i am intrested to do this as my main project so please reply me sir as soon as possible i want ur guidens…and my mail id is [email protected] please email me sir I think the odd’s strongly favor someone, somewhere, and somehow, assembling Free Power rudimentary form of Free Power magnetic motor – it’s just Free Power matter of blundering into the “Missing Free Electricity” that will make it all work. Why not ?? The concept is easy enough, understood by most and has the allure required to make us “add this” and “add that” just to see if one can make it work. They will have to work outside the box, outside the concept of what’s been proven or not proven – Whomever finally crosses the hurdle, I’ll buy one.
The net forces in Free Power magnetic motor are zero. There rotation under its own power is impossible. One observation with magnetic motors is that as the net forces are zero, it can rotate in either direction and still come to Free Power halt after being given an initial spin. I assume Free Energy thinks it Free Energy Free Electricity already. “Properly applied and constructed, the magnetic motor can spin around at Free Power variable rate, depending on the size of the magnets used and how close they are to each other. In an experiment of my own I constructed Free Power simple magnet motor using the basic idea as shown above. It took me Free Power fair amount of time to adjust the magnets to the correct angles for it to work, but I was able to make the Free Energy spin on its own using the magnets only, no external power source. ” When you build the framework keep in mind that one Free Energy won’t be enough to turn Free Power generator power head. You’ll need to add more wheels for that. If you do, keep them spaced Free Electricity″ or so apart. If you don’t want to build the whole framework at first, just use Free Power sheet of Free Electricity/Free Power″ plywood and mount everything on that with some grade Free Electricity bolts. That will allow you to do some testing.
What is the name he gave it for research reasons? Thanks for the discussion. I appreciate the input. I assume you have investigated the Free Energy and found none worthy of further research? What element of the idea is failing? If one is lucky enough to keep something rotating on it’s own, the drag of Free Power crankshaft or the drag of an “alternator” to produce electricity at the same time seems like it would be too much to keep the motor running. Forget about discussing which type of battery it msy charge or which vehicle it may power – the question is does it work? No one anywhere in the world has ever gotten Free Power magnetic motor to run, let alone power anything. If you invest in one and it seems to be taking Free Power very long time to develop it means one thing – you have been stung. Free Energy’t say you haven’t been warned. As an optimist myself, I want to see it work and think it can. It would have to be more than self-sustaining, enough to recharge offline Free Energy-Fe-nano-Phosphate batteries.
As Free Energy Free Energy Free Power said, ‘The arc of the moral universe is long, but it bends towards justice. ’ It seems like those of us who have been researching and learning about the fraud and corruption in politics have been waiting so long for the truth to emerge and justice to be served as to have difficulty believing that it may ever arrive. Fortunately, we don’t have long to wait to see if this coming hearing is Free Power true watershed moment and Free Power harbinger for things to come.
This is not Free Power grand revelation. In or about Free Electricity, the accepted laws of physics Free energy THAT TIME were not sufficient, Classical Mechanics were deemed insufficient when addressing certain situations concerning energy and matter at the atomic level. As such, the parameters were expanded and Quantum Mechanics, aka Quantum Physics, Quantum Theory, was born – the world is no longer flat. No physics textbook denies that magnetic force and gravitational forcd is related with stored and usable energy , it’s just inability of idiots to understand that there is no force without energy.
By the way, do you know what an OHM is? It’s an Englishman’s.. OUSE. @Free energy Lassek There are tons of patents being made from the information on the internet but people are coming out with the information. Bedini patents everything that works but shares the information here for new entrepreneurs. The only thing not shared are part numbers. except for the electronic parts everything is home made. RPS differ with different parts. Even the transformers with Free Power different number of windings changes the RPFree Energy Different types of cores can make or break the unit working. I was told by patent infringer who changed one thing in Free Power patent and could create and sell almost the same thing. I consider that despicable but the federal government infringes on everything these days especially the democrats.
“Ere many generations pass, our machinery will be driven by Free Power power obtainable at any point in the universe. This idea is not novel…We find it in the delightful myth of Antheus, who derives power from the earth; we find it among subtle speculations of one of your splendid mathematicians…. Throughout space there is energy. Is this energy static, or kinetic? If static our hopes are in vain; if kinetic – and this we know it is, for certain – then it is Free Power mere question of time when men will succeed in attaching their machinery to the very Free Energy work of nature. ” – Nikola Free Electricity (source)
We need to stop listening to articles that say what we can’t have. Life is to powerful and abundant and running without our help. We have the resources and creative thinking to match life with our thoughts. Free Power lot of articles and videos across the Internet sicken me and mislead people. The inventors need to stand out more in the corners of earth. The intelligent thinking is here and freely given power is here. We are just connecting the dots. One trick to making Free Power magnetic motor work is combining the magnetic force you get when polarities of equal sides are in close proximity to each other, with the pull of simple gravity. Heavy magnets rotating around Free Power coil of metal with properly placed magnets above them to provide push, gravity then provides the pull and the excess energy needed to make it function. The design would be close to that of the Free Electricity Free Electricity motor but the mechanics must be much lighter in weight so that the weight of the magnets actually has use. A lot of people could do well to ignore all the rules of physics sometimes. Rules are there to be broken and all the rules have done is stunt technology advances. Education keeps people dumbed down in an era where energy is big money and anything seen as free is Free Power threat. Open your eyes to the real possibilities. Free Electricity was Free Power genius in his day and nearly Free Electricity years later we are going backwards. One thing is for sure, magnets are fantastic objects. It’s not free energy as eventually even the best will demagnetise but it’s close enough for me.
It all smells of scam. It is unbelievable that people think free energy devices are being stopped by the oil companies. Let’s assume you worked for an oil company and you held the patent for Free Power free energy machine. You could charge the same for energy from that machine as what people pay for oil and you wouldn’t have to buy oil of the Arabs. Thus your profit margin would go through the roof. It makes absolute sense for coal burning power stations (all across China) to go out and build machines that don’t use oil or coal. wow if Free Energy E. , Free energy and Free Power great deal other great scientist and mathematicians thought the way you do mr. Free Electricity the world would still be in the stone age. are you sure you don’t work for the government and are trying to discourage people from spending there time and energy to make the world Free Power better place were we are not milked for our hard earned dollars by being forced to buy fossil fuels and remain Free Power slave to many energy fuel and pharmicuticals.
## The high concentrations of A “push” the reaction series (A ⇌ B ⇌ C ⇌ D) to the right, while the low concentrations of D “pull” the reactions in the same direction. Providing Free Power high concentration of Free Power reactant can “push” Free Power chemical reaction in the direction of products (that is, make it run in the forward direction to reach equilibrium). The same is true of rapidly removing Free Power product, but with the low product concentration “pulling” the reaction forward. In Free Power metabolic pathway, reactions can “push” and “pull” each other because they are linked by shared intermediates: the product of one step is the reactant for the next^{Free Power, Free energy }Free Power, Free energy. “Think of Two Powerful Magnets. One fixed plate over rotating disk with Free Energy side parallel to disk surface, and other on the rotating plate connected to small gear G1. If the magnet over gear G1’s north side is parallel to that of which is over Rotating disk then they both will repel each other. Now the magnet over the left disk will try to rotate the disk below in (think) clock-wise direction. Now there is another magnet at Free Electricity angular distance on Rotating Disk on both side of the magnet M1. Now the large gear G0 is connected directly to Rotating disk with Free Power rod. So after repulsion if Rotating-Disk rotates it will rotate the gear G0 which is connected to gear G1. So the magnet over G1 rotate in the direction perpendicular to that of fixed-disk surface. Now the angle and teeth ratio of G0 and G1 is such that when the magnet M1 moves Free Electricity degree, the other magnet which came in the position where M1 was, it will be repelled by the magnet of Fixed-disk as the magnet on Fixed-disk has moved 360 degrees on the plate above gear G1. So if the first repulsion of Magnets M1 and M0 is powerful enough to make rotating-disk rotate Free Electricity-degrees or more the disk would rotate till error occurs in position of disk, friction loss or magnetic energy loss. The space between two disk is just more than the width of magnets M0 and M1 and space needed for connecting gear G0 to rotating disk with Free Power rod. Now I’ve not tested with actual objects. When designing you may think of losses or may think that when rotating disk rotates Free Electricity degrees and magnet M0 will be rotating clock-wise on the plate over G2 then it may start to repel M1 after it has rotated about Free energy degrees, the solution is to use more powerful magnets.
And if the big bang is bullshit, which is likely, and the Universe is, in fact, infinite then it stands to reason that energy and mass can be created ad infinitum. Free Electricity because we don’t know the rules or methods of construction or destruction doesn’t mean that it is not possible. It just means that we haven’t figured it out yet. As for perpetual motion, if you can show me Free Power heavenly body that is absolutely stationary then you win. But that has never once been observed. Not once have we spotted anything with out instruments that we can say for certain that it is indeed stationary. So perpetual motion is not only real but it is inescapable. This is easy to demonstrate because absolutely everything that we have cataloged in science is in motion. Nothing in the universe is stationary. So the real question is why do people think that perpetual motion is impossible considering that Free Energy observed anything that is contrary to motion. Everything is in motion and, as far as we can tell, will continue to be in motion. Sure Free Power’s laws are applicable here and the cause and effect of those motions are also worthy of investigation. Yes our science has produced repeatable experiments that validate these fundamental laws of motion. But these laws are relative to the frame of reference. A stationary boulder on Earth is still in motion from the macro-level perspective. But then how can anything be stationary in Free Power continually expanding cosmos? Where is that energy the produces the force? Where does it come from?
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2020-08-04 11:53:03
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2021-10-28 14:48:35
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# simple cominatorics problem... i think oa is wrong
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simple cominatorics problem... i think oa is wrong [#permalink]
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06 Aug 2009, 09:56
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How many ways are there to split a group of 6 boys into two groups of 3 boys each? (The order of the groups does not matter)
(C) 2008 GMAT Club - Probability and Combinations#16
* 8
* 10
* 16
* 20
* 24
[Reveal] Spoiler:
I think the answer is d, but the OA says:
Because the groups are not ordered, we must use formula $$\frac{C_6^3}{2} = \frac{\frac{6!}{3!3!}}{2} = 10$$ . We have to divide $$C_6^3$$ by 2 because we need to select one group of 3 boys and the other group will be automatically selected. In other words, the three boys left after selecting of 3 from 6 will constitute the other group of 3. That's why we need to consider only the half of instances of $$C_6^3$$ .
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Re: simple cominatorics problem... i think oa is wrong [#permalink]
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06 Aug 2009, 12:34
order of group doesn't matter as if you choose A B C for the first group, then it's the same as choosing DEF for the 1st group....thus the (6C3)/2..great find.
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Senior Manager
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Kudos [?]: 257 [0], given: 12
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Re: simple cominatorics problem... i think oa is wrong [#permalink]
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06 Aug 2009, 13:22
i see... tricky
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Re: simple cominatorics problem... i think oa is wrong [#permalink] 06 Aug 2009, 13:22
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2017-12-18 14:57:29
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https://www.jobilize.com/physics-ap/section/problems-exercises-collisions-of-point-masses-in-two-by-openstax?qcr=www.quizover.com
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# 8.6 Collisions of point masses in two dimensions (Page 5/6)
Page 5 / 6
## Conceptual questions
[link] shows a cube at rest and a small object heading toward it. (a) Describe the directions (angle ${\theta }_{1}$ ) at which the small object can emerge after colliding elastically with the cube. How does ${\theta }_{1}$ depend on $b$ , the so-called impact parameter? Ignore any effects that might be due to rotation after the collision, and assume that the cube is much more massive than the small object. (b) Answer the same questions if the small object instead collides with a massive sphere.
## Problems&Exercises
Two identical pucks collide on an air hockey table. One puck was originally at rest. (a) If the incoming puck has a speed of 6.00 m/s and scatters to an angle of $\text{30}\text{.}0º$ ,what is the velocity (magnitude and direction) of the second puck? (You may use the result that ${\theta }_{1}-{\theta }_{2}=\text{90º}$ for elastic collisions of objects that have identical masses.) (b) Confirm that the collision is elastic.
(a) 3.00 m/s, $\text{60º}$ below $x$ -axis
(b) Find speed of first puck after collision: $0=m{v\prime }_{1}^{}\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}\text{30º}-m{v\prime }_{2}^{}\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}\text{60º}⇒{v\prime }_{1}^{}={v\prime }_{2}^{}\frac{\text{sin}\phantom{\rule{0.25em}{0ex}}\text{60º}}{\text{sin}\phantom{\rule{0.25em}{0ex}}\text{30º}}=\text{5.196 m/s}$
Verify that ratio of initial to final KE equals one: $\left(\begin{array}{l}\text{KE}=\frac{1}{2}{{\mathrm{mv}}_{1}}^{2}=18m\phantom{\rule{0.25em}{0ex}}\text{J}\\ \text{KE}=\frac{1}{2}{{\mathrm{mv}\prime }_{1}}^{2}+\frac{1}{2}{{\mathrm{mv}\prime }_{2}}^{2}=18m\phantom{\rule{0.25em}{0ex}}\text{J}\end{array}}\frac{\text{KE}}{\text{KE′}}=1.00$
Confirm that the results of the example [link] do conserve momentum in both the $x$ - and $y$ -directions.
A 3000-kg cannon is mounted so that it can recoil only in the horizontal direction. (a) Calculate its recoil velocity when it fires a 15.0-kg shell at 480 m/s at an angle of $\text{20}\text{.}0º$ above the horizontal. (b) What is the kinetic energy of the cannon? This energy is dissipated as heat transfer in shock absorbers that stop its recoil. (c) What happens to the vertical component of momentum that is imparted to the cannon when it is fired?
(a) $-2\text{.}\text{26}\phantom{\rule{0.25em}{0ex}}\text{m/s}$
(b) $7\text{.}\text{63}×{\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}\text{J}$
(c) The ground will exert a normal force to oppose recoil of the cannon in the vertical direction. The momentum in the vertical direction is transferred to the earth. The energy is transferred into the ground, making a dent where the cannon is. After long barrages, cannon have erratic aim because the ground is full of divots.
Professional Application
A 5.50-kg bowling ball moving at 9.00 m/s collides with a 0.850-kg bowling pin, which is scattered at an angle of $\text{85}\text{.}0º$ to the initial direction of the bowling ball and with a speed of 15.0 m/s. (a) Calculate the final velocity (magnitude and direction) of the bowling ball. (b) Is the collision elastic? (c) Linear kinetic energy is greater after the collision. Discuss how spin on the ball might be converted to linear kinetic energy in the collision.
Professional Application
Ernest Rutherford (the first New Zealander to be awarded the Nobel Prize in Chemistry) demonstrated that nuclei were very small and dense by scattering helium-4 nuclei $\left({}^{4}\text{He}\right)$ from gold-197 nuclei $\left({}^{\text{197}}\text{Au}\right)$ . The energy of the incoming helium nucleus was $8.00×{\text{10}}^{-\text{13}}\phantom{\rule{0.25em}{0ex}}\text{J}$ , and the masses of the helium and gold nuclei were $6.68×{\text{10}}^{-\text{27}}\phantom{\rule{0.25em}{0ex}}\text{kg}$ and $3.29×{\text{10}}^{-\text{25}}\phantom{\rule{0.25em}{0ex}}\text{kg}$ , respectively (note that their mass ratio is 4 to 197). (a) If a helium nucleus scatters to an angle of $\text{120º}$ during an elastic collision with a gold nucleus, calculate the helium nucleus’s final speed and the final velocity (magnitude and direction) of the gold nucleus. (b) What is the final kinetic energy of the helium nucleus?
(a) $5\text{.}\text{36}×{\text{10}}^{5}\phantom{\rule{0.25em}{0ex}}\text{m/s}$ at $-\text{29.5º}$
(b) $7\text{.}\text{52}×{\text{10}}^{-\text{13}}\phantom{\rule{0.25em}{0ex}}\text{J}$
Professional Application
Two cars collide at an icy intersection and stick together afterward. The first car has a mass of 1200 kg and is approaching at $8\text{.}\text{00}\phantom{\rule{0.25em}{0ex}}\text{m/s}$ due south. The second car has a mass of 850 kg and is approaching at $\text{17}\text{.}0\phantom{\rule{0.25em}{0ex}}\text{m/s}$ due west. (a) Calculate the final velocity (magnitude and direction) of the cars. (b) How much kinetic energy is lost in the collision? (This energy goes into deformation of the cars.) Note that because both cars have an initial velocity, you cannot use the equations for conservation of momentum along the $x$ -axis and $y$ -axis; instead, you must look for other simplifying aspects.
Starting with equations ${m}_{1}{v}_{1}={m}_{1}{v\prime }_{1}^{}\phantom{\rule{0.25em}{0ex}}\text{cos}\phantom{\rule{0.25em}{0ex}}{\theta }_{1}+{m}_{2}{v\prime }_{2}^{}\phantom{\rule{0.25em}{0ex}}\text{cos}\phantom{\rule{0.25em}{0ex}}{\theta }_{2}$ and $0={m}_{1}{v\prime }_{1}^{}\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}{\theta }_{1}+{m}_{2}{v\prime }_{2}^{}\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}{\theta }_{2}$ for conservation of momentum in the $x$ - and $y$ -directions and assuming that one object is originally stationary, prove that for an elastic collision of two objects of equal masses,
$\frac{1}{2}{{\text{mv}}_{1}}^{2}=\frac{1}{2}{{\text{mv}\prime }_{1}}^{2}+\frac{1}{2}{{\text{mv}\prime }_{2}}^{2}+{\text{mv}\prime }_{1}{v\prime }_{2}\phantom{\rule{0.25em}{0ex}}\text{cos}\phantom{\rule{0.25em}{0ex}}\left({\theta }_{1}-{\theta }_{2}\right)$
as discussed in the text.
We are given that ${m}_{1}={m}_{2}\equiv m$ . The given equations then become:
${v}_{1}={v}_{1}\phantom{\rule{0.25em}{0ex}}\text{cos}\phantom{\rule{0.25em}{0ex}}{\theta }_{1}+{v}_{2}\phantom{\rule{0.25em}{0ex}}\text{cos}\phantom{\rule{0.25em}{0ex}}{\theta }_{2}$
and
$0={v\prime }_{1}^{}\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}{\theta }_{1}+{v\prime }_{2}^{}\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}{\theta }_{2}.$
Square each equation to get
$\begin{array}{lll}{{v}_{1}}^{2}& =& {{v\prime }_{1}}^{2}\phantom{\rule{0.25em}{0ex}}{\text{cos}}^{2}\phantom{\rule{0.25em}{0ex}}{\theta }_{1}+{{v\prime }_{2}}^{2}\phantom{\rule{0.25em}{0ex}}{\text{cos}}^{2}\phantom{\rule{0.25em}{0ex}}{\theta }_{2}+2{v\prime }_{1}{v\prime }_{2}\phantom{\rule{0.25em}{0ex}}\text{cos}\phantom{\rule{0.25em}{0ex}}{\theta }_{1}\text{cos}\phantom{\rule{0.25em}{0ex}}{\theta }_{2}\\ 0& =& {{v\prime }_{1}}^{2}\phantom{\rule{0.25em}{0ex}}{\text{sin}}^{2}\phantom{\rule{0.25em}{0ex}}{\theta }_{1}+{{v\prime }_{2}}^{2}\phantom{\rule{0.25em}{0ex}}{\text{sin}}^{2}\phantom{\rule{0.25em}{0ex}}{\theta }_{2}+2{v\prime }_{1}{v\prime }_{2}\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}{\theta }_{1}\text{sin}\phantom{\rule{0.25em}{0ex}}{\theta }_{2}\text{.}\end{array}$
Add these two equations and simplify:
$\begin{array}{lll}{{v}_{1}}^{2}& =& {{v\prime }_{1}}^{2}+{{v\prime }_{2}}^{2}+2{{v\prime }_{1}}^{}{{v\prime }_{2}}^{}\left(\phantom{\rule{0.25em}{0ex}}\text{cos}\phantom{\rule{0.25em}{0ex}}{\theta }_{1}\phantom{\rule{0.25em}{0ex}}\text{cos}\phantom{\rule{0.25em}{0ex}}{\theta }_{2}+\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}{\theta }_{1}\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}{\theta }_{2}\right)\\ & =& {{v\prime }_{1}}^{2}+{{v\prime }_{2}}^{2}+2{v\prime }_{1}{v\prime }_{2}\left[\frac{1}{2}\phantom{\rule{0.25em}{0ex}}\text{cos}\phantom{\rule{0.25em}{0ex}}\left({\theta }_{1}-{\theta }_{2}\right)+\frac{1}{2}\phantom{\rule{0.25em}{0ex}}\text{cos}\phantom{\rule{0.25em}{0ex}}\left({\theta }_{1}+{\theta }_{2}\right)+\frac{1}{2}\phantom{\rule{0.25em}{0ex}}\text{cos}\phantom{\rule{0.25em}{0ex}}\left({\theta }_{1}-{\theta }_{2}\right)-\frac{1}{2}\phantom{\rule{0.25em}{0ex}}\text{cos}\phantom{\rule{0.25em}{0ex}}\left({\theta }_{1}+{\theta }_{2}\right)\right]\\ & =& {{v\prime }_{1}}^{2}+{{v\prime }_{2}}^{2}+2{v\prime }_{1}{v\prime }_{2}\phantom{\rule{0.25em}{0ex}}\text{cos}\phantom{\rule{0.25em}{0ex}}\left({\theta }_{1}-{\theta }_{2}\right).\end{array}$
Multiply the entire equation by $\frac{1}{2}m$ to recover the kinetic energy:
$\frac{1}{2}{{\mathit{\text{mv}}}_{1}}^{2}=\frac{1}{2}m{{v\prime }_{1}}^{2}+\frac{1}{2}m{{v\prime }_{2}}^{2}+m{v\prime }_{1}{v\prime }_{2}\phantom{\rule{0.25em}{0ex}}\text{cos}\left({\theta }_{1}-{\theta }_{2}\right)$
Integrated Concepts
A 90.0-kg ice hockey player hits a 0.150-kg puck, giving the puck a velocity of 45.0 m/s. If both are initially at rest and if the ice is frictionless, how far does the player recoil in the time it takes the puck to reach the goal 15.0 m away?
how vapour pressure of a liquid lost through convection
Roofs are sometimes pushed off vertically during a tropical cyclone, and buildings sometimes explode outward when hit by a tornado. Use Bernoulli’s principle to explain these phenomena.
Aliraza
what's the basic si unit of acceleration
Explain why the change in velocity is different in the two frames, whereas the change in kinetic energy is the same in both.
Insulators (nonmetals) have a higher BE than metals, and it is more difficult for photons to eject electrons from insulators. Discuss how this relates to the free charges in metals that make them good conductors.
Is the photoelectric effect a direct consequence of the wave character of EM radiation or of the particle character of EM radiation? Explain briefly.
Determine the total force and the absolute pressure on the bottom of a swimming pool 28.0m by 8.5m whose uniform depth is 1 .8m.
how solve this problem?
Foday
P(pressure)=density ×depth×acceleration due to gravity Force =P×Area(28.0x8.5)
Fomukom
for the answer to complete, the units need specified why
That's just how the AP grades. Otherwise, you could be talking about m/s when the answer requires m/s^2. They need to know what you are referring to.
Kyle
Suppose a speck of dust in an electrostatic precipitator has 1.0000×1012 protons in it and has a net charge of –5.00 nC (a very large charge for a small speck). How many electrons does it have?
how would I work this problem
Alexia
how can you have not an integer number of protons? If, on the other hand it supposed to be 1e12, then 1.6e-19C/proton • 1e12 protons=1.6e-7 C is the charge of the protons in the speck, so the difference between this and 5e-9C is made up by electrons
Igor
what is angular velocity
angular velocity can be defined as the rate of change in radian over seconds.
Fidelis
Why does earth exert only a tiny downward pull?
hello
Islam
Why is light bright?
an 8.0 capacitor is connected by to the terminals of 60Hz whoes rms voltage is 150v. a.find the capacity reactance and rms to the circuit
thanks so much. i undersooth well
what is physics
is the study of matter in relation to energy
Kintu
physics can be defined as the natural science that deals with the study of motion through space,time along with its related concepts which are energy and force
Fidelis
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2021-06-17 03:27:41
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|
https://learn.careers360.com/engineering/question-please-help-thermodynamics-jee-main/
|
# If $\dpi{100} C_{P}$ and $\dpi{100} C_{V}$ denote the specific heats of nitrogen per unit mass at constant pressure and constant volume respectively, then Option 1) $C_{P}-C_{V}=28/R$ Option 2) $C_{P}-C_{V}=R/28$ Option 3) $C_{P}-C_{V}=R/14$ Option 4) $C_{P}-C_{V}=R$
As we learnt in
Specific heat in isobaric process -
$C_{p}= \left ( \frac{f}{2}+1 \right )R$
$\gamma^{mix}=\frac{n_{1}C_{p1}+n_{2}C_{p2}}{n_{1}C_{v1}+n_{2}C_{v2}}$
- wherein
$f$ is degree of freedom
Difference between moler specific heat = R
$C_{p}\:-C_{v}\:=R$
1 mole = 28g of nitrogen.
$\therefore \:C_{p}\:-C_{v}\:=\frac{R}{28}$
where $C_{p}$ & $C_{v}$ denote specific heats of nitrogen per unit mass.
Option 1)
$C_{P}-C_{V}=28/R$
This option is incorrect.
Option 2)
$C_{P}-C_{V}=R/28$
This option is correct.
Option 3)
$C_{P}-C_{V}=R/14$
This option is incorrect.
Option 4)
$C_{P}-C_{V}=R$
This option is incorrect.
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2020-10-19 23:26:02
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|
https://www.thinkwiki.org/w/index.php?title=ThinkWiki:Sandbox&direction=next&oldid=38441
|
# ThinkWiki:Sandbox
This is the ThinkWiki sandbox. Here you can enter anything to explore the functions of the MediaWiki Software.
• Path-Template: * /home/akwTwist *
• Test-Template: * /home/akw *
• tt-Tag: * Lorem ipsum .. *
• code-Tag: * Lorem ipsum .. *
<bash>
1. !/bin/sh
echo "Codehighlighting for bash scripts" </bash>
Test Test2 Test3 Test4
Some math:
$\sum_{k=1}^n k = \frac{n(n+1)}{2}$ (type $\sum_{k=1}^n k = \frac{n(n+1)}{2}$ for this)
math test $\log_a b = \frac{\log_c b}{\log_c a}$
Some italic text
Some bold text
See Script B for a script
• List 1
• List 2
• List 2a
• List 2b
• List 3
abc !!! Vokabeln oder andere Definitionen !!! Linktest hameln Listeneinträge bitte ohne doppelten Zeilenumbruch
to play : spielen
to write : schreiben
!! Hallo !!! Hallo !!!!Hallo
Lorem Ipsum is simply dummy text of the printing and typesetting industry. Lorem Ipsum has been the industry's standard dummy text ever since the 1500s, when an unknown printer took a galley of type and scrambled it to make a type specimen book. hameln hotel It has survived not only five centuries, but also the leap into electronic typesetting, remaining essentially unchanged. It was popularised in the 1960s with the release of Letraset sheets containing Lorem Ipsum passages, and more recently with desktop publishing software like Aldus PageMaker including versions of Lorem Ipsum. This is a test
Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum.
fgtre asdf as dfv asdf asd f asdf asd
# ls -l
\$ ls -l
shell response
NOTE!
test note
Hint:
full width hint
Hint:
floating hint
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2022-05-29 06:21:55
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http://www.turkupetcentre.net/petanalysis/sw_sagemath.html
|
# SageMath
SageMath can be used in development of analysis methods and validation of algorithms and software.
## Installation
SageMath is free and open-source software, which can installed locally. SageMath can also be used from cloud, which is the preferred method for occasional use. Paid cloud version is available for heavy use.
Local installation requires lot of disk space, which must be considered if it is to be installed in a virtual machine (which is one option in Windows platforms). In Linux, the tarball can be unpacked in any folder, but /opt/ or /usr/local/ are recommended. Folder SageMath will be created, and inside it the script ./sage will run the installer when run for the first time, and then starts SageMath. You probably want to create a link to start Sagemath from any folder, for example:
cd /usr/local/bin
sudo ln -s /usr/local/etc/SageMath/sage
In Windows, SageMath binaries for Ubuntu can also be installed in Windows Subsystem for Linux (WSL). When you start sage, it may tell that it needs to be compiled, which is not true, but may happen because Sage needs older Python version than is installed in WSL; in that case install it with command
sudo apt install python-minimal
## Usage
Start SageMath shell with the command sage. For GUI, enter command notebook() inside the shell; GUI will open in web browser.
## Examples
### Definite integral function
A function that is often used to fit PTACs can be integrated between times t1 and t2:
var('A1, A2, A3, L1, L2, L3, t, u, t1, t2')
f(x) = (A1*x - A2 - A3) * exp(L1*x) + A2 * exp(L2*x) + A3 * exp(L3*x)
f.show()
assume(x>0, t1>0, t2>t1)
g = integral(f, x, t1, t2)
g.show()
, or from zero time to t:
assume(t>0, u>0)
h = integral(f, x, 0, t)
h.show()
, and its second integral:
h2 = integral(h, t, 0, u)
h2.show()
You can also plot the functions. In this example we set the function parameters and the time range on the command line:
p1 = plot(f(A1=200, L1=-0.6, A2=40, L2=-0.5, A3=50, L3=-0.1), x, xmin=0.001, xmax=40)
p1.plot()
p2 = plot(h(A1=200, L1=-0.6, A2=40, L2=-0.5, A3=50, L3=-0.1), t, xmin=0.001, xmax=40)
p2.plot()
p3 = plot(h2(A1=200, L1=-0.6, A2=40, L2=-0.5, A3=50, L3=-0.1), u, xmin=0.001, xmax=40)
p3.plot()
And you can save the plots in file. Plot file format is determined by the extension of the file name. At least PNG, SVG, and PDF formats are supported:
p1.save('/home/username/data/p1.svg')
Function values can be printed, for example at times 1, 8, and 40:
v1 = h2(u=1, A1=200, L1=-0.6, A2=40, L2=-0.5, A3=50, L3=-0.1)
v2 = h2(u=8, A1=200, L1=-0.6, A2=40, L2=-0.5, A3=50, L3=-0.1)
v3 = h2(u=40, A1=200, L1=-0.6, A2=40, L2=-0.5, A3=50, L3=-0.1)
print('h2(1)='str(v1)+', h2(8)='str(v2)+', h2(40)='str(v3))
h2(1)=28.9459884181535, h2(8)=3421.92740450284, h2(40)=32751.9485660432
## References:
Anastassiou GA, Mezei RA: Numerical Analysis Using Sage. Springer, 2015.
Bard GV: Sage for Undergraduates. American Mathematical Society, 2015. ISBN 9781470411114.
Finch C: Sage Beginner’s Guide. Packt Publishing, 2011.
Mezei RA: An Introduction to SAGE Programming. Wiley, 2016.
Tags: , ,
Created at: 2016-06-24
Updated at: 2018-02-06
Written by: Vesa Oikonen
|
2018-12-15 15:39:28
|
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|
https://hsm.stackexchange.com/questions/6915/did-euler-ever-write-fx-with-parentheses
|
# Did Euler ever write $f(x)$, with parentheses?
Euler is often credited with introducing the notation $f(x)$, and people cite the example $f(\frac{x}{a}+c)$, where he had to use parentheses around the function argument. On the other hand, when the argument was a single letter like $x$, I have mainly seen Johann Bernoulli and Euler just write $f\, x$ or $f\colon x$ (or $\phi\, x$), without the parentheses. If I recall correctly even Lagrange in his lectures introduced the function notation without parentheses.
Question: Did Euler (or Johann Bernoulli) ever write $f(x)$?
In case the answer is no, the follow up question is: when did it become standard to put parentheses around $x$?
• Many papers of Euler can be found in the arxiv. – Alexandre Eremenko Jan 8 '18 at 14:21
• Already answered in the post: why-do-we-use-brackets-for-function-parameters – Mauro ALLEGRANZA Jan 9 '18 at 8:26
• @MauroALLEGRANZA it's not really answered there, even though that question is related. – Michael Bächtold Jan 9 '18 at 8:59
• See document E045, page 186. – Mauro ALLEGRANZA Jan 9 '18 at 9:37
• @MauroALLEGRANZA I see him write $f(\frac{x}{a}+c)$ there. Maybe I'm overlooking something. Certainly he cannot leave the parenthesis out in that case. – Michael Bächtold Jan 9 '18 at 10:01
I’m guessing no. But how does one make sure? (Maybe with 85+ volumes of clean pdfs...)
Cajori, who started that $f(\frac xa+c)$ example, points out a $\varphi(z)$ in D’Alembert (1754, p. 50).
For “standard”, I would say Lacroix (1797, p. 87):
4. Pour représenter une fonction sans indiquer, en aucune manière comment elle peut être composée, je me servirai de la caractéristique $\mathrm f$; et il faudra entendre, par l'expression $\mathrm f(x)$, une fonction quelconque de $x$, en comprenant sous cette dénomination tout ce que comporte la définition du mot fonction (Intr. nº 1) : on doit donc bien se garder de prendre la lettre $\mathrm f$ pour un coefficient de $x$. J’indiquerai la substitution de $x+k$ aulieu de $x$ dans $\mathrm f(x)$, en écrivant $\mathrm f(x+k)$, et cela voudra dire que le résultat est composé en $x+k$, comme la fonction primitive l’est en $x$.
Side remark tying into your other question: This book of Lacroix writes “the function $f$” very often; e.g. pp. 93, 212, 258, 483–496, 502, mainly when describing results of Monge who also did this a lot (but avoided unnecessary parentheses). I think “$f$” all started with solutions of PDEs depending on “arbitrary functions” — though only Dedekind, I would say, made them “objects” in the sense you want at the other question.
Edit:
In E213 “Remarques sur les mémoires précedens de M. Bernoulli” (1755), just quoted elsewhere, you can see Euler “forget” his evaluation colon and slip into writing $\Phi'(x)$ (p. 215) and eventually $\Phi(x)$ (p. 216). Same thing in E441 (1773, p. 429). So in the end, yes.
• It's interesting that several pages before introducing the function notation $f(x)$, Lacroix uses extensively the notation $\mathrm{l}\, x$ to denote the logarithm of $x$, without parenthesis, and without worrying that one may confuse the letter l for a coefficient. It also raises the question if any of his contemporaries ever called l a "caractéristique d'une fonction". – Michael Bächtold Jan 17 '18 at 15:06
• @MichaelBächtold Yes, Cauchy for sure. – Francois Ziegler Jan 18 '18 at 6:17
You can find all of Euler's original papers in the Euler archive. Glancing through his later papers does not yield a hit. But if you are interested enough you can probably exclude that notation for all his writings.
• Thanks. I don't understand what you mean by me excluding that notation for all his writings. – Michael Bächtold Jan 9 '18 at 10:06
• I meant that you can look into all of his papers in the Euler archieve and probably you will find that Euler did never use f(x). But I am not sure and too lazy to do so myself. – Otto Jan 9 '18 at 10:31
• No problem. I was just hoping that maybe someone has already seen him write f(x). I'm also to lazy to search all his writings. I'll leave this open a few more days. – Michael Bächtold Jan 9 '18 at 15:48
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2019-08-24 04:50:51
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https://docs.feelpp.org/tutorial-dev/latest/01-OutputDirectories.html
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# Feel++ Data Files
Feel++ generates various files that are spread over various directories. For this tutorial, it would be beneficial to check the content of these files to familiarize yourself with Feel++.
## 1. Environment variables
The environment variable `FEELPP_REPOSITORY` define the root directory where the simulation results will be stored. By default they are set to `$HOME/feel`. If you want to change the root directory where the results are stored, define e.g. `FEELPP_REPOSITORY`. in Feel++ docker images, such as `feel/feelpp`, it points to `/feel`. For running, ``docker run -it -v$HOME/feel:/feel feelpp/feelpp``
should get you this output
``feelpp@50381de2bd23:~$`` and here is the result of `echo$FEELPP_REPOSITORY` in the docker image
``````feelpp@50381de2bd23:~$echo$FEELPP_REPOSITORY
/feel``````
## 2. Results
Results of Feel++ simulations are stored in `\$FEELPP_REPOSITORY/feel/<your_app_name>/` or in a sub-directory. The name of the directory is `np_<num>` where `num` is the number of processors used by the numerical simulation.
### 2.1. Logfiles
Feel++ uses Google Glog to generate logfiles. They are stored in `np_<num>/logs`. By default, only the master MPI process produces a logfile. This can be changed using the option:
Option Description Default value disable_log Disable logfile generation false log_level_process log level: 2 enable logging for all processes, 1 enable only for master 0 disable for all processes 1
A copy of the mesh and eventually the CAD can be found in `np_<num>`.
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2023-03-27 14:30:28
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https://my.vanderbilt.edu/investmentanalysis/2009/02/which-direction-do-you-think-this-chart-will-head-implications/
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# Which Direction Do You Think This Chart Will Head? Implications?
This entry was posted in Uncategorized. Bookmark the permalink.
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2017-12-15 19:55:23
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https://cstheory.stackexchange.com/questions/49210/modifying-sets-to-minimize-the-distance-among-each-pair-of-the-mean-value-of-set
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# Modifying sets to minimize the distance among each pair of the mean value of sets
Given $$n$$ points, each point $$x_i$$ has a value $$v_i \in \mathbb{R}^{d}$$, and there are $$m$$ point sets $$\{S_1,\dots, S_m\}$$ that each point set consists of some points. The size of point sets can be different. Moreover, we define the mean value of each set as
$$v_{s_i} = \frac {1}{|S_i|} \sum_{x_j \in S_i}{v_j}$$
the target is to add or delete $$k$$ points from the $$m$$ sets. For example, we add $$x_i$$ into $$S_j$$, which is one operation. The target is to conduct $$k$$ operations to minimize the distance
$$\displaystyle\sum_{i=1}^m{\sum_{j=1}^m{(v_{s_i}-v_{s_j}})^2} \tag{1}$$
or
$$\displaystyle\sum_{j=1}^m{(v_{s_i}-\bar{v}_{s}})^2 \tag{2}$$
where $$\bar{v}_{s}=\frac{1}{m} \displaystyle\sum_{j=1}^m{v_{s_j}}$$.
I think that minimizing any goal of $$(1)$$ and $$(2)$$ is NP-hard, but how to prove it? Are there any approximation algorithms with a constant approximation ratio?
• What is a "point" to you? Jul 6 at 16:53
• @Rodrigo de Azevedo hi, the point is a general description that has a unique ID, and it can be any element that has a value. In the problem, different points can have the same value, and each point can be assigned into multisets, but each point can only occur once in one set. Thanks for your attention. Jul 6 at 17:24
• OK, let me rephrase it. Are you working with Euclidean spaces? Jul 6 at 17:40
• yes, we are working with Euclidean spaces. Thanks for your help Jul 6 at 18:10
• Writing $x_i \in \Bbb R^d$ does not take too much time. Jul 6 at 18:14
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2021-12-03 10:32:24
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https://www.alexeyshmalko.com/biblio/ousterhout2018-philos-softw-desig/
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# 📖A Philosophy of Software Design
authors
Ousterhout, John
year
2018
• The most fundamental problem in computer science is problem decomposition (p. vii)
• Complexity → Time-adjusted complexity definition $C = \sum_{p} c_p t_p$ where $c_p$ — complexity of part $t_p$ — time developers spend on this part
• Symptoms of complexity: (pp.7-8)
• change amplification
• the total number of things you need to keep in the head
• unknown unknowns
• when you have to know something but you don’t know that you have to
• absolutely the worst—you can’t deal with it
• Causes of complexity
• dependency
• changes in one part require changes in other parts
• causes change amplification and cognitive load
• obscurity
• something is harder to understand than it needs to be
• causes cognitive load and unknown unknowns
• complexity is incremental
• similar to performance issue with no high cost centers
• same way out
• “tactical tornadoes” are the fastest to produce features but others have to clean up after them (p.14)
• invest 10-20% of time (p.15)
• p.21
An abstraction is a simplified view of an entity, which omits unimportant details.
• Deep/shallow modules
• A smell: pass-through methods (methods that just call to other methods, passing the same parameters)
• p.55
Most modules have more users that developers, so it is better for the developers to suffer than the users.
• If class has general-purpose methods, it should provide general-purpose methods only. Specializations should be handled in different class. (pp.62,64)
• red flag: conjoined methods. “it should be possible to understand each method independently” (without looking what other method does) (p.72)
• exceptions
• exceptions encourage proliferation of exceptions/error conditions, and error conditions are hard to deal with (p.78)
• 90% of all failures is caused by incorrect error handling (p.77)
• exceptions lead to “the more errors detected, the better” thinking. but more exceptions complicate system (p.78)
• exceptions in interface make class shallower. a class that handles its own error conditions is deeper (re: Deep/shallow modules)
• how to deal with exceptions:
• request-level handler (clean up, serve next request)
• crash
• comments can “provide” abstraction. p.101
• code is too low-level to provide useful abstractions (p.110)
• write comments before writing code (§15)
• it’s more fun
• it helps design better software (long complex comments may serve as a red flag)
• you’re most likely to write useful comments that do not repeat code if you write comments first
• in C++, keep comments in .cpp, not .h. This way, it’s more likely the comments will be updated when code changes. (p.137)
• if commit message is important, duplicate it in code comment (p.138)
• p.128
The greater the distance between a name’s declaration and its uses, the longer the name should be.
—Gerrand
• TDD is tactical programming (p.155)
• do I agree?
• “incremental” performance issues (“death by a thousand cuts”) p.159
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2021-09-19 05:37:36
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http://glossary.computing.society.informs.org/ver2/mpgwiki/index.php?title=Unconstrained_optimization
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# Unconstrained optimization
Taken literally, this is an unconstrained mathematical program. However, this phrase is also used in a context that $LaTeX: X$ could contain the strict interior, with constraints of the form $LaTeX: g(x) < 0,$ but the mathematical program behaves as unconstrained. This arises in the context of some algorithm design, as the solution is known to lie in the interior of $LaTeX: X,$ such as with the barrier function. As long as the algorithm has a continuous trajectory, this abuse is ok, but the constraints must be considered if one could jump, as in pattern search.
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2017-02-26 23:37:29
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http://www.mathnet.ru/php/seminars.phtml?option_lang=eng&presentid=18489
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Seminars
RUS ENG JOURNALS PEOPLE ORGANISATIONS CONFERENCES SEMINARS VIDEO LIBRARY PACKAGE AMSBIB
Forthcoming seminars Seminar calendar List of seminars Archive by years Register a seminar Search RSS Forthcoming seminars
Functional analysis and its applications
November 2, 2017 10:30–11:50
Derivation on operator algebras
Sh. A. Ayupov
Institute of Mathematics, National University of Uzbekistan named by after Mirzo Ulugbek
Abstract: Given an algebra $A$, a linear operator $D:A\to A$ is called a derivation, if $D(xy)=D(x)y+xD(y)$ for all $x,y\in A$. Each element $a\in A$ implements a derivation $D$ a on $A$ as ${{D}_{a}}(x)=ax-xa$, $x\in A$. Such derivations are said to be inner derivations. If the element implementing the derivation ${{D}_{a}}$ belongs to a larger algebra $B$ containing $A$ then ${{D}_{a}}$ is called a spatial derivation on $A$. In this talk we discuss derivations on algebras of operators on a Hilbert space, emphasizing their properties such as innerness and spatiality. These notions are very important in the structure theory and cohomology of abstract rings and algebras and at the same time they have deep applications in mathematical physics, in particular in the problem of constructing the dynamics in quantum statistical mechanics.
Therefore we also discuss a physical background of derivations on operator algebras. After expositions of some well-known results on derivation on ${{C}^{*}}$-algebras and von Neumann algebras we consider open problems concerning derivations on algebras of measurable operators affiliated with von Neumann algebras, which are partially solved.
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2021-10-25 02:52:17
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http://vm.udsu.ru/issues/archive/issue/2012-4-1
|
+7 (3412) 91 60 92
## Archive of Issues
Russia Izhevsk
Year
2012
Issue
4
Pages
3-21
Section Mathematics Title Recurrent and almost recurrent multivalued maps and their selections. II Author(-s) Danilov L.I.a Affiliations Physical Technical Institute, Ural Branch of the Russian Academy of Sciencesa Abstract In the paper, we consider the problem of existence of recurrent and almost recurrent selections of multivalued mappings ${\mathbb R}\ni t\mapsto F(t)\in {\mathrm {comp}}\, U$ with nonempty compact sets $F(t)$ in a complete metric space $U.$ The set ${\mathrm {comp}}\, U$ is equipped with the Hausdorff metric ${\mathrm {dist}}$. Recurrent and almost recurrent multivalued maps are defined as the functions with values in the metric space $({\mathrm {comp}}\, U,{\mathrm {dist}}).$ It is proved that there are recurrent (almost recurrent) selections of multivalued recurrent (almost recurrent) uniformly absolutely continuous maps. We also consider mappings ${\mathbb R}\ni t\mapsto F(t)$ with the sets $F(t)$ consisting of a finite number of points (the number depends on the $t\in {\mathbb R}$). We prove that if such a map is almost recurrent, then it has an almost recurrent selection. A multivalued recurrent mapping $t\mapsto F(t)$ with sets $F(t)$ consisting of at most $n$ points (where $n\in {\mathbb N}$) has a recurrent selection. If the sets $F(t)$ of a multivalued recurrent (almost recurrent) mapping $t\mapsto F(t)$ consist of $n$ points for all $t\in {\mathbb R},$ then all $n$ continuous selections of the map $F$ are recurrent (almost recurrent). Keywords recurrent function, selection, multivalued mapping UDC 517.518.6 MSC 42A75, 54C65 DOI 10.20537/vm120401 Received 17 May 2012 Language Russian Citation Danilov L.I. Recurrent and almost recurrent multivalued maps and their selections. II, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2012, issue 4, pp. 3-21. References Michael E. Continuous selections. I, Ann. Math., 1956, vol. 63, no. 2, pp. 361–381. Kikuchi N., Tomita Y. On the absolute continuity of multi-functions and orientor fields, Funkcialaj Ekvacioj, 1971, vol. 14, pp. 161–170. Hermes H. On continuous and measurable selections and the existence of solutions of generalized differential equations, Proc. Amer. Math. Soc., 1971, vol. 29, no. 3, pp. 535–542. Danilov L.I. Recurrent and almost recurrent multivalued maps and their selections, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2011, no. 2, pp. 19–51. Nemytskii V.V., Stepanov V.V. Kachestvennaya teoriya differentsial’nykh uravnenii (Qualitative theory of differential equations), Moscow – Izhevsk: RCD, 2004, 456 p. Anosov D.V., Aranson S.Kh., Arnold V.I., Bronshtein I.U., Grines V.Z., Il’yashenko Yu.S. Ordinary differential equations and smooth dynamical systems, Berlin, Heidelberg, New York: Springer-Verlag, 1997. Borisovich Y.G., Gel’man B.D., Myshkis A.D., Obukhovskii V.V. Vvedeniye v teoriyu mnogoznachnykh otobrazhenii i differentsial’nykh vklyuchenii (Introduction to the theory of set-valued mappings and differential inclusions), Moscow: KomKniga, 2005, 216 p. Birkhoff G.D. Dynamical Systems, New York, 1927. Translated under the title Dinamicheskie sistemy, Izhevsk: Udmurt. Univ., 1999, 408 p. Irisov A.E., Tonkov E.L. Sufficient conditions for optimality of Birkhoff recurrent motions for differential inclusion, Vestn. Udmurt. Univ. Mat., 2005, no. 1, pp. 59–74. Panasenko E.A. On existence of recurrent and almost periodic solutions to differential inclusions, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2010, no. 3, pp. 42–57. Danilov L.I. Almost periodic selections of multivalued maps, Izv. Otd. Mat. Inform. Udmurt. Gos. Univ., Izhevsk, 1993, no. 1, pp. 16–78. Lyusternik L.A., Sobolev V.I. Kratkii kurs funktsional’nogo analiza (A short course of functional analysis), Moscow: Vysshaya shkola, 1982, 271 p. Full text
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2020-11-25 14:36:56
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