text:
Online ISSN : 1880-7488
Print ISSN : 0514-5163
ISSN-L : 0514-5163

き裂長さデータの統計解析に基づく蒸気タービンケーシングの最適補修間隔と補修量予測法

ジャーナル フリー

2014 年 63 巻 8 号 p. 608-613

Statistical analyses of the field inspection database were conducted for steam turbine high pressure inner casings. The apparent reducing trends of crack amounts at nozzle box fit radius were exhibited as repair numbers increased. This trend may be caused by the blunting of the corner root radius due to the grind-off actions for cracks. The average and upper trends of maximum crack length with repeated inspection/repair numbers were expressed by Green function successfully, indicating initial increasing and subsequent decreasing trends. To estimate total crack length, log-normal distribution was applied to predict the individual crack length from the maximum crack length and the total crack length was obtained from the summation of distributed crack length over the total number of cracks. The average growth rates of maximum crack length against start-up cycles showed similar reducing trends as maximum crack length. Using the average crack growth rates of maximum crack length and total crack length as the function of the numbers of repeated repairs, optimum repair intervals were demonstrated as the combination of initial frequent repair followed by long intervals after 3rd or 4th repair. The expression of crack growth rate as the function of the number of repeated repairs could be also very useful for maintenance scheduling.


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# x and y are joinly discrete with joint probability mass function (pmf) (a) Find $E(XY)$

x and y are joinly discrete with joint probability mass function (pmf)

$$\begin{array}{|c|c|c|} \hline x/y & 0 & 1 \\ \hline 0& 0.2 & 0.1 \\ \hline 1 & 0.4 & 0.3 \\ \hline \end{array}$$

(a) Find $E(XY)$

(a) $E(XY) = \sum xy P_X(x) =$

The table gives you the probabilities $P(X=x, Y=y)$ for various values of $x$ and $y$. The formula of expectation you should be using here is $E[XY] = \sum_x \sum_y xy P(X=x, Y=y)$.

There are four terms in $E(XY) = \sum_{x=0}^1\sum_{y=0}^1 xy\,f_{X,Y}(x.y),$ only one of the non-zero, so $E(XY) = 1(0.3) = 0.3.$

Alternatively, $V = XY$ has PDF $P(V=0)=0.7,\,P(V=1) = 0.3,$ so $E(V) = E(XY)= 0.3.$

Maybe the next step is to find $Cov(X,Y) = E(XY) - E(X)E(Y) = 0.3 - (0.7)(0.4) = 0.02.$

Suppose the joint PDF were given by the following table.

$$\begin{array}{|c|c|c|} \hline x/y & 0 & 1 \\ \hline 0& 0.18 & 0.12 \\ \hline 1 & 0.42 & 0.28 \\ \hline \end{array}$$

Then $X$ and $Y$ would be independent and $Cov(X,Y) = 0.$

format: plain


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## Precalculus (6th Edition) Blitzer

The solution is $\underline{x=\frac{\pi }{6},\frac{11\pi }{6}}.$
The expression can be solved as follows: $3\cos x-6\sqrt{3}=\cos x-5\sqrt{3}$ Arrange the provided equation in such a way that the cosine functions are on one side and the real numbers are on the other side: \begin{align} & 3\cos x-\cos x=6\sqrt{3}-5\sqrt{3} \\ & 2\cos x=\sqrt{3} \\ & \cos x=\frac{\sqrt{3}}{2} \end{align} So, in the interval $\left[ 0,2\pi \right),$ the cosine function is $\frac{\sqrt{3}}{2}$ at $\frac{\pi }{6}\text{ and }\frac{11\pi }{6}$. So, \begin{align} & \cos x=\frac{\sqrt{3}}{2} \\ & x=\frac{\pi }{6} \end{align} And the other value is: \begin{align} & \cos x=\frac{\sqrt{3}}{2} \\ & =2\pi -\frac{\pi }{6} \\ & =\frac{11\pi }{6} \end{align} And the value of cos is positive in the fourth quadrant. The period of the sine function is $2\pi$, so the general solution of the equation is: $x=\frac{\pi }{6}+2n\pi$ The other value is: $x=\frac{11\pi }{6}+2n\pi$ Now, to get different solutions, $\text{put }n=0,1,2,3\ldots$ $\text{When }n=0$ \begin{align} & x=\frac{\pi }{6}+2n\pi \\ & =\frac{\pi }{6} \end{align} And the other solution will be: \begin{align} & x=\frac{11\pi }{6}+2n\pi \\ & =\frac{11\pi }{6} \end{align} When we put other values of $n$, the solution becomes outside the interval $\left[ 0,2\pi \right).$


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#### Further Information

In addition to our technical support (e.g. via chat), you’ll find resources on our website that may help you with your design using Dlubal Software.

Receive information including news, useful tips, scheduled events, special offers, and vouchers on a regular basis.

• ### How can I retroactively adjust my surface model with regard to the finite element size without the total number of elements becoming too high?

New FAQ 004136 EN-US

Basically, the surfaces of the model have the FE size that was preset in the general FE mesh settings.

You can add "FE Mesh Refinements" locally to a surface model.

This has the effect that the general finite size can be retained in the FE mesh settings, but the mesh is locally refined so that more accurate results can be expected from the calculation.

In this case, the FE mesh refinement can be performed on nodes, lines, and surfaces. The setting can be found in the "Edit" dialog boxes of the node, line, or surface.

By using the local FE mesh refinement, it is possible to keep or even increase the general FE mesh size. Thus, it is possible to keep the total number of elements in the model relatively low.

• ### How is friction set on a member end release via the COM interface?

New FAQ 004133 EN-US

Friction represents a nonlinearity and can therefore only be modified via the interface to the member end release hinge.

For this purpose, the member end release must be created first, if not already available. Then, the IMemberHinge interface is brought to the member end release and then to the nonlinearity (here IFriction ). Then , you can use the methods GetData and SetData to modify the data (here Friction ):

Sub SetMemberHingeFriction ()Dim model As RFEM5.modelSet model = GetObject (, "RFEM5.Model")model.GetApplication.LockLicenseOn Error GoTo eDim data As IModelDataSet data = model.GetModelDataDim hinge (0 To 0) As RFEM5.MemberHingehinge (0) .No = 1hinge (0) .RotationalConstantX = 1hinge (0) .RotationalConstantY = 2hinge (0) .RotationalConstantZ = 3hinge (0) .TranslationalConstantX = 4hinge (0) .TranslationalConstantY = 5hinge (0) .TranslationalConstantZ = 6hinge (0) .Comment = "Member Hinge 1"    hinge (0) .TranslationalNonlinearityX = FrictionATypedata.PrepareModificationdata.SetMemberHinges hingdata.FinishModification    'get interface for member'Dim imemhing As IMemberHingeSet imemhing = data.GetMemberHinge (1, AtNo)    'get interface for nonlinearity'Dim iFric As IFrictionSet iFric = imemhing.GetNonlinearity (AlongAxisX)    'get friction data'Dim fric As Frictionfric = iFric.GetData    fric.Coefficient1 = 0.3    'set friction datadata.PrepareModificationiFric.SetData fricdata.FinishModification        e: If Err.Number <> 0 Then MsgBox Err.Description,, Err.SourceSet data = Nothingmodel.GetApplication.UnlockLicenseSet model = NothingEnd Sub

For Coefficient Vy + Vz, Coeffcient2 is used to set the second coefficient. The translational spring in the Friction dialog box is controlled by the translational spring of the Member-End Hinge. In the concrete case, this is TranslationalConstantX for the x-direction (see Figure 01).

• ### I have modeled a continuous beam. One of the supports should only act from a deformation of 10 mm. How can I adjust such a nonlinearity?

New FAQ 003608 EN-US

The definition of slippage is recommended here. To do this, you have to set the partial effect in the & quot; Non-linearity & quot; edit dialog box of the dialog box & quot; Edit Nodal Support & quot ;. In the "Nonlinearity - Partial Effect" dialog box, you can now define a Slippage in the corresponding zone. The adjoining diagram is used for the check, see Figure 1.

• ### What is the difference between the materials Isotropic Plastic 1D and Isotropic Nonlinear Elastic 1D?

New FAQ 003604 EN-US

The difference between both material models is as follows:

In the Isotropic Nonlinear Elastic 1D material model , no plastic deformations are considered. This means that the material returns to its initial state when the load is released.

For the material model isotropic plastic 1D, the plastic deformation is considered.

For both material models, the nonlinear properties are defined in an additional dialog box. When entering data by means of a diagram, it is possible to define a distribution after the last step in both models.

For the material model Isotropic Nonlinear Elastic 1D, it is possible to enter the stress-strain diagram (different for the positive and negative zone) in an anti-metrical way, whereas for the isotropically plastic model 1D, only symmetric input is possible.

• ### What is the effect of the option "Divide the members by the nodes on the members" in the FE mesh settings?

New FAQ 003601 EN-US

With the option "Divide the members by the nodes on the members", a member is internally divided at the member locations where nodes are located.
The structure shown in Figure 1 includes two members, member S1 from node 1 to node 2 and member S2 from node 3 to node 4.
If the option "Divide the members by the nodes lying on the members" is activated, member S1 is internally divided at node 4. Thus, the member S1 is connected to the member S2, which you can also see in the deformation (Figure 2).
If the option "Divide the members by the nodes lying on the members" is deactivated, member S1 is not divided at node 4. Thus, member S1 is not connected to member S2. Figure 3 shows the deformation in this case.
The structure shown in Figure 4 includes the member S1 from node 1 to node 2. On the member, there is node 3, to which a nodal load P Z = 50 kN is applied. The nodal load only loads member S1 if the option "Divide Members via Nodes on Members" is selected (Figure 5). If this option is deactivated, member S1 is not loaded by this nodal load (Figure 6).
• ### What is the fastest way to model corrugated sheets in RFEM by means of surfaces?

New FAQ 003600 EN-US

Trapezoidal sheeting is already available in the cross-section library of RFEM. These cross-sections are primarily used for stabilization measures in the RF- / STEEL EC3 add-on module and as a template for modeling surfaces. A design as a member is thus not possible.

The easiest way is to first model the trapezoidal sheeting as a member. Subsequently, this member can be automatically converted into surfaces (see Figure 01). Thus, the geometry is obtained as a surface model. Curved trapezoidal sheets are also possible. If a curved member is split into surfaces, it is represented by straight segments. If you still want to use curved lines, you can simply rotate the contour lines accordingly. The exact procedure can be found in the video.
• ### How can I model a timber-concrete composite floor?

New FAQ 003589 EN-US

The detailed procedure can be found in the video. It indicates the most important features:
• ### I have modeled a floor slab with a downstand beam that intersects with a continuous column. How can I prevent the slab from resting on the downstand beam but only on the downstand beam?

New FAQ 003586 EN-US

To ensure that the floor rests only on the downstand beam in this situation (Figure 1), it is recommended to model a minimum opening within the floor surface, see Figure 2.

In order to keep the members or the lines of the members integrated in the surface, the function "Connect Lines / Members" has to be applied to the entire model once. Thus, it is ensured that the floor slab continues to rest on the downstand beams.

• ### During the plausibility check, I receive an error message about the overlapping of one or more polyline segments. How can I fix it?

New FAQ 003580 EN-US

The polyline displayed in Figure 1 was created in the order of Nodes 1 → 2 → 3 → 4 → 5. This is also visible in column B 'Node No.' Table '1.2 Lines'. Thus, the line segments between nodes 3, 4 and nodes 4, 5 are overlapping so that the error message in Figure 2 is displayed.

The order of the nodes must be changed so that no line segments could overlap. In this example, we want to specify the order of Nodes 1 → 2 → 3 → 5 → 4, as shown in Figure 3.

You can also use the 'Explode Polyline' function to decompose the polyline (Figure 4). Then, two lines are created between nodes 5 and 4. Then, delete a line. You can also use the 'Regenerate Model' function in the 'Tools' menu.

Both approaches are demonstrated in the video.

• ### How does the material model Plastic 1D work?

New FAQ 003578 EN-US

The plasticity for 1D elements currently only works in relation to the normal stresses in a member. This means that only interaction between axial force and moment is possible. The shear force interaction is not taken into account. In addition, the stresses from shear force are only calculated elastically.

When applying a plastic material model, it is also important to ensure a sufficient division of the elements, because a cross-section is internally generated at each Gauss point on the member element where the stress is calculated and a reduction of the stiffness to the re-distribution of the internal forces is performed, if necessary. If, for example, the number of divisions is increased, the model may become unstable because the redistributions of stresses can no longer be carried out and thus the cross-section's loading is too high.

It is generally recommended to use a division of '50' for member elements when using the plastic material model (see the figure).

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The Grand Locus / Life for statistical sciences

the Blog

Infinite expectations

In the first days of my PhD, I sincerely believed that there was a chance I would find a cure against cancer. As this possibility became more and more remote, and as it became obvious that my work would not mark a paradigm shift, I became envious of those few people who did change the face of science during their PhD. One of them is Andrey Kolmogorov, whose PhD work was nothing less than the modern theory of probability. His most famous result was the strong law of large numbers, which essentially says that random fluctuations become infinitesimal on average. Simply put, if you flip a fair coin a large number of times, the frequency that ‘tails’ turn up will be very close to the expected value 1/2.

The chaos of large numbers

Most fascinating about the strong law of large numbers is that it is a theorem, which means that it comes with hypotheses that do not always hold. There are cases that repeating a random experiment a very large number of times does not guarantee that you will get closer to the expected value — I wrote the gory detail on Cross Validated, for those interested...

Unpredictable?

What if I told you to choose a card at random? Simply choose one from a standard deck of 52 cards, think of a card, do not draw one from a real deck... Make sure you have one in mind before you read on.

The illusion

Assuming that you have a card in mind, I bet you chose the Ace of Spades. Of course, I don't know the card you were thinking of, but that is the one I bet on. Every textbook on probability says that I have a 1/52 chance of having guessed right. Actually, that is not quite true... About one out of 4 readers will choose the Ace of Spades, and one out of seven will choose the Queen of Hearts, as shown by a study of the researcher in psychology of magic Jay Olson and his collaborators.

In this experiment, is the choice of a card purely random? And what does random mean anyway? Even if purely random is not properly defined, most would agree that it means no information at all, or completely unpredictable. The outcome of the experiment above is clearly not what you would call purely random. But...

The chaos and the doubt

Probability is said to be born of the correspondence between Pierre de Fermat and Blaise Pascal, some time in the middle of the 17th century. Somewhat surprisingly, many texts retrace the history of the concept up until the 20th century; yet it has gone through major transformations since then. Probability always describes what we don't know about the world, but the focus has shifted from the world to what we don't know.

Henri Poincaré investigates in Science et Méthode (1908) why chance would ever happen in a deterministic world. Like most of his contemporaries, Poincaré believed in absolute determinism, there is no phenomenon without a cause, even though our limited minds may fail to understand or see it. He distinguishes two flavors of randomness, of which he gives examples.

If a cone stands on its point we know that it will fall but we do not know which way (...) A very small cause, which escapes us, determines a considerable effect that we can not but see, and then we say that this effect is due to chance.

And a little bit later he continues.

How do we represent a container filled with gas? Countless molecules...

Poetry and optimality

Claude Shannon was the hell of a scientist. His work in the field of information theory, (and in particular his famous noisy channel coding theorem) shaped the modern technological landscape, but also gave profound insight in the theory of probabilities.

In my previous post on statistical independence, I argued that causality is not a statistical concept, because all that matters to statistics is the sampling of events, which may not reflect their occurrence. On the other hand, the concept of information fits gracefully in the general framework of Bayesian probability and gives a key interpretation of statistical independence.

Shannon defines the information of an event with probability $Prob(A)$ as $-\log P(A)$. For years, this definition baffled me for its simplicity and its abstruseness. Yet it is actually intuitive. Let us call $\Omega$ the system under study and $\omega$ its state. You can think of $\Omega$ as a set of possible messages and of $\omega$ as the true message transmitted over a channel, or (if you are Bayesian) of $\Omega$ as a parameter set and $\omega$ as the true value of the parameter. We have total information about the system if we know $\omega$. If instead, all...

The fallacy of (in)dependence

In the post Why p-values are crap I argued that independence is a key assumption of statistical testing and that it almost never holds in practical cases, explaining how p-values can be insanely low even in the absence of effect. However, I did not explain how to test independence. As a matter of fact I did not even define independence because the concept is much more complex than it seems.

Apart from the singular case of Bayes theorem, which I referred to in my previous post, the many conflicts of probability theory have been settled by axiomatization. Instead of saying what probabilities are, the current definition says what properties they have. Likewise, independence is defined axiomatically by saying that events $A$ and $B$ are independent if $P(A \cap B) = P(A)P(B)$, or in English, if the probability of observing both is the product of their individual probabilities. Not very intuitive, but if we recall that $P(A|B) = P(A \cap B)/P(B)$, we see that an alternative formulation of the independence of $A$ and $B$ is $P(A | B) = P(A)$. In other words, if $A$ and $B$ are independent, observing...

The reverend’s gambit

Two years after the death of Reverend Thomas Bayes in 1761, the famous theorem that bears his name was published. The legend has it he felt the devilish nature of his result and was too afraid of the reaction of the Church to publish it during his lifetime. Two hundred and fifty years later, the theorem still sparkles debate, but among statisticians.

Bayes theorem is the object of the academic fight between the so-called frequentist and Bayesian schools. Actually, more shocking than this profound disagreement is the overall tolerance for both points of view. After all, Bayes theorem is a theorem. Mathematicians do not argue over the Pythagorean Theorem: either there is a proof or there isn't. There is no arguing about that.

So what's wrong with Bayes theorem? Well, it's the hypotheses. According to the frequentist, the theorem is right, it is just not applicable in the conditions used by the Bayesian. In short, the theorem says that if $A$ and $B$ are events, the probability of $A$ given that $B$ occurred is $P(A|B) = P(B|A) P(A)/P(B)$. The focus of the fight is the term $(P(B... Why p-values are crap I remember my statistics classes as a student. To do a t-test we had to carry out a series of tedious calculations and in the end look up the value in a table. Making those tables cost an enormous amount of sweat from talented statisticians, so you had only three tables, for three significance levels: 5%, 1% and 0.1%. This explains the common way to indicate significance in scientific papers, with one (*), two (**) or three (***) stars. Today, students use computers to do the calcultations so the star notation probably appears as a mysterious folklore and the idea of using a statistical table is properly unthinkable. And this is a good thing because computing those t statistics by hand was a pain. But statistical softwares also paved the way for the invasion of p-values in the scientific literature. To understand what is wrong with p-values, we will need to go deeper in the theory of statistical testing, so let us review the basic principles. Every statistical test consists of a null hypothesis, a test statistic (a score) and a decision rule — plus the often forgotten alternative hypothesis. A statistical test is an investigation protocol to... Drunk man walking Lotteries fascinate the human mind. In the The Lottery in Babylon, Jorge Luis Borges describes a city where the lottery takes a progressively dominant part in people’s life, to the extent that every decision, even life and death, becomes subject to the lottery. In this story, Borges brings us face to face with the discomfort that the concept of randomness creates in our mind. Paradoxes are like lighthouses, they indicate a dangerous reef, where the human mind can easily slip and fall into madness, but they also show us the way to greater understanding. One of the oldest paradoxes of probability theory is the so called Saint Petersburg paradox, which has been teasing statisticians since 1713. Imagine I offered you to play the following game: if you toss ‘tails’, you gain$1, and as long as you toss ‘tails’, you double your gains. The first ‘heads’ ends the spree and determines how much you gain. So you could gain $0,$1, $2,$4, \$8... with probability 1/2, 1/4, 1/8, 1/16, 1/32 etc. What is the fair price I can ask you to play the Saint Petersburg lottery?

Probability theory says that the...


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• Browse all
Search for scalar leptoquark pairs decaying to electrons and jets in $\bar{p}p$ collisions

The collaboration
Phys.Rev.Lett. 79 (1997) 4321-4326, 1997

Abstract (data abstract)
FERMILAB-COLLIDER. Search for first generation leptoquark pairs in pbar-p collisions at a centre-of-mass energy of 1.8 TeV. The results come from the (ee 2jet) channel from 123pb-1 integrated luminosity data taken during 1992-96. No candidate events are observed and 95% CL upper limits are given to the production cross sections.

• #### Table 1

Data from T 2

10.17182/hepdata.42172.v1/t1

No description provided.


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In [13]:
from datascience import *
path_data = '../../../../data/'
import matplotlib
matplotlib.use('Agg', warn=False)
%matplotlib inline
import matplotlib.pyplot as plots
plots.style.use('fivethirtyeight')
import numpy as np

### Sampling from a Population¶

The law of averages also holds when the random sample is drawn from individuals in a large population.

As an example, we will study a population of flight delay times. The table united contains data for United Airlines domestic flights departing from San Francisco in the summer of 2015. The data are made publicly available by the Bureau of Transportation Statistics in the United States Department of Transportation.

There are 13,825 rows, each corresponding to a flight. The columns are the date of the flight, the flight number, the destination airport code, and the departure delay time in minutes. Some delay times are negative; those flights left early.

In [14]:
united = Table.read_table(path_data + 'united_summer2015.csv')
united

Out[14]:
Date Flight Number Destination Delay
6/1/15 73 HNL 257
6/1/15 217 EWR 28
6/1/15 237 STL -3
6/1/15 250 SAN 0
6/1/15 267 PHL 64
6/1/15 273 SEA -6
6/1/15 278 SEA -8
6/1/15 292 EWR 12
6/1/15 300 HNL 20
6/1/15 317 IND -10

... (13815 rows omitted)

One flight departed 16 minutes early, and one was 580 minutes late. The other delay times were almost all between -10 minutes and 200 minutes, as the histogram below shows.

In [15]:
united.column('Delay').min()

Out[15]:
-16
In [16]:
united.column('Delay').max()

Out[16]:
580
In [17]:
delay_bins = np.append(np.arange(-20, 301, 10), 600)
united.hist('Delay', bins = delay_bins, unit = 'minute')

For the purposes of this section, it is enough to zoom in on the bulk of the data and ignore the 0.8% of flights that had delays of more than 200 minutes. This restriction is just for visual convenience; the table still retains all the data.

In [18]:
united.where('Delay', are.above(200)).num_rows/united.num_rows

Out[18]:
0.008390596745027125
In [19]:
delay_bins = np.arange(-20, 201, 10)
united.hist('Delay', bins = delay_bins, unit = 'minute')

The height of the [0, 10) bar is just under 3% per minute, which means that just under 30% of the flights had delays between 0 and 10 minutes. That is confirmed by counting rows:

In [8]:
united.where('Delay', are.between(0, 10)).num_rows/united.num_rows

Out[8]:
0.2935985533453888

### Empirical Distribution of the Sample¶

Let us now think of the 13,825 flights as a population, and draw random samples from it with replacement. It is helpful to package our code into a function. The function empirical_hist_delay takes the sample size as its argument and draws an empiricial histogram of the results.

In [20]:
def empirical_hist_delay(n):
united.sample(n).hist('Delay', bins = delay_bins, unit = 'minute')

As we saw with the dice, as the sample size increases, the empirical histogram of the sample more closely resembles the histogram of the population. Compare these histograms to the population histogram above.

In [22]:
empirical_hist_delay(10)

In [26]:
empirical_hist_delay(100)

The most consistently visible discrepancies are among the values that are rare in the population. In our example, those values are in the the right hand tail of the distribution. But as the sample size increases, even those values begin to appear in the sample in roughly the correct proportions.

In [27]:
empirical_hist_delay(1000)

### Convergence of the Empirical Histogram of the Sample¶

What we have observed in this section can be summarized as follows:

For a large random sample, the empirical histogram of the sample resembles the histogram of the population, with high probability.

This justifies the use of large random samples in statistical inference. The idea is that since a large random sample is likely to resemble the population from which it is drawn, quantities computed from the values in the sample are likely to be close to the corresponding quantities in the population.


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# Altitude Ratio

Geometry Level 4

$ABC$ is a triangle with a right angle at $A$. $M$ and $N$ are points on $BC$ such that $AM$ is the altitude, and $AN$ is the angle bisector of $\angle BAC$. If $\frac {CN} {NB} = 21$, what is $\frac {CM} {MB}$?

×


text:
• ### De-biased sparse PCA: Inference and testing for eigenstructure of large covariance matrices(1801.10567)

Jan. 31, 2018 math.ST, stat.TH, stat.ME
Sparse principal component analysis (sPCA) has become one of the most widely used techniques for dimensionality reduction in high-dimensional datasets. The main challenge underlying sPCA is to estimate the first vector of loadings of the population covariance matrix, provided that only a certain number of loadings are non-zero. In this paper, we propose confidence intervals for individual loadings and for the largest eigenvalue of the population covariance matrix. Given an independent sample $X^i \in\mathbb R^p, i = 1,...,n,$ generated from an unknown distribution with an unknown covariance matrix $\Sigma_0$, our aim is to estimate the first vector of loadings and the largest eigenvalue of $\Sigma_0$ in a setting where $p\gg n$. Next to the high-dimensionality, another challenge lies in the inherent non-convexity of the problem. We base our methodology on a Lasso-penalized M-estimator which, despite non-convexity, may be solved by a polynomial-time algorithm such as coordinate or gradient descent. We show that our estimator achieves the minimax optimal rates in $\ell_1$ and $\ell_2$-norm. We identify the bias in the Lasso-based estimator and propose a de-biased sparse PCA estimator for the vector of loadings and for the largest eigenvalue of the covariance matrix $\Sigma_0$. Our main results provide theoretical guarantees for asymptotic normality of the de-biased estimator. The major conditions we impose are sparsity in the first eigenvector of small order $\sqrt{n}/\log p$ and sparsity of the same order in the columns of the inverse Hessian matrix of the population risk.
• ### Inference in high-dimensional graphical models(1801.08512)

Jan. 25, 2018 math.ST, stat.TH, stat.ME
We provide a selected overview of methodology and theory for estimation and inference on the edge weights in high-dimensional directed and undirected Gaussian graphical models. For undirected graphical models, two main explicit constructions are provided: one based on a global method that maximizes the joint likelihood (the graphical Lasso) and one based on a local (nodewise) method that sequentially applies the Lasso to estimate the neighbourhood of each node. The proposed estimators lead to confidence intervals for edge weights and recovery of the edge structure. We evaluate their empirical performance in an extensive simulation study. The theoretical guarantees for the methods are achieved under a sparsity condition relative to the sample size and regularity conditions. For directed acyclic graphs, we apply similar ideas to construct confidence intervals for edge weights, when the directed acyclic graph is identifiable.
• ### Semi-parametric efficiency bounds for high-dimensional models(1601.00815)

Oct. 12, 2017 math.ST, stat.TH, stat.ME
Asymptotic lower bounds for estimation play a fundamental role in assessing the quality of statistical procedures. In this paper we propose a framework for obtaining semi-parametric efficiency bounds for sparse high-dimensional models, where the dimension of the parameter is larger than the sample size. We adopt a semi-parametric point of view: we concentrate on one dimensional functions of a high-dimensional parameter. We follow two different approaches to reach the lower bounds: asymptotic Cram\'er-Rao bounds and Le Cam's type of analysis. Both these approaches allow us to define a class of asymptotically unbiased or "regular" estimators for which a lower bound is derived. Consequently, we show that certain estimators obtained by de-sparsifying (or de-biasing) an $\ell_1$-penalized M-estimator are asymptotically unbiased and achieve the lower bound on the variance: thus in this sense they are asymptotically efficient. The paper discusses in detail the linear regression model and the Gaussian graphical model.
• ### Confidence regions for high-dimensional generalized linear models under sparsity(1610.01353)

Oct. 5, 2016 math.ST, stat.TH, stat.ME
We study asymptotically normal estimation and confidence regions for low-dimensional parameters in high-dimensional sparse models. Our approach is based on the $\ell_1$-penalized M-estimator which is used for construction of a bias corrected estimator. We show that the proposed estimator is asymptotically normal, under a sparsity assumption on the high-dimensional parameter, smoothness conditions on the expected loss and an entropy condition. This leads to uniformly valid confidence regions and hypothesis testing for low-dimensional parameters. The present approach is different in that it allows for treatment of loss functions that we not sufficiently differentiable, such as quantile loss, Huber loss or hinge loss functions. We also provide new results for estimation of the inverse Fisher information matrix, which is necessary for the construction of the proposed estimator. We formulate our results for general models under high-level conditions, but investigate these conditions in detail for generalized linear models and provide mild sufficient conditions. As particular examples, we investigate the case of quantile loss and Huber loss in linear regression and demonstrate the performance of the estimators in a simulation study and on real datasets from genome-wide association studies. We further investigate the case of logistic regression and illustrate the performance of the estimator on simulated and real data.
• ### Honest confidence regions and optimality in high-dimensional precision matrix estimation(1507.02061)

July 20, 2016 math.ST, stat.TH
We propose methodology for estimation of sparse precision matrices and statistical inference for their low-dimensional parameters in a high-dimensional setting where the number of parameters $p$ can be much larger than the sample size. We show that the novel estimator achieves minimax rates in supremum norm and the low-dimensional components of the estimator have a Gaussian limiting distribution. These results hold uniformly over the class of precision matrices with row sparsity of small order $\sqrt{n}/\log p$ and spectrum uniformly bounded, under a sub-Gaussian tail assumption on the margins of the true underlying distribution. Consequently, our results lead to uniformly valid confidence regions for low-dimensional parameters of the precision matrix. Thresholding the estimator leads to variable selection without imposing irrepresentability conditions. The performance of the method is demonstrated in a simulation study and on real data.
• ### Confidence intervals for high-dimensional inverse covariance estimation(1403.6752)

Aug. 11, 2015 math.ST, stat.TH, stat.ME
We propose methodology for statistical inference for low-dimensional parameters of sparse precision matrices in a high-dimensional setting. Our method leads to a non-sparse estimator of the precision matrix whose entries have a Gaussian limiting distribution. Asymptotic properties of the novel estimator are analyzed for the case of sub-Gaussian observations under a sparsity assumption on the entries of the true precision matrix and regularity conditions. Thresholding the de-sparsified estimator gives guarantees for edge selection in the associated graphical model. Performance of the proposed method is illustrated in a simulation study.


text:
#### THE GALACTIC CHAOS

The classical three-body problem arose in an attempt to understand the effect of the Sun on the Moon’s Keplerian orbit around the Earth. It has attracted the attention of some of the best physicists and mathematicians and led to the discovery of ‘chaos’. We survey the three-body problem in its historical context and use it to introduce several ideas and techniques that have been developed to understand classical mechanical systems.

The study of the three-body problem led to the discovery of the planet Neptune, it explains the location and stability of the Trojan asteroids and has furthered our understanding of the stability of the solar system. Quantum mechanical variants of the three-body problem are relevant to the helium atom and water molecule.

We consider the problem of three point masses (ma with position vectors ra for a = 1, 2, 3) moving under their mutual gravitational attraction. This system has 9 degrees of freedom, whose dynamics is determined by 9 coupled second order nonlinear ODEs:

As before, the three components of momentum,

three components of angular momentum,

and energy,

Wolfgang Pauli (1926) derived the quantum mechanical spectrum of the Hydrogen atom using the relation between E, L2 and A2 before the development of the Schrodinger equation. Indeed, if we postulate circular Bohr orbits which have zero eccentricity (A = 0) and quantized angular momentum L^2 = n^2(h/2pi)^2, then En = − (mα^2)/ (2(h/2pi)^2n^2) where α = e^2/4πϵ0 is the electromagnetic analogue of Gm1m2.

#### General solutions

There is no general analytical solution to the three-body problem given by simple algebraic expressions and integrals. Moreover, the motion of three bodies is generally non-repeating, except in special cases.

On the other hand, in 1912 the Finnish mathematician Karl Fritiof Sundman proved that there exists a series solution in powers of t1/3 for the 3-body problem. This series converges for all real t, except for initial conditions corresponding to zero angular momentum. (In practice the latter restriction is insignificant since such initial conditions are rare, having Lebesgue measure zero.)

An important issue in proving this result is the fact that the radius of convergence for this series is determined by the distance to the nearest singularity. Therefore, it is necessary to study the possible singularities of the 3-body problems. As it will be briefly discussed below, the only singularities in the 3-body problem are binary collisions (collisions between two particles at an instant) and triple collisions (collisions between three particles at an instant).

Collisions, whether binary or triple (in fact, any number), are somewhat improbable, since it has been shown that they correspond to a set of initial conditions of measure zero. However, there is no criterion known to be put on the initial state in order to avoid collisions for the corresponding solution. So Sundman’s strategy consisted of the following steps:

1. Using an appropriate change of variables to continue analyzing the solution beyond the binary collision, in a process known as regularization.
2. Proving that triple collisions only occur when the angular momentum L vanishes. By restricting the initial data to L ≠ 0, he removed all real singularities from the transformed equations for the 3-body problem.
3. Showing that if L ≠ 0, then not only can there be no triple collision, but the system is strictly bounded away from a triple collision. This implies, by using Cauchy’s existence theorem for differential equations, that there are no complex singularities in a strip (depending on the value of L) in the complex plane centered around the real axis (shades of Kovalevskaya).
4. Find a conformal transformation that maps this strip into the unit disc. For example, if s = t1/3 (the new variable after the regularization) and if |ln s| ≤ β, then this map is given by

Unfortunately, the corresponding series converges very slowly. That is, obtaining a value of meaningful precision requires so many terms that this solution is of little practical use. Indeed, in 1930, David Beloriszky calculated that if Sundman’s series were to be used for astronomical observations, then the computations would involve at least 108000000 terms.

#### n-body problem

The three-body problem is a special case of the n-body problem, which describes how n objects will move under one of the physical forces, such as gravity. These problems have a global analytical solution in the form of a convergent power series, as was proven by Karl F. Sundman for n = 3 and by Qiudong Wang for n > 3. However, the Sundman and Wang series converge so slowly that they are useless for practical purposes; therefore, it is currently necessary to approximate solutions by numerical analysis in the form of numerical integration or, for some cases, classical trigonometric series approximations. Atomic systems, e.g. atoms, ions, and molecules, can be treated in terms of the quantum n-body problem. Among classical physical systems, the n-body problem usually refers to a galaxy or to a cluster of galaxies; planetary systems, such as stars, planets, and their satellites, can also be treated as n-body systems. Some applications are conveniently treated by perturbation theory, in which the system is considered as a two-body problem plus additional forces causing deviations from a hypothetical unperturbed two-body trajectory.


text:
Language and Boolean Logic

1. Apr 15, 2010

skeptic2

The following is from regulations regarding setting up a Roth IRA

"Withdrawal of earnings prior to five years and age 59½ are subject to penalty and income tax."

Withdrawal of earnings (prior to five years) AND (prior to age 59½) are subject to penalty and income tax.

If I withdraw the earnings either prior to five years OR prior to age 59½, but not both, are they exempt from income tax?

2. Apr 15, 2010

lisab

Staff Emeritus
Boolean AND/OR confusion aside, that sentence doesn't make sense to me. "Withdrawal of earnings prior to five years..." Five years of what? Retirement? It's not clear what they mean.

(As an aside: I'm fairly comfortable with math, but put a $in front of the numbers and my brain flatlines!) 3. Apr 15, 2010 skeptic2 Checking on the IRS's website I found the following clarifications. 1. The five years is from Jan 1 of the year that the first contribution is made. 2. I found no statement similar to the above. Instead they provide a flowchart of all the possibilities showing whether taxes or penalties must be paid. In either case, prior to 5 years OR prior to age 59½ taxes and penalties apply. It is amazing how easy it is for language to misrepresent the meaning. Could this be because OR in common usage usaually means either/or and AND often means and/or? 4. Apr 15, 2010 rootX But isn't putting$ in front of a number signify that it is HEX

\$A04E

5. Apr 15, 2010

jobyts

The Roth IRA account has to be open for at least 5 years.


text:
Article | Open | Published:

# Aftershock sequences and seismic-like organization of acoustic events produced by a single propagating crack

Nature Communicationsvolume 9, Article number: 1253 (2018) | Download Citation

## Abstract

Brittle fractures of inhomogeneous materials like rocks, concrete, or ceramics are of two types: Nominally brittle and driven by the propagation of a single dominant crack or quasi-brittle and resulting from the accumulation of many microcracks. The latter goes along with acoustic noise, whose analysis has revealed that events form aftershock sequences obeying characteristic laws reminiscent of those in seismology. Yet, their origin lacks explanation. Here we show that such a statistical organization is not only specific to the multi-cracking situations of quasi-brittle failure and seismology, but also rules the acoustic events produced by a propagating crack. This simpler situation has permitted us to relate these laws to the overall scale-free distribution of inter-event time and energy and to uncover their selection by the crack speed. These results provide a comprehensive picture of how acoustic events are organized upon material failure in the most fundamental of fracture states: single propagating cracks.

## Introduction

Stress enhancement at defects makes the damage behavior observed at the continuum-level scale extremely dependent on material microstructure down to very-small scales. This results in large statistical fluctuations in the fracturing behavior at the macroscopic scale, which are difficult to control in practice. For homogeneous brittle solids under tension, the difficulty is tackled by reducing the problem down to that of the destabilization and further growth of a single pre-existing crack1. Linear elastic fracture mechanics then provides the relevant theoretical framework to describe crack propagation in homogeneous materials1, and the use of some concepts coming from out-of-equilibrium physics permits a global self-consistent approach of crack propagation in the presence of weak heterogeneities2. The problem becomes a priori different in heterogeneous materials for loading conditions stabilizing crack propagation (such as compression). In these situations of so-called quasi-brittle failure, the material starts accumulating diffuse damage through barely perceptible microfracturing events; then it collapses abruptly when a macroscopic crack percolates throughout the microcrack cloud3. Quasi-brittle failure can also be promoted in specimens upon tension by a higher degree of heterogeneity in the material4, lower strain rate5, and more active chemical environments6.

Today’s most widely used technique to probe damage evolution in quasi-brittle fracture consists in monitoring acoustic emission. This provides a sensitive non-intrusive method to detect microfracturing events and localize them in both time (μs resolution) and space (coarser resolution). A geophysical-scale analogy is seismicity analysis in the mitigation of earthquake hazard. In both cases, acoustic events (AE) display similar scale-free dynamics organized into mainshock (MS)–aftershock (AS) sequences characterized by a range of empirical scaling laws: First stated by Omori in 18947, and refined later by Utsu8, the AS frequency decays algebraically with time from MS. Next, the Gutenberg–Richter law asserted in 19449 that the event frequency decays as a power-law with energy (or equivalently the frequency decays exponentially with the event magnitude). In 1965, the Båth’s law10 affirmed that the difference in magnitude between a MS and its largest AS is constant, independent of the MS magnitude; the so-called AS productivity law11,12 states that the number of produced AS increases as a power-law with the energy of the triggering MS. Most recently Bak et al. (2002)13 showed that, once rescaled by the activity rate, the distribution of inter-event times obeys a unified scaling law. These laws are central in the implementation of probabilistic forecasting models for seismic hazard14.

These laws have proven of general validity, in natural13,15 and induced16,17 seismicity at the geophysical scale, and in quasi-brittle fracture experiments at the lab scale, either upon compression18,19,20 or caused by the release of a gas21. Yet, they remain empirical. They are usually seen as emergent properties for the collective dynamics of microcrack nucleation, structured by the long-range stress redistribution following each microfracturing event22,23. Still, the dynamics of a single peeling front propagating along a two-dimensional heterogeneous interface is governed by local and irregular jumps24,25, the size, occurrence time, and occurrence location of which share statistical similarities with that of earthquakes26. What if the time organization of events find its origin in the simpler and more tractable problem of a unique nominally brittle crack propagating in an heterogeneous solid?

Here, we analyze the time–energy organization of AE that accompanies the slow stable propagation of a single brittle crack throughout an artificial rock made of sintered monodisperse polystyrene beads (Methods). In the homogeneous parent polymer specimen, such a crack propagates continuously and regularly and no AE occur. On the other hand, increasing the microstructure scale (the diameter, d, of the sintered beads) unveils irregular burst-like dynamics and numerous AE accompanying the crack front’s movement. As in the multi-cracking situations of quasi-brittle fracture, the events form MS-AS sequences obeying the fundamental scaling laws of statistical seismology: the Omori–Utsu law, the productivity law and Båth’s law. Nonetheless, in this situation of single crack propagation, the above seismic laws are demonstrated to emerge directly from the scale-free statistics of energy (for the productivity law and Båth’s law) and from that of inter-event time (for the Omori–Utsu law) according to relations that have been unraveled, without further information on time–energy correlations (or spatio-temporal correlations).

## Results

### Selection of the activity rate

Figure 1a shows a typical time series of the AE observed for d = 583 μm and a mean crack speed $${\bar{v}} = 2.7\,\mu {\mathrm{m}}{\kern 1pt} {\mathrm{s}}^{ - 1}$$. Note the variety of sizes, as evidenced by using the logarithmic scale. Eight transducers spatially localize the AE sources inside the specimen (Fig. 1b and Supplementary Movie 1). Within the localization resolution (~5 mm), the sources gather along the moving crack front (Fig. 1b, c, Supplementary Movie 1) as expected for nominally brittle fracture. This nominally brittle characteristics has also been demonstrated in earlier work from the proportionality between the elastic power released at each time step and the instantaneous crack speed27. In the present experiments, AE result from the local jumps of the front as it suddenly depins from heterogeneities, and not from the collective nucleation of microcracks spreading throughout the solid as in quasi-brittle failure situations.

The cumulative number of produced AE increases continuously and linearly with crack length. Moreover, the proportionality constant, C, is independent of mean crack speed, $${\bar{v}}$$, over the region swept by the crack (Fig. 1d). This indicates that the mean number of AE produced as the crack propagates over a unit length is given by the number of heterogeneities met over this period: C ≈ H/d2, where H is the specimen thickness. The measured values, C = 53 ± 3 AE mm−1 for d = 583 μm and H = 15 mm, and C = 270 AE mm−1 for d = 223 μm and H = 15 mm, are in agreement with the values C ≈ 44 heterogeneities mm−1 and C ≈ 300 heterogeneities mm−1 expected from the preceding relation. As a result, the activity rate R (defined as the mean number of AE produced per unit time) is given by $$R \approx {\bar v} H/d^2$$, where $${\bar v}$$ is the mean crack speed (Supplementary Figs 1 and 2).

### The Gutenberg–Richter law and self-similarity

We now turn to the global statistical characterization of the AE time series. In all the experiments, the probability density function, P(E), decays as a power-law over nearly five decades up to an upper corner energy (Fig. 2a). It is well-fitted by:

$$P(E) \propto E^{ - \beta }{\mathrm{exp}}( - E/E_0),$$
(1)

with E ≥ Emin. The lower cutoff, Emin = 10−4, is the same in all our experiments. It is set by the sensitivity of the acquisition system. Conversely, the exponent β and the upper corner energy E0 depend on both crack speed (slightly) and material microstructure (more importantly). We will return at the end of this section to the analysis of these dependencies. Equation 1 is reminiscent of the Gutenberg–Richter law. Note, however, that the energy distributions observed in seismology often take the form of a pure power-law. Then, earthquake sizes are more commonly quantified by their magnitude, which is linearly related to the logarithm of the energy28: log10E = 1.5M + 11.8. The energy distribution takes the classical Gutenberg–Richter frequency–magnitude relation: log10N(M) = a − bM, where N(M) is the number of earthquakes per year with magnitude larger than M and a and b are constants. The b-value relates to the exponent β involved in Eq. 1 via: β = b/1.5 + 1.

Beyond the Gutenberg–Richter law, it has been demonstrated13,15 that the recurrence times, Δt, of earthquakes with energies above a threshold value bound Eth obey a unique universal distribution once time is rescaled with the rate of seismic activity over the considered energy range, R(Eth). Such a self-similar distribution is also observed in lab scale quasi-brittle fracture experiments19,20,21. The form of the rescaled distribution, f, depends on how the activity rate evolves with time21: For statistically stationary R(t), f(x) follows a gamma distribution15,21 while, in the presence of a trend (that is a slowly varying component in the time series), f(x) exhibits different power-law regimes19,20. Figure 2b shows Pt) for different Eth in a typical experiment and Fig. 2c shows the distribution after rescaling. The collapse and implied self-similarity are fulfilled. The scaled recurrence times obeys a gamma distribution:

$$P({\mathrm{\Delta }}t|E_{{\mathrm{th}}}) = R(E_{{\mathrm{th}}})f\left( {u = {\mathrm{\Delta }}tR(E_{{\mathrm{th}}})} \right),$$
(2)

with f(u) uγexp(−u/B) for u > b. This underpins a stationary statistics for the AE series. This distribution involves three parameters, which are interrelated (Supplementary Note 1 and Supplementary Eq. 6): The exponent γ and the two rescaled time scales b and B.

### Aftershock sequences and seismic laws

The next step is to identify the AS sequences and to characterize their time–energy organization. In seismology, there exists powerful declustering methods to separate earthquakes into independent (background or MS) and dependent (offspring or AF) earthquakes29. Most of these methods are based on the spatio-temporal proximity of the events. Here, we adopted a procedure19,20,21 used in compressive fracture experiments, where spatial information is not available and considered as MS all AE with energies in a predefined interval. The AS sequence following each of these MS is then defined as all subsequent AE, until an event of energy equal or larger than that of the MS is encountered.

Figure 3a shows the mean number of AS, NAS, triggered by a MS of energy EMS, in a typical fracture experiment. The productivity law is fulfilled, and NAS goes as a power-law with EMS as long as EMS is not too large (below a crossover energy value Ec). The curve remains unchanged after having permuted randomly the energy between the events (that is having attributed to each event i the energy E j of another event j chosen randomly), and having arbitrary set the time step to unity (that is having arbitrary set the time occurrence of the event i to t i  = i + 1). This indicates that the productivity law, here, simply emerges from the Gutenberg–Richter distribution of the AE energy, without any further information on their time organization. Calling $$F(E) = {\int}_{E_{{\mathrm{min}}}}^E P(u){\mathrm{d}}u$$ the cumulative distribution of energy, we then expect (Supplementary Note 2):

$$N_{{\mathrm{AS}}}(E_{{\mathrm{MS}}}) = F(E_{{\mathrm{MS}}})/(1 - F(E_{{\mathrm{MS}}})),$$
(3)

which compares very well with the experimental curve (Fig. 3a). When β is larger than unity and the exponential cutoff in Eq. 1 can be neglected, this expression takes a simple scaling form (Supplementary Note 2): NAS(EMS) ≈ (EMS/Emin)α with α = β − 1. Note that the measured curve NAS versus EMS a priori depends on the declustering method, that is on the algorithm used to decompose the catalog into AS sequences. It was checked that applying a different procedure does not affect significantly the form of this curve (Supplementary Fig. 3).

Båth’s law states that the relative difference ΔM in magnitude ($$M = {\mathrm{log}}_{10}E$$) between the MS and its largest AS is constant, independent of the MS energy. Figure 3b demonstrates this law is actually true here, as long as EMS is smaller than the crossover value Ec defined from the productivity law. Above Ec, ΔM decays exponentially with EMS. As for the productivity law, permuting randomly the events and setting arbitrary the time step to unity do not modify the curve. This implies that Båth’s law finds its origin in the distribution of individual AE energy, without requiring further information on their overall sequencing. This picture is different from that provided in epidemic-type aftershock sequence (ETAS) models14,30, where the series are built using a stochastic branching process and the Båth's law emerges from the correlations induced by the branching31,32. Here, extreme event theory permits to compute the statistics of the largest AS energy from the sequence triggered by a MS of prescribed energy, to compute its mean value, and finally to compute ΔM and its variations with EMS (Supplementary Note 3 and Supplementary Eq. 10). The predicted curve compares quite well with the experimental curve (Fig. 3b). As for productivity law, ΔM(EMS/Emin|β) takes a simpler form when the energy distribution is a simple power law P(E) Eβ (Supplementary Note 3, Supplementary Eq. 12 and Supplementary Fig. 4).

We finally address the Omori–Utsu law, which states that the number of AS per unit time, RAS, decays algebraically with the elapsed time since the MS occurrence, tMS: RAS(t) = R0/(1 + (t − tMS)/τ)p, where R0 and τ are characteristic rates and times, and p defines the Omori exponent. In our experiments, RAS(t|EMS) is computed by binning the AS events over t − tMS, and subsequently averaging the so-obtained curves over all MS with energy falling into the prescribed interval. Figure 3c shows the so-obtained curves. The algebraic decay predicted by Omori is fulfilled, over almost five decades. The Omori–Utsu exponent p is independent of EMS. Conversely, the prefactor increases with EMS.

As $$N_{{\mathrm{AS}}} = {\int}_{t_{{\mathrm{MS}}}}^\infty R_{{\mathrm{AS}}}(t|E_{{\mathrm{MS}}}){\mathrm{dt}}$$, making the Omori–Utsu law consistent with the productivity law yields either R0 or τ to be proportional to NAS(EMS). The former scaling proves to be wrong while the second yields a perfect collapse of the curves (Fig. 3d). The collapse also reveals an exponential cutoff in the Omori–Utsu law, which is finally written as:

$$R_{{\mathrm{AS}}}(t|E_{{\mathrm{MS}}}) = \frac{{R_0}}{{\left( {1 + \frac{{t - t_{{\mathrm{MS}}}}}{{\tau _{{\mathrm{min}}}N_{{\mathrm{AS}}}(E_{{\mathrm{MS}}})}}} \right)^p}}{\mathrm{exp}}\left( { - \frac{{t - t_{{\mathrm{MS}}}}}{{\tau _0N_{{\mathrm{AS}}}(E_{{\mathrm{MS}}})}}} \right)$$
(4)

The four constants, the Omori–Utsu exponent p, the lower and upper time scales τmin and τ0, and the characteristic activity rate R0 are interrelated (Supplementary Note 4 and Supplementary Eq. 17).The very same law holds for the foreshock (FS) rate R FS (t|EMS) versus time to MS, tMS − t (Supplementary Fig. 5). This symmetry along time reversal is a consequence from the fact that the AE time series, here, are stationary.

Note finally that permuting randomly the AE energy in the initial series does not modify the curve in Fig. 3d. Hence, the Omori–Utsu law and time dependency of RAS(t|EMS) do not emerge from correlations between time occurrence and energy, but simply results from the scale-free distribution Pt); the dependency with EMS, for its part, only intervenes in RAS(t|EMS) through NAS(EMS). As a consequence, the parameters at play in Eq. 4 relates to those in Eq. 2 (Supplementary Note 5, Supplementary Figs. 6 and 7, and Supplementary Eq. 18). The equivalence between the Omori–Utsu law and the scale-free distribution of Pt) observed here differs from what is reported in compressive fracture or seismicity (where the Omori–Utsu law implies the power-law distribution for inter-event times, but the reciprocal is not true19,33).

### Effect of crack speed

The Gutenberg–Richter law, the unified scaling law for inter-event time, the productivity law, Båth’s law and the Omori–Utsu law occur in all our experiments, irrespectively of the crack speed $${\bar v}$$. Conversely the underlying parameters vary with $${\bar v}$$. As demonstrated above, the productivity law and Båth’s law are direct consequences of the Gutenberg–Richter distribution of energies, and the Omori–Utsu law for AS is equivalent to the power-law distribution of inter-event times. As a consequence, analyzing the effect of crack speed on the Gutenberg–Richter law and the Omori–Utsu law is sufficient to fully characterize its effect on the AE time–energy organization and the five associated seismic laws.

The lower cutoff for energy, Emin = 10−4, is independent of $${\bar v}$$ and is set by the sensitivity of the acquisition system. The Gutenberg–Richter exponent β logarithmically decreases with $${\bar v}$$, from about 1 to 0.9 as $${\bar v}$$ goes from 10−6 ms−1 to 10−3 ms−1 (Fig. 4a). This evolution can be due to the overlap of some AE, $${\bar v}$$, and which increases with increasing $${\bar v}$$, and which has been demonstrated34 to lower the value of the effective exponent in systems containing temporal correlations. This may also be compared with other observations on quasi-brittle fracture experiments in rocks, which evidence a decrease of the Gutenberg–Richter b-value (analog to β) with the loading rate5 or stress intensity factor6. Note finally that, within the errorbars, the corner energy $$E_0 \simeq 40$$ does not evolve significantly with $${\bar v}$$. The existence of such a corner energy for the Gutenberg–Richter law might find its origin in the finite size and/or the limited volume of material. Still, changing the microstructure length-scale d while keeping the specimen dimensions constant also significantly affects both E0 and the form of the cutoff function (Supplementary Fig. 8). The way the acoustic waves attenuate within the material (which depends on d) might then be a parameter to consider here.

Concerning the Omori–Utsu law (Eq. 4), increasing $${\bar v}$$ yields a significant logarithmic increase of the exponent p, from about 1.1 to 1.6 as $${\bar v}$$ goes from 10−6 ms−1 to 10−3 ms−1 (Fig. 4b). Conversely, it does not affect the characteristic time $$\tau _{{\mathrm{min}}} \sim 0.05\,{\mathrm{s}}$$. The latter closely resembles to the duration of the largest AE. The curve saturation observed for t − tMS ≤ τmin is interpreted as the consequence of missing AS in the catalog right after the MS; their waveform having been drown in that of the MS. This mechanism is analog to the problem of short time aftershock incompleteness (STAI) documented in seismology35,36 and responsible to some bias in the estimation of τmin (generally referred to as the c-value in the seismicity context). Finally, the upper corner time scale τ0 significantly decays with $${\bar v}$$. This decrease can be predicted (Supplementary Note 6 and Supplementary Eq. 22), since τ0 is set by the upper time scale of the inter-event distribution and the activity rate is set by the crack speed $$R \approx {\bar v}H/d^2$$. Neglecting the slight increase of p with $${\bar v}$$, this yields $$\tau _0 \propto 1/{\bar v}^{1/(2 - p)}$$, which is compatible with the observations (Magenta line in Fig. 4c).

The above values correspond to materials with a microstructure length-scale d = 583 μm. It was checked that changing (reducing) d does not change the picture: Scale-free statistics for energy and inter-event time, together with aftershock sequences obeying the productivity law, Båth’s law and the Omori–Utsu law remain true regardless of the value of d. Conversely, the value of the exponents and the form of the cutoff function are material dependent and significantly evolve with specimen microstructure (Supplementary Fig. 8).

## Discussion

We have characterized here the statistical organization of the AE produced by a single crack propagating in a brittle heterogeneous material. The events form MS-AS sequences obeying the fundamental scaling laws of statistical seismology: The productivity law relating the AS number with the MS energy, Båth’s law relating the magnitude of the largest AS with that of the MS, and the Omori–Utsu law relating the AS frequency with the time elapsed since MS. Hence, these laws are not specific to the multicracking situations of compressive quasi-brittle fracture or seismology, but extend to the far simpler situation of a single, slowly propagating, opening crack. In the latter case, they are direct consequences of the individual scale-free statistics of the energies (for the productivity law and Båth’s law) and of inter-event times (for the Omori–Utsu law), without requiring the presence of additional time–energy correlations; Supplementary Fig. 9 provides a more in-depth analysis of these. In this context, it is worth recalling that the propagation of a peeling front along an heterogeneous interface has been reported to be governed by irregular depinning jumps with power-law distributed sizes and inter-event times24,26. It might be interesting to check whether these jumps also form MS-AS sequences according to the fundamental seismic laws, and whether these actually relate to the individual distributions of energies and waiting times as anticipated here (Supplementary Eq. 8 for productivity law, Supplementary Eq. 12 and Supplementary Fig. 4 for Båth’s law, Supplementary Eq. 18 for the Omori–Utsu law). Finally, our finding severely constrains, in this present situation of a single propagating crack, the design of probabilistic forecasting models for the occurrence, and energy of future AE events based on the scaling laws of seismology.

The origins of these laws can be discussed. Over the past years, the avalanche dynamics or crackling noise37 exhibited by a tensile crack propagating in an heterogeneous solid has found a formulation in terms of a critical depinning transition of a long-range elastic manifold in a random potential25,38,39,40. Within this approach, the area swept by the crack front during a depinning event exhibits a scale-free distribution25,41, along with the total elastic energy released within the sample during the event; in the nominally brittle fracture experienced here, these two quantities are proportional and the proportionality constant is equal to the fracture energy27. As a consequence, both productivity law and Båth’s law are anticipated. Note, however, that there is no one-to-one relationship between the depinning events defined above and the acoustic events analyzed here. In particular, earlier work27 has permitted, on the same artificial rocks, to measure the exponent β′ characterizing the scale-free distribution of depinning events. It was found to be significantly higher: $$\beta {\prime} \simeq 1.4$$ for $${\bar v}= 2.7\,\mu m\,s^{-1}$$. It is finally worth to mention that depinning models predict that, at vanishing driving rate, depinning events are randomly triggered in time, with exponential distribution for the inter-event time42, in apparent contradiction with the scale-free distribution observed here on AE waiting times. However, it has been recently shown42 how the application of a finite thresholding divides each true depinning avalanche into a correlated burst of disconnected sub-avalanches; the waiting times separating these correspond to the “hidden” parts below the threshold. A similar mechanism might be invoked here, where each depinning avalanche leads to a correlated burst of AE emitted by the successive points of strong acceleration/deceleration encountered during the avalanche considered.

The uncovering of the relations between the scale-free statistics of inter-event time and energy and the seismic laws characterizing the organization of AS sequences has been made possible since, in the experiments here, the time series are stationary. Surprisingly, the so-obtained relations are also compatible with observations reported on compressive fracture experiments: Our findings yield a relation α = β − 1 between productivity and the Gutenberg–Richter exponent, which compares very well with what was observed in ethanol-dampened charcoal21: {β = 1.30,α = 0.28 ± 0.01}, in slowly compressed wood20: {β = 1.40 ± 0.03,α ≈ 0.3}20 and slowly compressed Vycor19: {β = 1.40 ± 0.05,α = 0.33 ± 0.07}; Our findings also yield a prediction on how the magnitude difference, ΔM, between the largest AS and its triggering MS depends on β (Supplementary Fig. 4), and in particular that ΔM(β = 1.3) = 1, which is compatible with what was reported in ethanol-dampened charcoal21. In other words, the inter-relations between the seismic laws unraveled here in a single crack propagation situation and for a stationary time series seem to remain valid in the much more complex situations of compressional fracture, involving the collective nucleation of numerous microcracks and non-stationary time series. Conversely, the relation α = β − 1 is not valid for earthquakes: For instance, analysis of the seismicity catalog for Southern California has yielded12 β = 1.72 ± 0.07 and α ≈ 0.5, which does not fulfill the relation α = β − 1. Let us recall in this context that, in the Earth, the boundary loading conditions are not forced as in lab experiments but may themselves be emergent properties from the fracturing system.

## Methods

### Synthesis of artificial rocks

The artificial rocks were obtained by sintering polystyrene beads by means of the procedure described in43 and briefly summarized herein. First, a mold filled with monodisperse polystyrene beads  (Dynoseeds from Microbeads SA) of diameter d was heated up to T = 105 °C (90% of the temperature at glass transition). Second, a slowly linearly increasing compressive stress was applied while keeping T = 105 °C, up to a prescribed value P. Both P and T were then kept constant for one hour to achieve the sintering. Third, the system was unloaded and the sample was taken out of the mold, while keeping T = 105 °C to avoid thermal shocks. Fourth, the sample was cooled down to ambient temperature at a rate slow enough to avoid residual stress. This procedure provides artificial rocks with homogeneous microstructure, the porosity and length-scale of which are set by P and d, respectively. In all the experiments reported here, P was chosen large enough (larger than 1 MPa) to have negligible rock porosity (less than 1%), regardless of d. This ensures a nominally brittle fracture with a single intergranular crack propagating in between the sintered grains (Supplementary Fig. 10). The heterogeneity length-scale is directly set by d. The heterogeneity contrast is mainly set by the small out-of-plane distortions due to the disordered nature of the grain joint network, which induces mixed mode fracture at the very local scale. The contrast hence remains small (weak heterogeneity limit), not sufficient to promote microcracking and quasi-brittle fracture27,43. For all experiments, except those underpinning Supplementary Fig. 8, the nominal diameter of beads prior sintering is d = 583 μm and the standard deviation around is 28 μm. This diameter is large enough to ensure global crackling dynamics at finite driving rate44. In Supplementary Fig. 8, the first series of experiments was carried out with beads of nominal diameter d = 233 μm and standard deviation 6.2 μm, and the second series of experiments with beads of nominal diameter d = 24 μm and standard deviation 4 μm. Table 1 provides a synthesis of the samples and parameters to be associated with the different experiments analyzed here.

### Experimental arrangement for the fracture tests

Stable tensile cracks were driven by the wedge splitting fracture set-up described in27. Parallelepiped samples of size 140 × 125 × 15 mm3 in the propagation, loading, and thickness directions were machined from the obtained artificial rocks. An additional 30 × 40 mm2 rectangular notch was cut out on one of the two lateral edges and a 10 mm-long seed crack was introduced in its middle. This crack is loaded in tension by pushing a triangular steel wedge (semi-angle 15°) in the notch at a constant velocity, Vwedge. Two steel blocks with rollers coming in between the wedge and notch ensure the damage processes at the crack tip to be the sole dissipation source for mechanical energy in the system. During each experiment, the force f(t) applied by the wedge was monitored in real-time by means of a S-type Vishay cell force, and the instantaneous specimen stiffness k(t) = f(t)/Vwedge × t was deduced. From this signal and the knowledge of the variation of k with crack length c in such a geometry (obtained by finite element simulations), the instantaneous crack length (spatially averaged over specimen thickness) was obtained and the instantaneous spatially averaged crack speed was deduced (see27 for details). Its mean value, $${\bar v}$$ over the considered range for acoustic analysis was tuned by modulating the wedge speed.

### Monitoring of acoustic events

The acoustic emission was collected at 8 different locations via 8 piezoacoustic transducers. The signals were preamplified, band filtered, and recorded by a PCI-2 acquisition system (Europhysical Acoustics) at 40 MSampless−1. An acoustic event (AE) is defined to start at the time ti when the preamplified signal V(t) goes above a prescribed threshold (40 dB), and to stop when V(t) decreases below this threshold. The minimal time interval between two successive events is 402 μs. This interval breaks down into two parts: The hit definition time (HDT) of 400 μs and the the hit lockout time (HLT) of 2μs. The former sets the minimal interval during which the signal should not exceed the threshold after the event initiation to end it and the latter is the interval during which the system remains deaf after the HDT to avoid multiple detections of the same event due to reflexions. Each so-detected AE is characterized by two quantities: Its occurrence time identified with ti and its energy defined as the square of the maximum value V2(t) between ti and tf; we have verified that the results reported here do not change if we choose instead to define the energy as the integral of V2(t) over the duration of the event (Supplementary Fig. 11). From the knowledge of the wave speed, cW, in the material (measured using the pencil lead break procedure: cW = 2048 ms−1) and the arrival time at each of the 8 transducers, it is also possible to localize spatially the sources of emitted AE (Supplementary Movie 1). The spatial accuracy is set by the main frequency f of the AE waveform. This frequency was measured to vary from f = 40 kHz to f = 130 kHz depending on the analyzed pulse, yielding an overall spatial accuracy $$\delta x \sim f/c_{\mathrm{W}} \sim 5\,{\mathrm{mm}}$$.

### Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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## Acknowledgements

We thank Thierry Bernard for technical support, and Hervé Bercegol and Vadim Nikolayev for fruitful discussions. We also thank Francois Daviaud, Hugues Chaté, Francois Ladieu, Cindy Rountree, and Julien Scheibert for the careful reading of the manuscript. Funding through ANR project MEPHYSTAR (ANR-09-SYSC-006-01) and by “Investissements d’Avenir” LabEx PALM (ANR-10-LABX-0039-PALM) is also gratefully acknowledged.

## Author information

### Affiliations

1. #### Service de Physique de l’Etat Condensé, CEA, CNRS, Université Paris-Saclay, CEA Saclay, 91191, Gif-sur-Yvette, Cedex, France

• Jonathan Barés
• , Alizée Dubois
• , Lamine Hattali
•  & Daniel Bonamy
2. #### Laboratoire de Mécanique et Génie Civil Université de Montpellier CNRS, 163 rue Auguste Broussonnet, 34090, Montpellier, France

• Jonathan Barés
3. #### Laboratoire FAST, Université Paris-Sud, CNRS, Université Paris-Saclay, F-91405, Orsay, France

• Lamine Hattali
4. #### Laboratoire de Tribologie et Dynamique des Systemes, CNRS, Ecole Centrale de Lyon, 36, Avenue Guy de Collongue, 69134, Ecully, Cedex, France

• Davy Dalmas

### Contributions

J.B., D.D., and D.B. conceived the experiment; J.B., M.L.H., D.D., and D.B. performed the experiments; J.B. A.D., and D.B. analyzed the data; D.B. wrote the manuscript; all authors reviewed the manuscript.

### Competing interests

The authors declare no competing interests.

### Corresponding author

Correspondence to Daniel Bonamy.


text:
# Prove Vector Quadruple Product with Levi-Civita/Index Notation

1. Sep 6, 2014

### johnnydoejr

I'm asked to prove the following using Levi-Civita/index notation:
$(\mathbf{a \times b} )\mathbf{\times} (\mathbf{c}\times \mathbf{d}) = [\mathbf{a,\ b, \ d}] \mathbf c - [\mathbf{a,\ b, \ c}] \mathbf d \$

I'm able to prove it using triple product identities, but I'm completely stuck with the index notation. I was previously able to prove Lagrange's Identity with index notation, but applying similar concepts I just get stuck on the first step with the quadruple product.

Using the same first step of proving Lagrange's identity, I transformed $(\mathbf{a \times b} )$ into $\varepsilon_{ijk} a^j b^k$ and $(\mathbf{c \times d} )$ into $\varepsilon_{imn} c^m d^n$ but then I'm just left with $(\varepsilon_{ijk} a^j b^k) \mathbf{\times} (\varepsilon_{imn} c^m d^n)$ which is seemingly unhelpful.

I also tried letting AxB = W and CxD = Z and transforming WxZ to index notation. Then I tried to 'un-nest' the original cross products in index notation, but it quickly ended up in a place where I couldn't understand what the different indexes represented.

Any help would be appreciated. Thanks.

2. Sep 6, 2014

I am not entirely sure how you defined $[\mathbf{a},\mathbf{b},\mathbf{c}]$. Probably as a determinant? Either way, you must have $[\mathbf{a},\mathbf{b}, \mathbf{c}] = \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})$. That might be helpful.

There is also an identity that relates $\epsilon_{ijk}\epsilon_{lmk}$ to Kronecker deltas. It might also prove useful.

3. Sep 7, 2014

### johnnydoejr

Yes, [a,b,c]=a⋅(b×c). I'm sure it will be helpful, but only after I can get past the initial steps.

I'm aware of the Kronecker delta identity you refer to, I used it to prove the scalar quadruple product/Lagrange's Identity as part of the same assignment. I don't doubt that that will be involved as well, but again, only after I can get past this initial step.

I don't know how to combine $(\varepsilon_{ijk} a^j b^k) \mathbf{\times} (\varepsilon_{imn} c^m d^n)$ without the index notation blowing up on me to a point where I don't understand it. It could just be that I'm approaching it from the wrong direction in the one step I did make in re-writing the two inner cross products in index notation.

4. Sep 7, 2014

### johnnydoejr

Thought I had a breakthrough after referencing another forum post. Was able to work through this:

\begin{align*} [(A\times B)\times(C\times D)]_{i} &= \varepsilon_{ijk}(A\times B)_{j}(C\times D)_{k} \\ &= \varepsilon_{ijk} \varepsilon_{jmn} A_{m}B_{n} \varepsilon_{kpq} C_{p}D_{q} \\ &= -\varepsilon_{jik} \varepsilon_{jmn} A_{m}B_{n} \varepsilon_{kpq} C_{p}D_{q} \\ &= -(\delta_{im}\delta_{kn} - \delta_{in}\delta_{km})A_{m}B_{n} \varepsilon_{kpq} C_{p}D_{q}\\ &= (-A_{i}B_{k} + A_{k}B_{i}) \varepsilon_{kpq} C_{p}D_{q}\\ &= -A_{i} \varepsilon_{kpq} B_{k}C_{p}D_{q} + B_{i} \varepsilon_{kpq} A_{k}C_{p}D_{q} \\ &= (\mathbf{A} \cdot \mathbf{C} \times \mathbf{D})\mathbf{B} - (\mathbf{B} \cdot \mathbf{C} \times \mathbf{D})\mathbf{A}\end{align*}

This has to be the right approach...too many things went right while I was working through it, but then I came to then end and compared it to the identity:

$(\mathbf{A} \cdot \mathbf{B} \times \mathbf{D})\mathbf{C} - (\mathbf{A} \cdot \mathbf{B} \times \mathbf{C})\mathbf{D}$

So close, yet so far. Anybody see any mistakes? My brain is fried.

5. Sep 7, 2014

### vela

Staff Emeritus
You didn't make a mistake, but to get the identity you want, try contracting $\varepsilon_{ijk}$ with $\varepsilon_{kpq}$ instead.

6. Sep 7, 2014

### johnnydoejr

Thanks, that does it.


text:
# Trigonometry sin cot CAST

5. (a) For the triangle, a=10, b=12, B=60°, labelled by convention, find all the angles and sides.

(b) Solve the following, ????????^2?=1, express your solution in degrees in the range ?180°???180°, and in radians in the range 0°???2?rad.

If someone could do these and show working out that would be great!

note from me (the cot is squared it just does not show the cot to the power of 2 very well)

Answers can be viewed only if
1. The questioner was satisfied and accepted the answer, or
2. The answer was disputed, but the judge evaluated it as 100% correct.


text:
Deepak Scored 45->99%ile with Bounce Back Crack Course. You can do it too!

# The diameters of circles (in mm) drawn in a design are given below:

Question:

The diameters of circles (in mm) drawn in a design are given below:

Solution:

Here, N = 100, h = 4

Let the assumed mean, A, be 42.5.

Mean, $\bar{x}=A+\frac{\sum_{i=1}^{5} f_{i} y_{i}}{N} \times h=42.5+\frac{25}{100} \times 4=43.5$

$\operatorname{Variance}\left(\sigma^{2}\right)=\frac{h^{2}}{N^{2}}\left[N \sum_{i=1}^{3} f_{i} y_{i}^{2}-\left(\sum_{i=1}^{3} f_{i} y_{i}\right)^{2}\right]$

$=\frac{16}{10000}\left[100 \times 199-(25)^{2}\right]$

$=\frac{16}{10000}[19900-625]$

$=\frac{16}{10000} \times 19275$

$=30.84$

$\therefore$ Stan dard deviation $(\sigma)=5.55$


text:
# Difference between tuple and row matrix

In Munkres Analysis on Manifolds the author uses the word ''tuple spaces'' to refer to a special sort of vector space. Further down on the same page (page 6) he discusses the linear isomorphism that maps a tuple to a row matrix.

I am greatly confused by this as I do not understand the difference between a tuple and a row matrix and also I do not know what a tuple space is. I do know linear algebra and I do know the definition of a vector space.

Please could someone help me understand this?

• I have the same question as you. I do not have access to Munkres book at this time. It would be very helpful to me to know exactly what Munkres writes as regards to, Further down on the same page (page 6) he discusses the linear isomorphism that maps a tuple to a row matrix.'' Is it possible for you to provide these details? – Michael Levy Jun 28 '19 at 16:19

An $n$-tuple is just a list of $n$ numbers $x_k$ $\>(1\leq k\leq n)$, written as $$(x_1,x_2,\ldots, x_n)\ .\tag{1}$$ Unless you want to do linear algebra with it you can leave it at that.

In linear algebra tuples are used for various purposes and become part of a certain algebraic technique called matrix algebra. Depending on the purpose, it is convenient to arrange a tuple $(1)$ as an $(n\times 1)$-matrix, or column vector, like so: $$\left[\matrix{x_1\cr x_2\cr \vdots \cr x_n}\right]\ ,\tag{2}$$ and sometimes it is more appropriate to arrange it as $(1\times n)$-matrix, or row vector, like so: $$[x_1\ x_2\ \cdots \ x_n]\ .\tag{3}$$ There is no such thing as an isomorphism involved with this; it's just a typographical convenience. (Of course, as you go on in linear algebra you'll understand better which vectors are by their nature column vectors and which ones should be written as row vectors.)

As column vectors $(2)$ take up much space in text they are sometimes written as $[x_1\ x_2\ \cdots \ x_n]'$, or similar. A good rule of thumb is the following: When an $n$-tuple $(1)$ is abbreviated by $x$ then the letter $x$ denotes as well the column vector $(2)$, and one writes $x'$ when the row vector $(3)$ is meant.

• Thank you. Then I can ignore the paragraph in the book that discusses a linear isomorphism between tuples and vectors? – blue Apr 13 '14 at 11:17
• You can ignore "the linear isomorphism that maps a tuple to a row matrix", but you cannot ignore the isomorphism that maps "abstract" vectors to tuples after a basis has been chosen. – Christian Blatter Apr 13 '14 at 11:50
• @ChristianBlatter Why can you ignore that there is an isomorphism between the two spaces? Aren't they different vector spaces defined differently but an there is an isomorphism? And could you please explain again in detail the second part of your comment? Thanks a lot. – LearningMath Sep 25 '15 at 22:15
• @notorious: Writing $[2\ {-3}\ 5\ 7]$ instead of $(2,{-3},5,7)$ does not exploit an isomorphism; it is just a useful typographical convention. On the other hand it is a fundamental fact of linear algebra that any finitely generated real vector space is isomorphic to the "standard model" ${\mathbb R}^n$ for some $n\geq0$. – Christian Blatter Sep 26 '15 at 8:58
• @ChristianBlatter That was what I was asking about. We have the "standard model" or the vector space of n-tuples called $\mathbb{R}^n$. And then, there is another vector space of the row matrices and another one of column matrices. I consider these three as different vector spaces but we can establish an isomorphism between them. Because the notion "vector" is defined as a member of a vector space. The "standard model" is just another vector space in the pool of vector spaces and it happens that its members are n-tuples. Is my understanding right? – LearningMath Sep 28 '15 at 7:32

I think the point of the author is simply saying that you can understand $\mathbb R^n$ as a set of $n$-tuples, but also as a set of $n$-rows or $n$-columns. It doesn't matter how you look at it, all three views are isomorphic.

There is a formal difference between the tupel space $$\mathbb R^n:=\mathbb R \times \dots \times \mathbb R$$ and the space of row vectors $\mathbb R^{1,n}$, which is defined as the space 'numbers arranged in a row' (think arranged in a spreadsheet). In the book you linked, the difference is emphasized by using different kinds of brackets: $(x_1,\dots,x_n)$ in the former, $[x_1,\dots,x_n]$ in the latter case.

The spaces are defined differently first. Then one shows that they are isomorphic. This means in terms of linear algebra they are equivalent.

• (1) I found angle-brackets used to denote ordered pairs of objects (tuples?) and Bourbaki-style structures: $\textbf{Mycategory}=\langle X, A(X), \circ \rangle$. From wikipedia, using angles or parenthesis is a matter of taste, but maybe there is something more that I missed? (2) Are the comma-separated objects in sequent calculus (i.e. $A_1, A_2, \dots, A_n \vdash C_1, C_2, \ldots, C_m$) isomorphic to tuples as well? – Niriel May 3 '15 at 17:24


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For any $s \geq t$, note that \begin{align*} r_s = r_t + \sigma\int_t^s dW_u + \int_t^s \theta_u du. \end{align*} Then, \begin{align*} \int_t^T r_s ds &= (T-t)r_t + \sigma\int_t^T\int_t^s dW_u ds + \int_t^T \int_t^s\theta_u du ds\\ &=(T-t)r_t + \sigma\int_t^T\int_u^T ds\, dW_u +\int_t^T\int_u^T\theta_u ds du\\ &=(T-t)r_t + \sigma\int_t^T (T-u)...
Let $$Z_t = \exp(-X_t)$$ with $$X_t = \sigma(T-t)W_t+\sigma\int_0^tW_sds+\int_0^Tf(0,u)du+\int_0^t\int_s^T\alpha(s,u)du ds$$ and $W_t$ a standard Brownian motion, along with the usual assumptions. We can write $X_t=f(t,W_t)$ and apply Itô's lemma to get: $$dX_t = \frac{\partial f}{\partial t}(t,W_t) dt + \frac{\partial f}{\partial W_t} (t,W_t) dW_t + \... 5 This is known as the classical Leibniz rule. The link sends to Wikipedia, where you can find a proof. It allows to differentiate under the integral sign. A general statement of the formula is:$$\text{d}\left(\int_{g(x)}^{h(x)}f(x,s)\text{d}s\right)=h'(x)f(x,h(x))\text{d}x-g'(x)f(x,g(x))\text{d}x+\int_{g(x)}^{h(x)}\text{d}f(x,s)\text{d}s4 Keeping it simple, you know HJM SDE gives the dynamics of an instantaneous forward referencing a fixed maturity T, f\left(t, T\right), but there is a continuum of such maturities - the whole forward curve as a function of T. You can have each forward driven by a different brownian for example, so in general the HJM approach will be infinite dimensional. ... 3 OK, your framework on this is right. Long-term yields embed "expectations" about future short rates. That is: what do I think I'll get if I sat for T years with cash in the bank for swaps, or the same rolling Govvie Bills with respect to Bonds? Plus a cherry-on-top called "term premium". One can rationalise the existence of this in lots of intuitive ways, ... 3 Based on this question, for the Hull-White model of the form \begin{align*} dr_t = (\theta(t)-a r_t) dt + \sigma dW_t, \end{align*} where a and \sigma are constants, a(t) is a deterministic function, and W_t is a standard Brownian motion, the price at time t of a zero-coupon bond with maturity T and unit face value is given by \begin{align*} P(t, ... 3 You almost get there. However, you ca not conclude that \rho^2 is a constant based on (10). Note that, from your (7) and (8), \begin{align*} \frac{\rho(z_t)^2}{\beta} e^{\beta \tau} (e^{\beta \tau} - 1) = -h'(\tau)+e^{\beta \tau}h'(0). \end{align*} Taking derivative with respect to \tau on both sides, we obtain that \begin{align*} \frac{\rho(... 3 Here we provide another answer using Ito's calculus. It appears involved, but it also has its own interest. Given the short rate dynamics \begin{align*} dr_t = \nu(r_t, t) dt + \rho(r_t, t) dW_t, \end{align*} we define the function \begin{align*} g(x, t, T) = -\ln E\left(e^{-\int_t^T r_s ds} \,\big|\, r_t = x\right). \end{align*} The forward rate f(t, T) ... 2 If you have a simple instrument, short rate models capture all the key variance, but they impose structure on the shape of forward volatility curves (and, usually, forward tilt) that is often far from realistic. If you have instruments whose value is sensitive to what might happen with tilt or forward volatility, you need a multifactor model like HJM. ... 2 I am not sure about this specific algorithmic implementation, but I am a bit confused by your indexes and suspect you might be as well (e.g. M not defined, you're showing cases of i looping when it seems you mean j). I think it would be useful to revisit the basics: Let D_t \in (0,1] be the present value factor for a cash flow at time t. By ... 2 This is my first ever answer so please bear with me. Apologies in advance for terrible formatting. Also fyi, you have some typos in your post that may be making things more confusing. The issue here is that \int_{0}^{t}\int_{s}^{T}\sigma(s,u)dudW_s is itself a stochastic process and trying to take its partial derivative with respect to time (which you ... 2 We assume that \begin{align*} d \ln P(t,T) = \mu(t, T) dt - \sigma (t,T) dW(t). \end{align*} Then, \begin{align*} \ln P(t,T) = \ln P(0,T) + \int_0^t \mu(s, T) ds - \int_0^t \sigma (s,T) dW(s). \end{align*} Moreover, \begin{align*} f(t, T) &= -\frac{\partial\ln P(t,T)}{\partial T} \\ &= -\frac{\partial\ln P(0,T)}{\partial T} - \int_0^t \frac{\partial\... 2 In the real world, many people believe that the yield of a government bond consists of an expectations part plus a term premium. However financial models such as HJM are built in a “risk -neutral” probability measure, with respect to which every asset is the expectation of future cash flows without regard to any term premium. This is done to ensure the ... 2 It is just an application of the Leibniz integral rule, written in differential form. Please see here: https://en.m.wikipedia.org/wiki/Leibniz_integral_rule Capital T is constant, t is changing, so the second term on the right hand side is the exchange of integral and differential, the first term on the right hand side is the function value at the lower ... 1 Note that \begin{align*} f(t, T) = f(0, T) + \int_0^t\alpha(u,T)du+\int_0^t\sigma e^{-a(T-u)}dW_u, \end{align*} where, based on this question, \begin{align*} f(0, T) = \int_0^T \theta(u) e^{-a(T-u)} du - \frac{\sigma^2}{2a^2}\big(e^{-a T} -1\big)^2 + e^{-a T} r_0. \end{align*} Note also that \begin{align*} \int_0^t\alpha(u,T)du &= \int_0^t\sigma(u,T)\... 1 In the typical HJM model, during a single time step forward rates evolve according to their individual volatilities and according to pairwise correlations which can be specified. That arrangement is arbitrage free within the time step. In addition, different time steps are independently generated. This latter feature ensures that the model does not ... 1 OK, so I think I have figured it out. I assumed that we need to use Ito's lemma here, however, it seems the authors mean to use Ito's isometry, which must be used to prove below equality\mathbb{E}_Q\Big(e^{-\sigma\int_t^T (T-u)dW_u} \mid r_t\Big)=e^{\frac{\sigma^2}{2}\int_t^T(T-u)^2 du}$$We know that for normal-distibuted random variable X (with mean ... 1 Your issue is that you misinterpreted the NA criterion, it reads:$$ a_t(x) \triangleq \sum_{i=1}^{\infty} \left(b^i_t(x) \int_0^x (b_t^i(u))^T du\right)e_i,  where $b^i_t$ denotes the $i^{th}$ column of the volatility matrix $b_t$, $^T$ the transpose and $e_i$ the $i^{th}$ standard basis vector in $\ell^1$. In other words the $i^th$ coordinate of the ...


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## Smooth atlas

### Set

 context $\langle M,T\rangle$ … second-countable Hausdorff space context $n\in \mathbb N$ definiendum $A\in$ it inclusion $A\subseteq$ atlas ($\langle M,T\rangle,n$) forall $\langle V,\phi\rangle,\langle W,\psi\rangle\in A$ postulate $\phi\circ\psi^{-1}$ … smooth

A priori “$\phi\circ\psi^{-1}$” in the postulate doesn't make sense as their domains/codomains will not much. Here, really, one must choose functions with appropriately restricted domain.

#### Universal property

For a smooth atlas there are smooth coordinate changes on $\mathbb R^n$.

#### Idea

For a smooth atlas there are smooth coordinate changes on $\mathbb R^n$.