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# How to view beth number as ordinal?

The following is the snippet of the paragraph given in my lecture note:

(Note: $\beth$ is the beth-number.)

(When I introduced $\beth_\alpha$ I told you that it was to be a particular size of infinity; specifically, the cardinality of $S_\alpha$. Now I am suggesting that you can think of $\beth_\alpha$ as an ordinal (and, in particular, an initial ordinal). Isn’t there something weird going on here? In fact, there’s nothing to worry about. It’s just that initial ordinals can be used to do double-duty: although they are ordinals, they can also be used to represent the cardinalities of sets (and, in particular, the cardinalities of ordinals). So, for instance, $\omega$ is an ordinal, but it can also be used to represent a particular cardinality, and when it’s playing that role it is often called $\beth_{0_o}$, rather than $\omega$.)

If we think of the beths as ordinals, we can start characterizing some seriously big infinities. $\beth_{\beth_\omega}$, for example, is much bigger than anything we had talked about so far. And $\beth_{\beth_{\beth_\omega}}$ is bigger still.

My question is what does $\beth_{\beth_\omega}$ and $\beth_{\beth_{\beth_\omega}}$ correspond to? I can't really understand how to view beth as ordinals.

You can view $\beth_0$ as the ordinal $\omega$, but it is more difficult to understand what other beth numbers look like. (One could argue that seing what $\omega$ looks like isn't that easy either, but looking like is not a matematical notion after all.)

For instance, $\beth_1 = 2^{\aleph_0}$ is hard to describe. We know that $\omega_1 \leq \beth_1$ so $\beth_1$ contains every countable ordinal, but it is hard to say more because $\omega_1 = \beth_1$ is undecidable in ZFC (it is the continuum hypothesis). In fact, it is consistent with ZFC that all aleph numbers up to (and excluding) $\aleph_{2^{\omega}}$ are in $\beth_1$.

The situation with $\beth_{\beth_{\omega}}$ and $\beth_{\beth_{\beth_{\omega}}}$ should be even more complicated.

So the answer in my opinion is we don't know what beth-numbers other than $\beth_0$ look like as ordinals; we can only get some insight if we study a specific model of ZFC where CH is settled in some way.

In any case, in mathematics, it isn't always possible to visualize properties, and one has to rely on the language instead.

• if $\beth_1$ contained all $\aleph_\alpha$ for $\alpha < 2^{\omega}$ then it would contain $\aleph_{2^\omega}$ as well. It is consistent with ZFC that $\beth_1$ contains all aleph numbers up to and including $\aleph_{2^\omega}$, but we can't exclude the limit. – Deedlit Aug 16 '15 at 22:48
• @Deedlit No, that's incorrect - it's very easy to show that $\aleph_{2^\omega}\ge\beth_1$, so in fact $\beth_1$ can never contain $\aleph_{2^\omega}$. However, it is consistent that $\beth_1=\aleph_{2^\omega}$, in which case $\beth_1$ contains all the $\aleph$s up to $\aleph_{2^\omega}$, but not $\aleph_{2^\omega}$ itself. – Noah Schweber May 6 '16 at 0:33
• Oh right - I must have confused "contains" with "is greater than or equal to". – Deedlit May 6 '16 at 6:32


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Often it’s desirable to have local copies of third-party data sets. Fetching data on the fly from remote sources can be a great strategy, but for speed or other reasons it may be better to have local copies. This is particularly common in environmental and other sciences that deal with large data sets (e.g. satellite or global climate model products). Bowerbird is an R package for maintaining a local collection of data sets from a range of data providers.

A comprehensive introduction to bowerbird can be found at https://ropensci.github.io/bowerbird/articles/bowerbird.html, along with full package documentation.

## Installing

install.packages("devtools")
library(devtools)
install_github("ropensci/bowerbird",build_vignettes=TRUE)

## Usage overview

### Configuration

Build up a configuration by first defining global options such as the destination on your local file system:

library(bowerbird)
my_directory <- "~/my/data/directory"
cf <- bb_config(local_file_root = my_directory)

Bowerbird must then be told which data sources to synchronize. Let’s use data from the Australian 2016 federal election, which is provided as one of the example data sources:

my_source <- bb_example_sources("Australian Election 2016 House of Representatives data")

## add this data source to the configuration
cf <- bb_add(cf, my_source)

Once the configuration has been defined and the data source added to it, we can run the sync process. We set verbose = TRUE here so that we see additional progress output:

status <- bb_sync(cf, verbose = TRUE)

Congratulations! You now have your own local copy of your chosen data set. This particular example is fairly small (about 10MB), so it should not take too long to download. Details of the files in this data set are given in the status$files object: status$files

At a later time you can re-run this synchronization process. If the remote files have not changed, and assuming that your configuration has the clobber parameter set to 0 (“do not overwrite existing files”) or 1 (“overwrite only if the remote file is newer than the local copy”) then the sync process will run more quickly because it will not need to re-download any data files.

## Data source definitions

The blueant package provides a suite of bowerbird data source definitions themed around Southern Ocean and Antarctic data, and includes a range of oceanographic, meteorological, topographic, and other environmental data sets.

format: plain


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As I discussed in my previous post, records of global temperatures over the last few decades figure prominently in the debate over the climate effects of CO2 emitted by burning fossil fuels. I am interested in what this data says about which of the reasonable positions in this debate is more likely to be true —  the warmer’ position, that CO2 from burning of fossil fuels results in a global increase in temperatures large enough to have quite substantial (though not absolutely catastrophic) harmful effects on humans and the environment, or  the lukewarmer’ position, that CO2 has some warming effect, but this effect is not large enough to be a major cause for worry, and does not warrant imposition of costly policies aimed at reducing fossil fuel consumption.

A recent focus of this debate has been whether temperature records show a pause’ (or hiatus’) in global warming over the last 10 to 20 years (or at least a slowdown’ compared to the previous trend), and if so, what it might mean. Lukewarmers might interpret such a pause as evidence that other factors are comparable in importance to CO2, and can temporarily mask or exaggerate its effects, and hence that naively assuming the warming from 1970 to 2000 is primarily due to CO2 could lead one to overestimate the effect of CO2 on temperature.

Whether you sees a pause might, of course, depend on which data set of global temperatures you look at. These data sets are continually revised, not just by adding the latest observations, but by readjusting past observations.

Here are the yearly average land-ocean temperature anomaly data from 1955 to 2014 from the Goddard Institute for Space Studies (GISS), in the version before and after July of this year:

The old version shows signs of a pause or slowdown after about 2000, which has largely disappeared in the new version.  Unsurprisingly, the revision has engendered some controversy. I should note that the difference is not really due to GISS itself, but rather to NOAA, from whom GISS gets the sea surface temperatures used.

Many people pointing to a pause look at the satellite temperature data from UAH, which starts in 1979.  Below, I show it on the right, with the new GISS data from 1979 on the left, both in yearly (top) and monthly (bottom) forms:

Two things can be noted from these plots. First, the yearly UAH data (top right) can certainly be seen as showing roughly constant temperatures since somewhere between 1995 and 2000, apart from short-term variability. However, if one so wishes, one can also see it as showing a pretty much constant upward trend, again with short-term variability. Looking at the monthly UAH data (bottom right) gives a much stronger impression of a pause, since fitting a straight line to the monthly data leads to most points after about 2007 being under the line, while those before then back to about 2001 are mostly above the line, which is what one would expect if there is a pause at the end — see the plot below of the least-squares fitted line and its residuals:

The (new) GISS data also gives more of an impression of a slowdown with monthly rather than yearly data:

There are two issues with looking at monthly data, however. The first is that although both GISS and UAH data effectively have a seasonal adjustment — anomalies for each month are from a baseline for that month in particular — the seasonal effects actually vary over the years, introducing possible confusion.  I’ll try fitting a model that handles this in a later post, but for now sticking to the yearly data avoids the problem.  The second issue is that one can see a considerable amount of autocorrelation’ in the monthly data.  This brings us to the crucial question of what one should really be asking when considering whether there is a pause (or a slowdown) in the temperature data.

To some extent, talk of a pause’ by lukewarmers is for rhetorical effect — look, no warming for 15 years! — as a counter to the rhetoric of the warmers — see how much the planet has warmed since 1880!  — with such rhetoric by both sides being only loosely related to any valid scientific argument.  However, one should try as much as possible to interpret both sides as making sensible arguments.

In this respect, note that the lukewarmers are certainly not claiming that the pause shows that although CO2 had a warming effect up until the year 2000, it stopped having a warming effect after 2000, so we don’t have to worry now.  I doubt that anyone in the entire world believes such a thing (which is saying a lot considering what some people do believe).

Instead, the sensible lukewarmer interpretation of a pause’ would be that the departures from the underlying trend in the temperature time series have a high degree of positive autocorrelation — that the departure from trend in one year is likely to be similar to the departures from trend of recent years. (Alternatively, some lukewarmers might think that there are deterministic or stochastic cycles, with periods of decades or more.) The effect of high autocorrelation is to make it harder to infer the magnitude of the true underlying trend from a relatively short series of observations.

The problem can be illustrated with simulated data sets, which I’ve arranged to look vaguely similar to the GISS data from 1955 to 2014 (though to avoid misleading anyone, I label the x-axis from 1 to 60 rather than 1955  to 2014).

I start by generating a series of 20000 values with high autocorrelation that will be added as residuals to a linear trend. I do this by summing a Gaussian series with autocorrelations that slowly decline to zero at lag 70, a slightly non-Gaussian series with autocorrelations that decline more quickly, and a series of independent Gaussian values.  The R code is as follows:

set.seed(1) n0 <- 20069 fa <- c(1,0.95,0.9,0.8/(1:67)^0.8); fa <- fa/sum(fa) fb <- exp(-(0:69)/2.0); fb <- fb/sum(fb) xa <- filter(rnorm(n0),fa); xa <- xa[!is.na(xa)] xb <- filter(rt(n0,5),fb); xb <- xb[!is.na(xb)] xc <- rnorm(length(xb)) xresid <- 0.75*xa + 0.08*xb + 0.06*xc

Here are the first 1500 values of this residual series:

Here are the autocorrelations estimated from the entire simulated residual series:

The autocorrelation time’ shown above is one plus twice the sum of autocorrelations at lag 1 and up. It is the factor by which the effective sample size is less than it would be if the points were independent. With an autocorrelation time of 13 as above, for example, a data set of 60 points is equivalent to about 5 independent points.

I then split this long residual series into chunks of length 60, to each of which I added a trend with slope 0.01, and then shifted it to have sample mean of zero. Here are the first twenty of the 333 series that resulted:

The slope of the least-squares fit line is shown above each plot. As one can see, some slope estimates are almost twice the underlying trend of 0.01, while other slopes are much less than the underlying trend. Here is the histogram of slope estimates from all 333 series of length 60, along with the lower bound of the 95% confidence interval for the slope, computed assuming no autocorrelation:

Ignoring autocorrelation results in the true slope of 0.01 being below the lower bound of the 95% confidence interval 24% of the time (ten times what should be the case).

What is even more worrying is that looking at the residuals from the regression often shows only mild autocorrelation. Here are the autocorrelation (and autocorrelation time) estimates for the first 20 series:

One can compare these estimates with the plot of true residual autocorrelation above, and the true autocorrelation time of 13.

To see the possible relevance of this simulation to global temperature data, here are old and new GISS global temperature anomaly series (from 1955), centred and relabeled as for the simulated series, along with simulated series B and L from above:

It is worrying that the GISS series do not appear much different from the simulated series, which substantially overestimate the trend.

The real significance of a pause’ or slowdown’ in temperatures is that it would be evidence of such high autocorrelation, whose physical basis could be internal variability in the climate system, or the influence of external factors that themselves exhibit autocorrelation. Looking for a pause’ may not be the best way of assessing whether autocorrelation is a big problem. But direct estimation of long-lag autocorrelations from relative short series is not an easy problem, and may be impossible without making strong prior assumptions regarding the form of the autocorrelation function.

Accordingly, I’ll now go back to looking at whether one can see a pause in the GISS and UAH temperature data, while keeping in mind that the point of this is to see whether high autocorrelation is a problem. I’ll look only at the yearly data, though as noted above, a pause or slowdown may be more evident in the monthly data.

Here are the old and new versions of the GISS data, from 1955 through 2014, with least-squares regression lines fitted separately to data before 1970, from 1970 to 2001, and after 2001. In the top plots, the fits are required to join up; in the bottom plots, there may jumps as well as slope changes at 1970 and 2001.

In the two top plots, the estimated slopes after 2001 are smaller than the slopes from 1970 to 2001, but the differences are not statistically significant (p-values about 0.3, assuming independent residuals). In the bottom two plots, the slopes before and after 2001 differ substantially, with the differences being significant (p-values of 0.003 and 0.018, assuming independent residuals). However, one might wonder whether the abrupt jumps are physically plausible.

Next, let’s look at the UAH data, which starts in 1979, along with the (new) GISS data from that date for comparison, and again consider a change in slope and/or a jump in 2001:

Omitting the data from 1970 to 1978 decreases the pre-2001 slope of the GISS data, lessening the contrast with the post-2001 slope. For the UAH data, the difference in slopes before and after 2001 is quite noticeable. However, for the top UAH plot, the difference is not statistically significant (p-value 0.19, assuming independent residuals). For the bottom plot, the two-sided p-value is 0.08. Based on the comparison with the GISS data, however, one might think that both differences would have been significant if data back to 1970 had been available.

There is a cherry-picking’ issue with all the above p-values, however. The selection of 2001 as the point where the slope changes was made by looking at the data. One could try correcting for this by multiplying the p-values by the number of alternative choices of year, but this number is not clear. In a long series one would expect the slope to change at other times as well, as indeed seems to have happened in 1970. One could try fitting a general model of multiple change-points’, but this seems inappropriately elaborate, given that the entire exercise is a crude way of testing for long-lag autocorrelation.

I have, however, tried out a Bayesian analysis, comparing a model with a single linear trend, a model with a trend that changes slope at an unknown year (between 1975 and 2010), a model with both a change in slope and a jump (at an unknown year), and a model in which the trend is a constant apart from a jump (at an unknown year).  I selected informative priors for all the parameters, as is essential when comparing models in the Bayesian way by marginal likelihood, and computed the marginal likelihoods (and posterior quantities) by importance sampling from the prior (a feasible method for this small-scale problem).  See the R code linked to below for details.

Here are the results of these four Bayesian models, shown as the posterior average trend lines:

In the last plot, note that the model has an abrupt step up at some year, but the posterior average shows a more gradual rise, since the year of the jump is uncertain. The log marginal likelihoods for the four models above are 16.0, 15.4, 15.7, and 14.4. If one were to (rather artificially) assume that these are the only four possible models, and that they have equal prior probabilities, the posterior probabilities of the four models would be 39%, 23%, 30%, and 9%.

I emphasize again that the exercise of looking for a pause’ or slowdown’ is really a crude way of looking for evidence of long-lag autocorrelation. The quantitative results should not be taken too seriously. Nevertheless, the conclusion I reach is that this data does not produce a definitive yes or no answer to whether there is a pause, even in the UAH data, for which a pause seems most evident. A few years more data might (or might not) be enough to make the situation clearer. Analysis of monthly data might also give a more definite result. Note, however, that lack of definite evidence of a pause’ is not the same as no pause’. It is not reasonable to assume a lack of long-lag autocorrelation absent definite evidence to the contrary, since the presence of such autocorrelation is quite plausible a priori.

In my previous post, I had said that this next post would examine two papers `debunking’ the pause, but it’s gotten too long already, so I’ll leave that for the post after this. I’ll then look at what can be learned by looking at monthly data, and by modeling some known effects on temperature (such as volcanic activity).


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# show that $n\Upsilon_{n-1} \equiv -1 \pmod{n}$ iff $n$ is prime

The Bernoulli numbers $$B_n$$. where all numbers $$B_n$$ are zero with odd index $$n>1$$. first values are given by $$B_{0} = 1$$ , $$B_{1} = -1/2$$, $$B_{2} = 1/6$$, $$B_{3} = -1/30$$.

Agoh conjecture: let $$n$$ be a positive integer with $$n \ge 2$$, then

$$nB_{n-1} \equiv -1 \pmod{n}\iff n\text{ is prime}$$

The idea of $$\Upsilon$$ number comes from my power sum formula and negative values of zeta function

Definition let $$m$$ be a non-negative integer $$\zeta(-m)=(-1)^{m}\frac{B_{m+1}}{m+1}=\sum_{b=1}^{m+1} \Upsilon_b\sum_{i=0}^{b-1} (-1)^{i}(b-i)^{m}\binom{b-1}i$$

so we can calculate values of $$\Upsilon_b$$ , if we substitute value $$\zeta (0)=-1/2$$ gives $$\Upsilon_1=-1/2$$.

Again we can calculate next value using or substituting previous values of $$\Upsilon_b$$.

first values are given by \begin{align*} \Upsilon_1&=-\frac{ 1}{2},\\ \Upsilon_2&=\frac{ 5}{12},\\ \Upsilon_3&=-\frac{ 3}{8},\\ \Upsilon_4&=\frac{ 251}{720},\\ \Upsilon_5&=-\frac{ 95}{288},\\ \Upsilon_6&=\frac{ 19087}{60480},\\ \Upsilon_7&=-\frac {5257}{17280}. \end{align*}

hypothesis: let $$n$$ be a positive integer with $$n \ge 2$$, then

$$n\Upsilon_{n-1} \equiv -1 \pmod{n} \iff n\text{ is prime}$$

Else show that hypothesis is equivalence to agoh's conjectures

Else show that if n is prime then $$n\Upsilon_{n-1} \equiv -1 \pmod{n}$$

Steven Clark had verified conjecture $$n\Upsilon_{n-1} \equiv -1 \pmod{n}\iff n\in \mathbb{P}$$ for $$2 \leq n \leq 500$$

More simply

So $$D_{m,m+1}=m!$$

$$\Upsilon_{m+1}=\frac{(-1)^{m}B_{m+1}-(m+1)(\Upsilon_1 D_{m,1}+\Upsilon_2 D_{m,2}+\cdots+\Upsilon_m D_{m,m})}{(m+1)!}$$

Please see my post on MO

• Are you familiar with Giuga's conjecture? It's in the same territory. en.wikipedia.org/wiki/Agoh–Giuga_conjecture Aug 10, 2019 at 6:48
• @Gerry_Myerson yes I am giuga's and agoh's conjectures are equivalence Aug 10, 2019 at 7:02
• So, what evidence do you have for your hypothesis? Aug 10, 2019 at 7:10
• @Gerry_Myerson Based on given $\Upsilon_n$ values I have assumed my hypothesis is true but I am working on a program for further observation Aug 10, 2019 at 10:41
• I deleted one of my earlier comments after I found an error in my code. Assuming no more errors in my code, I've now verified your conjecture $n\,\Upsilon_{n-1}\equiv -1\,(mod\ n)\,\iff\,n\in \mathbb{P}$ for $2\le n\le 500$. My interest in this question is partially motivated by my conjectured formulas for $\zeta(-j)$ in my question math.stackexchange.com/q/3262873. Aug 26, 2019 at 19:41

Assuming no errors in my code, I have now verified the conjecture $$n\,\Upsilon_{n-1}\equiv -1\,(mod\ n)\,\iff\,n\in \mathbb{P}$$ for $$2\le n\le 1000$$.
I attached an image of my Mathematica code to provide others the opportunity to verify correctness of my code and use it to explore the conjecture for a larger range of $$n$$ values if so desired.


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Public Group

# Getting 0xcccccccc error

## Recommended Posts

I don't know why this is happening, in the image below you can see my current situation.

Basically I have an Utility.h header which contains wathever random stuff I think is useful at the time but don't yet deserves it's own separate file, so inside of it I've put a BaseFont class and defined it in place since I am not sure it will stay into Utility.h for long (Font is an alias for BaseFont because I wanted the internal variable name for the font to be Font, and that would collide with the class name, so that's why I have it setup like this) and right below this class, still in the header, I have a global object constructed from it so that I can use it everywhere for debug since pretty much all my .cpp #include "Utility.h". Now when I close the game I get an error that says "Breakout.exe stop working" and I tracked it down to the BaseFont destructor. On the right of the image you can see the local variable situation at that break point, which doesn't looks good...

I have some assumptions regarding the cause but I am not quite sure, can you tell me what its going wrong exactly?

Edited by MarcusAseth

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Your class doesn't comply with the rule of three, which means that if anyone copies it, you'll end up with a double deletion bug in the destructor, so that could be it.

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2 minutes ago, Hodgman said:

Your class doesn't comply with the rule of three, which means that if anyone copies it, you'll end up with a double deletion bug in the destructor, so that could be it.

Added the code below, still in the same situation, must be something else

	BaseFont(const BaseFont&) = delete;
BaseFont& operator=(const BaseFont&) = delete;
BaseFont(BaseFont&&) = delete;
BaseFont& operator=(BaseFont&&) = delete;

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App::~App()
{
if (Window) { SDL_DestroyWindow(Window); }
if (Renderer) { SDL_DestroyRenderer(Renderer); }
for (auto& elem : Textures)
{ SDL_DestroyTexture(elem.second); }

if (TTF_WasInit()) { TTF_Quit(); };
SDL_Quit();
}

I think the "global static" Font Arial lifetime has to do with the problem, since I am guessing the object is probably still around after TTF_Quit() is called (which happen after main() returns basically.

But still, I tested by removing the TTF_Quit() and it improves since I don't get the "Breakout.exe stop working" anymore, but according to the debbugger when we hit that breakpoint in the image in the first post, the situation is still the same, the memory of "this" being 0xcccccccc... any idea why is that?

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8 minutes ago, MarcusAseth said:

the memory of "this" being 0xcccccccc... any idea why is that?

0xcc is microsoft's debug mode marker for uninitialized stack memory. You'll notice others over time too -- 0xcd is uninitialized heap memory, 0xdd/0xfeee is free'ed memory, 0xfd is guard regions to check for overruns... So this is an indication that 'this' is pointing to a region of the stack that's never been used

What's the call-stack when the crash happens? How are you instantiating the BaseFont class?

 Taking another look at your code, a side issue is that you never initialize the 'Font' member, but read from it before ever writing to it (via "if (Font)"). This is extremely dangerous and is only working due to luck at the moment.

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I apologize, actually it appears that if I step forward into the code, the 0xcccccccc turns itself into a reasonable address when it hit the problem (as shown in the image below), and the problem I think it was calling a TTF function after TTF_Quit().

It's just strange that didn't show me the proper "this" address at the breakpoint and even after stepping forward twice, only changed into it when it hits the error, is this something that happens debugging static variables?!

Anyway I still have the problem of  TTF_Quit() being called before the font are closed...how should I do in order to have that font global? I should like wrap it into a "global static FontManager" in which I register and request fonts and it takes care of freeying all the fonts before calling TTF_Quit()? Seems the only way I think :\

Edited by MarcusAseth

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Why would a Font not be freed before TTF_Quit is called? Isn't that happening right at the end of your program?

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The Font is a static object created in an included header, the App object is created inside of main(), I think it has to do with it.

I placed 2 breakpoints, one on App destructor and one on the Font destructor, App's destructor get fired first :\

So that's why I thought they should be wrapped together in another class "FontManager", so that I can keep the global object behaviour for the Font and correctly order the cleanup inside of a single destructor

Edited by MarcusAseth

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Don't create code that runs before/after main. That means don't create global objects with non-trivial constructors/destructors. It only causes headaches.

If you really need a global font (hint: you don't, but whatever), then create the font object itself inside main, and create a global pointer to it.

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7 minutes ago, Hodgman said:

If you really need a global font (hint: you don't, but whatever)

Got it

I should probably pack it in a vector<Font> into the App class . Is ok to have it on the stack though? Or should I use unique_ptr for those as well? Font sounds like a lightweight thing, but I don't want to make assumptions...

Edited by MarcusAseth

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As others have noted, this could be called "classical" or "Galilean" relativity, which is not what the Einstein theory is that most people associate with the word. Assuming this is ...
• 1,506
1 vote

### Is there anything truly "stationary" in the universe?

The word "stationary" in the modern (post-Newton) physics requires a reference frame: Object A is stationary relative to e.g. a non-rotating frame attached to the center of mass of the ...
• 6,528
1 vote

### Is there anything truly "stationary" in the universe?

As far as is understood, there is not absolute sense of "stationary" so the question you link is not even really well defined. All that could be asked is "do the galaxies move relative ...
• 1,506
1 vote

### Confusing definition of proper time – which is correct?

A 'propre' French word for own, propre time later polluted as proper time of an object is the time that is measured by a clock which is at rest w.r.t. the object, not only at rest but is must be ...

Only top scored, non community-wiki answers of a minimum length are eligible


text:
# How do you use the product rule to differentiate y=(x+1)^2(2x-1)?

Mar 26, 2018

So I also need to use chain rule on ${\left(x + 1\right)}^{2}$

#### Explanation:

$\frac{\mathrm{dy}}{\mathrm{dx}} = u ' v + v ' u$
$u ' = 2 \left(x + 1\right) \cdot 1$
$v ' = 2$

$u = {\left(x + 1\right)}^{2}$
$v = \left(2 x - 1\right)$

subbing into the product rule.

$\frac{\mathrm{dy}}{\mathrm{dx}} = 2 \left(2 x + 1\right) \cdot \left(2 x - 1\right) + 2 {\left(x + 1\right)}^{2}$

$\frac{\mathrm{dy}}{\mathrm{dx}} = 2 \left(4 {x}^{2} - 1\right) + 2 \left({x}^{2} + 2 x + 1\right)$

$\frac{\mathrm{dy}}{\mathrm{dx}} = 8 {x}^{2} - 2 + 2 {x}^{2} + 4 x + 2$

$\frac{\mathrm{dy}}{\mathrm{dx}} = 10 {x}^{2} + 4 x$

Mar 26, 2018

$\frac{\mathrm{dy}}{\mathrm{dx}} = 2 x {\left(x + 1\right)}^{2} + 2 \left(x + 1\right) \left(2 x - 1\right)$
or
$\frac{\mathrm{dy}}{\mathrm{dx}} = 2 {x}^{3} + 8 {x}^{2} + 4 x - 2$

#### Explanation:

We know that a product is to things multiplied by each other so ${\left(x + 1\right)}^{2}$ and $\left(2 x - 1\right)$ are separate products

$u = {\left(x + 1\right)}^{2}$
$u ' = 2 \left(x + 1\right) \cdot 1$

$v = 2 x - 1$
$v ' = 2 x$

The product rule is $\frac{\mathrm{dy}}{\mathrm{dx}} = u v ' + v u '$

so it is

$\frac{\mathrm{dy}}{\mathrm{dx}} = 2 x {\left(x + 1\right)}^{2} + 2 \left(x + 1\right) \left(2 x - 1\right)$

simplified

dy/dx=2(x+1) ((x(x+1)+(2x-1))
$\frac{\mathrm{dy}}{\mathrm{dx}} = \left(2 x + 2\right) \left({x}^{2} + x + 2 x - 1\right)$
$\frac{\mathrm{dy}}{\mathrm{dx}} = \left(2 x + 2\right) \left({x}^{2} + 3 x - 1\right)$

Further simplification

$\frac{\mathrm{dy}}{\mathrm{dx}} = 2 {x}^{3} + 6 {x}^{2} - 2 x + 2 {x}^{2} + 6 x - 2$
$\frac{\mathrm{dy}}{\mathrm{dx}} = 2 {x}^{3} + 8 {x}^{2} + 4 x - 2$


text:
Fitting multiple pKa's

I have three histidine residues that are next to each other in a protein. When I compute the fraction of unprotonated states for each individual histidine as a function of $$\mathrm{pH}$$ my titration curve has a non-Henderson–Hasselbalch (HH) shape. Hence I can't fit it using the HH equation for 1 site:

$$\frac{1}{1 + 10^{n(\mathrm{p}K_\mathrm{a} - \mathrm{pH})}}$$

in order to solve for the Hill coefficient $$n$$ and $$\mathrm{p}K_\mathrm{a}.$$

Obviously the three histidine $$\mathrm{p}K_\mathrm{a}$$'s are dependent on each other since they are very close to one another and are probably sharing the proton. So how do I go about fitting the curves? Should I sum the total unprotonated states in all three histidines and fit the curve with the three site equation

$$\frac{1}{1 + 10^{n_1(\mathrm{p}K_\mathrm{a1} - \mathrm{pH})}} + \frac{1}{1 + 10^{n_2(\mathrm{p}K_\mathrm{a2} - \mathrm{pH})}} + \frac{1}{1 + 10^{n_3(\mathrm{p}K_\mathrm{a3} - \mathrm{pH})}}?$$

When I use this equation I still don't get a good fit and I get unreasonable $$\mathrm{p}K_\mathrm{a}$$'s $$(-1.4, 7.7, -1.3).$$ Any suggestions?

As requested below, the matlab code I used for the fit is the following:

ft=fittype('(1/(1+10.^(a1*(p1-x))))+(1/(1+10.^(a2*(p2-x))))+(1/(1+10.^(a3*(p3-x))))', 'independent','x','dependent','y');
opts = fitoptions('Method','NonlinearLeastSquares');
[fitresult, gof] = fit( xData, yData, ft, opts );
h = plot(fitresult,xData,yData)

I posted values for p1, p2, p3 which should report $$\mathrm{p}K_\mathrm{a1},$$ $$\mathrm{p}K_\mathrm{a2}$$ and $$\mathrm{p}K_\mathrm{a3}.$$

• I suggest you post the code for the matlab function, at least. The fitting parameters you reportedly obtained do not match the shape of the curve in the figure. – Buck Thorn Dec 13 '19 at 6:50
• I second the comment above - please provide more detail of what and how you are doing the fitting? Are you fitting to the n values and allowing them to vary freely? It would make more sense to fit directly to the concentrations of 3H, 2H, H and no H with the constraint that they must all sum to 1. I would also worry about the effect of very high and very low pH on your protein. Is it stable across that range? The behavior should be similar to what one gets with a triprotic acid such as citric acid. For example, look at the chemistry libretext entry on "titration of a polyprotic weak acid". – Andrew Dec 13 '19 at 14:07
• As pointed out in one of the answers, your equation as written should reach a maximum of three at high pH not one. You need to put a correction factor to get from three to a scale of 0 to 1. More generally, fitting to that exponential form is extremely difficult with non-linear regression, even for a single HH equation. Replacing those denominators with the relevant expressions in terms of Ka’s and [H] will make the fit more robust. Even with that, fitting six independent parameters accurately with that small dataset will be hard. I would try setting all Hill coeffs to 1 on first try. – Andrew Dec 13 '19 at 20:42
• Do you have additional measurements than the 13 blue points represent? I'm not a biochemist, but to determine out three $pK_a$ (and not only one) with one measurement per unit of pH appears to me as coarse data, especially with the jumps for pH ~5 (in comparison to pH ~4 and ~6) and for pH ~7 in reference to the interpolation in red. As a suggestion: What about three runs each with measurements in increments of 0.1 units of pH, ploting both a) fraction of unprotonated in function of pH and b) the first derivative (change of fraction of unprotonated over change of pH value? – Buttonwood Dec 13 '19 at 21:34

Given your hypothesis that the three $$\mathrm{p}K_\mathrm{a}$$'s are dependent on one another and that experimentally you do not seem to be able to identify which histidines are protonated and which are not at any given time, it would seem that the simplest model to use is that of a triprotic acid such as citric acid. The triple HH equation model you used is only appropriate for three completely independent $$\mathrm{p}K_\mathrm{a}$$'s (such that the $$\mathrm{p}K_\mathrm{a}$$ at site 1 is unchanged by the protonation state of site 2).

In the dependent $$\mathrm{p}K_\mathrm{a}$$ model, we have four species (vs the eight in your model): $$\ce{H3P},$$ $$\ce{H2P},$$ $$\ce{HP}$$ and $$\ce{P},$$ where "$$\ce{P}$$" is the protein. There are three "apparent $$\mathrm{p}K_\mathrm{a}$$'s" none of which represents a true $$\mathrm{p}K_\mathrm{a}$$ of a specific site. They instead correlate to the combination of possible reactions that give a change in the overall protonation state. For example, the first $$\mathrm{p}K_\mathrm{a}$$ represents the change in state from $$\ce{H3P}$$ to $$\ce{H2P},$$ regardless of which of the three sites loses a proton.

We can then write four equations starting with mass balance:

\begin{align} [\text{total protein}] &= [\ce{H3P}] + [\ce{H2P}] + [\ce{HP}] + [\ce{P}]\tag{1}\\ K_\mathrm{a1} &= \frac{[\ce{H}][\ce{H2P}]}{[\ce{H3P}]}\tag{2}\\ K_\mathrm{a2} &= \frac{[\ce{H}][\ce{HP}]}{[\ce{H2P}]}\tag{3}\\ K_\mathrm{a3} &= \frac{[\ce{H}][\ce{P}]}{[\ce{HP}]}\tag{4} \end{align}

We can also define the fraction of sites unprotonated as

$$\phi = \frac{3[\ce{P}]+2[\ce{HP}]+[\ce{H2P}]}{3[\text{total protein}]}\tag{5}$$

It's relatively straightforward to use the four above equations to get expressions for each of the species concentrations in terms of $$[\ce{H}]$$ and $$[\text{total protein}]$$ only. Substituting those expressions into the bottom equation will give you an equation to which to fit your data.

If you post your data in tabular form, others can play around with different fitting approaches and give feedback.

• I understand that it might be an improvement but I am not sure this is entirely appropriate because the protons are not quite equivalent (the microscopic pKas are not identical at any given pH). – Buck Thorn Dec 13 '19 at 21:54
• They don’t need to be equivalent. In fact, if they are completely independent and have widely spaced pka’s, the apparent pka’s will converge on the microscopic pka’s – Andrew Dec 13 '19 at 22:03
• I took a liberty to brush up formatting a little and unify notations such as "[total protein]" and "total protein concentration", HH and H-H etc. for clarity. Also, I changed $[\ce{H3}]$ to $[\ce{H3P}]$ in eqn. (1). – andselisk Dec 14 '19 at 7:49

It makes no sense to solve for different pKa´s. They are constants specific to the acid groups of Histidine. Furthermore, the HH-equation describes the fraction of (in your case) dissociated acid in one acid/base equilibria. So if you sum the three steps, your Y-axis for one thing, has to go from 0 to 3. Just off the top of my head, it seems a better solution to multiply the HH-equations: At a given pH, a fraction of His has dissociated the first proton, and out of this fraction another fraction has given of one more proton etc.

The data look as if you have simply done a titration of histidine, with a first pKa at around 6, and another one at around 8.5 - 9. This is exactly what is expected from the tabulated values for histidine : pKa = 6.04 and 9.09. Why are you not satisfied about these measurements ? Your results are in perfect agreement with predictions form the tabulated values. It is no use trying to fit your points with the strange analytical curve that you propose.

• I believe the 9.09 pKa you are referring to is for the amino terminus. Within a polypeptide chain, this proton is not acidic, as the amine is now part of an amide bond. – Andrew Dec 13 '19 at 14:10


text:
# Any techniques for evading frequency based crypt analysis without encrypting?

Are there any general techniques for evading frequency based crypt analysis without using any encryption techniques?

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Why would you need to evade cryptanalysis period if you aren't using encryption? –  mikeazo Feb 17 '12 at 15:17
Your question isn't really clear. What are you trying to do? –  Paŭlo Ebermann Feb 17 '12 at 19:00
Compression tends to flatten frequencies, but is of course designed to be undone and thus has no crypto value by itself. –  GregS Feb 18 '12 at 17:52

Have a look at "Chaffing and winnowing: Confidentiality without encryption", RL Rivest, CryptoBytes (RSA laboratories), 1998. You can find the paper on scholar.google.com

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Yes i looked at this , thx ! but that doesnot help me . –  sashank Feb 18 '12 at 8:33
@sashank but that doesnot help me – That comment really confuses me… if the answer was not helpful to you, why did you accept it? –  e-sushi May 28 at 3:19

No.

Frequency-based cryptanalysis aims at breaking an encryption technique; hence evading frequency-based cryptanalysis implies an encryption technique.

Because frequency-based cryptanalysis is a poor attack, even some ridiculously weak encryption technique may evade frequency-based cryptanalysis. For example, putting the symbols alphabetically on two rings, using that as a correspondence table, and sliding the rings by one step after each use evades frequency-based cryptanalysis, at least in basic form.

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text:
• ### Anomalous quantum-critical spin dynamics in YFe2Al10(1801.03659)

Feb. 14, 2018 cond-mat.str-el
We report results of a muon spin relaxation ($\mu$SR) study of YFe$_2$Al$_{10}$, a quasi-2D nearly-ferromagnetic metal in which unconventional quantum critical behavior is observed. No static Fe$^{2+}$ magnetism, with or without long-range order, is found down to 19~mK\@. The dynamic muon spin relaxation rate~$\lambda$ exhibits power-law divergences in temperature and magnetic field, the latter for fields that are too weak to affect the electronic spin dynamics directly. We attribute this to the proportionality of $\lambda(\omega_\mu,T)$ to the dynamic structure factor~$S(\omega_\mu,T)$, where $\omega_\mu \approx 10^5$--$10^7~\mathrm{s}^{-1}$ is the muon Zeeman frequency. These results suggest critical divergences of $S(\omega_\mu,T)$ in both temperature and frequency. Power-law scaling and a 2D dissipative quantum XY (2D-DQXY) model both yield forms for $S(\omega,T)$ that agree with neutron scattering data ($\omega \approx 10^{12}~\mathrm{s}^{-1}$). Extrapolation to $\mu$SR frequencies agrees semi-quantitatively with the observed temperature dependence of $\lambda(\omega_\mu,T)$, but predicts frequency independence for $\omega_\mu \ll T$ in extreme disagreement with experiment. We conclude that the quantum critical spin dynamics of YFe$_2$Al$_{10}$ are not well understood at low frequencies.
• ### Reply to "Comment on `Discovery of slow magnetic fluctuations and critical slowing down in the pseudogap phase of YBa$_2$Cu$_3$O$_y$' "(1707.00069)

July 1, 2017 cond-mat.supr-con
We reply to the objections raised in a recent Comment (arXiv:1706.03023) regarding our observation of slow magnetic fluctuations and critical slowing down of magnetic fluctuations in the pseudogap phase of YBa$_2$Cu$_3$O$_y$ by zero-field and longitudinal-field muon spin relaxation (arXiv:1703.06799).
• ### Discovery of slow magnetic fluctuations and critical slowing down in the pseudogap phase of YBa$_2$Cu$_3$O$_y$(1703.06799)

Evidence for intra-unit-cell (IUC) magnetic order in the pseudogap region of high-$T_c$ cuprates below a temperature $T^\ast$ is found in several studies, but NMR and $\mu$SR experiments do not observe the expected static local magnetic fields. It has been noted, however, that such fields could be averaged by fluctuations. Our measurements of muon spin relaxation rates in single crystals of YBa$_2$Cu$_3$O$_y$ reveal magnetic fluctuations of the expected order of magnitude that exhibit critical slowing down at $T^\ast$. These results are strong evidence for fluctuating IUC magnetic order in the pseudogap phase.
• ### Upper critical magnetic field of $Ln$O$_{0.5}$F$_{0.5}$BiS$_2$ ($Ln$ = La, Nd) superconductors at ambient and high pressure(1611.03926)

Nov. 12, 2016 cond-mat.supr-con
The upper critical field $H_{c2}$ of polycrystalline samples of $Ln$O$_{0.5}$F$_{0.5}$BiS$_{2}$ ($Ln$ = La, Nd) at ambient pressure (tetragonal structure) and high pressure (HP) (monoclinic structure) have been investigated via electrical resistivity measurements at various magnetic fields up to 8.5 T. The $H_{c2}$($T$) curves for all the samples show an uncharacteristic concave upward curvature at temperatures below $T_c$, which cannot be described by the conventional one-band Werthamer-Helfand-Hohenberg theory. For the LaO$_{0.5}$F$_{0.5}$BiS$_{2}$ sample under HP, as temperature is decreased, the upper critical field $H_{onset}$, estimated from the onset of the superconducting transitions, increases slowly between 4.9 and 5.8 T compared with the slope of $H_{onset}$($T$) below 4.9 T and above 5.8 T. This anomalous behavior reveals a remarkable similarity in superconductivity between LaO$_{0.5}$F$_{0.5}$BiS$_{2}$ samples measured under HP and synthesized under HP, although the crystal structures of the two samples were reported to be different. The experimental results support the idea that local atomic environment, which can be tuned by applying external pressure and can be quenched to ambient pressure via high temperature-pressure annealing, is possibly more essential to the enhancement of $T_c$ for BiS$_2$-based superconductors than the structural phase transition. On the other hand, such anomalous behavior is very subtle in the case of NdO$_{0.5}$F$_{0.5}$BiS$_{2}$ under HP, suggesting that the anisotropy of the upper critical field in the $ab$-plane and the possible lattice deformation induced by external pressure is weak. This explains why the pressure-induced enhancement of $T_c$ for NdO$_{0.5}$F$_{0.5}$BiS$_{2}$ is not as large as that for LaO$_{0.5}$F$_{0.5}$BiS$_{2}$.
• ### Study of Localized Character of 4f Electrons and Ultrasonic Dispersions in SmOs4Sb12 by High-Pressure High-Frequency Ultrasonic Measurements(1608.01066)

Aug. 3, 2016 cond-mat.str-el
We present high-frequency ultrasonic measurements on the filled skutterudite SmOs$_4$Sb$_{12}$ under hydrostatic pressure. The results clarify that the 4$f$ electrons in this compound transform from delocalized at ambient pressure to localized at high pressures with a crossover pressure of approximately 0.7 GPa. This drastic change in the 4$f$ electrons under pressure is apparently related to the non-Fermi liquid state, which appears in an intermediate-pressure range of 0.5-1.5 GPa. The results or our analysis strongly suggest that the ferro-octupolar interaction becomes dominant at high pressure. Moreover, we report the pressure dependence of the ultrasonic dispersion, which is due to rattling, over a wide range of ultrasonic frequencies up to 323 MHz. The drastic change in the ultrasonic dispersions and the frequency dependent elastic anomaly in the $C_{11}$ mode at lower temperatures imply a possible coupling between rattling phonons and 4$f$ electrons.
• ### Quantum Criticality and Inhomogeneous Magnetic Order in Fe-doped alpha-YbAlB4(1605.07993)

May 25, 2016 cond-mat.str-el
The intermediate-valent polymorphs $\alpha$- and $\beta$-YbAlB$_4$ exhibit quantum criticality and other novel properties not usually associated with intermediate valence. Iron doping induces quantum criticality in $\alpha$-YbAlB$_4$ and magnetic order in both compounds. We report results of muon spin relaxation ($\mu$SR) experiments in the intermediate-valent alloys $\alpha$-YbAl$_{1-x}$Fe$_x$B$_4$, $x = 0.014$ and 0.25. For $x = 0.014$ we find no evidence for magnetic order down to 25 mK\@. The dynamic muon spin relaxation rate $\lambda_d$ exhibits a power-law temperature dependence $\lambda_d \propto T^{-a}$, $a = 0.40(4)$, in the temperature range 100 mK--2 K, in disagreement with predictions by theories of antiferromagnetic (AFM) or valence quantum critical behavior. For $x = 0.25$, where AFM order develops in the temperature range 7.5--10 K, where we find coexistence of meso- or macroscopically segregated paramagnetic and AFM phases, with considerable disorder in the latter down to 2 K.
• ### Investigation of superconducting and normal-state properties of the filled-skutterudite system PrPt$_{4}$Ge$_{12-x}$Sb$_{x}$(1603.02357)

March 8, 2016 cond-mat.supr-con
We report a study of the superconducting and normal-state properties of the filled-skutterudite system PrPt$_{4}$Ge$_{12-x}$Sb$_x$. Polycrystalline samples with Sb concentrations up to $x =$ 5 were synthesized and investigated by means of x-ray diffraction, electrical resistivity, magnetic susceptibility, and specific heat measurements. We observed a suppression of superconductivity with increasing Sb substitution up to $x =$ 4, above which, no signature of superconductivity was observed down to 140 mK. The Sommerfeld coefficient, $\gamma$, of superconducting specimens decreases with increasing $x$ up to $x =$ 3, suggesting that superconductivity may depend on the density of electronic states in this system. The specific heat for $x =$ 0.5 exhibits an exponential temperature dependence in the superconducting state, reminiscent of a nodeless superconducting energy gap. We observed evidence for a weak "rattling" mode associated with the Pr ions, characterized by an Einstein temperature $\Theta_{\mathrm{E}} \sim$ 60 K for 0 $\leq x \leq$ 5; however, the rattling mode may not play any role in suppressing superconductivity.
• ### Landau Renormalizations of Superfluid density in Heavy Fermion Superconductor CeCoIn5(1410.2955)

The formation of heavy fermion bands can occur by means of the conversion of a periodic array of local moments into itinerant electrons via the Kondo effect and the huge consequent Fermi-liquid renormalizations. Leggett predicted for liquid $^3$He that Fermi-liquid renormalizations change in the superconducting state, leading to a temperature dependence of the London penetration depth~$\Lambda$ quite different from that in the BCS theory. Using Leggett's theory, as modified for heavy fermions, it is possible to extract from the measured temperature dependence of $\Lambda$ in high quality samples both Landau parameters $F_0^s$ and $F_1^s$; this has never been accomplished before. A modification of the temperature dependence of the specific heat $C_\mathrm{el}$, related to that of $\Lambda$, is also expected. We have carefully determined the magnitude and temperature dependence of $\Lambda$ in CeCoIn$_5$ by muon spin relaxation rate measurements to obtain $F_0^s = 36 \pm 1$ and $F_1^s = 1.2 \pm 0.3$, and find a consistent change in the temperature dependence of electronic specific heat $C_\mathrm{el}$. This, the first determination of $F_1^s$ with a value~$\ll F_0^s$ in a heavy fermion compound, tests the basic assumption of the theory of heavy fermions, that the frequency dependence of the self-energy is much more important than its momentum dependence.
• ### Muon spin rotation and relaxation in Pr$_{1-x}$Nd$_x$Os$_4$Sb$_{12}$: Paramagnetic states(1406.0903)

June 3, 2014 cond-mat.str-el
Positive-muon ($\mu^+$) Knight shifts have been measured in the paramagnetic states of Pr$_{1-x}$Nd$_x$Os$_4$Sb$_{12}$ alloys, where $x =$ 0, 0.25, 0.45, 0.50, 0.55, 0.75, and 1.00. In Pr-substituted NdOs$_4$Sb$_{12}$ ($x \le$ 0.75), but not in NdOs$_4$Sb$_{12}$, Clogston-Jaccarino plots of $\mu^+$ Knight shift~$K$ versus magnetic susceptibility~$\chi$ exhibit an anomalous saturation of $K(\chi)$ at $\sim-$0.5% for large susceptibilities (low temperatures), indicating a reduction of the coupling strength between $\mu^+$ spins and $4f$ paramagnetism for temperatures $\lesssim$ 15~K. We speculate that itinerant Pr$^{3+}$ quadrupolar excitations, invoked to mediate the superconducting Cooper-pair interaction, might modify the $\mu^+$-$4f$ ion indirect spin-spin interaction.
• ### Muon spin rotation and relaxation in Pr$_{1-x}$Nd$_x$Os$_4$Sb$_{12}$: Magnetic and superconducting ground states(1404.4090)

Muon spin rotation and relaxation ($\mu$SR) experiments have been carried out to characterize magnetic and superconducting ground states in the Pr$_{1-x}$Nd$_x$Os$_4$Sb$_{12}$ alloy series. In the ferromagnetic end compound NdOs$_4$Sb$_{12}$ the spontaneous local field at positive-muon ($\mu^+$) sites below the ordering temperature $T_C$ is greater than expected from dipolar coupling to ferromagnetically aligned Nd$^{3+}$ moments, indicating an additional indirect RKKY-like transferred hyperfine mechanism. For 0.45 $\le x \le$ 0.75, $\mu^+$ spin relaxation rates in zero and weak longitudinal applied fields indicate that static fields at $\mu^+$ sites below $T_C$ are reduced and strongly disordered. We argue this is unlikely to be due to reduction of Nd$^{3+}$ moments, and speculate that the Nd$^{3+}$-$\mu^+$ interaction is suppressed and disordered by Pr doping. In an $x$ = 0.25 sample, which is superconducting below $T_c$ = 1.3 K, there is no sign of "spin freezing" (static Nd$^{3+}$ magnetism), ordered or disordered, down to 25 mK. Dynamic $\mu^+$ spin relaxation is strong, indicating significant Nd-moment fluctuations. The $\mu^+$ diamagnetic frequency shift and spin relaxation in the superconducting vortex-lattice phase decrease slowly below $T_c$, suggesting pair breaking and/or possible modification of Fermi-liquid renormalization by Nd spin fluctuations. For 0.25 $\le x \le$ 0.75, the $\mu$SR data provide evidence against phase separation; superconductivity and Nd$^{3+}$ magnetism coexist on the atomic scale.
• ### Probing the superconductivity of PrPt4Ge12 through Ce substitution(1403.5329)

We report measurements of electrical resistivity, magnetic susceptibility, specific heat, and thermoelectric power on the system Pr1-xCexPt4Ge12. Superconductivity is suppressed with increasing Ce concentration up to x = 0.5, above which there is no evidence for superconductivity down to 1.1 K. The Sommerfeld coefficient {\gamma} increases with increasing x from 48 mJ/mol K^2 up to 120 mJ/mol K^2 at x = 0.5, indicating an increase in strength of electronic correlations. The temperature dependence of the specific heat at low temperatures evolves from roughly T^3 for x = 0 to e^(-\Delta /T) behavior for x = 0.05 and above, suggesting a crossover from a nodal to a nodeless superconducting energy gap or a transition from multiband to single-band superconductivity. Fermi-liquid behavior is observed throughout the series in low-temperature magnetization, specific heat, and electrical resistivity measurements.
• ### Superconductivity, magnetic order, and quadrupolar order in the filled skutterudite system Pr$_{1-x}$Nd$_{x}$Os$_4$Sb$_{12}$(1008.5198)

Aug. 31, 2010 cond-mat.str-el
Superconductivity, magnetic order, and quadrupolar order have been investigated in the filled skutterudite system Pr$_{1-x}$Nd$_{x}$Os$_4$Sb$_{12}$ as a function of composition $x$ in magnetic fields up to 9 tesla and at temperatures between 50 mK and 10 K. Electrical resistivity measurements indicate that the high field ordered phase (HFOP), which has been identified with antiferroquadruoplar order, persists to $x$ $\sim$ 0.5. The superconducting critical temperature $T_c$ of PrOs$_4$Sb$_{12}$ is depressed linearly with Nd concentration to $x$ $\sim$ 0.55, whereas the Curie temperature $T_{FM}$ of NdOs$_4$Sb$_{12}$ is depressed linearly with Pr composition to ($1-x$) $\sim$ 0.45. In the superconducting region, the upper critical field $H_{c2}(x,0)$ is depressed quadratically with $x$ in the range 0 $<$ $x$ $\lesssim$ 0.3, exhibits a kink at $x$ $\approx$ 0.3, and then decreases linearly with $x$ in the range 0.3 $\lesssim$ $x$ $\lesssim$ 0.6. The behavior of $H_{c2}(x,0)$ appears to be due to pair breaking caused by the applied magnetic field and the exhange field associated with the polarization of the Nd magnetic moments, in the superconducting state. From magnetic susceptibility measurements, the correlations between the Nd moments in the superconducting state appear to change from ferromagnetic in the range 0.3 $\lesssim$ $x$ $\lesssim$ 0.6 to antiferromagnetic in the range 0 $<$ $x$ $\lesssim$ 0.3. Specific heat measurements on a sample with $x$ $=$ 0.45 indicate that magnetic order occurs in the superconducting state, as is also inferred from the depression of $H_{c2}(x,0)$ with $x$.
• ### Infrared spectroscopy and nano-imaging of the insulator-to-metal transition in vanadium dioxide(0904.0294)

We present a detailed infrared study of the insulator-to-metal transition (IMT) in vanadium dioxide (VO2) thin films. Conventional infrared spectroscopy was employed to investigate the IMT in the far-field. Scanning near-field infrared microscopy directly revealed the percolative IMT with increasing temperature. We confirmed that the phase transition is also percolative with cooling across the IMT. We present extensive near-field infrared images of phase coexistence in the IMT regime in VO2. We find that the coexisting insulating and metallic regions at a fixed temperature are static on the time scale of our measurements. A novel approach for analyzing the far-field and near-field infrared data within the Bruggeman effective medium theory was employed to extract the optical constants of the incipient metallic puddles at the onset of the IMT. We found divergent effective carrier mass in the metallic puddles that demonstrates the importance of electronic correlations to the IMT in VO2. We employ the extended dipole model for a quantitative analysis of the observed near-field infrared amplitude contrast and compare the results with those obtained with the basic dipole model.
• ### Incoherent non-Fermi liquid scattering in a Kondo lattice(0705.2592)

One of the most notorious non-Fermi liquid properties of both archetypal heavy-fermion systems [1-4] and the high-Tc copper oxide superconductors [5] is an electrical resistivity that evolves linearly with temperature, T. In the heavy-fermion superconductor CeCoIn5 [5], this linear behaviour was one of the first indications of the presence of a zero-temperature instability, or quantum critical point. Here, we report the observation of a unique control parameter of T-linear scattering in CeCoIn5, found through systematic chemical substitutions of both magnetic and non-magnetic rare-earth, R, ions into the Ce sub-lattice. We find that the evolution of inelastic scattering in Ce1-xRxCoIn5 is strongly dependent on the f-electron configuration of the R ion, whereas two other key properties -- Cooper-pair breaking and Kondo-lattice coherence -- are not. Thus, T-linear resistivity in CeCoIn5 is intimately related to the nature of incoherent scattering centers in the Kondo lattice, which provides insight into the anomalous scattering rate synonymous with quantum criticality [7].
• ### Mott transition in VO2 revealed by infrared spectroscopy and nano-imaging(0801.1171)

Electrons in correlated insulators are prevented from conducting by Coulomb repulsion between them. When an insulator-to-metal transition is induced in a correlated insulator by doping or heating, the resulting conducting state can be radically different from that characterized by free electrons in conventional metals. We report on the electronic properties of a prototypical correlated insulator vanadium dioxide (VO2) in which the metallic state can be induced by increasing temperature. Scanning near-field infrared microscopy allows us to directly image nano-scale metallic puddles that appear at the onset of the insulator-to-metal transition. In combination with far-field infrared spectroscopy, the data reveal the Mott transition with divergent quasiparticle mass in the metallic puddles. The experimental approach employed here sets the stage for investigations of charge dynamics on the nanoscale in other inhomogeneous correlated electron systems.
• ### Ambient-pressure bulk superconductivity deep in the magnetic state of CeRhIn5(0712.0382)

Specific heat, magnetic susceptibility and electrical transport measurements were performed at ambient pressure on high-quality single crystal specimens of CeRhIn5 down to ultra-low temperatures. We report signatures of an anomaly observed in all measured quantities consistent with a bulk phase transition to a superconducting state at T_c=110 mK. Occurring far below the onset of antiferromagnetism at T_N=3.8 K, this transition appears to involve a significant portion of the available low-temperature density of electronic states, exhibiting an entropy change in line with that found in other members of the 115 family of superconductors tuned away from quantum criticality.
• ### Multiple ordered phases in the filled skutterudite compound PrOs$_{4}$As$_{12}$(cond-mat/0512325)

March 7, 2006 cond-mat.str-el
Magnetization, specific heat, and electrical resistivity measurements were made on single crystals of the filled skutterudite compound PrOs$_{4}$As$_{12}$. Specific heat measurements indicate an electronic specific heat coefficient $\gamma$ $\sim 50-200$ mJ/mol K$^{2}$ at temperatures 10 K $\leq T \leq 18$ K, and $\sim 1$ J/mol K$^{2}$ for $T \leq 1.6$ K. Magnetization, specific heat, and electrical resistivity measurements reveal the presence of two, or possibly three, ordered phases at temperatures below $\sim 2.3$ K and in fields below $\sim 3$ T. The low temperature phase displays antiferromagnetic characteristics, while the nature of the ordering in the other phase(s) has yet to be determined.
• ### Ferromagnetism and possible heavy fermion behavior in single crystals of NdOs$_4$Sb$_{12}$(cond-mat/0412713)

Single crystals of the filled-skutterudite compound NdOs$_4$Sb$_{12}$ have been investigated by means of electrical resistivity, magnetization, and specific heat measurements. The NdOs$_4$Sb$_{12}$ crystals have the LaFe$_4$P$_{12}$-type cubic structure with a lattice parameter of 9.3 \AA. Possible heavy-fermion behavior is inferred from specific heat measurements, which reveal a large electronic specific heat coefficient $\gamma \approx 520$ mJ/mol-K$^2$, corresponding to an effective mass $m^* \sim$ 98 $m_e$. Features related to a ferromagnetic transition at {$\sim$ 0.9 K} can be observed in electrical resistivity, magnetization and specific heat. Conventional Arrott-plot analysis indicates that NdOs$_4$Sb$_{12}$ conforms to mean-field ferromagnetism.
• ### Superconductivity and crystalline electric field effects in the filled skutterudite series Pr(Os$_{1-x}$Ru$_x$)$_4$Sb$_{12}$(cond-mat/0307059)

X-ray powder diffraction, magnetic susceptibility $\chi(T)$, and electrical resistivity $\rho(T)$ measurements were made on single crystals of the filled skutterudite series Pr(Os$_{1-x}$Ru$_x$)$_4$Sb$_{12}$. One end of the series ($x = 0$) is a heavy fermion superconductor with a superconducting critical temperature $T_{c} = 1.85$ K, while the other end ($x = 1$) is a conventional superconductor with $T_{c} \approx 1$ K. The lattice constant $a$ decreases approximately linearly with increasing Ru concentration $x$. As Ru (Os) is substituted for Os (Ru), $T_{c}$ decreases nearly linearly with substituent concentration and exhibits a minimum with a value of $T_{c} = 0.75$ K at $x = 0.6$, suggesting that the two types of superconductivity compete with one another. Crystalline electric field (CEF) effects in $\chi_\mathrm{dc}(T)$ and $\rho(T)$ due to the splitting of the Pr$^{3+}$ nine-fold degenerate Hund's rule $J = 4$ multiplet are observed throughout the series, with the splitting between the ground state and the first excited state increasing monotonically as $x$ increases. The fits to the $\chi_\mathrm{dc}(T)$ and $\rho(T)$ data are consistent with a $\Gamma_{3}$ doublet ground state for all values of x, although reasonable fits can be obtained for a $\Gamma_{1}$ ground state for $x$ values near the end member compounds ($x = 0$ or $x = 1$).
• ### Electrical resistivity and magnetization measurements on the heavy fermion superconductor PrOs4Sb12(cond-mat/0303405)

The filled skutterudite compound PrOs4Sb12, the first example of a Pr-based heavy fermion superconductor, displays superconductivity with $T_c\sim 1.85$ K and has an effective mass $m^* \sim$ {50 $m_e$}, where $m_e$ is the free electron mass. For magnetic fields above 4.5 T, sharp features in the normal state electrical resistivity, magnetization, specific heat, and thermal expansion data suggest the occurrence of a phase transition at high fields. This high field ordered phase in the normal state may originate from a combination of crystalline electric field enhanced Zeeman splitting and quadrupolar ordering. We present an investigation of the electrical resistivity and magnetization of PrOs4Sb12 as a function of temperature between {350 mK} and {3.5 K} and magnetic field up to {18 T}. The data reveal a detailed phase boundary of the high field ordered phase as well as the lower critical field $H_{c1}$ and the onset field of the peak effect in the superconducting state of PrOs4Sb12.
• ### A new heavy fermion superconductor: the filled skutterudite compound PrOs$_4$Sb$_{12}$(cond-mat/0303370)

The filled skutterudite compound \PrOsSb{} exhibits superconductivity below a critical temperature $T_\mathrm{c} = 1.85$ K that develops out of a nonmagnetic heavy Fermi liquid with an effective mass $m^{*} \approx 50 m_\mathrm{e}$, where $m_\mathrm{e}$ is the free electron mass. Analysis of magnetic susceptibility, specific heat, electrical resistivity and inelastic neutron scattering measurements within the context of a cubic crystalline electric field yields a Pr$^{3+}$ energy level scheme that consists of a $\Gamma_{3}$ nonmagnetic doublet ground state that carries an electric quadrupole moment, a low lying $\Gamma_{5}$ triplet excited state at $\sim 10$ K, and $\Gamma_{4}$ triplet and $\Gamma_{1}$ singlet excited states at much higher temperatures. The superconducting state appears to be unconventional and to consist of two distinct superconducting phases. An ordered phase of magnetic or quadrupolar origin occurs at high fields and low temperatures, suggesting that the superconductivity may occur in the vicinity of a magnetic or electric quadrupolar quantum critical point.
• ### Heavy fermion superconductivity in the filled skutterudite compound PrOs$_4$Sb$_{12}$(cond-mat/0303371)

The filled skutterudite compound \PrOsSb{} has been found to exhibit superconductivity with a critical temperature $T_\mathrm{c} = 1.85$ K that develops out of a heavy Fermi liquid with an effective mass $m^{*} \approx 50 m_\mathrm{e}$. The current experimental situation regarding the heavy fermion state, the superconducting state, and a high field, low temperature phase that is apparently associated with magnetic or quadrupolar order in \PrOsSb{} is briefly reviewed herein.
• ### Superconductivity and the high field ordered phase in the heavy fermion compound PrOs$_4$Sb$_{12}$(cond-mat/0303373)

Superconductivity is observed in the filled skutterudite compound \PrOsSb{} below a critical temperature temperature $T_\mathrm{c} = 1.85$ K and appears to develop out of a nonmagnetic heavy Fermi liquid with an effective mass $m^{*} \approx 50 m_\mathrm{e}$, where $m_\mathrm{e}$ is the free electron mass. Features associated with a cubic crystalline electric field are present in magnetic susceptibility, specific heat, electrical resistivity, and inelastic neutron scattering measurements, yielding a Pr$^{3+}$ energy level scheme consisting of a $\Gamma_{3}$ nonmagnetic doublet ground state, a low lying $\Gamma_{5}$ triplet excitied state at $\sim 10$ K, and much higher temperature $\Gamma_{4}$ triplet and $\Gamma_{1}$ singlet excited states. Measurements also indicate that the superconducting state is unconventional and consists of two distinct superconducting phases. At high fields and low temperatures, an ordered phase of magnetic or quadrupolar origin is observed, suggesting that the superconductivity may occur in the vicinity of a magnetic or quadrupolar quantum critical point.
• ### Magnetic Relaxation in the Peak Effect Region of CeRu$_2$(cond-mat/0301281)

The different pinning strengths of the flux line lattice in the peak effect (PE) region of a polycrystalline sample of CeRu$_2$ with a superconducting transition temperature {$T_c = 6.1$ K} have been probed by means of magnetization measurements with a SQUID magnetometer as the temperature $T$ and the magnetic field $H$ are varied. Magnetic relaxation measurements were used to monitor the flux line dynamics in the PE region. For {$T < 4.5$ K} and $H < H_P$, where $H_P$ is the field where the magnetization reaches a maximum in the PE region, the relaxation rate was found to be significantly larger in the descending-field branch of the PE than it is in other sections of the PE region. For {$T \geq 4.5$ K}, the relaxation rate in the entire PE region is so large that the magnetization reached a stable (equilibrium) value within {$10^4$ s}. This experimentally determined stable state appears as an increase of the magnetization in the PE region and has a dome shape superimposed on a linear interpolation through the PE region. It was also found that the PE in CeRu$_2$ can be suppressed by rapid thermal cycling of the sample between {10 K} and {300 K} four times. The reversible magnetization after the PE has been suppressed coincides with the linear interpolation through the PE region, in contrast to the behavior of the equilibrium magnetization when the PE is present. PACS number: 74.25.Qt, 74.70.Ad
• ### Low temperature specific heat of the heavy fermion superconductor PrOs$_4$Sb$_{12}$(cond-mat/0207225)

We report the magnetic field dependence of the low temperature specific heat of single crystals of the first Pr-based heavy fermion superconductor PrOs$_4$Sb$_{12}$. The low temperature specific heat and the magnetic phase diagram inferred from specific heat, resistivity and magnetisation provide compelling evidence of a doublet ground state and hence superconductivity mediated by quadrupolar fluctuations. This establishes PrOs$_4$Sb$_{12}$ as a very strong contender of superconductive pairing that is neither electron-phonon nor magnetically mediated.


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 TMF, 1994, Volume 99, Number 1, Pages 103–120 (Mi tmf1570)

Limits of applicability of the tight binding approximation

A. L. Mironov, V. L. Oleinik

St. Petersburg State University, Faculty of Physics

Abstract: The mathematical foundation of the tight binding approximation is given. Let $\lambda _0$ be a negative energy level of a real potential $q$. Then there exists an energy band for a one-dimensional chain with period $2T$ of the identical atoms which lies near $\lambda _0$. We study this band when $T$ tends to infinity.

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English version:
Theoretical and Mathematical Physics, 1994, 99:1, 457–469

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Citation: A. L. Mironov, V. L. Oleinik, “Limits of applicability of the tight binding approximation”, TMF, 99:1 (1994), 103–120; Theoret. and Math. Phys., 99:1 (1994), 457–469

Citation in format AMSBIB
\Bibitem{MirOle94} \by A.~L.~Mironov, V.~L.~Oleinik \paper Limits of applicability of the tight binding approximation \jour TMF \yr 1994 \vol 99 \issue 1 \pages 103--120 \mathnet{http://mi.mathnet.ru/tmf1570} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1302462} \zmath{https://zbmath.org/?q=an:0849.34073} \transl \jour Theoret. and Math. Phys. \yr 1994 \vol 99 \issue 1 \pages 457--469 \crossref{https://doi.org/10.1007/BF01018800} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1994PU13400009}

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This publication is cited in the following articles:
1. V. A. Geiler, V. V. Demidov, “Spectrum of three-dimensional landau operator perturbed by a periodic point potential”, Theoret. and Math. Phys., 103:2 (1995), 561–569
2. Yu. P. Chuburin, “On approximation of the “Membrane” Schrödinger operator by the “Crystal” operator”, Math. Notes, 62:5 (1997), 648–654
3. A. L. Mironov, V. L. Oleinik, “Limits of applicability of the tight binding approximation for complex-valued potential function”, Theoret. and Math. Phys., 112:3 (1997), 1157–1171
4. Oleinik, VL, “Analysis of the dispersion equation for the Schrodinger operator on periodic metric graphs”, Waves in Random Media, 14:2 (2004), 157
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# How to solve for acceleration and position when only time and displacement is known

## Homework Statement

A rocket blasts off with constant acceleration straight upward. After 3 seconds the rocket is at a height of 80m. (a) what are the magnitude and direction of the rocket's acceleration? (b) what is its velocity at the time

## Homework Equations

$v=v_0+at$
$v^2=V_0^2+2a\Delta x$
$a=v^2-v_0/(2*x)$
$x=x_0+v_0t+\frac{1}{2}at^2$

## The Attempt at a Solution

None, because I was unsure how to proceed as using these equations would result in one unknown (either v or a)

Last edited:

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You are missing one equation, the that gives the position as a function of time.

lewando
Homework Helper
Gold Member

The 4th equation looks promising!

Yeah i realize that now, plugged in and got acceleration-now i need to just plug acceleration in to get velocity! thanks

PeterO
Homework Helper

Yeah i realize that now, plugged in and got acceleration-now i need to just plug acceleration in to get velocity! thanks
With constant acceleration from rest, the average velocity is one half of the final velocity.

This rocket covered 80m in 3 seconds, so average velocity is easy to find. Double that and you have the final velocity.

Once you have that final velocity you can easily get acceleration since you know by how much the velocity changed in 3 seconds.


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# hw3solns - SIMULATIONS PROBLEM 5 Problem 2 For this problem...

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SIMULATIONS: PROBLEM 5 Problem 2 For this problem, very low capacitance of 0.001fF was used to avoid invalid dB values for the S-parameters during simulation. As mentioned in the email, this was for SIMULATION ONLY. S (2, 1) nearly remains the same and S (1, 2), S (2, 2) are almost zero. S (1, 1) increases at high frequencies because of Cgs.
Problem 4 Problem 3 To calculate MAG, a non-zero S12is required and for this purpose, a very small Cgd= 0.001fF was used. The device model