text:
# All Questions

106 views

### Counter Mode with a sequence of zeros bits plaintext, is it secure?

I had a quiz last week in computer security course. There was a confusing question that I am still looking for a good and clear answer. First, I know that counter mode with a good block cipher is ...
383 views

### Generation of a cyclic group of prime order

I am trying to implement a cyclic group generator in Java, but I am running into some issues. In many cryptosystems, the following phrase is expressed during the key generation stage. Let G be ...
53 views

### What does K stand for in frequency analysis

It was from frequency analysis lecture and the professor mentioned but I cannot recall. The K value for English was .067, German was .076. The lowest K was Russian which was .056. When the frequency ...
197 views

### How fast would a polynomial time factoring algorithm compute?

I know factoring is the chief means of breaking RSA keys. I know an algorithm that runs in polynomial time would be able to break an RSA key pair "quickly". But how quickly is "quickly"? Note, I'm not ...
37 views

### ssh-keygen bit why not use 4095 or 4097? [duplicate]

All examples of using ssh-keygen I have seen has always been some power of 2 (2048, 4096). ssh-keygen -t rsa -b 4096 Why use those numbers? Why not use 3333? ...
70 views

### Weak Boneh-Boyen Signature in Composite order group

In the paper Ring Signatures of Sub-linear Size Without Random Oracles, authors have remarked that the scheme is instantiable in composite order setting, too. I am attaching the following reference ...
276 views

### Encrypting with Davies-Meyer hash function scheme

So I need to to illustrate Davies-Meyer hash function for a message “123” with 1-byte blocks. I can use the extended ASCII representation where each character is represented as a byte. And I can pick ...
558 views

### Who first published the interest of more than two prime factors in RSA?

Multi-prime RSA is now a well known technique: it uses $k>2$ distinct secret prime factors in the public RSA modulus, with the advantage that, using the CRT, we can gain a speed boost in ...
50 views

### Automatic generation of secure passwords with the least inconvenience for a user

I'm working on a web site for a private company that should allow them to upload files, which will be later retrieved by their affiliates. The site will be available from a public Internet using ...
87 views

### Is BBS used for generation of keys for any modem cryptosystem?

Is there any (current) cryptosystem or digital signature or cryptographic primitive that uses the BBS PRG in order to generate its secret keys?
152 views

For Bitcoin's ECDSA curve (secp256k1, where $a=0$, $b=7$), why can't the generator point's first coordinate be $x=0$? That is, the point on the curve would be $(0,y)$ where $y$ satisfies $y^2 = 0^3 + ... 1answer 293 views ### SHA1 Partial Collision I'm working on a little python script that reads in two fairly similar English sentences (they're in text files - reading them in) and I'm trying to force a SHA1 partial collision (e.g. first 32 bits ... 1answer 170 views ### CPA security of a stateless and deterministic encryption system Why can no stateless and deterministic encryption system be IND-CPA secure? Is there a formal proof for it? 1answer 43 views ### Using Permutation polynomial to compute a MAC Is the following MAC secure? For a block$y_i$in a file, we defined a MAC as follows:$Mac_i:PRF(k,i) \cdot g^{y_i \cdot r_i} \bmod p$. Where$p$is a prime number,$g \in \mathbb{G}$,$PRF(k,i)$is ... 0answers 23 views ### Kryptos K2 keyword derivation [duplicate] So I've been doing some reading on Kryptos, and to be honest the keyword for K2, ABSCISSA, has a pretty weak derivation. (The method using the eee's). Isn't there a better way to come to that? I dont ... 2answers 104 views ### RSA generation of private key using public key In RSA private key generation e*d ≡ 1 mod φ e is public, also n is public. How to prove mathematically, generation of ... 1answer 143 views ### PRF based on the GGM construction What's the differences between the concepts “pseudorandom generator” and “pseudorandom number generator”? In fact, I want to implement a pseudorandom function based on GGM's construction at ... 1answer 146 views ### Noise bound in FHE over the integers I'm studying the paper Fully Homomorphic Encryption over the Integers by Marten van Dijk, Craig Gentry, Shai Halevi and Vinod Vaikuntanathan. I have questions about the proof of Lemma A.1. In page ... 0answers 149 views ### Python implementation of a blind signature scheme which doesn't involve RSA RSA seems a bit creepy after the Snowden revelations and i'm looking for a simple python based blind signature library to fiddle around with. So far i've been unable to find anything. What am i ... 2answers 837 views ### Can RSA encryption produce collisions? In RSA, a message is encrypted by$m^e \pmod N$.$N$is the modulus,$m$is the message and$e$is the public exponent. (I know that$m$should not be greater than$N$.) My question is, can$m^e$be ... 1answer 464 views ### What is the difference between a hash function and a pseudorandom function? [duplicate] Read the title. I've seen in RFCs that some MAC functions are called "pseudorandom functions". What are those? How are they different than hash functions? Why can't a hash function be used instead? 1answer 257 views ### Why does the DES crypto algorithm NOT use 2 rounds? Now, if we were to go round by round, you could give a distinct reason for not using a single round since after just one round, the right half of the text comes directly, as-is, to form the left half ... 3answers 4k views ### For Diffie-Hellman, must g be a generator? Due to a number of recently asked questions about Diffie-Hellman, I was thinking this morning: must$g$in Diffie-Hellman be a generator? Recall the mathematics of Diffie-Hellman: Given public ... 0answers 54 views ### Decrypt a message which is encrypted using XOR? [duplicate] This is a puzzle asked in a contest. Given that encryption , decryption happens as per following rule/code: ... 1answer 111 views ### Definition of ciphertext security If I got it right, chosen ciphertext security implies also CPA security. In other words, attacker can submit plaintexts to the challenger (along with ciphertexts). I do not understand why ... 1answer 96 views ### Negative exponents in Shoup's threshold RSA? I'm trying to implement threshold RSA operations, starting with decryption based on Peeters, R., Nikova, S., & Preneel, B. (2008). Practical RSA Threshold Decryption for Things That Think. ... 1answer 102 views ### Proof that$gcd(e, \lambda(N)) = 1 \hspace{1mm} \Longleftrightarrow \hspace{1mm} gcd(e, \varphi(N)) = 1$What is the proof for the fact that$gcd(e, \lambda(N)) = 1 \hspace{1mm} \Longleftrightarrow \hspace{1mm} gcd(e, \varphi(N)) = 1$Where:$N = P * Q$where$P$and$Q$are both primes.$\varphi(N)$... 1answer 79 views ### Post processing of AAD and len A||C in a hardware AES GCM implementation I'm new to GCM and I need to implement it in hardware, using FPGA. The data bus is 640 bits, so I will use 5 adder/multiplier blocks in parallel. The message size and AAD size are constant. My design ... 1answer 67 views ### Non-repudiation and digital signature of a dishonest participant Let's assume a dishonest Alice who sends, encrypts & digital signs a message to Bob. Bob stores the decrypted message and the digital signature in a database. However Alice is a bad girl and ... 3answers 995 views ### Kryptos : K1. What is the origin of the “palimpsest” keyword? I'm studying the Kryptos sculpture with its cryptographic puzzles K1 to K4. I understand that the keyword "palimpsest" was reverse-engineered using the tableau (and brute-force computer processing), ... 0answers 151 views ### Parallel Pollard's Rho: Number of distinguished points When using the parallel version of Pollard's Rho algorithm for discrete logs, each processor performs its own random walk to find distinguished points, and reports the starting point and the ... 1answer 326 views ### Can public key be recovered from ciphertext & encrypted private key? I'd like to implement something like a write-once public/private encrypted shared secret (no better quick description for the lack of terminology knowledge). I guess, I'm trying to implement HSM. The ... 1answer 186 views ### XOR cipher with three different ciphertexts and repeated key, key length known. How do I find the plaintexts? Let us say we have three different plaintexts (all alphabets, A-Z):$x$,$y$and$z$, each of length$21$. Let the key,$a$, be also of length$21$. Now, what we have is$x \oplus a$,$y \oplus a$... 2answers 146 views ### The perfect way of using IV in CTR mode I understand that it is necessary to use the same IV for both encryption and decryption in the CTR mode. I'm thinking about the case when I concatenate the secret ... 1answer 57 views ### Why is it a quadratic equation? In Groth-Sahai NIZK proof system, they have defined something called Quadratic Equation in$\mathbb{Z}_n$as shown below. But, my idea of quadratic equation was a second order polynomial equation in a ... 2answers 75 views ### One-way function definition I cannot understand why a one-way function$f$is defined in this way$\text{Pr}(f(A(f(x))) = f(x)) < \frac{1}{p(n)}$and not$\text{Pr}(A(f(x)) = x) < \frac{1}{p(n)}$where$A$is a ... 1answer 128 views ### One time pad probability I'm trying to solve a question about one time pads but I'm not sure if my assumptions are correct. I am to assume that I have a one time pad with perfect secrecy. This one time pad is used to ... 1answer 47 views ### Do any stream ciphers with aperiodic keystreams exist? Exactly what it says on the tin. I can imagine constructing such keystreams from: The binary expansions of irrational numbers Chaotic systems like the logistic map or the Lorenz attractor. The ... 1answer 12k views ### Decrypt files with original file CTB-Locker [closed] I have problem called CTB-Locker. It encrypted all of my files on computer and since I have lot of documents that are very important I am in problems! As I read online CTB-Locker uses "elliptical ... 1answer 371 views ### Merkle hash tree updates It seems that merkle hash tree (MHT) traversals have been discussed somewhat in the literature, but there does not appear to be much written on inserting, deleting, and updating leaves. Is this lack ... 0answers 32 views ### Reducing key shares in Damgård-Dupont threshold RSA I'm working on understanding and implementing Damgård, I., & Dupont, K. (2005). Efficient Threshold RSA Signatures with General Moduli and No Extra Assumptions. Public Key Cryptography-PKC ... 2answers 85 views ### Usage of GF(p^m) fields, where p != 2$GF(2^m)$Galois fields are widely used in different cryptographic algorithms, for example, in Rijndael. However,$GF(p^m)$fields are possible with any prime$p$, not only 2, but$GF(2^m)$fields ... 0answers 311 views ### Elliptic Curve based blind signature implementation I want to use Elliptic Curve based blind signature scheme for my research. There is no proper implementation of ECC-based blind signatures. Can someone describe to me which things I need to follow ... 0answers 218 views ### Is there a flaw in this ECC blind signature scheme? Recently I've found the following work on the internet: An ECC-Based Blind Signature Scheme The paper claims to be an ECDSA blind signature however it seems that their scheme has a flaw in it. The ... 2answers 455 views ### Can AES in CCM or GCM counter mode interoperate with AES in “plain” counter mode (CTR)? I am exploring the use of Windows CNG to replace some OpenSSL-based code that takes advantage of AES in counter mode. From the outside, everything should look the same after the switch. The section ... 0answers 28 views ### Proving the complement property of DES? [duplicate] I'm trying to show that for a given plaintext (P), key (K), and cipher (C) in the DES algorithm: ... 2answers 998 views ### DES Encryption Algorithm all 64 bits for key instead of 56 bits Would a DES algorithm that uses all 64 bits for the key instead of just the 56 bits be more secure? I have been thinking about it but those 8 bits used for parity are very useful and but including ... 1answer 63 views ### Is there a reference that prove that the AES Key Schedule generate random looking round keys? Starting from uniformly random generated AES master key, is there a reference that prove that an specific roundkey can be considered as uniformly random generated as well ? 1answer 13k views ### Many time pad attack [duplicate] I've already sent my correct solution to a homework exercise from Dan Boneh's Introduction to Cryptography class on Coursera: "Let us see what goes wrong when a stream cipher key is used more than ... 2answers 245 views ### Achieving 32-bit verification code with 16-bit CRC? [closed] I am programming an embedded chip that has a hardware 16-bit CRC module. I have to protect some data bytes$d_0,d_1,...,d_{n-1}\$ against corruption caused by sudden loss of power; a 32-bit CRC would ...

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## Combinatorics and Graph Theory

•  Nicholas Ovenhouse, MSU
•  Log-Canonical Coordinates for Poisson Brackets
•  03/21/2017
•  4:10 PM - 5:00 PM
•  C304 Wells Hall

In symplectic geometry, Darboux's theorem gives so-called 'canonical' local coordinates, in which the Poisson bracket (obtained from the symplectic structure) takes a particularly simple form: all brackets of coordinate functions are constants. We investigate whether something analogous is true for Poisson varieties (when the coordinate change is only allowed to be rational functions), and give a negative answer in the case of log-canonical coordinates.

## Contact

Department of Mathematics
Michigan State University

format: plain


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# Generate grid coordinates from PySCF DFT package for plotting grid points

As can be read in the pyscf.dft documentation, there should be the possibility of generating the coordinates of the grid for numerical integration. According to the documentation, one can generate the coordinates with the get_partition function, which is part of the pyscf.dft.gen_grid module.

My question is the following: If the get_partition function returns the mesh grid coordinates and the corresponding grid weights, how can one directly access and manipulate the generated data (printing, shaping arrays, etc.)? I tried to print the grid coordinates, which are returned as an array according to the documentation, but it is not specified under which name the grid coordinates are returned.

• Can you explain in more detail what you want to know? It seems to me like you already have the answer: get_partition gives you the coordinates of the grid points and the weights (returned arg 1 and 2 respectively). Is your question how to integrate quantities on the generated grid? Jun 29 at 14:31

What I usually do is I generate the grid with pyscf.dft.gen_grid as you mentioned, for example

def __init__(self, mol, calc):
self._grid = pyscf.dft.gen_grid.Grids(self._mol)
self._grid.level = 8 # level of accuracy of the grid. 8 is pretty dense, you might be OK with the default (maybe 5?)
self._grid.build()

Doing this I will have access to self._grid.coords and self._grid.weights. Then all you need to do is apply whatever function you want to the coordinates and take the dot product of this quantity with the weights to do a quadrature

desired_quantity = apply_function(self._grid.coords)
res = self._grid.coords * self._grid.weights


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# Urgent! C++ project is kicking my butt.

#### chinesebarbiedoll

Joined Nov 9, 2011
9
I am having problems with my code. For example when I execute my code and type 1 in the one goes to coin return when it should go into the slot. I can't figure out what's wrong with my code. Any help would be extremely appreciated seeing as how the program is due tomorrow.

Here is the code:
Rich (BB code):
#include<iostream>
#include <conio.h>
#include <iomanip> //use for setw
using namespace std;
int main()
{
int intnum, quarters, dimes, nickels;
double decnum, money, slot=0.00, coinreturn=0.00;
bool flag=true;

do
{
// setprecision(2) so that the 0.00 will appear in the executable.
cout << setprecision(2) << fixed;
slot=0.00;
coinreturn=0.00;

do
{
system("cls");
cout<<endl;
cout << setw(35) << "ICE COLD COLA";
cout << endl;
cout << setw(33) << ".75 cents";
cout << endl;
cout << endl;
cout <<"         Deposit -- dollar, quarters, dimes, or nickels:";
cout << endl;
cout << setw(22) << "( 1" << "    ," << setw(6)<< " .25" << "   ,"
<<  "  .10" << " ," << setw(8) << " .05" << " )";
cout << endl;
cout << endl;
cout << setw(16) << "Slot"<<" [" << slot <<" ]" <<" :";
cout << endl;
cout << endl;
cout << endl;
cout << endl;
cout << setw(14) << "Coin Return" <<" [" << coinreturn <<" ]" <<" :";
cout << endl;
cout << endl;
cout << setw(10) << "     Enter Coin" << "[" << "------" <<"]" << " ";
cin>> money;
cout<<endl<<endl;

/* makes it so that when .05, .10, .25, and $1.00 is entered into the machine the money will be added together and shown in the slot Any amount of money that is not equal to .05, .10, .25, and$1.00
will be placed in the coin return.
*/

if (money == 1.00 || money == 0.25 || money == 0.10 || money == 0.05)
slot += money;
else
coinreturn += money;

}while(slot <= .75 && money != -1);

cout<<"   Have a nice day!"<<endl;

decnum = (slot - .75);

//use static_cast to convert a double into an integer

intnum = static_cast<int>((decnum +.005) * 100);

quarters = intnum/25;
intnum = intnum%25;

dimes = intnum/10;
intnum = intnum%10;

nickels = intnum/5;
intnum = intnum%5;

cout<< endl<< endl<< "  Your change is: " <<endl;
cout<<"\t"<< quarters <<" Quarter(s)"<<endl;
cout<<"\t"<< dimes <<" Dime(s)"<<endl;
cout<<"\t"<< nickels<< " Nickel(s)"<<endl;

system("pause");

}while(money != -1);

cout<<endl<<endl<<endl;

}

#### t06afre

Joined May 11, 2009
5,936
hmm have you done some debugging. Like done some single-stepping on your program, setting Breakpoints and so on.

#### TheFox

Joined Apr 29, 2009
66
If I were you, I'd break down the interface into some functions, so it'll be tidier.

#### Josato

Joined Mar 8, 2011
17
Yes, if I was you, I would also print out the value of 'money' before you compare it, and after you compared it to see where it is gone.

Perhaps there is some problem if you are comparing a double. Also, I would suggest you post your question to stack overflow, they are very helpful and more importantly quick to reply there.

A quick workaround might be to use integers instead of doubles until you figure it out.

#### Josato

Joined Mar 8, 2011
17
Also the thing about floats and doubles is that they store the number quite weirdly by working out the decimal as a fraction (not that you care).

#### ho69e3k

Joined Nov 13, 2011
7
@chinesebarbiedoll
I think the problem here is the conditions in while and if conditions.. If you could only fix them..

Here is the solution to your problem in when you type 1 then goes to slot..:
Rich (BB code):
 do
{
system("cls");
cout<<endl;
cout << setw(35) << "ICE COLD COLA";
cout << endl;
cout << setw(33) << ".75 cents";
cout << endl;
cout << endl;
cout <<"         Deposit -- dollar, quarters, dimes, or nickels:";
cout << endl;
cout << setw(22) << "( 1" << "    ," << setw(6)<< " .25" << "   ,"
<<  "  .10" << " ," << setw(8) << " .05" << " )";
cout << endl;
cout << endl;
cout << setw(16) << "Slot"<<" [" << slot <<" ]" <<" :";
cout << endl;
cout << endl;
cout << endl;
cout << endl;
cout << setw(14) << "Coin Return" <<" [" << coinreturn <<" ]" <<" :";
cout << endl;
cout << endl;
cout << setw(10) << "     Enter Coin" << "[" << "------" <<"]" << " ";
cin>> money;
cout<<endl<<endl;

/* makes it so that when .05, .10, .25, and $1.00 is entered into the machine the money will be added together and shown in the slot Any amount of money that is not equal to .05, .10, .25, and$1.00
will be placed in the coin return.
*/

if (money == 1.00 || money == 0.25 || money == 0.10 || money == 0.05)
slot += money;
else
coinreturn += money;

}while(slot <= 1.00 );
Actually another problem occurred in this program when I did it..
I have a change of 5 quarters, 0 dimes and 0 nickels..
But I think you can easily fix the problem..,


text:
AI ALIGNMENT FORUMAF

(...) the term technical is a red flag for me, as it is many times used not for the routine business of implementing ideas but for the parts, ideas and all, which are just hard to understand and many times contain the main novelties.
- Saharon Shelah

For my most of my writing see my short-forms (new shortform, old shortform)

Sequences

Singular Learning Theory

Sorted by New

Wiki Contributions

Yeah follow-up posts will definitely get into that!

To be clear: (1) the initial posts won't be about Crutchfield work yet - just introducing some background material and overarching philosophy (2) The claim isn't that standard measures of information theory are bad. To the contrary! If anything we hope these posts will be somewhat of an ode to information theory as a tool for interpretability.

Adam wanted to add a lot of academic caveats - I was adamant that we streamline the presentation to make it short and snappy for a general audience but it appears I might have overshot ! I will make an edit to clarify. Thank you!

I agree with you about the importance of Kolmogorov complexity philosophically and would love to read a follow-up post on your thoughts about Kolmogorov complexity and LLM interpretability:)

Concept splintering in Imprecise Probability: Aleatoric and Epistemic Uncertainty.

There is a general phenomena in mathematics [and outside maths as well!] where in a certain context/ theory  we have two equivalent definitions  of a concept  that become inequivalent when we move to a more general context/theory . In our case we are moving from the concept of probability distributions to the concept of an imprecise distribution (i.e. a convex set of probability distributions, which in particular could be just one probability distribution). In this case the concepts of 'independence' and 'invariant under group action' will splinter into inequivalent concepts

Example (splintering of Indepence) In classical probability theory there are three equivalent ways to state that a distribution is independent

1.

2.

3.

In imprecise probability these notions split into three inequivalent notions. The first is 'strong independence' or 'aleatoric independence'. The second and third are called 'irrelevance', i.e. knowing  does not tell us anything about  [or for 3 knowing  does not tell us anything about ].

Example (splintering of invariance). There are often debates in foundations of probability, especially subjective Bayesian accounts about the 'right' prior. An ultra-Jaynesian point of view would argue that we are compelled to adopt a prior invariant under some symmetry if we do not posses subjective knowledge that breaks that symmetry ['epistemic invariance'], while a more frequentist or physicalist point of view would retort that we would need evidence that the system in question is in fact invariant under said symmetry ['aleatoric invariance']. In imprecise probability the notion of invariance under a symmetry splits into a weak 'epistemic' invariance and a strong 'aleatoric' invariance. Roughly spreaking, latter means that each individual distribution in the convex set is invariant under the group action while the former just means that the convex set is closed under the action

The point isn't about goal misalignment but capability generalisation. It is surprising to some degree that just selecting on reproductive fitness through its proxies of being well-fed, social status etc humans have obtained the capability to go to the moon. It points toward a coherent notion & existence of 'general intelligence' as opposed to specific capabilities.

Thank you for writing this post; I had been struggling with these considerations a while back. I investigated going full paranoid mode but in the end mostly decided against it.

I agree theoretical insight on agency and intelligence have a real chance of leiding to capability gains. I agree on the government spy threat model as being unlikely. I would like to add however that if say MIRI builds a safe AGI prototype - perhaps based on different principles than systems used by adversaries it might make sense for an (ai-assisted) adversary to trawl through your old blogposts.

Byrnes has already mentioned the distinction between pioneers and median researchers. Another aspect that your threat models don't capture is: research that builds on your research. Your research may end up in a very long chain of theoretical research only a minority of which you have contributed. Or the spirit if not the letter of your ideas may percolate through the research community. Additionally, the alignment field will almost certainly become very much larger raising both the status of John and the alignment field in general. Over longer timescales I expect percolation to be quite strong.

Even if approximately nobody reads your or know of your works the insights may very well become massively signalboosted by other alignment researchers (once again I expect the community to explode in size within a decade) and thereby end up in a flashy demo.

All-in-all these and other considerations let me to the conclusion that this danger is very real. That is there is a significant minority of possible worlds in which early alignment researchers tragically contribute to DOOM.

However, I still think on the whole most alignment researchers should work in the open. Any solution to alignment will most likely come from a group (albeit-small) of people. Working privately massively hampers collaboration. It makes the community look weird and makes it way harder to recruit good people. Also, for most researchers it is difficult to support themselves financially if they can't show their work. As by far the most likely doom scenario is some company/government simply building AGI without sufficient safeguards because either there is no alignment solution or they are simply unaware of it/it ote it I conclude that the best policy in expected value is to work mostly publicly*.

*Ofc if there is a clear path to capability gain keeping it secret might be the best.

EDIT: Cochran has a comical suggestion

Georgy Flerov was a young nuclear physicist in the Soviet Union who ( in 1943) sent a letter to Stalin advocating an atomic bomb project. It is not clear that Stalin read that letter, but one of Flerov’s arguments was particularly interesting: he pointed out the abrupt and complete silence on the subject of nuclear fission in the scientific literature of the US, UK, and Germany – previously an extremely hot topic.

Stopping publications on atomic energy ( which happened in April 1940) was a voluntary effort by American and British physicists. But that cessation was itself a signal that something strategically important was going on.

Imagine another important discovery with important strategic implications: how would you maximize your advantage ?

Probably this is only practically possible if your side alone has made the discovery. If the US and the UK had continued publishing watered-down nuclear research, the paper stoppage in Germany would still have given away the game. But suppose, for the moment, that you have a monopoly on the information. Suddenly stopping closely related publications obviously doesn’t work. What do you do?

You have to continue publications, but they must stop being useful. You have to have the same names at the top ( an abrupt personnel switch would also be a giveaway) but the useful content must slide to zero. You could employ people that A. can sound like the previous real authors and B. are good at faking boring trash. Or, possibly, hire people who are genuinely mediocre and don’t have to fake it.

Maybe you can distract your rivals with a different, totally fake but extremely exciting semiplausible breakthrough.

Or – an accidental example of a very effective approach to suppression. Once upon a time, around 1940, some researchers began to suspect that duodenal ulcers were caused by a spiral bacterium. Some physicians were even using early antibiotics against them, which seemed to work. Others thought what they were seeing might be postmortem contamination. A famous pathologist offered to settle the issue.

He looked, didn’t see anything, and the hypothesis was buried for 40 years.

But he was wrong: he had used the wrong stains.

So, a new (?) intelligence tactic for hiding strategic breakthroughs: the magisterial review article.

Just wanted to chime in to say that this feels important and I'm happy you brought it up.

Daniel Kokotaljo and I agreed on the following bet: I paid Daniel $1000 today. Daniel will pay me$1100 inflation adjusted if there is no AGI in 2030.

Ramana Kumar will serve as the arbiter. Under unforeseen events we will renegotiate in good-faith.

As a guideline for 'what counts as AGI' I suggested the following, to which Daniel agreed:

"the Arbiter agrees with the statement "there is convincing evidence that there is an operational Artificial General Intelligence"  on 6/7/2030"

Defining an artificial general intelligence is a little hard and has a strong 'know it when you see it vibe' which is why I'd like to leave it up to Ramana's discretion.

We hold these properties to be self-evident requirements for a true Artificial General Intelligence:

1. be able to equal or outperform any human on virtually all relevant domains, at least theoretically

-> there might be e.g. physical tasks that it is artificially constrained from completing because it is lacks actuators for instance - but it should be able to do this 'in theory'. again I leave it up to the arbiter to make the right judgement call here.

2. it should be able to asymptotically outperform or equal human performance for a task with equal fixed data, compute, and prior knowledge

3. it should autonomously be able to formalize vaguely stated directives into tasks and solve these (if possible by a human)

4. it should be able to solve difficult unsolved maths problems for which there are no similar cases in its dataset

(again difficult, know it when you see it)

5. it should be immune / atleast outperform humans against an adversarial opponent (e.g. it shouldn't fail Gary Marcus style questioning)

6. outperform or equals humans on causal & counterfactual reasoning

7. This list is not a complete enumeration but a moving goalpost (but importantly set by Ramana! not me)

-> as we understand more about intelligence we peel off capability layers that turn out to not be essential /downstream of 'true' intelligence.

Importantly, I think near-future ML systems to be start to outperform humans in virtually all (data-rich) clearly defined tasks (almost) purely on scale but I feel that an AGI should be able to solve data-poor, vaguely defined tasks, be robust to adversarial actions, correctly perform counterfactual & causal reasoning and be able to autonomously 'formalize questions'.

Are we supposed to know who Yafa is?

I am not able to ascertain the truth value of the relevant sentences with or without assistance. I am a human if that helps


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# A cubic box with an edge length of 4 inches holds a ball

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17 Sep 2014, 10:44
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A cubic box with an edge length of 4 inches holds a ball with a radius of 2 inches, such that the ball touches the box on all six sides (it is tangent to all six sides). What is the distance, in inches, between the center of the ball and a corner of the box?

A 6√3
B 4√3
C 2√3
D 2√2
E √2

Does GMAT test us on such questions?

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Re: A cubic box with an edge length of 4 inches holds a ball  [#permalink]

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17 Sep 2014, 11:39
JarvisR wrote:
A cubic box with an edge length of 4 inches holds a ball with a radius of 2 inches, such that the ball touches the box on all six sides (it is tangent to all six sides). What is the distance, in inches, between the center of the ball and a corner of the box?

A 6√3
B 4√3
C 2√3
D 2√2
E √2

Does GMAT test us on such questions?

Given that the sphere is inscribed into the cube.

The length of a diagonal of a cube equals to $$\sqrt{4^2+4^2+4^2}=4\sqrt{3}$$.

The distance between the center of the ball and a corner of the box will simply be half of the diagonal, so $$2\sqrt{3}$$.

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17 Sep 2014, 11:42
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Re: A cubic box with an edge length of 4 inches holds a ball  [#permalink]

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30 Jul 2016, 12:02
JarvisR wrote:
A cubic box with an edge length of 4 inches holds a ball with a radius of 2 inches, such that the ball touches the box on all six sides (it is tangent to all six sides). What is the distance, in inches, between the center of the ball and a corner of the box?

A 6√3
B 4√3
C 2√3
D 2√2
E √2

Does GMAT test us on such questions?

diagonal = √3a------>4√3
so half upto centre=2√3
Ans C

Experts please move the question in PS section...
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Re: A cubic box with an edge length of 4 inches holds a ball  [#permalink]

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30 Jul 2016, 13:19
When approaching this problem I noticed that the sphere touches the six edges of the box. This means I don't need additional information about the box as I already knew the radius. I also noted that calculating the distance between the center of the sphere and the corner of the box is the same thing as calculating the diagonal of a cube with each side equal to one radius of the sphere).

Given the radius of 2 I calculate the diagonal of the cube with side length 2. As the formula for the diameter of a cube is "side length*sqrt(3)". The answer had to be C. (sidelength = 2 was given in q-stem)

It would be interesting to see a question such as this one, where the test makers leave room at the top and the bottom but allow the sphere to be tangent to the up-right sides. This could likely be used in a data sufficiency question in different ways. Does anyone know about any questions like the one I described?

Best wishes.
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Re: A cubic box with an edge length of 4 inches holds a ball  [#permalink]

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15 Oct 2017, 10:27
The distance between the center of the ball and a corner of the box is in fact half the long diagonal of this cube.

Long diagonal of a cube = s√3

so long diagonal of this particular cube is 4√3, half of which is 2√3, so C is the right answer.
Re: A cubic box with an edge length of 4 inches holds a ball &nbs [#permalink] 15 Oct 2017, 10:27
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text:
# The perimeter of parallelogram is 238 cm. The ratio of the two adjacent sides is 3:4. What are the lengths of the four sides of the parallelogram?

May 17, 2018

$51 , 68 , 54 , 68$

#### Explanation:

Since opposite sides of a parallelogram are equal, then we can say the sides are in the ratio 3:4:3:4. Multiplying into 238, we get the lengths
$51 , 68 , 54 , 68$
(As there are 14 parts, each part is equal to 17)


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# Understanding the logic behind particular derivative

I have

$\frac{\partial (f(x) g(x))}{\partial x}$=$g(x) f'(x)+f(x) g'(x)$,

I need to differentate this function with respect to x.

$f(x)=(x+1) (x+2)^2 (x+3)^3 (x+4)^4$

However I do not see the logic using the product rule.

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Why not? $f(x)$ in the question is the product of several functions. –  Quang Hoang Jul 28 '14 at 3:56
$f$ is a product of two function $g(x)=(x+1)(x+2)^2$ and $h(x)=(x+3)^3(x+4)^4$ that are themselves products. With method, this is not difficult to apply. Since it is difficult to be good in Calculus without raising one's sleeves and compute, you should give it a try, and we will correct it. Alternatively, it is not very difficult to show that for a product of $3$ functions $(fgh)'=f'gh+fg'h+fgh'$ or for a product of $4$ functions $(fghk)'=f'ghk+fg'hk+fgh'k+fghk'$. –  Taladris Jul 28 '14 at 4:04
Note also that for a function of one variable, the notation $\frac{\partial f}{\partial x}$ is considered incorrect and is reserved to function of several variables. Usual notations are $\frac{d f}{d x}$ or just $f'(x)$. I would also avoid to mix notations for the same thing and write $(fg)'=f'g+fg'$ (or the equivalent with the $d$-notation). –  Taladris Jul 28 '14 at 4:08
How do you compute $2 \cdot 3 \cdot 4 \cdot 5$ if the only thing you know is how to multiply two numbers? –  Hurkyl Jul 28 '14 at 4:17

The product rule states that: $(f_1(x)f_2(x))' = f_1'(x)f_2(x) + f_1(x)f_2'(x)$. This works in general for $n$ functions: $$(f_1(x) \cdots f_n(x))' = f_1'(x)f_2(x)\cdots f_n(x) + f_1(x)f_2'(x)\cdots f_n(x) + \cdots + f_1(x)\cdots f_n'(x)$$ The prime $'$ runs through all of the functions. Writing in a more compact way, we can say that: $$\left(\prod_{i=1}^n f_i(x)\right)' = \sum_{i = 1}^n \left(f_i'(x) \prod_{j = 1, j \neq i}^n f_j(x)\right)$$ You can prove this easily by induction on $n$. Now, applying this on your specific problem, we have: $$f'(x) = (x+1)'(x+2)^2(x+3)^3(x+4)^4 + (x+1)\left((x+2)^2\right)'(x+3)^3(x+4)^4 + \cdots+(x+1)(x+2)^2(x+3)^3\left((x+4)^4\right)'$$ In each factor, you use the chain rule.

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It seems worth noting that the familiar power rule is itself an instance of this, e.g. $$\dfrac{d}{dx}x^3 = \dfrac{dx}{dx}\cdot x \cdot x+x\cdot\dfrac{dx}{dx}\cdot x+ x \cdot x \cdot \dfrac{dx}{dx} =3x^2$$ (at least for natural powers) –  Semiclassical Jul 28 '14 at 4:17
Nice comment. We can also prove that $(x^n)' = nx^{n-1}$ by using induction and the product rule for two functions. –  Ivo Terek Jul 28 '14 at 4:19
Do not see what you mean by "in each factor you use the chain rule" –  ALEXANDER Jul 28 '14 at 4:47
@ALEXANDER for example, doing $$\left((x+3)^3\right)' = 3(x+3)^2 \cdot 1$$ instead of writing everything, like: $$(x+3)^3 = x^3 + 3x^2(3) + 3x(3^2) + 3^3$$ and then differentiating it. –  Ivo Terek Jul 28 '14 at 4:54
Okey, perfect I see! Thank you! –  ALEXANDER Jul 28 '14 at 4:55

For this example, it will be useful to use logarithmic differentiation. Let $y$ be our function. Then $$\ln y=\ln(x+1)+2\ln(x+2)+3\ln(x+3)+4\ln(x+4).$$ Differentiate. We get $$\frac{1}{y}\frac{dy}{dx}=\frac{1}{x+1}+\frac{2}{x+2}+\frac{3}{x+3}+\frac{4}{x+4}.$$

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HINT

You must do that step by step.

$f(x)=(x+1) (x+2)^2 (x+3)^3 (x+4)^4$

$$f'(x)=((x+1))' ((x+2)^2 (x+3)^3 (x+4)^4)+((x+1)) ((x+2)^2 (x+3)^3 (x+4)^4)'$$

$$((x+2)^2 (x+3)^3 (x+4)^4)'=((x+2)^2)' ((x+3)^3(x+4)^4)+((x+2)^2) ((x+3)^3(x+4)^4)'$$

$$((x+3)^3(x+4)^4)'=((x+3)^3)'((x+4)^4)+((x+3)^3)((x+4)^4)'$$

now you go from the last to the first.

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Are you saying that the derivative of a product is the product of derivatives? –  Taladris Jul 28 '14 at 4:09
@Taladris did a terrible mistake. I fixed that. –  Shabbeh Jul 28 '14 at 4:13
OK: no problem. I can revert my downvote :) –  Taladris Jul 28 '14 at 4:24


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Unlike a piano or the voices of a choir, the guitar (in standard tuning) has difficulty playing the chords as stacks of thirds, which would require the left hand to span too many frets,[40] particularly for dominant seventh chords, as explained below. If in a particular tuning chords cannot be played in closed position, then they often can be played in open position; similarly, if in a particular tuning chords cannot be played in root position, they can often be played in inverted positions. A chord is inverted when the bass note is not the root note. Additional chords can be generated with drop-2 (or drop-3) voicing, which are discussed for standard tuning's implementation of dominant seventh chords (below).
MIDI converters use a hexaphonic guitar signal to determine pitch, duration, attack, and decay characteristics. The MIDI sends the note information to an internal or external sound bank device. The resulting sound closely mimics numerous instruments. The MIDI setup can also let the guitar be used as a game controller (i.e., Rock Band Squier) or as an instructional tool, as with the Fretlight Guitar.
A chord is inverted when the bass note is not the root note. Chord inversion is especially simple in M3 tuning. Chords are inverted simply by raising one or two notes by three strings; each raised note is played with the same finger as the original note. Inverted major and minor chords can be played on two frets in M3 tuning.[56][74] In standard tuning, the shape of inversions depends on the involvement of the irregular major-third, and can involve four frets.[75]

Once you've got your categories narrowed down, then you can start getting into the nitty-gritty differences between strings. For instance, electric guitars will give you the choice between nickel (for authentic vintage sound) and stainless steel (for maximum durability). Some string manufacturers have exotic material options with their own unique characteristics, like Ernie Ball's Slinky Cobalt strings. With so many subtle differences separating guitar strings, you owe it to yourself to browse carefully and look at all the choices on the table before making a decision.
When all is said and done, the only way to know for sure what a certain set of strings will sound like on your guitar is to take them for a spin. For that reason, it's not a bad idea to try out a few different ones that interest you to see which ones you like best - then you can stock up! However you like to approach your string shopping, the amazing selection you'll find here is sure to satisfy.
Octo Music Studio teaches piano and guitar lessons to adults and children 5 years and older.  Our lessons are personalized and are exciting this results in the fastest possible progress for all students. We use the best computer guided music teaching programs to teach and are ranked in the top best 18 best music studios in Brooklyn, NY. Our piano or guitar teachers have over 30 years experience. Learn Gospel,...
Though they may not be as exciting of a topic as instruments or amplifiers, choosing the right type of strings for your guitar is incredibly important. Guitar strings have different responses and different feels, and when the right type of string is used it’s a huge asset to your playing. Likewise, when the wrong type of string is used it can be incredibly detrimental.
Solid body seven-string guitars were popularized in the 1980s and 1990s. Other artists go a step further, by using an eight-string guitar with two extra low strings. Although the most common seven-string has a low B string, Roger McGuinn (of The Byrds and Rickenbacker) uses an octave G string paired with the regular G string as on a 12-string guitar, allowing him to incorporate chiming 12-string elements in standard six-string playing. In 1982 Uli Jon Roth developed the "Sky Guitar", with a vastly extended number of frets, which was the first guitar to venture into the upper registers of the violin. Roth's seven-string and "Mighty Wing" guitar features a wider octave range.[citation needed]
Once you've got your categories narrowed down, then you can start getting into the nitty-gritty differences between strings. For instance, electric guitars will give you the choice between nickel (for authentic vintage sound) and stainless steel (for maximum durability). Some string manufacturers have exotic material options with their own unique characteristics, like Ernie Ball's Slinky Cobalt strings. With so many subtle differences separating guitar strings, you owe it to yourself to browse carefully and look at all the choices on the table before making a decision.

The Beatles668 tabs 637 visualizations1 Elvis Presley542 tabs 410 visualizations2 Nirvana513 tabs 360 visualizations3 Eagles139 tabs 349 visualizations4 Frank Sinatra387 tabs 348 visualizations5 Misc Soundtrack1160 tabs 279 visualizations6 Misc Cartoons897 tabs 265 visualizations7 Green Day650 tabs 254 visualizations8 Metallica382 tabs 247 visualizations9 Johnny Cash441 tabs 221 visualizations10
I strongly recommend beginner guitar players to use the Uberchord app (click for free download) for practicing chord progressions and chord changes, and use the real-time feedback to improve your playing skills. While, I’ll help you expedite the process of grabbing chords confidently on the neck and get you on your way to playing along expertly with your favourite band, or better yet, running a band of your own.
Open tunings improve the intonation of major chords by reducing the error of third intervals in equal temperaments. For example, in the open-G overtones tuning G-G-D-G-B-D, the (G,B) interval is a major third, and of course each successive pair of notes on the G- and B-strings is also a major third; similarly, the open-string minor-third (B,D) induces minor thirds among all the frets of the B-D strings. The thirds of equal temperament have audible deviations from the thirds of just intonation: Equal temperaments is used in modern music because it facilitates music in all keys, while (on a piano and other instruments) just intonation provided better-sounding major-third intervals for only a subset of keys.[65] "Sonny Landreth, Keith Richards and other open-G masters often lower the second string slightly so the major third is in tune with the overtone series. This adjustment dials out the dissonance, and makes those big one-finger major-chords come alive."[66]
Stacking the C-major scale with thirds creates a chord progression  Play (help·info), traditionally enumerated with the Roman numerals I, ii, iii, IV, V, vi, viio. Its major-key sub-progression C-F-G (I-IV-V) is conventional in popular music. In this progression, the minor triads ii-iii-vi appear in the relative minor key (Am)'s corresponding chord progression.
Electric guitars and bass guitars have to be used with a guitar amplifier and loudspeaker or a bass amplifier and speaker, respectively, in order to make enough sound to be heard by the performer and audience. Electric guitars and bass guitars almost always use magnetic pickups, which generate an electric signal when the musician plucks, strums or otherwise plays the instrument. The amplifier and speaker strengthen this signal using a power amplifier and a loudspeaker. Acoustic guitars that are equipped with a piezoelectric pickup or microphone can also be plugged into an instrument amplifier, acoustic guitar amp or PA system to make them louder. With electric guitar and bass, the amplifier and speaker are not just used to make the instrument louder; by adjusting the equalizer controls, the preamplifier, and any onboard effects units (reverb, distortion/overdrive, etc.) the player can also modify the tone (aka timbre or "colour") and sound of the instrument. Acoustic guitar players can also use the amp to change the sound of their instrument, but in general, acoustic guitar amps are used to make the natural acoustic sound of the instrument louder without changing its sound that much.
The teen and tween years are the make or break years for most people, in deciding whether they’re going to stick with a childhood hobby or interest, for life. As you go through puberty, and begin preparing for life as an adult, and responsibilities pile on your plate, the first things to go tend to be the personal interests you’re passionate about, because they don’t pay the bills. Also, because they’re not an official part of your school curriculum. In other words, what was once a must-have becomes a nice-to-have.
The ratio of the spacing of two consecutive frets is {\displaystyle {\sqrt[{12}]{2}}} (twelfth root of two). In practice, luthiers determine fret positions using the constant 17.817—an approximation to 1/(1-1/ {\displaystyle {\sqrt[{12}]{2}}} ). If the nth fret is a distance x from the bridge, then the distance from the (n+1)th fret to the bridge is x-(x/17.817).[15] Frets are available in several different gauges and can be fitted according to player preference. Among these are "jumbo" frets, which have much thicker gauge, allowing for use of a slight vibrato technique from pushing the string down harder and softer. "Scalloped" fretboards, where the wood of the fretboard itself is "scooped out" between the frets, allow a dramatic vibrato effect. Fine frets, much flatter, allow a very low string-action, but require that other conditions, such as curvature of the neck, be well-maintained to prevent buzz.
Most electric guitar bodies are made of wood and include a plastic pick guard. Boards wide enough to use as a solid body are very expensive due to the worldwide depletion of hardwood stock since the 1970s, so the wood is rarely one solid piece. Most bodies are made from two pieces of wood with some of them including a seam running down the center line of the body. The most common woods used for electric guitar body construction include maple, basswood, ash, poplar, alder, and mahogany. Many bodies consist of good-sounding, but inexpensive woods, like ash, with a "top", or thin layer of another, more attractive wood (such as maple with a natural "flame" pattern) glued to the top of the basic wood. Guitars constructed like this are often called "flame tops". The body is usually carved or routed to accept the other elements, such as the bridge, pickup, neck, and other electronic components. Most electrics have a polyurethane or nitrocellulose lacquer finish. Other alternative materials to wood are used in guitar body construction. Some of these include carbon composites, plastic material, such as polycarbonate, and aluminum alloys.

Acoustic guitars form several notable subcategories within the acoustic guitar group: classical and flamenco guitars; steel-string guitars, which include the flat-topped, or "folk", guitar; twelve-string guitars; and the arched-top guitar. The acoustic guitar group also includes unamplified guitars designed to play in different registers, such as the acoustic bass guitar, which has a similar tuning to that of the electric bass guitar.
###### Instructor ProfileArlen RothThe King of All Guitar TeachersMusic lesson pioneer Arlen Roth is the quintessential guitarist. An accomplished and brilliant musician — and one of the very few who can honestly say he’s done it all — Roth has, over the course of his celebrated 35-year career, played on the world’s grandest stages, accompanied many of the greatest figures in modern music and revolutionized the concept of teaching guitar. Read More...Lessons Wes Montgomery-style Octaves

Learning guitar is a lot easier when you have a step-by-step system to follow. Guitar Tricks lessons are interconnected and organized to get slightly harder as you progress. You watch a video lesson, play along, and then click a “Next” button to go to the next lesson. Lessons have multiple camera angles, guitar tabs, jam tracks and everything else you need to learn.
This is by far the best online instruction. The fact you receive instruction from an established professional musician who wants to hear and see you play and has the enthusiasm to want you to become the best guitar player/musician you can be speaks volumes about the level and dedication he has towards his students. This is a great value and I recommend it to everyone I know who is learning guitar.
Electric guitars and bass guitars have to be used with a guitar amplifier and loudspeaker or a bass amplifier and speaker, respectively, in order to make enough sound to be heard by the performer and audience. Electric guitars and bass guitars almost always use magnetic pickups, which generate an electric signal when the musician plucks, strums or otherwise plays the instrument. The amplifier and speaker strengthen this signal using a power amplifier and a loudspeaker. Acoustic guitars that are equipped with a piezoelectric pickup or microphone can also be plugged into an instrument amplifier, acoustic guitar amp or PA system to make them louder. With electric guitar and bass, the amplifier and speaker are not just used to make the instrument louder; by adjusting the equalizer controls, the preamplifier, and any onboard effects units (reverb, distortion/overdrive, etc.) the player can also modify the tone (aka timbre or "colour") and sound of the instrument. Acoustic guitar players can also use the amp to change the sound of their instrument, but in general, acoustic guitar amps are used to make the natural acoustic sound of the instrument louder without changing its sound that much.
After teaching guitar and music theory to thousands of students over past three decades, I thought that I had basically 'seen it all' when it comes to guitar instruction. Then I discovered Justin’s website, and man was I impressed! Justin’s caring spirit, attention to detail, vast knowledge base, and especially his lucid, laidback and nurturing style, allow students to fall in love with the learning process. You see, it’s not enough to simply find out how to play a few cool licks or chords. A truly great teacher will make you fall in love with the process of discovery so that you can unlock the best within you. Justin is one of these great teachers, and I highly recommend justinguitar.com to anyone who wants to tap into their best selves.
At MI, you learn from a proven curriculum taught by the best guitar instructors in the world, augmented by visiting artists’ seminars, concerts, and lessons from some of the greatest players in contemporary music. At our guitar music college, you get to network with other players, find your creative voice, and get the training you need to become the player you have always dreamed of being.


text:
# Post Leads To Avatar Change... A Bug!

It's probably nothing but you should know that the "post" button on top of your homepage is hyper-linked to https://brilliant.org/avatar/change/ [just hover your mouse over it to see]. If you click on that button, you see a drop-down menu for posting notes and problems. But when you right-click on that button and try to open it in a new tab, you are redirected to the avatar change page.

This isn't that important and people are unlikely to notice this if they don't have the habit of opening everything in a new tab [or hovering their mouse over everything]. However, it does look odd and I think you should fix this.

You probably wanted the profile pictures to be hyper-linked, didn't you? Because that would make sense.

Just another harmless bug I thought you should know about.

Edit: Now it's fixed.

Note by Mursalin Habib
7 years, 6 months ago

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

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It is fixed for me..Working fine. Only when I hover my mouse over the post button, It shows in the bottom ""avatar change"", but when clicked on, it pops up a menu showing problem posting and note posting. When clicked on any one, it goes to the right page.

- 7 years, 6 months ago

That is what I said.

If you click on that button, you see a drop-down menu for posting notes and problems. But when you right-click on that button and try to open it in a new tab, you are redirected to the avatar change page.

You have to right click on the post button and open the link in a new tab to be redirected to the avatar change page.

- 7 years, 6 months ago

you are right!

- 7 years, 6 months ago


text:
main-content

## Über dieses Buch

For this edition, a number of typographical errors and minor slip-ups have been corrected. In addition, following the persistent encouragement of Olga Oleinik, I have added a new chapter, Chapter 25, which I titled "Recent Results." This chapter is divided into four sections, and in these I have discussed what I consider to be some of the important developments which have come about since the writing of the first edition. Section I deals with reaction-diffusion equations, and in it are described both the work of C. Jones, on the stability of the travelling wave for the Fitz-Hugh-Nagumo equations, and symmetry-breaking bifurcations. Section II deals with some recent results in shock-wave theory. The main topics considered are L. Tartar's notion of compensated compactness, together with its application to pairs of conservation laws, and T.-P. Liu's work on the stability of viscous profiles for shock waves. In the next section, Conley's connection index and connection matrix are described; these general notions are useful in con structing travelling waves for systems of nonlinear equations. The final sec tion, Section IV, is devoted to the very recent results of C. Jones and R. Gardner, whereby they construct a general theory enabling them to locate the point spectrum of a wide class of linear operators which arise in stability problems for travelling waves. Their theory is general enough to be applica ble to many interesting reaction-diffusion systems.

## Inhaltsverzeichnis

### Chapter 1. Ill-Posed Problems

Abstract
Problems involving differential equations usually come in the following form: we are given an equation for the unknown function u, P(u) = f, on a domain Ω together with some “side” conditions on u. For example, we may require that u assumes certain preassigned values on ∂Ω, or that u is in L 2(Ω), or that u is in class C k in Ω. At first glance, it would seem that any of these extra conditions are quite reasonable, and that one is as good as the other. However, we shall see that this is far from being true, and that whichever additional supplementary conditions one assigns is intimately connected with the form of equation.
Joel Smoller

### Chapter 2. Characteristics and Initial-Value Problems

Abstract
Roughly speaking, characteristics are curves which carry information. They are particularly relevant in the study of “initial-value” problems ; that is, in solving partial differential equations, in which the solution surface is required to assume prescribed values “initially.” Such a problem presupposes the existence of a distinguished coordinate, ξ, where the equation ξ, = 0 defines the “initial ” surface. Of course, as we have seen in the last chapter, one needs some kind of compatibility between the equation and the initial surface. The notion of characteristic serves to classify and make more precise these intuitive ideas.
Joel Smoller

### Chapter 3. The One-Dimensional Wave Equation

Abstract
In this chapter we study a simple but quite interesting equation for which the initial-value problem is well-posed. The ideas which we introduce here will be used in various places throughout the book, albeit at a “higher dialectical ” level. The equation is derived from physical considerations, and in the case we consider here, the solution u(x, t), may be thought of as describing the position of a vibrating string at a point x at a time t.
Joel Smoller

### Chapter 4. Uniqueness and Energy Integrals

Abstract
We shall extend the method of energy integrals to more general second-order (hyperbolic) operators. This “energy” method is a basic technique in the modern theory of partial differential operators, and in the course of our development, we shall establish some interesting and important classical inequalities.
Joel Smoller

### Chapter 5. Holmgren’s Uniqueness Theorem

Abstract
There is a well-known theorem, called the Cauchy-Kowaleski theorem, which asserts that there exists a unique analytic solution of an analytic initial-value problem. Here, by an analytic initial-value problem, we mean a problem in which everything (the terms in the equation, the initial data, and the initial hypersurface), is analytic in a neighbourhood of a point (see [Ga]). The possibility is thereby left open as to whether there can exist a nonanalytic solution to this problem. Holmgren’s uniqueness theorem denies this possibility. We shall also find this result useful in Chapter 6 where we shall apply it to determine qualitative information on domains of dependence. For this reason, we shall prove a rather general version of the theorem.
Joel Smoller

### Chapter 6. An Initial-Value Problem for a Hyperbolic Equation

Abstract
We consider the equation for the homigeneous operator P:
$$P(u) \equiv \sum\limits_{{|\alpha | = m}} {{{a}_{\alpha }}{{D}^{\alpha }}u = 0,\;\quad x = (t,\xi ) \in {{R}_{ + }} \times {{R}^{n}},}$$
(6.1)
with initial data
$$D_{0}^{j}u(0,\xi ) = {{\psi }_{j}}(\xi ),\quad 0 \leqslant j < m.$$
(6.2)
We assume that each a a is constant, and that the hyperplane t = 0 is noncharacteristic with respect to P.
Joel Smoller

### Chapter 7. Distribution Theory

Abstract
Consider the partial differential equation (in R 2), u xy = f. If φ 0 2 (R), then multiplying both sides of this equation by φ, and integrating by parts, gives ∫ xy = ∫ . Now suppose that u is not necessarily smooth, but that this last equation holds for each φ∈ C 0 (R 2). We could then say that f is the “weak” mixed derivative of u. We could actually go one step further and consider the linear functional on C 0 (R 2) defined by
$$\phi \to \int\limits_{{{{R}^{2}}}} {u{{\phi }_{{xy}}}.}$$
Joel Smoller

### Chapter 8. Second-Order Linear Elliptic Equations

Abstract
Solutions of elliptic equations represent steady-state solutions; i.e., solutions which do not vary with time. They often describe the asymptotic states achieved by solutions of time-dependent problems, as t → ∞. Physically speaking, all the “rough spots” smooth out by the time this steady state is achieved.
Joel Smoller

### Chapter 9. Second-Order Linear Parabolic Equations

Abstract
Parabolic equations arise in diffusion processes, and more generally in “irreversible” time-dependent processes. Mathematically, this is reflected in the fact that the equations are not invariant under the reversal of time ; i.e., under the transformation t → —t. This means that knowledge about the “past” is lost as time increases. For example, there may be dissipation effects which lead to an increase in entropy and a consequent loss of information.
Joel Smoller

### Chapter 10. Comparison Theorems and Monotonicity Methods

Abstract
In this chapter we begin to study nonlinear partial differential equations. The results which we obtain here all follow from the maximum principles which were obtained in Chapters 8 and 9. We shall show how they apply to nonlinear elliptic and parabolic partial differential equations. As a first application, we will use the strong maximum principles to prove comparison theorems ; i.e., pointwise inequalities between different solutions. These say, roughly, that if u and v are two solutions, and if uv on ∂D, it follows that u d v on D. Such theorems can be quite useful in obtaining qualitative information about solutions. For example, comparison theorems are often used to obtain information about the asymptotic behavior of solutions of parabolic equations as t→ + ∞. As a second application of the maximum principle, we shall show how it can be used to prove existence theorems. This is the method of “upper” and “lower” solutions, the solution being the limit of a monotone iteration scheme, where the monotonicity is a consequence of the maximum principle.
Joel Smoller

### Chapter 11. Linearization

Abstract
There is a well-known theorem in ordinary differential equations, going back to Poincaré, which states that the stability of a rest point can be inferred from “linearization.” More precisely, if one considers the ordinary differential equation in ℝn, u′ = f(u), and ū is a rest point (so that/(ū) = 0), then if the differential (matrix) dfū) has all of its eigenvalues in the left-half plane, Re z x003C; 0, it follows that ū is asymptotically stable ; i.e., if u 0 is near ū, then the solution of the equation through u 0 tends to ū as t → + ∞. It is the main purpose of this chapter to prove an analogous theorem for partial differential equations, one which is sufficiently general to include systems of reaction-diffusion (parabolic) equations. In this context, the equation u′ = f(u) is replaced by an abstract equation of the form u t = Au + f(u), where u takes values in a Banach space B; i.e., for each t, u(t) is in B, and A is a linear operator. The main example is the case where A is a linear elliptic operator. The “rest points” ū, in this setting now are solutions of the equation Au + f(u) = 0, and the linearized operator becomes A + df(ū). We shall show that if the spectrum of this operator lies in the left-half plane, then again one can conclude that ū is asymptotically stable. This leads us quite naturally to a study of the spectrum of such linear operators. We shall undertake this study in §A, and in §B we shall prove a linearized stability theorem. The techniques which we develop here will be applied to specific problems in later chapters. In §C we shall give a useful extension of a result obtained in §A ; this is the celebrated Krein-Rutman theorem
Joel Smoller

### Chapter 12. Topological Methods

Abstract
The invention of modern topology goes back to Poincaré, who was led to it in his study of the differential equations of celestial mechanics. Its development was taken over, for quite a while, by people who interestingly enough, seemed to have completely forgotten its origins. Perhaps this really was necessary in order that the subject develop rapidly. In any case, already in the twenties and thirties, people like Morse, Leray, Schauder, and others, were applying topological methods to differential equations, It is our purpose here to explain the relevance of some of these techniques to nonlinear differential equations.
Joel Smoller

### Chapter 13. Bifurcation Theory

Abstract
Many problems in mathematics, and its applications to theoretical physics, chemistry, and biology, lead to a problem of the form
$$f(\lambda ,{\text{x}}){\text{ = 0}},$$
(13.1)
where f is an operator on R x B 1 into B2, with B 1 B 2 Banach spaces. For example, (13.1) could represent a system of differential or integral equations, depending on a parameter λ. We are interested in the structure of the solution set ; namely, the set
$${{f}^{ - }}1(0){\text{ = }}\;\{ (\lambda ,{\text{x}}) \in R \times {{B}_{1}}:f(\lambda ,x) = 0\}$$
(13.2)
.
Joel Smoller

### Chapter 14. Systems of Reaction — Diffusion Equations

Abstract
In recent years, systems of reaction-diffusion equations have received a great deal of attention, motivated by both their widespread occurrence in models of chemical and biological phenomena, and by the richness of the structure of their solution sets. In the simplest models, the equations take the form
$$\frac{\partial }{u}\partial t{\text{ = D}}\Delta u + f(u),\quad x \in \Omega \subset {{R}^{k}},\quad t > 0,$$
(14.1)
where u ∈ ℝn, D is an n x n matrix, and f(u) is a smooth function. The combination of diffusion terms together with the nonlinear interaction terms, produces mathematical features that are not predictable from the vantage point of either mechanism alone. Thus, the term DΔu acts in such a way as to “dampen” u, while the nonlinear function f(u) tends to produce large solutions, steep gradients, etc. This leads to the possibility of threshold phenomena, and indeed this is one of the interesting features of this class of equations.
Joel Smoller

### Chapter 15. Discontinuous Solutions of Conservation Laws

Abstract
In this chapter we shall begin the study of quasi-linear systems of the form
$${{u}_{t}}{\text{ + f}}{{({\text{u}})}_{{\text{x}}}}{\text{ = 0}},$$
(15.1)
where, and. We assume that the vector-valued function f is C 2 in some open subset Ω c ℝn. These equations are commonly called conservation laws in analogy to the examples of such systems which arise in physics ; see the examples below.
Joel Smoller

### Chapter 16. The Single Conservation Law

Abstract
In this chapter we shall obtain precise mathematical results on the existence and uniqueness of solutions for a single conservation law. In addition we shall also study the asymptotic behavior of our constructed solution. The existence problem will be attacked via a finite-difference method. Thus we shall replace the given differential equation by a finite-difference approximation depending on mesh parameters Ax and At. For every such pair (Δx, Δt) we shall construct a solution of the finite-difference equation, and we shall then obtain estimates which enable us to pass to the limit as the mesh parameters tend to zero in a certain definite way. The estimates which we obtain will be in the sup-norm and in the total variation-norm of the approximants, both sets of estimates being independent of the mesh parameters. It is worth noting that we are forced into obtaining bounds on the variation of the approximants, rather than (the usually encountered) bounds on derivatives, since the latter bounds would imply via the standard compactness criteria, that the limit would be continuous ; we know that this is not generally true.
Joel Smoller

### Chapter 17. The Riemann Problem for Systems of Conservation Laws

Abstract
We shall begin this chapter by studying the Riemann problem for an important class of equations which we designate as p-systems. This class includes as a special case the equations of isentropic as well as isothermal gas dynamics. For these equations one can give a complete solution of the Riemann problem for any two constant states. In §B we shall study the general properties of shocks and rarefaction waves, while in §C we shall solve the Riemann problem for general hyperbolic systems of conservation laws, but only when the initial states are sufficiently close.
Joel Smoller

### Chapter 18. Applications to Gas Dynamics

Abstract
In earlier sections, we have written the equations of gas dynamics in several different forms (see equations (18.3) and (18.3)′ of this chapter, and Examples 1 and 2 in Chapter 15). In many standard texts, it is shown that they are all equivalent for classical solutions ; i.e., they determine the same classical solutions. What we shall do here first is to prove that the computation of eigenvalues, Riemann invariants, and the genuine nonlinearity or linear degeneracy of the characteristic fields, are all independent of our choice of equations. That is, they are invariant under coordinate changes.
Joel Smoller

### Chapter 19. The Glimm Difference Scheme

Abstract
We consider a general system of conservation laws
$${{u}_{t}}{\text{ + }}f{{(u)}_{x}} = 0,\quad x \in R,\;\quad t > 0,$$
(19.1)
where u = (u 1,⋯,u n), with initial data
$$u(x,0) = {{u}_{0}}(x),\quad x \in R.$$
(19.2)
The system (19.1) is assumed to be hyperbolic and genuinely nonlinear in each characteristic field, in some open set U ⊂ ℝn (see Definition 17.7). We let λ1(u) < ⋯ < λn(u) denote the eigenvalues of df(u). Concerning u o(x), we assume that T.V.(u 0) is sufficiently small, where by T.V.(-) we mean the total variation. With these assumptions, we shall show that the above problem has a solution which exists for all t > 0.
Joel Smoller

### Chapter 20. Riemann Invariants, Entropy, and Uniqueness

Abstract
There are better results known for pairs of conservation laws than for systems with more than two equations. We have already seen an example of this in the last chapter ; namely, the interaction estimates are stronger when n = 2 than when n> 2. This was due to the existence of a distinguished coordinate system called Riemann invariants, which in general exists only for two equations. We shall study the implications one can draw using these coordinates. It turns out that the equations take a particularly nice form when written in terms of the Riemann invariants, and using this we can prove that for genuinely nonlinear systems, global classical solutions generally do not exist. (We only know this now for a single conservation law ; see Chapter 15, §B.)
Joel Smoller

### Chapter 21. Quasi-Linear Parabolic Systems

Abstract
We have studied second-order quasi-linear parabolic systems in Chapter 14, where it was assumed that the equations admitted a bounded invariant region. For the gas dynamics equations with all of the dissipative mechanisms taken into account (viscosity and thermal conductivity), and for various models of these, there may exist invariant regions, but they are usually unbounded. Thus we cannot conclude that the solution is a-priori bounded, and global existence theorems become more difficult to prove. One way to overcome this problem is to obtain “energy” inequalities in the unknown function and its derivatives, in a manner somewhat analogous to what we have done for linear hyperbolic equations in Chapter 4. In order to obtain these estimates for nonlinear equations, certain additional restrictions must be imposed : small data, special forms of the equations, restrictions on the data at infinity, and so on.
Joel Smoller

### Chapter 22. The Conley Index

Abstract
In any theoretical investigation of a real physical system, one is always forced to make simplifying assumptions concerning the true nature of the system. Since such idealizations are inevitable, it is reasonable to inquire as to how far one can go in this direction and still obtain satisfactory results. In certain cases, for example the motion of the planets, the equations of celestial mechanics provide a quite accurate model of the real physical system. In other situations, such as ecological or chemical interactions, or the study of large scale atmospheric phenomena, one either writes down certain reasonable relations between the quantities involved and their rates of change, or one tremendously reduces the number of actual equations involved. If such “leaps of faith” are to be of any use, it is necessary to study “rough” equations in “rough” terms. This in a nutshell, is our aim in this chapter and the next one. In other words, we want to fit these vague notions into a precise mathematical framework.
Joel Smoller

### Chapter 23. Index Pairs and the Continuation Theorem

Abstract
In this chapter we shall consider the Conley index from a more general point of view, one which allows us to apply the theory to a wide variety of equations including in particular, systems of reaction-diffusion equations. For such equations, it is not at all clear that the equations even define a flow. To get around such problems, we introduce the concept of a local flow and develop the theory in this setting. Roughly speaking, a local flow is a subset of the underlying space which is locally invariant for positive time; one thinks of a subspace of a function space, say L 2, which is invariant under the equations for small t > 0.
Joel Smoller

### Chapter 24. Travelling Waves

Abstract
The Conley index is a double-edged sword : if it is ever shown to be nontrivial, then this implies the existence of an orbit which stays in the isolating neighborhood for all time ; in this sense it gives an existence theorem. On the other hand, being a Morse-type index, it also carries stability information concerning the isolated invariant set. In this chapter we shall illustrate both of these properties for a special class of solutions of partial differential equations called travelling waves.
Joel Smoller

### Chapter 25. Recent Results

Abstract
In this chapter we shall present a summary of what could be considered as some of the most significant results which have appeared since the appearance of the original edition of this book. Of course, limitations of space (and time !) forces us to give only statements of major results, and very brief outlines of some of their proofs. This chapter will be divided into four sections, the first two corresponding to Parts II and III of the text, and the last two related to Part IV of the text. The numbered references correspond to the new reference list given at the end of this chapter.
Joel Smoller

### Backmatter

Weitere Informationen


text:
# Combining Tableaux and Algebraic Decision Procedures for Dealing with Qualified Number Restrictions in Description Logics

• Published 2001

#### Abstract

This paper investigates an optimization technique for reasoning with qualified number restrictions in the description logic ALCQHR+ . We present a hybrid architecture where a standard tableaux calculus is combined with a procedure deciding the satisfiability of linear (in)equations derived from qualified number restrictions. The advances are demonstrated by… (More)


text:
+0

# Polynomials

0
111
1
+222

Let $$f(x) = x^4-3x^2 + 2$$ and $$g(x) = 2x^4 - 6x^2 + 2x -1$$. What is the degree of $$f(x) \cdot g(x)$$

Apr 18, 2021


text:
# Worksheet on Subtraction of Money

Practice the questions given in the worksheet on subtraction of money by using without conversion and by conversion method (without regrouping and with regrouping).

Note: Arrange  the amount of rupees and paise in columns and then subtract the amounts to find the difference between the two given amounts.

1. Convert into paise and subtract without regrouping:

(i) Rs 38.72 - Rs 16.32

(ii) Rs 36.17 - Rs 24.15

(iii) Rs 68.93 - Rs 16.32

(iv) Rs 60.29 - Rs 20.25

(v) Rs 95.56 - Rs 32.41

(vi) Rs 75.35 - Rs 75.23

2. Convert into paise and subtract by regrouping:

(i) Rs 85.82 - Rs 67.93

(ii) Rs 67.31 - Rs 38.29

(iii) Rs 75.34 - Rs 36.26

(iv) Rs 92.64 - Rs 80.05

(v) Rs 36.13 - Rs 10.23

(vi) Rs 59.05 - Rs 49.10

3. Find the difference without conversion and without regrouping:

(i) Rs 40.41 from Rs 65.52

(ii) Rs 12.20 from Rs 23.94

(iii) Rs 28.05 from Rs 89.95

(iv) Rs 53.90 from Rs 94.90

(v) Rs 79.05 from Rs 99.95

(vi) Rs 53.65 from Rs 67.85

4. Regroup and subtract.

(i) Rupees forty-seven and thirty paise from rupees seventy-eight and twenty-five paise.

(ii) Rupees sixty-two and eighty-five paise from rupees ninety-seven and fifty-two paise.

(iii) Rupees sixty-nine and eighty-six paise from rupees ninety-eight and sixty-five paise.

(iv) Rupees twenty and fifty paise from rupees forty-seven and forty-five paise.

(v) Rupees thirty-five and eighty-nine paise from rupees forty-three and fifty-seven paise.

5. Subtract the following:

(i) Rs. 27.85 from Rs. 95.37

(ii) Rs. 0.57 from Rs. 1.00

(iii) Rs. 23.72 from Rs. 133.54

(iv) Rs. 69.05 from Rs. 100.75

(v) Rs. 100.01 from Rs. 200.92

(vi) Rs. 0.85 from Rs. 1.05

(vii) Rs. 0.95 from Rs. 5.00

6. Subtract the following arranging in columns of rupees and paise:

(i) Rs. 0.05 from Rs. 15.00

(ii) Rs. 0.75 from Rs. 5.00

(iii) Rs. 1.36 from Rs. 5.00

(iv) Rs. 85.62 from Rs. 110.75

(v) Rs. 58.38 from Rs. 100.00

(vi) Rs. 11.05 from Rs. 70.00

(vii) Rs. 122.49 from Rs. 215.18

(viii) Rs. 139.84 from Rs. 532.54

Word Problems on Subtraction of Money:

• ### Word Problems on Multiplying Money | Word Problems Involving Money

Read the questions given in the word problems on multiplying money. We need to understand the statement and multiply the amount of money as ordinary numbers with two digit numbers. 1. Max buys 25 books at the rate of \$ 252.55 per book. How much did he spend?

• ### Preparing a Bill | Bill Preparation Processes | Shopping Bill | Bills

A shopkeeper prepares a bill for different items purchased by the customer. A bill usually has details of items being sold, quantity, rate and the total cost. So, when we get a bill, we must check whether the quantity, price and the amount for each item have been entered

• ### Use of Decimal to Represent Money | Write Rupees and Paise as Decimal

We use decimals everyday while dealing with money. While writing rupee and paise, we use point to separate rupee from paise. We call this dot the decimal point. We can write rupees and paise as decimal numbers. Paise is always written as a 2-digit number.

• ### Conversion of Money | Convert the Amount from Rupees to Paisa

In conversion of money we will learn to convert the amount from rupees to paisa and from paisa to rupees. Conversion from rupees to paisa and from paisa to rupees is based on the basic fact that Re. 1.00 = 100 paisa and 100 paisa = Re. 1.00

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Browse Questions

A photon of wavelength $\;\lambda\;$ is scattered from an electron , which was at rest .The wavelength shift $\;\bigtriangleup \lambda\;$ is three times of $\;\lambda\;$ and the angle of scattering $\;\theta\;$ is $\;60^{0}\;$ . The angle at which the electron recoiled is $\;\phi\;$ .The value of $\;tan \phi\;$ is : (electron speed is much smaller than the speed of light)

$(a)\;0.16\qquad(b)\;0.22\qquad(c)\;0.25\qquad(d)\;0.28$

I dont know how to solve the question...