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# Tag Info

## Hot answers tagged sparsity

13

"Is there any exact, i.e. numerical, definition for sparsity?" And by numerical, I understand both computable, and practically "usable". My take is that: not yet, as least, there is no consensus, yet there are some worthy contenders. The first option "count only non zero terms" is precise, but inefficient (sensitive to numerical approximation and noise, and ...

5

A couple of reference works offer an exaplanation: A neurological interpretation described in Scholarpedia Stanford's Unsupervised Feature Learning and Deep Learning tutorial If we look at the definition of the term in the context of dictionary learning, for example in K-SVD: An Algorithm for Designing Overcomplete Dictionaries for Sparse Representation, ...

5

As you correctly noted compressed sensing, compressive sampling, sparse sampling all mean the same thing. Some authors also call it sparse sensing. The idea behind compressed sensing is that a sparse signal can be recovered from very few linear measurements. In symbols, if $\mathbf x$ is $N\times 1$ sparse$^\ddagger$ vector, and $\mathbf A$ is an $M\times N$ ...

5

Basically your problem is called Blind Deconvolution. It means we want to estimate both the operator and the input given the output. You model is Linear Time Invariant Operator so we have LTI Blind Deconvolution. In general blind deconvolution is ill poised problem. So we need to make assumptions about the model. The more assumptions the better the chance ...

4

I am sorry I cannot comment your answer due to my low reputation. Gini and your suggested sparsity ratio ($l_1(x)/l_2(x)$) both give me the same value for $\lambda$. But The problem I still see is that I cannot take into account how well the vector is solving the equation $Ax-y$. I would like to combine the residuum $l_1(A\hat{x}-y)$ and the sparsity $l_1(\... 4 Sparsity concept is extensively being used in computer vision and image processing. The Idea is that natural image can be pretty sparse when it is transformed to different bases. this bases can be predefined, e.g. FFT,DCT or can be learned from the image, e.g. sparse coding. Here are a few well know example of algorithms that uses the sparsity assumption on ... 3 It can easily solved by the Gradient Descent Framework with one adjustment in order to take care of the$ {L}_{1} $norm term. Since the$ {L}_{1} $norm isn't smooth you need to use the concept of Sub Gradient / Sub Derivative. When you integrate Sub Gradient instead of Gradient into the Gradient Descent Method it becomes the Sub Gradient Method. In the ... 3 Let's think about it in a different way - Generate Noise from a Dictionary. Let's create a Dictionary$ A \in \mathbb{R}^{m \times n} $where each of its rows is normalized (Has Euclidean Norm of$ 1 $) and generated by a Gaussian Random Vector. Now, let's create$ N $random vector$ {\left\{ {r}_{i} \right\}}_{i = 1}^{N} $by: $${r}_{i} = A {g}_{i}$$ ... 3 The term sparse, as you mention, refers to the fact that some "signal", usually represented by a vector$x$contains mostly zero or negligible values and only a few non-zero or significant values. In fact, the situation with neglible vs significant values which is not strictly sparse is often referred to as "compressible", see e.g. Candès & Wakin 2008 (... 3 Take a sine-like signal$s$. In the appropriate Fourier$\mathcal{F}$domain, it is represented by two "peaks", the other coefficients being zero. Fourier is a sparse representation for sines or close-to-sine signals. Conversely, a zero signal, except for a few values, is sparse in its original domain. In narrow sense, a sparse representation of data is a ... 2 Norms like$\ell_p$,$p \ge 1$, or quasi-norms ($0<p< 1$) are all$1$-homogeneous:$\ell_p(\lambda x) = |\lambda|\ell_p( x)$. Which is not the case for the$\ell_0$count measure, which is scale invariant: a signal and its scaled versions possess the same sparsity index, as they have the same quantity of zero values. I have been thinking a lot about ... 2 Compressive Sensing is an approach to reconstruct sparse signals from incomplete set of measurements. In doing so, we need to know$Ψ$to recover$x$, don't we? yes, we do. But if we know$Ψ$, why don't we just measure$z=Ψ^{−1}x=Ψ^{−1}Ψα=a$, and store the value and index of the non-zero components of$z (=α)$? Compressive Sensing as the name ... 2 Sparsity covers a wide range of concepts. It characterizes an object (a signal, a system, a function) for which their exists a representation (exact or approximate) whose dimension (number of parameters, degrees of freedom) is much lower than the inherent dimension of the object. For instance, let us first consider 1.000.000 points$(x_i,y_i)$, acquired to ... 2 A couple of example areas: Sonar beamforming - in many cases there are a small number of targets Radar processing - a radar image can be decomposed to a background and sparse set of point like targets, or a small number of moving targets. Radar tomography - This application uses multiple radar passes at slightly different elevations to extract elevation ... 2 I will start the explanation from the compression viewpoint. There are two main types: lossless compression, and lossy compression. Noise, at least divergence or loss from the original data, arises only with lossy compression. When one considers the original data as the "clean" reference, lossy compression adds a amount of loss related (generally vaguely ... 2 Here are two interpretations of why sparse representations are useful for denoising: 1) Your clean image is well represented by your dictionary (usually a wavelet or 2D-DCT dictionary), but the noise isn't. This means that when calculating your sparse representation from the noisy signal, you will only capture the clean signal, and not the noise. 2) ... 2 How you parameterize your sparsity will depend on your application. The authors of that paper, in a paragraph on page 231 say: which is why they clump the coefficients together in$P$blocks of start time$t_{B_k}$of duration$n_{B_k}$. For one impulse response this is shown in their figure 1(b). The overall sparsity will depend on the reflections and ... 2 It seems to me you have a little misassumption here. During the sampling only$\Phi$matrix is applied to the signal$x$, which is resulted in measurements vector$y$,$y=Φx. Later, in reconstruction we use sparsity assumption to reconstruct the signal through solving problems of this form (there are other forms of solution): $$\min|x|_1 s.t. |\Phi ... 2 MATLAB has a curve-fitting toolbox where you pass a function of your choice to the fit function. Hence, the number of terms is totally up to you in this case. If you do not have a curve-fitting toolbox, you could try to find a free one or write a fit function by yourself (e.g. with fminsearch). However, this is not the fastest algorithm and also you would ... 2 Usually the classic problem is given by:$$\begin{align*} \arg \min_{x} \quad & \frac{1}{2} {\left\| A x - b \right\|}_{2}^{2} \\ \text{subject to} \quad & {\left\| x \right\|}_{0} \leq k \end{align*}$Where$ {\left\| \cdot \right\|}_{0} $is the Cardinality Measure which counts the number of non zero elements in the argument. The above is NP ... 2 Why would you add the constraint of being Orthonormal Dictionary? It doesn't make sense in the context of what you ask. First we need to define resolution. If you mean the grid to be denser than indeed what you need is to create the DFT matrix of zero padded signal and take the subset which you're interested in as you wrote in your question. If you define ... 2 Update I would like to rephrase the original problem with slightly different notation and claim that there is no need to measure the coherence of the product of the circulant matrix A and the DCT matrix B as asked by the OP. In the compressive sensing setting, the coherence of a sensing matrix$A$is useful when we are using it for constructing ... 2 You can approach this problem as a special case of the "$k$-simple bounded signal" class described in (Donoho & Tanner, 2010 - Precise Undersampling Theorems ), see page 2, Example 3. Particularly, your signal is a "0-simple" signal, i.e. your values are either 0 or some constant. The problem can easily be scaled to 0 or "some constant" instead of 0 or 1.... 1 If the signal$x$is already sparse in the original/time domain, then it does not need to be transformed, in other words the transform$\mathcal{J}$can be taken to be the identity. So$x$and$X=\mathcal{J}x$are equal. What we want is to recover, from observed$y$, the unknown$x$, via the recovery of$X$, since$x=\mathcal{J}^{-1}X$. In that case,$A$is ... 1 The question of the existence of a sparse basis of noise is closely related to the question of the effective dimensionality of the noise subspace. First, it is important to realise that noise is a process, and not a signal. You can think of a full characterisation of any kind of noise by means of a function$p: S \to \mathbb{R}^+_0 $that maps a signal$s$... 1 Due to the non-smoothness of the$l_1$norm, the algorithm is called subgradient descent. Because the you are looking for a solution that has a lot of zeros in it, you are still going to have to evaluate sub-gradients around points where elements of$\mathbf{x}$are zero. In fact most of the algorithms effectively treat elements below a certain threshold as ... 1 I haven't read the paper, but from the information you provided it appears the appropriate definition is the first one you gave, i.e., the "entrywise" matrix norm. The text you quoted refers to the norms being "elementwise" which is equivalent to being "entrywise". One way to look at this norm is as the$l_1\$ equivalent of the Frobenius norm. Essentially, ...

1

No. What did you expect? For sparse algorithm to work, the signal has to be sparse. If it's sparse after a integral transform, that doesn't mean it's sparse under a different transform. Also note that according to this answer, the benefit of the sparse fft only apply to very sparse signals - a factor of at least about 2000 between the zero and non-zero ...

1

Since you have tagged your question compressive-sensing, I would guess that you are interested in a compressive sensing solution to this. In principle, this could work by taking a number (less than 64) of random linear combinations (measurements) of your length 64 vector and transmitting these. At the receiver, the fact that this vector has a sparse ...

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


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## Optimizing the Dirichlet hyperparameters

One of the things you’ll notice in papers describing generative models of documents using a Dirichlet prior is to simply fix the Dirichlet hyperparameter $\alpha$ that controls the distributions of topic mixtures for each document. This isn’t ideal when you wish to then compute the probability of an unseen document $p(\mathbf{w}_d | \alpha, \beta)$ because a fixed $\alpha$ encodes no knowledge of the distribution of topic-mixtures over documents in the training corpus.

In the appendix of the journal paper by Blei et. al [1], we find a procedure to learn the $\alpha$ that maximizes the following log-likelihood where $\theta_j$ are the document specific topic mixtures and $\alpha$ the dirichlet hyperparameter.

$\displaystyle \Lambda := \log \prod_j p(\theta_j | \alpha) = \\ N \log \Gamma(\sum \alpha_i) - N\sum \log \Gamma(\alpha_i) + \sum_{i,j} (\alpha_i -1) \log \theta_{ij}$

The expression is maximized using the Newton-Raphson method, which requires computing the derivative of $\Lambda$ and the Hessian (the matrix of second-order derivatives) – with respect to $\alpha$.

## Derivative of $\Lambda$

Making use of the digamma function $\psi(x) = \frac{\partial \log \Gamma(x)}{\partial x}$ the partial derivatives with respect of each component of $\alpha$ gives

$\displaystyle \frac{\partial \Lambda}{\partial \alpha_r} = N \psi(\sum \alpha_i) - N \psi(\alpha_r) + \sum_j \log \theta_{jr}$

## The Hessian

The Hessian component-wise with respect of $\alpha_r$ and $\alpha_s$ is

$\displaystyle \frac{\partial \log \Lambda}{\partial \alpha_r \alpha_s} = N \psi'(\sum \alpha_i) - \delta_{rs} N \psi'(\alpha_r)$

where $\delta_{rs}$ is the Kronecker delta. Note that this Hessian matrix can be written in the following form

$\displaystyle H = \text{diag}(h) + \mathbf{1}z\mathbf{1}^T$

where $\text{diag}(h)$ is the diagonal matrix containing the second-order derivatives with respect to $\alpha_i$s across the diagonal and zero elsewhere; $z = \psi'(\sum \alpha_i)$; and $\mathbf{1}$ is vector of $1$s. The inverse of a matrix of his form is given by the Matrix Inversion Lemma which states

$\displaystyle (A - BD^{-1}C)^{-1} = \frac{A^{-1} + A^{-1}BCA^{-1}}{D - CA^{-1}B}$

In our case, $H^{-1}$ is given by

$\displaystyle H^{-1} = \text{diag}(h)^{-1} - \frac{\text{diag}(h)^{-1} \mathbf{1}\mathbf{1}^T \text{diag}(h)^{-1}}{ z^-1 + \sum_{j=1}^k h_j^{-1}} \\ H^{-1}_{ij} = \text{diag}(h)^{-1}_{ij} - \frac{h^{-1}(h^{-1})^T}{ z^{-1} + \sum_j h^{-1}_j}$

## The upside

We are now ready to compute the new guess for the next iteration in the Newton-Raphson method: $\alpha_{new} = \alpha_{old} - H^{-1}(\alpha_{old}) g(\alpha_{old})$. Given the gradients $g$ evaluated at the old $\alpha$s, the new $\alpha$ are given by

$\displaystyle \alpha_{old} - (H^{-1} g)_i = \alpha_{old} - \sum_j H^{-1}_{ij} g_j \\ = \alpha_{old} - h^{-1}g_i - \frac{\sum_j (h^{-1} (h^{-1})^T)_{ij} g_j}{ z^{-1} + \sum_j h^{-1}_j} \\ = \alpha_{old} - \frac{1}{h^{-1}} \left( g_i - \frac{ \sum_j \frac{g_j}{h_j} }{z^{-1} + \sum_j h^{-1}_j} \right)$

The reason for the special attention given to the form of the Hessian in this problem si that it requires the computation of only the values $h_i$ and $g_i$ which amount to $2k$ values which is only linear in $k$ (the dimension of $\alpha$) to the otherwise $O(k^3)$ required for a full-blown matrix inversion of $H^{-1}$.

[1] David M. Blei and Andrew Y. Ng and Michael I. Jordan and John Lafferty. 2003. “Latent Dirichlet Allocation.” Journal of Machine Learning Research.

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# System of 2 equations

Tags:
1. Jan 16, 2017

### akaliuseheal

1. The problem statement, all variables and given/known data

It's a system of 2 equations.

2. Relevant equations

3. The attempt at a solution
My attempt is not worth writing here.
Results I got using Microsoft Mathematics without showing me step by step.

2. Jan 16, 2017

### Math_QED

Your attempt is worth it! It shows us you made effort to solve the question.

You have that $y$ must be equal to both $\frac{1}{750} + \frac{8}{x}$ and $\frac{3}{2500} + \frac{12}{x}$

Thus you need to solve for $x$: $\frac{1}{750} + \frac{8}{x} = \frac{3}{2500} + \frac{12}{x}$ and then substitute your answer for $x$ to obtain the value for $y$

3. Jan 16, 2017

### stevendaryl

Staff Emeritus
The most straight-forward way to solve a system of equations (although not always the most efficient way) is:
• Use the first equation to solve for one variable in terms of the rest
• In the second equation, replace that variable by its value (found in step 1). This will give you an equation involving one fewer variable.
• Now, solve for a second variable.
• Continue for as many equations as you have (which should be the same as the number of variables)
In your case, the first equation already gives you $y$ as a function of $x$. So just use that value of $y$ in the second equation, and see what you get.

4. Jan 16, 2017

### akaliuseheal

Okay, so I did manage to solve it. I was stuck for some time on fractions. Thanks


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Korean J Financ Stud Search

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 Korean Journal of Financial Studies 2009;38(1):53-75. Published online March 31, 2009.
 Diversification and Bank Performance: International Evidence from 66 Countries Jae Hwa Lee, Byung Jin Kwak, Kwang Woo Park, Rae Soo Park 겸업화와 은행의 경영성과 이재화, 곽병진, 박광우, 박래수 Abstract The deregulation in the financial services industry makes commercial banks around the world face increased competition and change their managerial activities tremendously. A banks typical reaction to the new environment is to widen the range of their products and services, i.e., diversification of income sources. As a result, the ratio of banks non-interest income to operating revenue increases in many countries. Many studies question the implications of this change on banks performance and risk. Especially, supervisory authorities concern about this issue for the safety and soundness of the banking system. Prior studies, mostly based on US banks, on the effect of the increase in non-interest income on banks performance and risk show mixed results. Some studies show that the improvement of performance and a potential for risk reduction can be obtained by non-bank activities, but others report that there exists some increase in volatility of accounting returns and risk. This paper examines the effects of diversification on bank performance and risk by analyzing 3,220 banks from 66 countries. We find that diversification may improve bank`s capital adequacy, and earnings performance, but deteriorate the asset quality and increase the volatility of earnings and default risk. It appears that the non-interest income from various activities may be more volatile than interest income and banks may be less cautious to manage their asset compared to the interest focused activity. Key Words: 겸업화,경영성과,비이자수익,상업은행,위험,Bank Performance,Commercial Bank,Diversification,Non-interest Income,Risk


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## Activities: Scheduler

mod_scheduler
Maintained by Henning Bostelmann
An appointment scheduler for planning face-to-face meetings between teachers and students. Teachers can set up time slots, and students can choose one of these on Moodle.
3550 sites
99 fans
Current versions available: 9

The Scheduler module helps you in scheduling appointments with your students. Teachers specify time slots for meetings, students then choose one of them on Moodle. Teachers in turn can record the outcome of the meeting - and optionally a grade - within the scheduler.

Group scheduling is supported; that is, each time slot can accomodate several students, and optionally it is possible to schedule appointments for entire groups at the same time.

Please note that minor updates to this module will normally be released on github only.

For help and discussion about the module, including feature requests etc., please use the Scheduler forum on moodle.org or the bug tracker. Please don't use the comment section below for that, queries might go unanswered.

For an earlier version supporting Moodle up to release 1.9, see the Scheduler 1.9 module.

### Contributors

• Wed, Sep 26, 2018, 7:05 PM
Hi Henning,
It would be great if we can bind it with an existing activity. Say if I can bind it to a quiz. So that a student can book his slot for the quiz. When he attempts a quiz he should be asked to schedule it. If he booked a slot then he should be restricted to that slot only. Is it possible? If so can we expect to have such a feature any time soon?
• Thu, Nov 8, 2018, 5:34 AM
Dear Henning,

Long time no see, how are you? My manager decides to continue with the Moodle LMS development, I am not sure whether the current scheduler does the following features:

1. When creating recurring slots, attach the students to all recurring slots. It seems that the current scheduler does not support it. When creating recurring slots, only the slot information is inputted, not the students.

2. There is no button to delete the created "recurring" slots. For example, on Apple iPhone's calendar, when creating recurring activities, upon delete one of the recurring activities, the calendar app asks whether the user would like to delete this one and its following activities that belong to the same recurring activities. Currently, it looks like the scheduler only has "Delete selected slots", "Delete my unused slots" and "Delete all my slots".

If the above features are not supported right now, is there a chance that you will find time to develop them?

Thank you!

Kevin
• Thu, Nov 8, 2018, 8:31 AM
@Kevin Lu

1. I agree with you: we sometimes create recurring slots for a given teacher/student pair, it'd be a time saver to be able to assign a student on the "add repeated slots" page.

2. Alternately to your proposal (if it seems too complex to develop), I suggest a feature enabling to group edit slots; first by selecting a number of slots, then by showing a page similar to the "add single slot" page. Inputting a value in a field would result in inputting that value in the same field for all selected slots.

If you add this feature requests in the tracker I'll make sure to vote for them ;)

Sébastien
• Thu, Nov 8, 2018, 9:22 AM
@Sebastien Jaffredo

Thank you for your support. How do I add these wanted features to the tracker?
• Thu, Nov 8, 2018, 10:23 AM
Hi Kevin

The bug tracker is here:
https://tracker.moodle.org/issues/?jql=project%20%3D%20CONTRIB%20AND%20component%20%3D%20%22Module%3A+Scheduler%22

You'll have to register to post and follow issues.
First check the current feature requests to see if someone hasn't already made the same request, in which case you'll want to vote for it and follow it. If the request is not exactly what you have in mind, you may add your comment to the discussion.

• Fri, Nov 30, 2018, 11:03 AM
Thanks for the tool it has proved extremely helpful. I just have one query surrounding the functionality. when the teacher/tutor/educator books in a students, no email appears to be sent out to them informing them of the booking.

This might prove to be a beneficial option for educators whom want to take greater control of the scheduler activity, and relieving them from the task of having to email or inform students verbally that they have booked them.
• Tue, Apr 16, 2019, 6:08 PM
Hi,

First of all, thank you for providing this tool, however I am wondering if it would be possible to use it in a different way as it was first meant to be. Let me explain what we intend to do : we would like to have a Moodle space dedicated to pedagogical meetings between students and teachers OR between students and peer-students OR even perhaps between teachers and peer-teachers.
Slots could then be created eitherway by teachers AND students. The idea is that any user, disregarding their role(s) on the platform (be them teachers or students), could equally register online for a meeting or organize one themselves so that any other peer-student / teacher could attend such a meeting.
I have been having a deep look at the various permissions and overriding possibilities within the course and within each of the 2 schedulers I have added, (one is meant for "Registering for a meeting" while the other should be used for "Planning a meeting"). It seems like having the permission to add slots in the scheduler automatically takes out the permission to register to any slot.
Any idea to help us solve this dead-end situation ?

Thanks for helping,
Best,

Yannick
• Fri, May 17, 2019, 5:44 PM
Hi,

Using 3.5.0 on Moodle 3.5.3. Can you confirm a suspicion I have in that if a teacher sets up a schedule and assigns students to the slots, the students don't get an email about the appointment being made? That is what we are seeing. Notifications are on, they do get reminder emails, it appears in their calendar, just not getting an initial booking email.

Thanks.

Glenn.
• Fri, May 17, 2019, 6:07 PM
@Glenn Gidden: That is correct. No e-mail is supposed to go out in this case. E-mail notifications are sent: (a) to the teacher, when a student books or cancels a slot; (b) to the student, when the teacher cancels a slot. See here for details: https://docs.moodle.org/36/en/Scheduler_module#Messages_and_notifications
• Fri, May 17, 2019, 6:09 PM
@Yannick Vrolant: I'm afraid that Scheduler is not designed to do that. It is designed for a situation where teachers offer slots, and students book them.
• Fri, May 17, 2019, 6:58 PM
@Henning Bostelmann: Thanks for confirming that. We tried offering slots for students to book, but found better appointment coverage if teachers booked them in, so a confirmation email in that case would have been good for us. We'll just advise staff/students accordingly.
• Mon, May 20, 2019, 1:04 PM
@Henning Bostelmann: Thank you for your work developing the scheduler. In the Scheduler setup Options: Mode, students can be limited to a specified number of appointments in this scheduler, up to 10. For our situation, "up to 12" would be ideal as it would match the number required in the semester. Is there an edit we can make to the plugin code that would change the upper limit in the drop down menu to 12?
• Tue, May 21, 2019, 12:01 AM
Dear Scheduler users,

for any questions regarding usage, feature requests, bugs, etc. please see the Scheduler forum:

https://moodle.org/mod/forum/view.php?id=7139

I might now be answering them here.

Best wishes
Henning
• Thu, May 21, 2020, 2:42 AM
I keep getting "Guests can't do anything here."
I'm on Moodle 3.8 with the latest Scheduler.
• Wed, Jul 1, 2020, 4:13 AM
Hello and thank you for this great plugin. In this COVID-19 era, we are forced to have online office hours. Currently we are posting Zoom links into the location field. But that requires students and teachers to copy the link and paste in another browser tab. So I tried creating a URL activity and then use the Auto linking Filter. But that still does not work. We are trying to make it as easy as possible for students and teachers to access the Zoom link. Could you update your code on line 783 of the render dot php to $rowdata[] = format_text($slot->location, FORMAT_HTML); That will make auto linking work. That would also avoid us having to make the change each time we update the plugin. Thank you.


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# Tag Info

## Hot answers tagged options

12

We created the SABR model because we realized that (a) option values were nonlinear in the volatility, and (b) volatilities are stochastic. This means that if one had an option (or portfolio of options) which have positive gamma in the volatility dimension, on average we'd make money from fluctuations in the volatility, and we'd lose money with negative vol-...

7

If they were a bank, or insurer, utility etc, then some regulator would likely encourage them do everything that others do, whether they like it or not, or whether it makes any sense. But if no regulator tells them to... and if if they don't have exposure to interest rates beyond 1 year... well, let's look at USD 1 year swap rate, for example, https://fred....

6

This is more of a math question than a quant question. Under Black Scholes dynamics (assuming $r=0$ for simplicity), as everyone knows we have $$C=SN(d_1)-KN(d_2)$$. In this case, we are interested in large negative $d$, since $lnS$ is large and negative. There is an asymptotic series for $N(x)$ whose first term for large negative x is $$N(x)=-\phi(x)/x$$,...

6

Let $P(t, T)$ be the price at time $t$ of a zero-coupon bond with maturity $T$ and unit face value. Consider the pricing of the caplet with payoff $(L(t_1; t_1, t_2)-K)^+$ at time $t_1$, where $0<t_1 < t_2$ and, for $0\le s \le t_1$, \begin{align*} L(s; t_1, t_2) = \frac{1}{t_2-t_1}\left(\frac{P(s, t_1)}{P(s, t_2)}-1\right) \end{align*} is the forward ...

5

In a past life, I was an equity strategist at a sell-side bulge bracket firm. In 2008 (obviously) the bank decided to take a long hard look at the funding costs of its derivatives books. So they appointed an MD in IBD/corporate finance who would obviously lack “skin in that game”, and I was his chosen “research guy” (deliberately not then from a fixed income ...

4

I assume you want to price a derivative product that pays $\int_0^T\ln S_tdt$ at maturity time $T$, from time $t=0$. I'll ignore generalization to time $t$ because it is trivial (split the integral in two, before and after $t$ as you did). The first trick it to do an integration by part on $\ln S_t dt$: $d(t\ln S_t) = \ln S_t dt + \frac{t}{S_t}dS_t + \frac{t}... 3 The problem seems to be that you forgot the mean of the process. If$ds_t = rs_tdt + \sigma dW_t^\mathbb Q$, then the solution of the SDE is given by $$s_T = s_te^{r(T-t)} + \sigma\int_t^Te^{r(T-u)}dW^\mathbb Q_u.$$ Since the last integral is Gaussian, the distribution of the terminal price is given by $$s_T \sim\mathrm N\left(s_te^{r(T-t)}, \frac{\sigma^2}{... 3 If you sell a call option, in the language of vol trading you are short gamma and vega, and long theta. So yes, as the underlying wiggles around, if you keep delta-hedged by trading it you will lose some money. This can alternatively be seen as gamma losses (coming from the second-order move in the underlying, which you have not hedged) or as coming from the ... 2 It would be an Asian strike call option, with the Asianing being computed over some period [0;\tau]. Not a big deal that the average is not computed from 0 to T. It even seems more natural to be done that way in practice. Before \tau, pricing would be similar to your option Asianing until T. For t>=\tau, you already know the value of the Asian ... 2 I would add to the answers here, the concept of margin of error. If your swaptions and options are short dated the effect of discounting them is going to be minimal, for a one year option at 1% OIS you have a 0.99 discount factor, i.e a 1% error in valuation if you substitute a discount factor at 1.00 and ignore discounting. If your models are similarly ... 2 Since dividends and interest rates mature annually and the time of expiry is two years, we can model the option as a two-step binomial tree, structured as the one in figure: Data from OP question: S_0 = 45 current price of the underlying S; \hspace{2.5cm} \sigma = 0.2\;\text{per annum} volatility of S; r = 0.02\;\text{per annum} interest rate; ... 2 Achieving gamma neutrality refers to your whole portfolio situation. If you have a portfolio P made up of n_S shares of stock S, and of n_1, n_2 option calls C_1, C_2 on S (options 1 and 2 differ in strike price or expiration), pursuing a gamma hedging strategy would imply to achieve neutrality on the Delta \Delta_P and Gamma \Gamma_P ... 2 Case I Let us consider a derivative with a payoff H(L(T_{f},T_{S},T_{E})) which is paid at time T_{p}. Note that: T_{f} - LIBOR fixing date; T_{S} - LIBOR start date; T_{E} - LIBOR maturity date; T_{p} - derivative payment date. Also, T_{f}=T_{S}=t_{1} and T_{E}=T_{p}=t_{2} in the question. In your first case H(L(T_{f},T_{S},T_{E}))=(L(T_{... 2 Most, but not all, columns on the typical risk report will double if you double the notional ; and can be also netted for two different trades. Moneyness (how far the is the option from being at the money) is usually expressed as how much the underlying needs to move, so it does not depend on the notional. Market data (implied vol, interest rates, yields, ... 2 To extend @d_797's answer, then this stems from Jensens inequality (see this): For a function V: I \rightarrow \mathbb{R} for I being an interval in \mathbb{R}, then V is convex if it satisfies:$$ V(S_1 \lambda + (1-\lambda)S_2) \leq \lambda V(S_1)+(1-\lambda)V(S_2)$$for any two points$S_1,S_2 \in \mathbb{R}$and$\lambda\in [0,1]$. Now, if$V(\cdot)...

1

I'm not 100% sure, please double-check. I think that both in the ITM and the OTM case (requested), a model-free answer cannot exist. In particular, the rate at which: ITM: $C(S_0) \rightarrow S_0$ as $S_0 \rightarrow \infty$ and OTM: $C(S_0) \rightarrow 0$ as $S_0 \rightarrow 0$ depends on the model-specific risk-neutral transition density $p^Q(S_T, T | ... 1 Edit: the original question didn't specify "model independence" and so the below focuses on the BS framework. Also, i focused on speed of convergence rather than order of convergence: I will try to update my answer with some thoughts on order of convergence later on. Not sure this answers your question, but the speed at which the option prices ... 1 The definition of linear is just the usual definition, it would imply that$V(S) = aS$for some constant$a$(on an interval like$[S_1,S_2]$). The definitions of convexity, concavity are more the "first-principles" definition and are equivalent to conditions related to positivity or negativity of$V''(S)$that one sees in introductory calculus. (... 1 I will try to provide an answer to your questions: In retrospect, I do not believe that there is any industry standard for calculating historical volatility (I could be wrong on that part). As long as you have a consistent (and unbiased) estimator of variance (quadratic variation) you're set. As described in the original article of the Yang-Zhang estimator, ... 1 On an expiration basis, your put protected long underlying makes money above$80 and you have a locked in loss of \$5 below \$75. Note that long underlying plus long put is synthetically equal to a long call. Pretending no carry cost or dividend, your position is the same as buying the \$75 call for \$5 and the P&L is the same as stated above. There's ...

1

I am actually more interested in it from the other perspective. If we have a price shock what is the likely IV change that will affect the options pricing? If you're using Python, I would recommend the Mibian library (http://code.mibian.net/). You can simulate a price shock by increasing the volatility parameter (which is HISTORICAL volatility in this case) ...

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


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# Math Help - Complex number problem

1. ## Complex number problem

for any complex number z prove that - I z I < or equal to Re (z) < or equal to I z I

2. ## Re: Complex number problem

Originally Posted by purpeil
for any complex number z prove that $- | z |\le Re (z) \le |z|$
If each of $a~\&~b$ is a real number: \begin{align*} a^2 &\le a^2+b^2\\|a| &\le \sqrt{a^2+b^2}\end{align*} .

Let $a=\text{Re}(z)~\&~b=\text{Im}(z)$ you are done.


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# South Korea's FX reserves dip to $94.44bn SOUTH KOREA - The South Korea's foreign exchange reserves dipped by$890 million from a month earlier to $94.44 billion as of late Mar 2001, falling for the fourth straight month, the Bank of Korea (BOK) said Apr 2. The central bank attributed the drop to the country's debt payment totaling$690 million to the International Monetary Fund and the dollar's steep rise against the yen and the euro and the yen, which lowered the overall reserves in dollar terms.

The reserves peaked at \$96.3 billion


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# 4.1: Counting via Bijections

It can be hard to figure out how to count the number of outcomes for a particular problem. Sometimes it will be possible to find a different problem, and to prove that the two problems have the same number of outcomes (by finding a bijection between their outcomes). If we can work out how to count the outcomes for the second problem, then we’ve also solved the first problem! This may seem blatantly obvious intuitively, but this technique can provide simple solutions to problems that at first glance seem very difficult.

This technique of counting a set (or the number of outcomes to some problem) indirectly, via a different set or problem, is the bijective technique for counting. We begin with a classic example of this technique.

Example $$\PageIndex{1}$$

How many possible subsets are there, from a set of $$n$$ elements?

Solution

One approach would be to figure out how many $$0$$-element subsets there are, how many $$1$$-element subsets, etc., and add up all of the values we find. This works, but there are so many pieces involved that it is prone to error. Also, the value will not be easy to calculate once $$n$$ gets reasonably large.

Instead, we imagine creating a table. The columns are indexed by the elements of the set, so there are $$n$$ columns. We index the rows by the subsets of our set (one per row). In each entry of the table, we place a $$1$$ if the subset that corresponds to this row contains the element that corresponds to this column. Here’s an example of such a table when the set is $$\{x, y, z\}$$:

 $$x$$ $$y$$ $$z$$ $$0$$ $$0$$ $$0$$ $$0$$ $$\{x\}$$ $$1$$ $$0$$ $$0$$ $$\{y\}$$ $$0$$ $$1$$ $$0$$ $$\{z\}$$ $$0$$ $$0$$ $$1$$ $$\{x,y\}$$ $$1$$ $$1$$ $$0$$ $$\{x,z\}$$ $$1$$ $$0$$ $$1$$ $$\{y,z\}$$ $$0$$ $$1$$ $$1$$ $$\{x,y,z\}$$ $$1$$ $$1$$ $$1$$

As you can see, the pattern of $$1$$s and $$0$$s is different in each row of the table, since the elements of each subset are different. Furthermore, any pattern of $$1$$s and $$0$$s that has length $$3$$, appears in some row of this table.

This is not a coincidence. In general, we can define a bijection between the binary strings of length $$n$$, and the subsets of a set of $$n$$ elements, as follows. We already know by the definition of cardinality, that there is a bijection between our set of $$n$$ elements, and the set $$\{1, . . . , n\}$$, so we’ll actually define a bijection between the subsets of $$\{1, . . . , n\}$$ and the binary strings of length $$n$$. Since the composition of two bijections is a bijection, this will indirectly define a bijection between our original set, and the binary strings of length $$n$$.

Given a subset of $$\{1, . . . , n\}$$, the binary string that corresponds to this subset will be the binary string that has $$1$$s in the $$i^{th}$$ position if and only if $$i$$ is in the subset. This tells us how to determine the binary string from the subset. We can also reverse (invert) the process. Given a binary string of length $$n$$, the corresponding subset of $$\{1, . . . , n\}$$ will be the subset whose elements are the positions of the $$1$$s in the binary string.

Although we haven’t directly proven that this map from subsets to binary strings is both one-to-one and onto, an invertible function must be a bijection, so the fact that we were able to find an inverse function does prove that this map is a bijection. (You should check that you agree that the function we’ve claimed as an inverse really does invert the original function.)

Now, our imaginary table wouldn’t be much use if we actually had to write it out. In order to write it out, we would need to know all of the subsets of our set already; and if we knew them all, we could certainly count them! Fortunately, we do not need to write it out. Instead, we use the bijection we have just defined. Rather than count the number of subsets of an $$n$$-set, we count the number of binary strings of length $$n$$. We can do this using just the multiplication rule! In each position, there is either a zero or a one, so there are $$2$$ choices for each of the $$n$$ positions. Hence, there are $$2^n$$ binary strings of length $$n$$.

We conclude that $$2^n$$ subsets can be formed from a set of cardinality $$n$$.

In some ways, we’ve actually been using this idea pretty much every time we’ve come up with more than one way to solve a problem. Implicitly, finding a different way of thinking of the problem is equivalent to finding a bijection between the solutions to these different approaches.

Example $$\PageIndex{2}$$

How many ways are there to choose ten people from a group of $$30$$ men and $$30$$ women, if the group must include at least one woman?

Solution

Attacking this problem directly will get ugly. We would have to consider separately the cases of including one woman, two women, etc., all the way up to ten women, in our group, and add all of the resulting terms together. Instead, we note that there is an obvious bijection (the identity map) between groups that do include at least one woman, and groups that do not include exactly zero women.

The latter is relatively easy to figure out: there are $$\binom{60}{10}$$ possible groups of ten people that could be chosen from the $$60$$ people. Of these, there are $$\binom{30}{10}$$ groups that include zero women (since the members of any such group must be chosen entirely from the $$30$$ men). Therefore, the number of groups that do not include exactly zero women, is $$\binom{60}{10} - \binom{30}{10}$$.

Thanks to our bijection, we conclude that the number of groups that can be chosen, that will include at least one woman, is also $$\binom{60}{10} - \binom{30}{10}$$.

Exercise $$\PageIndex{1}$$

The following problems should help you in working with the bijective technique for counting.

1. We define a structure that is like a subset, except that any element of the original set may appear $$0$$, $$1$$, or $$2$$ times in the structure. How many of these structures can we form from the set $$\{1, . . . , n\}$$?
2. Find a bijection between the coefficient of $$x^r$$ in $$(1 + x)^n$$, and the number of $$r$$-combinations of an $$n$$-set.
3. Find a bijection between the number of ways in which three different dolls can be put into ten numbered cribs, and the number of ways in which ten Olympic contenders can win the medals in their event.


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[4ede1e]: itpp / base / algebra / svd.h Maximize Restore History

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 /*! * \file * \brief Definitions of Singular Value Decompositions * \author Tony Ottosson * * ------------------------------------------------------------------------- * * Copyright (C) 1995-2010 (see AUTHORS file for a list of contributors) * * This file is part of IT++ - a C++ library of mathematical, signal * processing, speech processing, and communications classes and functions. * * IT++ is free software: you can redistribute it and/or modify it under the * terms of the GNU General Public License as published by the Free Software * Foundation, either version 3 of the License, or (at your option) any * later version. * * IT++ is distributed in the hope that it will be useful, but WITHOUT ANY * WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS * FOR A PARTICULAR PURPOSE. See the GNU General Public License for more * details. * * You should have received a copy of the GNU General Public License along * with IT++. If not, see . * * ------------------------------------------------------------------------- */ #ifndef SVD_H #define SVD_H #include namespace itpp { /*! * \ingroup matrixdecomp * \brief Get singular values \c s of a real matrix \c A using SVD * * This function calculates singular values \f$s\f$ from the SVD * decomposition of a real matrix \f$A\f$. The SVD algorithm computes the * decomposition of a real \f$m \times n\f$ matrix \f$\mathbf{A}\f$ so * that * \f[ * \mathrm{diag}(\mathbf{U}^T \mathbf{A} \mathbf{V}) = \mathbf{s} * = \sigma_1, \ldots, \sigma_p * \f] * where \f$\sigma_1 \geq \sigma_2 \geq \ldots \sigma_p \geq 0\f$ are the * singular values of \f$\mathbf{A}\f$. * Or put differently: * \f[ * \mathbf{A} = \mathbf{U} \mathbf{S} \mathbf{V}^T * \f] * where \f$\mathrm{diag}(\mathbf{S}) = \mathbf{s} \f$ * * \note An external LAPACK library is required by this function. */ bool svd(const mat &A, vec &s); /*! * \ingroup matrixdecomp * \brief Get singular values \c s of a complex matrix \c A using SVD * * This function calculates singular values \f$s\f$ from the SVD * decomposition of a complex matrix \f$A\f$. The SVD algorithm computes * the decomposition of a complex \f$m \times n\f$ matrix \f$\mathbf{A}\f$ * so that * \f[ * \mathrm{diag}(\mathbf{U}^H \mathbf{A} \mathbf{V}) = \mathbf{s} * = \sigma_1, \ldots, \sigma_p * \f] * where \f$\sigma_1 \geq \sigma_2 \geq \ldots \sigma_p \geq 0\f$ * are the singular values of \f$\mathbf{A}\f$. * Or put differently: * \f[ * \mathbf{A} = \mathbf{U} \mathbf{S} \mathbf{V}^H * \f] * where \f$\mathrm{diag}(\mathbf{S}) = \mathbf{s} \f$ * * \note An external LAPACK library is required by this function. */ bool svd(const cmat &A, vec &s); /*! * \ingroup matrixdecomp * \brief Return singular values of a real matrix \c A using SVD * * This function returns singular values from the SVD decomposition * of a real matrix \f$A\f$. The SVD algorithm computes the decomposition * of a real \f$m \times n\f$ matrix \f$\mathbf{A}\f$ so that * \f[ * \mathrm{diag}(\mathbf{U}^T \mathbf{A} \mathbf{V}) = \mathbf{s} * = \sigma_1, \ldots, \sigma_p * \f] * where \f$\sigma_1 \geq \sigma_2 \geq \ldots \sigma_p \geq 0\f$ are the * singular values of \f$\mathbf{A}\f$. * Or put differently: * \f[ * \mathbf{A} = \mathbf{U} \mathbf{S} \mathbf{V}^T * \f] * where \f$\mathrm{diag}(\mathbf{S}) = \mathbf{s} \f$ * * \note An external LAPACK library is required by this function. */ vec svd(const mat &A); /*! * \ingroup matrixdecomp * \brief Return singular values of a complex matrix \c A using SVD * * This function returns singular values from the SVD * decomposition of a complex matrix \f$A\f$. The SVD algorithm computes * the decomposition of a complex \f$m \times n\f$ matrix \f$\mathbf{A}\f$ * so that * \f[ * \mathrm{diag}(\mathbf{U}^H \mathbf{A} \mathbf{V}) = \mathbf{s} * = \sigma_1, \ldots, \sigma_p * \f] * where \f$\sigma_1 \geq \sigma_2 \geq \ldots \sigma_p \geq 0\f$ * are the singular values of \f$\mathbf{A}\f$. * Or put differently: * \f[ * \mathbf{A} = \mathbf{U} \mathbf{S} \mathbf{V}^H * \f] * where \f$\mathrm{diag}(\mathbf{S}) = \mathbf{s} \f$ * * \note An external LAPACK library is required by this function. */ vec svd(const cmat &A); /*! * \ingroup matrixdecomp * \brief Perform Singular Value Decomposition (SVD) of a real matrix \c A * * This function returns two orthonormal matrices \f$U\f$ and \f$V\f$ * and a vector of singular values \f$s\f$. * The SVD algorithm computes the decomposition of a real \f$m \times n\f$ * matrix \f$\mathbf{A}\f$ so that * \f[ * \mathrm{diag}(\mathbf{U}^T \mathbf{A} \mathbf{V}) = \mathbf{s} * = \sigma_1, \ldots, \sigma_p * \f] * where the elements of \f$\mathbf{s}\f$, \f$\sigma_1 \geq \sigma_2 \geq * \ldots \sigma_p \geq 0\f$ are the singular values of \f$\mathbf{A}\f$. * Or put differently: * \f[ * \mathbf{A} = \mathbf{U} \mathbf{S} \mathbf{V}^T * \f] * where \f$\mathrm{diag}(\mathbf{S}) = \mathbf{s} \f$ * * \note An external LAPACK library is required by this function. */ bool svd(const mat &A, mat &U, vec &s, mat &V); /*! * \ingroup matrixdecomp * \brief Perform Singular Value Decomposition (SVD) of a complex matrix \c A * * This function returns two orthonormal matrices \f$U\f$ and \f$V\f$ * and a vector of singular values \f$s\f$. * The SVD algorithm computes the decomposition of a complex \f$m \times n\f$ * matrix \f$\mathbf{A}\f$ so that * \f[ * \mathrm{diag}(\mathbf{U}^H \mathbf{A} \mathbf{V}) = \mathbf{s} * = \sigma_1, \ldots, \sigma_p * \f] * where the elements of \f$\mathbf{s}\f$, \f$\sigma_1 \geq \sigma_2 \geq * \ldots \sigma_p \geq 0\f$ are the singular values of \f$\mathbf{A}\f$. * Or put differently: * \f[ * \mathbf{A} = \mathbf{U} \mathbf{S} \mathbf{V}^H * \f] * where \f$\mathrm{diag}(\mathbf{S}) = \mathbf{s} \f$ * * \note An external LAPACK library is required by this function. */ bool svd(const cmat &A, cmat &U, vec &s, cmat &V); } // namespace itpp #endif // #ifndef SVD_H


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Exceptional Lie algebra

In mathematics, an exceptional Lie algebra is a complex simple Lie algebra whose Dynkin diagram is of exceptional (nonclassical) type.[1] There are exactly five of them: ${\displaystyle {\mathfrak {g}}_{2},{\mathfrak {f}}_{4},{\mathfrak {e}}_{6},{\mathfrak {e}}_{7},{\mathfrak {e}}_{8}}$; their respective dimensions are 14, 52, 78, 133, 248.[2] The corresponding diagrams are:[3]

In contrast, simple Lie algebras that are not exceptional are called classical Lie algebras (there are infinitely many of them).

Construction

There is no simple universally accepted way to construct exceptional Lie algebras; in fact, they were discovered only in the process of the classification program. Here are some constructions:

• § 22.1-2 of (Fulton & Harris 1991) give a detailed construction of ${\displaystyle {\mathfrak {g}}_{2}}$ .
• Exceptional Lie algebras may be realized as the derivation algebras of appropriate nonassociative algebras.
• Construct ${\displaystyle {\mathfrak {e}}_{8}}$  first and then find ${\displaystyle {\mathfrak {e}}_{6},{\mathfrak {e}}_{7}}$  as subalgebras.
• Tits has given a uniformed construction of the five exception Lie algebras.[citation needed]

References

1. ^ Fulton & Harris, Theorem 9.26.
2. ^ Knapp, Appendix C, § 2.
3. ^ Fulton & Harris, § 21.2.
• Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103.
• N. Jacobson, "Exceptional Lie algebras" (1971)


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11,357
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Madore's $$\psi$$ function is an ordinal collapsing function introduced by David Madore (under the alias "Gro-Tsen" on Wikipedia)[1].

## Definition

Madore's $$\psi$$ function is defined as follows:

Let $$\omega$$ be the smallest transfinite ordinal and $$\Omega$$ be the smallest uncountable ordinal. Then,

$$C_0(\alpha) = \{0, 1, \omega, \Omega\}$$

$$C_{n+1}(\alpha) = \{\gamma + \delta, \gamma\delta, \gamma^{\delta}, \psi(\eta) | \gamma, \delta, \eta \in C_n (\alpha); \eta < \alpha\}$$

$$C(\alpha) = \bigcup_{n < \omega} C_n (\alpha)$$

$$\psi(\alpha) = \min\{\beta < \Omega|\beta \notin C(\alpha)\}$$

In other words, $$\psi(\alpha)$$ is the least ordinal number less than $$\Omega$$ which cannot be generated from the ordinals $$0, 1, \omega, \Omega$$ using any finite combination of addition, multiplication, exponentiation, or the $$\psi$$ function itself (with the restriction of applying $$\psi(\eta)$$ only for ordinals $$\eta < \alpha$$.

The limit of this function is the Bachmann-Howard ordinal.

## Examples

• $$\psi(0)=$$$$\varepsilon_0$$
• $$\psi(1)=\varepsilon_1$$
• $$\psi(2)=\varepsilon_2$$
• $$\psi(\omega)=\varepsilon_\omega$$
• $$\psi(\psi(0))=\psi(\varepsilon_0)=\varepsilon_{\varepsilon_0}$$
• $$\psi(\psi(\psi(0)))=\psi(\psi(\varepsilon_0))=\psi(\varepsilon_{\varepsilon_0})=\varepsilon_{\varepsilon_{\varepsilon_0}}$$
• $$\psi(\zeta_0)=$$$$\zeta_0$$
• $$\psi(\zeta_0+1)=\zeta_0$$
• $$\psi(\eta_0)=\zeta_0$$
• $$\psi(\Omega)=\zeta_0$$
• $$\psi(\Omega+1)=\varepsilon_{\zeta_0+1}$$
• $$\psi(\Omega+2)=\varepsilon_{\zeta_0+2}$$
• $$\psi(\Omega+\omega)=\varepsilon_{\zeta_0+\omega}$$
• $$\psi(\Omega+\zeta_0)=\varepsilon_{\zeta_02}$$
• $$\psi(\Omega+\zeta_02)=\varepsilon_{\zeta_03}$$
• $$\psi(\Omega+\zeta_0\omega)=\varepsilon_{\zeta_0\omega}$$
• $$\psi(\Omega+\zeta_0^2)=\varepsilon_{\zeta_0^2}$$
• $$\psi(\Omega+\varepsilon_{\zeta_0+1})=\varepsilon_{\varepsilon_{\zeta_0+1}}$$
• $$\psi(\Omega+\varepsilon_{\varepsilon_{\zeta_0+1}})=\varepsilon_{\varepsilon_{\varepsilon_{\zeta_0+1}}}$$
• $$\psi(\Omega+\zeta_1)=\zeta_1$$
• $$\psi(\Omega+\zeta_1+1)=\zeta_1$$
• $$\psi(\Omega2)=\zeta_1$$
• $$\psi(\Omega2+1)=\varepsilon_{\zeta_1+1}$$

## Fundamental sequences

Now we assign a fundamental sequence for each limit ordinal below the Bachmann-Howard ordinal. The fundamental sequence for an ordinal number $$\alpha$$ with cofinality $$\beta$$ is a strictly increasing sequence $$(\alpha[\eta])_{\eta<\beta}$$ with a length of $$\beta$$ and whose limit is $$\alpha$$, where $$\alpha[\eta]$$ is the $$\eta$$-th element of the sequence. If $$\alpha$$ is a countable limit ordinal (i.e. $$\alpha$$ is a limit ordinal less than $$\Omega$$) then $$\text{cof}(\alpha)=\omega$$. The first uncountable ordinal $$\Omega$$ is the least ordinal whose cofinality greater than $$\omega$$ since $$\text{cof}(\Omega)=\Omega$$.

First we must define the normal forms for expressions given by all relevant functions:

$$\alpha=_{NF}\alpha_1+\alpha_2+\cdots+\alpha_n$$ iff $$\alpha=\alpha_1+\alpha_2+\cdots+\alpha_n$$ and $$\alpha_1\geq\alpha_2\geq\cdots\geq\alpha_n$$

$$\alpha=_{NF}\omega^\beta$$ iff $$\alpha=\omega^\beta$$ for some $$\beta<\alpha$$

$$\alpha=_{NF}\Omega^\beta\gamma$$ iff $$\alpha=\Omega^\beta\gamma$$ for some $$\gamma<\Omega$$

$$\alpha=_{NF}\psi(\beta)$$ iff $$\alpha=\psi(\beta)$$ for some $$\beta\in C(\beta)$$

Then for these limit ordinals, when written in normal form, we can assign the fundamental sequences as follows:

1) if $$\alpha=\alpha_1+\alpha_2+\cdots+\alpha_n$$ then $$\text{cof} (\alpha)= \text{cof} (\alpha_n)$$ and $$\alpha[\eta]=\alpha_1+\alpha_2+\cdots+(\alpha_n[\eta])$$

2) if $$\alpha=\omega^\beta$$ and $$\beta$$ is a countable limit ordinal then $$\alpha[n]=\omega^{\beta[n]}$$

3) if $$\alpha=\omega^\beta$$ and $$\beta=\gamma+1$$ then $$\alpha[n]=\omega^\gamma n$$

4) if $$\alpha=\psi(0)$$ then $$\alpha[0]=1$$ and $$\alpha[n+1]=\omega^{\alpha[n]}$$

5) if $$\alpha=\psi(\beta+1)$$ then $$\alpha[0]=\psi(\beta)+1$$ and $$\alpha[n+1]=\omega^{\alpha[n]}$$

6) if $$\alpha=\Omega^{\beta}\gamma$$ and $$\text{cof} (\gamma)=\omega$$ then $$\text{cof} (\alpha)= \omega$$ and $$\alpha[\eta]=\Omega^{\beta}(\gamma[\eta])$$

7) if $$\alpha=\Omega^{\beta+1}(\gamma+1)$$ then $$\text{cof} (\alpha)=\Omega$$ and $$\alpha[\eta]=\Omega^{\beta+1}\gamma+\Omega^\beta\eta$$

8) if $$\alpha=\Omega^\beta(\gamma+1)$$ and $$\text{cof}(\beta)\geq\omega$$ then $$\text{cof}(\alpha)= \text{cof}(\beta)$$ and $$\alpha[\eta]=\Omega^\beta\gamma+\Omega^{\beta[\eta]}$$

9) if $$\alpha=\varepsilon_{\Omega+1}$$ then $$\text{cof} (\alpha)=\omega$$ and $$\alpha[0]=1$$ and $$\alpha[n+1]=\Omega^{\alpha[n]}$$

10) if $$\alpha=\psi(\beta)$$ and $$\text{cof}(\beta)=\omega$$ then $$\text{cof} (\alpha)=\omega$$ and $$\alpha[n]=\psi(\beta[n])$$

11) if $$\alpha=\psi(\beta)$$ and $$\text{cof}(\beta)=\Omega$$ then $$\text{cof} (\alpha)=\omega$$ and $$\alpha[0]=1$$ and $$\alpha[n+1]=\psi(\beta[\alpha[n]])$$

For example, the ordinal $$\psi(\Omega^{\Omega^2+\Omega3})$$ has the following fundamental sequence (using rules 1, 7, 8, 10)

$$\psi(\Omega^{\Omega^2+\Omega3})[0]=1$$,

$$\psi(\Omega^{\Omega^2+\Omega3})[1]=\psi(\Omega^{\Omega^2+\Omega2+1})$$,

$$\psi(\Omega^{\Omega^2+\Omega3})[2]=\psi\left(\Omega^{\Omega^2+\Omega2+\psi(\Omega^{\Omega^2+\Omega2+1})}\right)$$,

and so on.

Assigning fundamental sequences for any ordinal collapsing function is vital for its use in the fast-growing hierarchy, slow-growing hierarchy and Hardy hierarchy.

## Ordinal Notation

Unlike other standard ordinal collapsing functions, there seems to be no canonical ordinal notation associated to Madore's function. In order to create a computable large number by using fast-growing hierarchy applied to the system of fundamental sequences defined in set theory, we need to fix an ordinal notation associated to it.

## References

1. "Ordinal Collapsing Function". Wikipedia. Retrieved 2014-08-29.


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# Eighty Three Days Until Exam FM

I scheduled my exam yesterday, for exactly eighty-three days from now.  I am well over the hump with studying.  I can work exam level problems on all of the interest theory material, but I have a long way to go with derivatives markets.  So, my goal until the end of December is to keep working interest theory exam problems, filling in missing bits when I find them, while I pour over the derivatives markets material.  By January I will be taking practice tests every other day, while I review solutions on the off days.

Most recently, I am working with duration and convexity.  This is good, meaty stuff, with many different levels of understanding.  I will be doing a post on it tomorrow.  I will also be posting on how to best arrive at a correct answer on these computationally complex problems, as well as posting some computation exercises.

# The Cards

Cutting them up:

I have owned this paper cutter for 25 years.  At that time, I did not own much but the clothes on my back.  How would I have survived without a paper cutter to make little books and cards and such?

Things that I gained by making these cards:

1. A strengthened knowledge of FM fundamentals.
2. Increased fluency with LaTeX.
3. A creative project which kept me focused on the subject at hand.
4. A nice little pile of finished cards.
5. Time to think about the pluses and minuses of virtual products versus actual.

# Exam FM, Chapter 1 Flashcards

I just posted my nearly final version of the first chapter of flashcards for exam FM.  I added some questions on geometric progressions, and some on force of interest, so now there are 226 cards.  I know that there are still a few holes, but I will worry about that later.  I could probably come up with 300 cards for the first chapter very easily.  But, having put together several sets of study cards, I find that I add fewer cards for subsequent topics.  In almost any subject, the most vital information to commit permanently to memory is at the beginning.

I tried to use a sampling of the different terminologies.  For instance, I used the accumulation function $a(t)$ as well as the FV, PV terminology.  When I look at the solutions for the exam sample questions, I see both types used, so I suppose that it is possible for either to appear in an examination question.

For most formulas, I give several types of numerical examples.  It is important to see the relationships at work.  To see the animals in their native habitat.

In a later post, I will describe how to use these cards to best effect.  I do intend them as a means of permanently learning the material.

# Another Update on Exam FM Flashcards

I updated my Financial Mathematics cards again this morning. The link from a couple of days ago reflects the updates. There are now 190 cards.

The cards are in the form of a PDF file, to be printed on letter size paper (8.5 x 11 in). Many printers do not print the front and back of each page in good alignment, so I made sure not to place text too near the borders of the back of the cards.  Simply print the cards out, cut a 1cm border from around the page, then cut the cards out.  Whoops!  I have mixed inches and centimeters.  I will straighten out the unit issue on the next version, with a 1/2 inch border.

I am trying to really fill out most of the fundamentals.  This makes the cards useful not just for someone studying for the actuary exams, but also for anyone learning the basics of finance.

Let me know how the cards work for you, or if you find any errors.

I have been working hard today.  Here is a PDF of the first draft of my exam FM flashcards.  So far, they just cover “chapter 1” stuff like Interest Theory and Lump Sums.  I think that there are about 110 cards so far.  I will soon be fleshing out the missing Interest Theory topics, then I will add the Annuity cards that I have made.

I am delighted to have used LaTeX to design the cards.  They are 2 cards by 5 cards on a letter size page.  This is a standard card size, which you can buy paper for at the store.  I will add some cut marks when I figure out how.

More tomorrow.

# Update on Card Formatting

Yesterday, I posted this flash card as an example:

When I did my repetitions this morning, I realized that this card, and several similar ones that I created yesterday, are too complicated.  Cards should only test one very small item of knowledge. (Read SuperMemo 20 Rules of Formatting Knowledge)  The cards from yesterday test not only the function of the “a angle n” function, but also require a calculation with this function.  A much better card is as so:

The wording of the card now makes it obvious that I am not looking for a numeric answer.  The purpose of this card is simply to help the mind to recognize and recall a common relationship that involves the present value of the annuity immediate.

# The Annuity Symbol in LaTeX and Anki

As you know, one of my favorite creative projects is to create Anki flashcards using the LaTeX markup language.  I have written about some of the process on this blog.  Originally, I needed to install MiKTeX on my system.  Now, Anki can interpret the language when it sees it.

My recent problem has been to format the symbols for annuities.  That looks like so:

Which is read as “a angle n”, and:

Which is read as “a double dot angle n.”

It gets really annoying to write and interpret things like “a double dot angle n;”  that is where the proper symbols come in.

First, I discovered that there is an \actuarialangle command which has been recently added to the MiKTeX library.  Open the Package Manager on MiKTeX, and you may install it.  To activate it in Anki, you need to use some code, in the usepackage line in the preamble of your cards.  See below.

I also discovered the lifecon package on the web.  This contains all sorts of actuarial symbols, but you need to know how to add a package to MiKTeX directly with a .sty file.  I had a little trouble with this task.

The alternative to both of the above solutions is to add some code into the preamble of your document, that defines a new LaTeX command.  There are several versions of this floating around.  The following is what I used:

\DeclareRobustCommand{\lcroof}[1]{
\hbox{\vtop{\vbox{%
\hrule\kern 1pt\hbox{%
$\scriptstyle #1$%
\kern 1pt}}\kern1pt}%
\vrule\kern1pt}}
\DeclareRobustCommand{\angle}[1]{
_{\lcroof{#1}}}

In Anki:

1. Hit the button to add a note.
2. Hit the button to choose the note type.
3. Select “Options”
4. Cut and paste the above code into the header, just under “\usepackage{amssymb,amsmath}”
5. Now you may use the \angle command in your cards.

Here is an example:

I put things on cards that I figure I should be able to solve nearly instantaneously.  The notes on the bottom of the card are to refresh my mind, if I have forgotten.  The numerical answer is not really very important, the real thing is reading the problem, and visualizing a solution.  Real test problems, and real life problems, are much more complex than the ones I put on my cards.  But the component parts need to be solved quickly, with confidence.

I am so happy now that I have annuity symbols appearing correctly.  I think that I will go make some flashcards.

# What worked, What didn’t

Studying the Easy Stuff

I didn’t spend enough time with the “easy” stuff. On the exam, the problems surrounding the algebra of events were real chestnuts. Likewise, the first time that I took the exam I encountered an “easy” normal curve problem with a clever twist of algebra. This time, there were two normal distribution problems with the same difficult twist. Study the easy stuff.

One and a Half Years of Study

Actually, two winters worth of studying. My first winter of studying suffered from a lack of study materials. The only book that I had was the free Finan book, plus whatever statistics books were on my shelf (Hogg and Tanis, Casella and Berger). I mostly worked problems from the Online Math Tests Home Page. This was not nearly enough of a variety of problems. Also, my first crack at studying required much reviewing of calculus topics.

The second winter worth of studying was more varied, and more productive. For me, one and a half years was just the right amount of study time. (During this same period, I also bought and renovated a dilapidated house, worked lots of 55 hour weeks at my job, had major surgery, and was reasonably available for my friends, family, pets, and spouse.) For exam FM, I am planning 1 year.

Actex Manual (Broverman)

The exercises in this book are difficult and varied. I will get the Actex manual for FM.

Guo Book

This book provided me with some variety of material when I needed it, and got me thinking about highly efficient solutions. I may get the book for FM.

Finan Book

This book was a good start for me. There are no worked solutions for the exercises, which can become frustrating. I am now started on the book for FM.

Coaching Actuaries

This was a great help at the last minute, in getting a feel for the real test. For the next exam, I am going to get a one month subscription, two months before my exam.

Having Some Background with the Material

Thinking that I had some sort of familiarity with probability was a handicap for the first exam. It encouraged me to not be as thorough with the “easy” stuff as I should have been.

Fortunately, for exam FM/2, I will not have the same problem. I have no background with the material, so I am starting everything from the beginning.

Memorization

My memory work deserves some real thought, and a post of its own, to follow later.

In short, I am glad that I put as much time as I did into memory work, although it is hard to place its role in passing the exam. The biggest payoff is in the work that I performed in turning complex topics into little digestible examples. As a result, the elementary distributions now have a permanent place in my consciousness.

Early Mornings

Early mornings work for me. I stuck with a wake up time of 3:30 am, or at latest 4:30 am, for more than 6 months. I had time each mornings to complete the most important study tasks, so any additional study time after work was bonus study time. I could study in the afternoons in the coffee shop, and not worry if I lost some study time in socializing

# Sorry, More Technical Stuff

Currently, I am coming to grips with the law of total variance.

In words, the variance of X equals the variance of the expected value of X, given Y, plus the expected value of the variance of X, given Y.

In symbols, Var(X) = Var(E(X|Y)) + E(Var(X|Y).

By reading the verbal definition, one can see that the logic is convoluted. Given two probability distributions, it can be tricky to see how to apply the law. Once the law is applied, there are additional tricky steps of logic involving the independence of variables.

From the SOA/CAS sample problems:

A motorist makes 3 driving errors, each independently resulting in an accident with probability 0.25.  Each accident results in a loss that is exponentially distributed with mean 0.80.  Losses are mutually independent and independent of the number of accidents.  The motorists insurer reimburses 70% of each loss.  Find the variance of the total unreimbursed loss.

The above item illustrates the technical problem that I have been having when using Spaced Repetition Software to schedule repetitions of complex material.

When I encounter a difficult item, I spend time exploring the given solution, and alternate solutions. Eventually, I move on to another item. Most likely, I am not entirely comfortable with the material, and I would like to see the same material every day for a while, to approach it with different solutions. The way that the defaults on SRS software are set up, as soon as you start rating an item anything other than the most difficult setting, the item starts getting pushed off way into the future. After a week or two weeks, I have entirely forgotten many of the finer details of the item, and it is almost as if I am starting from scratch.  For difficult material, it is beneficial to see the same items every day, or every 2nd or 3rd day.

The solution to this problem is to reset the forgetting index. In Anki, I have now set my forgetting index to 3%. This task is done by downloading the shared “Forgetting Index” plugin. In the “File” menu, select “Download Shared Plugin”, and find the forgetting index plugin.

The result is that I can now look at problems on a nearly daily basis before they start whizzing into the future. Each time I look at the problem with fresh eyes, I observe new things.

In the last week, I finally have my Spaced Repetition Software working to its utmost. I use it to schedule repetitions of difficult exam problems, and of memorization items. These are two fundamentally different tasks, so it is important to set up Anki (the software that I am now using), in two different ways.  After spending time every day for a week with the above “unreimbursed loss” problem, I look at it, and think “piece of cake.”  Now, when I encounter other law of total variance situations, I have several good comparison problems stashed away in my head.

# Tons of Problems

68 more days until SOA/CAS Exam P/1.  Although I got up late today, I still managed to work tons of problems.  My goal until test time is to do at least 30 problems each day.

I have quickly realized that using spaced repetition for difficult problems is an entirely different process from using it for simple memorization items.  If I am having any trouble at all in remembering the flow of logic of the solution, I need to repeat the problem daily, or even twice daily, until I am really sure that I understand every little twist.  Then, I start rating the item so that it comes up on a spaced repetition schedule.  It is the old truism of SRS:  you can’t learn it until you understand it:

1. Do not learn if you do not understandTrying to learn things you do not understand may seem like an utmost nonsense. Still, an amazing proportion of students commit the offence of learning without comprehension. Very often they have no other choice! The quality of many textbooks or lecture scripts is deplorable while examination deadlines are unmovable.

If you are not a speaker of German, it is still possible to learn a history textbook in German. The book can be crammed word for word. However, the time needed for such “blind learning” is astronomical. Even more important: The value of such knowledge is negligible. If you cram a German book on history, you will still know nothing of history.

The German history book example is an extreme. However, the materials you learn may often seem well structured and you may tend to blame yourself for lack of comprehension. Soon you may pollute your learning process with a great deal of useless material that treacherously makes you believe “it will be useful some day”.

2. Learn before you memorizeBefore you proceed with memorizing individual facts and rules, you need to build an overall picture of the learned knowledge. Only when individual pieces fit to build a single coherent structure, will you be able to dramatically reduce the learning time. This is closely related to the problem comprehension mentioned in Rule 1: Do not learn if you do not understand. A single separated piece of your picture is like a single German word in the textbook of history.

Taken from the SuperMemo website.

Using SRS for studying difficult problems violates several of the other principles of SRS, but it still a simple way to schedule the problems, and to monitor progress.

More tomorrow 🙂


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# zbMATH — the first resource for mathematics

## Tesi, Alberto

Compute Distance To:
 Author ID: tesi.alberto Published as: Tesi, A.; Tesi, Alberto
 Documents Indexed: 79 Publications since 1988, including 1 Book 1 Contribution as Editor
all top 5

#### Co-Authors

 0 single-authored 38 Vicino, Antonio 29 Genesio, Roberto 13 Chesi, Graziano 13 Garulli, Andrea 9 Forti, Mauro 8 Basso, Michele 7 Di Marco, Mauro 5 Zappa, Giovanni 4 Bianchini, Gianni 4 Innocenti, Giacomo 4 Villoresi, Francesca 3 Battistelli, Giorgio 3 Giovanardi, Lorenzo 3 Selvi, Daniela 2 de Angeli, Andrea 2 Torrini, G. 1 Abed, Eyad H. 1 Bagni, Giorgio Tomaso 1 Bartlett, Andrew C. 1 Evangelisti, A. 1 Falugi, Paola 1 Johnson, Russell Allan 1 Mari, Daniele 1 Milanese, Mario 1 Mocenni, Chiara 1 Nistri, Paolo 1 Pagni, P. J. 1 Parenti, Alessandro 1 Tesi, Pietro 1 Wang, Hua O.
all top 5

#### Serials

 16 IEEE Transactions on Automatic Control 13 International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 10 Automatica 8 Systems & Control Letters 4 IEEE Transactions on Circuits and Systems. I: Fundamental Theory and Applications 2 Kybernetika 2 International Journal of Robust and Nonlinear Control 2 Nonlinear Dynamics 2 Lecture Notes in Control and Information Sciences 1 International Journal of Control 1 Chaos, Solitons and Fractals 1 IEEE Transactions on Circuits and Systems 1 SIAM Journal on Control and Optimization 1 International Journal of Adaptive Control and Signal Processing 1 IEE Proceedings. Part D. Control Theory and Applications 1 Bollettino della Unione Matematica Italiana. Serie VIII. Sezione B. Articoli di Ricerca Matematica 1 Dynamics of Continuous, Discrete & Impulsive Systems. Series B. Applications & Algorithms 1 Journal of Applied Mathematics 1 IEEE Transactions on Circuits and Systems I: Regular Papers
all top 5

#### Fields

 62 Systems theory; control (93-XX) 21 Ordinary differential equations (34-XX) 18 Dynamical systems and ergodic theory (37-XX) 8 Linear and multilinear algebra; matrix theory (15-XX) 8 Biology and other natural sciences (92-XX) 6 Information and communication theory, circuits (94-XX) 4 Operations research, mathematical programming (90-XX) 2 Functions of a complex variable (30-XX) 2 Mechanics of particles and systems (70-XX) 1 General and overarching topics; collections (00-XX) 1 Numerical analysis (65-XX) 1 Computer science (68-XX) 1 Statistical mechanics, structure of matter (82-XX) 1 Geophysics (86-XX)

#### Citations contained in zbMATH Open

63 Publications have been cited 940 times in 727 Documents Cited by Year
New conditions for global stability of neural networks with application to linear and quadratic programming problems. Zbl 0849.68105
Forti, Mauro; Tesi, Alberto
1995
Harmonic balance methods for the analysis of chaotic dynamics in nonlinear systems. Zbl 0765.93030
Genesio, R.; Tesi, A.
1992
Polynomially parameter-dependent Lyapunov functions for robust stability of polytopic systems: an LMI approach. Zbl 1365.93365
Chesi, G.; Garulli, A.; Tesi, A.; Vicino, A.
2005
Harmonic balance analysis of period-doubling bifurcations with implications for control of nonlinear dynamics. Zbl 0859.93028
Tesi, A.; Abed, E. H.; Genesio, R.; Wang, H. O.
1996
Homogeneous polynomial forms for robustness analysis of uncertain systems. Zbl 1218.93002
Chesi, Graziano; Garulli, Andrea; Tesi, Alberto; Vicino, Antonio
2009
Homogeneous Lyapunov functions for systems with structured uncertainties. Zbl 1079.93036
Chesi, G.; Garulli, A.; Tesi, A.; Vicino, A.
2003
Robust stability of time-varying polytopic systems via parameter-dependent homogeneous Lyapunov functions. Zbl 1111.93064
Chesi, G.; Garulli, A.; Tesi, A.; Vicino, A.
2007
Solving quadratic distance problems: an LMI-based approach. Zbl 1364.90240
Chesi, Graziano; Garulli, Andrea; Tesi, Alberto; Vicino, Antonio
2003
A convex approach to a class of miminum norm problems. Zbl 0934.93028
Chesi, Graziano; Tesi, Alberto; Vicino, Antonio; Genesio, Roberto
1999
LMI-based computation of optimal quadratic Lyapunov functions for odd polynomial systems. Zbl 1056.93059
Chesi, G.; Garulli, A.; Tesi, A.; Vicino, A.
2005
A frequency method for predicting limit cycle bifurcations. Zbl 0890.34033
Basso, M.; Genesio, R.; Tesi, A.
1997
Characterizing the solution set of polynomial systems in terms of homogeneous forms: An LMI approach. Zbl 1039.93021
Chesi, G.; Garulli, A.; Tesi, A.; Vicino, A.
2003
Chaos prediction in nonlinear feedback systems. Zbl 0754.93024
Genesio, R.; Tesi, A.
1991
A frequency approach for analyzing and controlling chaos in nonlinear circuits. Zbl 0848.93029
Genesio, Roberto; Tesi, Alberto; Villoresi, Francesca
1993
Computation of nonconservative stability perturbation bounds for systems with nonlinearly correlated uncertainties. Zbl 0714.93035
Vicino, A.; Tesi, A.; Milanese, M.
1990
An overview of extremal properties for robust control of interval plants. Zbl 0771.93064
Dahleh, M.; Tesi, A.; Vicino, A.
1993
Robust absolute stability of Lur’e control systems in parameter space. Zbl 0732.93062
Tesi, A.; Vicino, A.
1991
Robust stability of state-space models with structured uncertainties. Zbl 0705.93061
Tesi, A.; Vicino, A.
1990
Bifurcations and oscillatory behavior in a class of competitive cellular neural networks. Zbl 1090.37565
Di Marco, M.; Tesi, A.; Forti, M.
2000
A harmonic balance approach for chaos prediction: Chua’s circuit. Zbl 0874.94042
Genesio, R.; Tesi, A.
1992
Robustness in identification and control. Workshop, Siena, Italy, July 30 - August 2, 1998. Zbl 0918.00041
1999
On the computation of characteristic multipliers for predicting limit cycle bifurcations. Zbl 0940.34023
Torrini, G.; Genesio, R.; Tesi, A.
1998
Integrating identification and qualitative analysis for the dynamic model of a lagoon. Zbl 1058.92044
Garulli, A.; Mocenni, C.; Vicino, A.; Tesi, A.
2003
On the robust Popov criterion for interval Lur’e system. Zbl 0787.93077
Dahleh, M.; Tesi, A.; Vicino, A.
1993
A new method to analyze complete stability of PWL cellular neural networks. Zbl 1090.37567
Forti, M.; Tesi, A.
2001
Robust analysis of LFR systems through homogeneous polynomial Lyapunov functions. Zbl 1365.93364
Chesi, G.; Garulli, A.; Tesi, A.; Vicino, A.
2004
On bifurcation control in time delay feedback systems. Zbl 0938.93542
Basso, M.; Evangelisti, A.; Genesio, R.; Tesi, A.
1998
Frequency response of uncertain systems with interval plants. Zbl 0786.93019
Bartlett, Andrew C.; Tesi, Alberto; Vicino, Antonio
1993
An LMI-based technique for robust stability analysis of linear systems with polynomial parametric uncertainties. Zbl 1138.93393
Chesi, Graziano; Garulli, Andrea; Tesi, Alberto; Vicino, Antonio
2005
The Łojasiewicz exponent at an equilibrium point of a standard CNN is $$1/2$$. Zbl 1192.37116
Forti, Mauro; Tesi, Alberto
2006
Results on the stability robustness of systems with state space perturbations. Zbl 0645.93045
Genesio, R.; Tesi, A.
1988
Robustness analysis for linear dynamical systems with linearly correlated parametric uncertainties. Zbl 0704.93062
Tesi, A.; Vicino, A.
1990
On the stability domain estimation via a quadratic Lyapunov function: Convexity and optimality properties for polynomial systems. Zbl 0870.34057
Tesi, A.; Villoresi, F.; Genesio, R.
1996
Clockwise property of the Nyquist plot with implications for absolute stability. Zbl 0746.93067
Tesi, A.; Vicino, A.; Zappa, Giovanni
1992
Kharitonov segments suffice for frequency response analysis of interval plant-controller families. Zbl 0837.93053
Tesi, A.; Vicino, A.
1992
Models of complex dynamics in nonlinear systems. Zbl 0877.93002
Genesio, Roberto; Tesi, Alberto; Villoresi, Francesca
1995
On system decomposition for synchronizing chaos. Zbl 0875.94003
Tesi, Alberto; de Angeli, Andrea; Genesio, Roberto
1994
Complex behavior analysis in quadratic jerk systems via frequency domain Hopf bifurcation. Zbl 1193.34081
Innocenti, G.; Tesi, A.; Genesio, R.
2010
On optimal stabilization of periodic orbits via time delayed feedback control. Zbl 0939.93014
Basso, M.; Genesio, R.; Giovanardi, L.; Tesi, A.; Torrini, G.
1998
Absolute stability of analytic neural networks: an approach based on finite trajectory length. Zbl 1371.93151
Forti, M.; Tesi, A.
2004
A new fast algorithm for robust stability analysis of control systems with linearly dependent parametric uncertainties. Zbl 0684.93026
Tesi, A.; Vicino, A.
1989
Global $$H_{\infty}$$ controllers for a class of nonlinear systems. Zbl 1365.93130
Bianchini, Gianni; Genesio, Roberto; Parenti, Alessandro; Tesi, Alberto
2004
On the $$D$$-stability problem for real matrices. Zbl 0936.15014
Johnson, Russell; Tesi, Alberto
1999
Controller design for extending periodic dynamics of a chaotic $$\text{CO}_{2}$$ laser. Zbl 0901.93032
Basso, M.; Genesio, R.; Tesi, A.
1997
Robust stability and performance of interval plants. Zbl 0763.93067
Dahleh, M.; Tesi, A.; Vicino, A.
1992
A convex approach to the characterization of the frequency response of ellipsoidal plants. Zbl 0991.93027
Chesi, G.; Garulli, A.; Tesi, A.; Vicino, A.
2002
Synthesis of robust strictly positive real systems with $$l_2$$ parametric uncertainty. Zbl 1011.93086
Bianchini, Gianni; Tesi, Alberto; Vicino, Antonio
2001
Enhancing strict positive realness condition on families of polynomials by filter design. Zbl 0848.93038
Tesi, Alberto; Zappa, Giovanni; Vicino, Antonio
1993
Control techniques for chaotic dynamical systems. Zbl 0809.34058
Genesio, Roberto; Tesi, Alberto
1993
Convexity properties of polynomials with assigned root location. Zbl 0814.93053
Tesi, Alberto; Vicino, Antonio; Zappa, Giovanni
1994
An LMI approach to constrained optimization with homogeneous forms. Zbl 0974.93022
Chesi, G.; Tesi, A.; Vicino, A.; Genesio, R.
2001
The output stabilization of SISO bilinear systems. Zbl 0655.93054
Genesio, R.; Tesi, A.
1988
Adaptive disturbance attenuation via logic-based switching. Zbl 1297.93103
Battistelli, Giorgio; Mari, Daniele; Selvi, Daniela; Tesi, Alberto; Tesi, Pietro
2014
An LMI-based controller synthesis for periodic trajectories in a class of nonlinear systems. Zbl 1364.93253
Basso, Michele; Genesio, Roberto; Tesi, Alberto
2002
On absolute stability of convergence for nonlinear neural network models. Zbl 1203.93186
Di Marco, Mauro; Forti, Mauro; Tesi, Alberto
2003
Secure communications based on discrete time chaotic systems. Zbl 1043.94503
Argenti, F.; De Angeli, A.; Del Re, E.; Genesio, R.; Pagni, P.; Tesi, A.
1997
Robust strict prositive realness: New results for interval plant- controller families. Zbl 0786.93076
Vicino, Antonio; Tesi, Alberto
1993
Design criteria for robust strict positive realness in adaptive schemes. Zbl 0808.93030
Tesi, A.; Vicino, A.; Zappa, Giovanni
1994
On the effect of neuron activation gain on robustness of complete stability. Zbl 1065.92002
Di Marco, M.; Forti, M.; Nistri, P.; Tesi, A.
2004
Robust switching control: stability analysis and application to active disturbance attenuation. Zbl 1390.93413
Battistelli, Giorgio; Selvi, Daniela; Tesi, Alberto
2017
Feedback of SISO bilinear systems. Zbl 0651.93034
Genesio, R.; Tesi, A.
1988
On limit cycles in feedback polynomial systems. Zbl 0662.93053
Genesio, R.; Tesi, A.
1988
Robustness bounds for classes of structured perturbations. Zbl 0727.93047
Vicino, A.; Tesi, A.
1989
Robust switching control: stability analysis and application to active disturbance attenuation. Zbl 1390.93413
Battistelli, Giorgio; Selvi, Daniela; Tesi, Alberto
2017
Adaptive disturbance attenuation via logic-based switching. Zbl 1297.93103
Battistelli, Giorgio; Mari, Daniele; Selvi, Daniela; Tesi, Alberto; Tesi, Pietro
2014
Complex behavior analysis in quadratic jerk systems via frequency domain Hopf bifurcation. Zbl 1193.34081
Innocenti, G.; Tesi, A.; Genesio, R.
2010
Homogeneous polynomial forms for robustness analysis of uncertain systems. Zbl 1218.93002
Chesi, Graziano; Garulli, Andrea; Tesi, Alberto; Vicino, Antonio
2009
Robust stability of time-varying polytopic systems via parameter-dependent homogeneous Lyapunov functions. Zbl 1111.93064
Chesi, G.; Garulli, A.; Tesi, A.; Vicino, A.
2007
The Łojasiewicz exponent at an equilibrium point of a standard CNN is $$1/2$$. Zbl 1192.37116
Forti, Mauro; Tesi, Alberto
2006
Polynomially parameter-dependent Lyapunov functions for robust stability of polytopic systems: an LMI approach. Zbl 1365.93365
Chesi, G.; Garulli, A.; Tesi, A.; Vicino, A.
2005
LMI-based computation of optimal quadratic Lyapunov functions for odd polynomial systems. Zbl 1056.93059
Chesi, G.; Garulli, A.; Tesi, A.; Vicino, A.
2005
An LMI-based technique for robust stability analysis of linear systems with polynomial parametric uncertainties. Zbl 1138.93393
Chesi, Graziano; Garulli, Andrea; Tesi, Alberto; Vicino, Antonio
2005
Robust analysis of LFR systems through homogeneous polynomial Lyapunov functions. Zbl 1365.93364
Chesi, G.; Garulli, A.; Tesi, A.; Vicino, A.
2004
Absolute stability of analytic neural networks: an approach based on finite trajectory length. Zbl 1371.93151
Forti, M.; Tesi, A.
2004
Global $$H_{\infty}$$ controllers for a class of nonlinear systems. Zbl 1365.93130
Bianchini, Gianni; Genesio, Roberto; Parenti, Alessandro; Tesi, Alberto
2004
On the effect of neuron activation gain on robustness of complete stability. Zbl 1065.92002
Di Marco, M.; Forti, M.; Nistri, P.; Tesi, A.
2004
Homogeneous Lyapunov functions for systems with structured uncertainties. Zbl 1079.93036
Chesi, G.; Garulli, A.; Tesi, A.; Vicino, A.
2003
Solving quadratic distance problems: an LMI-based approach. Zbl 1364.90240
Chesi, Graziano; Garulli, Andrea; Tesi, Alberto; Vicino, Antonio
2003
Characterizing the solution set of polynomial systems in terms of homogeneous forms: An LMI approach. Zbl 1039.93021
Chesi, G.; Garulli, A.; Tesi, A.; Vicino, A.
2003
Integrating identification and qualitative analysis for the dynamic model of a lagoon. Zbl 1058.92044
Garulli, A.; Mocenni, C.; Vicino, A.; Tesi, A.
2003
On absolute stability of convergence for nonlinear neural network models. Zbl 1203.93186
Di Marco, Mauro; Forti, Mauro; Tesi, Alberto
2003
A convex approach to the characterization of the frequency response of ellipsoidal plants. Zbl 0991.93027
Chesi, G.; Garulli, A.; Tesi, A.; Vicino, A.
2002
An LMI-based controller synthesis for periodic trajectories in a class of nonlinear systems. Zbl 1364.93253
Basso, Michele; Genesio, Roberto; Tesi, Alberto
2002
A new method to analyze complete stability of PWL cellular neural networks. Zbl 1090.37567
Forti, M.; Tesi, A.
2001
Synthesis of robust strictly positive real systems with $$l_2$$ parametric uncertainty. Zbl 1011.93086
Bianchini, Gianni; Tesi, Alberto; Vicino, Antonio
2001
An LMI approach to constrained optimization with homogeneous forms. Zbl 0974.93022
Chesi, G.; Tesi, A.; Vicino, A.; Genesio, R.
2001
Bifurcations and oscillatory behavior in a class of competitive cellular neural networks. Zbl 1090.37565
Di Marco, M.; Tesi, A.; Forti, M.
2000
A convex approach to a class of miminum norm problems. Zbl 0934.93028
Chesi, Graziano; Tesi, Alberto; Vicino, Antonio; Genesio, Roberto
1999
Robustness in identification and control. Workshop, Siena, Italy, July 30 - August 2, 1998. Zbl 0918.00041
1999
On the $$D$$-stability problem for real matrices. Zbl 0936.15014
Johnson, Russell; Tesi, Alberto
1999
On the computation of characteristic multipliers for predicting limit cycle bifurcations. Zbl 0940.34023
Torrini, G.; Genesio, R.; Tesi, A.
1998
On bifurcation control in time delay feedback systems. Zbl 0938.93542
Basso, M.; Evangelisti, A.; Genesio, R.; Tesi, A.
1998
On optimal stabilization of periodic orbits via time delayed feedback control. Zbl 0939.93014
Basso, M.; Genesio, R.; Giovanardi, L.; Tesi, A.; Torrini, G.
1998
A frequency method for predicting limit cycle bifurcations. Zbl 0890.34033
Basso, M.; Genesio, R.; Tesi, A.
1997
Controller design for extending periodic dynamics of a chaotic $$\text{CO}_{2}$$ laser. Zbl 0901.93032
Basso, M.; Genesio, R.; Tesi, A.
1997
Secure communications based on discrete time chaotic systems. Zbl 1043.94503
Argenti, F.; De Angeli, A.; Del Re, E.; Genesio, R.; Pagni, P.; Tesi, A.
1997
Harmonic balance analysis of period-doubling bifurcations with implications for control of nonlinear dynamics. Zbl 0859.93028
Tesi, A.; Abed, E. H.; Genesio, R.; Wang, H. O.
1996
On the stability domain estimation via a quadratic Lyapunov function: Convexity and optimality properties for polynomial systems. Zbl 0870.34057
Tesi, A.; Villoresi, F.; Genesio, R.
1996
New conditions for global stability of neural networks with application to linear and quadratic programming problems. Zbl 0849.68105
Forti, Mauro; Tesi, Alberto
1995
Models of complex dynamics in nonlinear systems. Zbl 0877.93002
Genesio, Roberto; Tesi, Alberto; Villoresi, Francesca
1995
On system decomposition for synchronizing chaos. Zbl 0875.94003
Tesi, Alberto; de Angeli, Andrea; Genesio, Roberto
1994
Convexity properties of polynomials with assigned root location. Zbl 0814.93053
Tesi, Alberto; Vicino, Antonio; Zappa, Giovanni
1994
Design criteria for robust strict positive realness in adaptive schemes. Zbl 0808.93030
Tesi, A.; Vicino, A.; Zappa, Giovanni
1994
A frequency approach for analyzing and controlling chaos in nonlinear circuits. Zbl 0848.93029
Genesio, Roberto; Tesi, Alberto; Villoresi, Francesca
1993
An overview of extremal properties for robust control of interval plants. Zbl 0771.93064
Dahleh, M.; Tesi, A.; Vicino, A.
1993
On the robust Popov criterion for interval Lur’e system. Zbl 0787.93077
Dahleh, M.; Tesi, A.; Vicino, A.
1993
Frequency response of uncertain systems with interval plants. Zbl 0786.93019
Bartlett, Andrew C.; Tesi, Alberto; Vicino, Antonio
1993
Enhancing strict positive realness condition on families of polynomials by filter design. Zbl 0848.93038
Tesi, Alberto; Zappa, Giovanni; Vicino, Antonio
1993
Control techniques for chaotic dynamical systems. Zbl 0809.34058
Genesio, Roberto; Tesi, Alberto
1993
Robust strict prositive realness: New results for interval plant- controller families. Zbl 0786.93076
Vicino, Antonio; Tesi, Alberto
1993
Harmonic balance methods for the analysis of chaotic dynamics in nonlinear systems. Zbl 0765.93030
Genesio, R.; Tesi, A.
1992
A harmonic balance approach for chaos prediction: Chua’s circuit. Zbl 0874.94042
Genesio, R.; Tesi, A.
1992
Clockwise property of the Nyquist plot with implications for absolute stability. Zbl 0746.93067
Tesi, A.; Vicino, A.; Zappa, Giovanni
1992
Kharitonov segments suffice for frequency response analysis of interval plant-controller families. Zbl 0837.93053
Tesi, A.; Vicino, A.
1992
Robust stability and performance of interval plants. Zbl 0763.93067
Dahleh, M.; Tesi, A.; Vicino, A.
1992
Chaos prediction in nonlinear feedback systems. Zbl 0754.93024
Genesio, R.; Tesi, A.
1991
Robust absolute stability of Lur’e control systems in parameter space. Zbl 0732.93062
Tesi, A.; Vicino, A.
1991
Computation of nonconservative stability perturbation bounds for systems with nonlinearly correlated uncertainties. Zbl 0714.93035
Vicino, A.; Tesi, A.; Milanese, M.
1990
Robust stability of state-space models with structured uncertainties. Zbl 0705.93061
Tesi, A.; Vicino, A.
1990
Robustness analysis for linear dynamical systems with linearly correlated parametric uncertainties. Zbl 0704.93062
Tesi, A.; Vicino, A.
1990
A new fast algorithm for robust stability analysis of control systems with linearly dependent parametric uncertainties. Zbl 0684.93026
Tesi, A.; Vicino, A.
1989
Robustness bounds for classes of structured perturbations. Zbl 0727.93047
Vicino, A.; Tesi, A.
1989
Results on the stability robustness of systems with state space perturbations. Zbl 0645.93045
Genesio, R.; Tesi, A.
1988
The output stabilization of SISO bilinear systems. Zbl 0655.93054
Genesio, R.; Tesi, A.
1988
Feedback of SISO bilinear systems. Zbl 0651.93034
Genesio, R.; Tesi, A.
1988
On limit cycles in feedback polynomial systems. Zbl 0662.93053
Genesio, R.; Tesi, A.
1988
all top 5

#### Cited by 1,109 Authors

 29 Chesi, Graziano 26 Cao, Jinde 26 Tesi, Alberto 17 Vicino, Antonio 15 Di Marco, Mauro 15 Forti, Mauro 15 Peres, Pedro L. D. 14 Genesio, Roberto 14 Oliveira, Ricardo C. L. F. 11 Chen, Guanrong 11 Moiola, Jorge Luis 11 Zhang, Jiye 10 Cui, Baotong 10 Huang, Lihong 9 Pancioni, Luca 8 Grazzini, Massimo 7 Garulli, Andrea 7 Milanese, Mario 7 Mohamad, Sannay 7 Park, Juhyun (Jessie) 6 Chen, Tianping 6 de Oliveira, Maurício Carvalho 6 Lu, Wenlian 6 Rakkiyappan, Rajan 6 Singh, Vimal Pratap 6 Song, Qiankun 6 Sun, Yeong-Jeu 6 Zhao, Hongyong 5 Abed, Eyad H. 5 Balasubramaniam, Pagavathigounder 5 Cerone, Vito 5 Hu, Tingshu 5 Innocenti, Giacomo 5 Lacerda, Márcio J. 5 Lam, James 5 Luo, Runzi 5 Noorani, Mohd Salmi Mohd 5 Regruto, Diego 5 Šiljak, Dragoslav D. 5 Wang, Zhanshan 5 Zheng, Chengde 4 Ahn, Choon Ki 4 Álamo, Teodoro 4 Basso, Michele 4 Chen, Maoyin 4 Gopalsamy, Kondalsamy 4 Han, Zhengzhi 4 Hashim, Ishak 4 Li, Chuandong 4 Li, Xuemei 4 Liang, Jinling 4 Mocenni, Chiara 4 Oliveira, José J. 4 Robbio, Federico I. 4 Scherer, Carsten W. 4 Wang, Hua O. 4 Xiao, Min 4 Zheng, Wei Xing 3 Ahmadi, Mohamadreza 3 Alvarez-Vázquez, Lino Jose 3 Alvarez, Joaquín 3 Arik, Sabri 3 Azar, Ahmad Taher 3 Bhattacharyya, Shankar Prashad 3 Camacho, Eduardo Fernández 3 Fernández, Francisco Javier 3 Ghilardi, Chiara 3 Hoon Joo, Young 3 Huang, Chuangxia 3 Hwang, Chyi 3 Kaslik, Eva 3 Keel, Lee H. 3 Lee, Donghwan 3 Li, Panshuo 3 Li, Yongkun 3 Liu, Derong 3 Lou, Xuyang 3 Lu, Renquan 3 Martinez Palhares, Reinaldo 3 Momeni, Hamidreza 3 Nistri, Paolo 3 Oishi, Yasuaki 3 Pang, Guochen 3 Papachristodoulou, Antonis 3 Peaucelle, Dimitri 3 Piga, Dario 3 Ponce, Enrique 3 Razza, Valentino 3 Seiler, Peter J. 3 Shang, Yun 3 Sharma, B. B. 3 Sun, Yongzheng 3 Tan, Manchun 3 Tang, Wansheng 3 Tian, Yuping 3 Vaidyanathan, Sundarapandian 3 Verduzco, Fernando 3 Wang, Jun 3 Wang, Likui 3 Wang, Linshan ...and 1,009 more Authors
all top 5

#### Cited in 113 Serials

 101 Automatica 66 International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 60 Chaos, Solitons and Fractals 34 International Journal of Control 33 Nonlinear Dynamics 29 Systems & Control Letters 25 Neural Networks 24 Applied Mathematics and Computation 24 International Journal of Robust and Nonlinear Control 22 Journal of the Franklin Institute 17 Communications in Nonlinear Science and Numerical Simulation 15 Nonlinear Analysis. Real World Applications 13 International Journal of Systems Science 13 European Journal of Control 10 Kybernetika 10 Mathematics and Computers in Simulation 9 Neural Computation 9 Journal of Applied Mathematics 8 Journal of Mathematical Analysis and Applications 8 Mathematical Problems in Engineering 8 Asian Journal of Control 7 Journal of Computational and Applied Mathematics 7 Mathematical and Computer Modelling 6 Computers & Mathematics with Applications 6 International Journal of Circuit Theory and Applications 5 Fuzzy Sets and Systems 5 Circuits, Systems, and Signal Processing 5 Physica D 5 Complexity 5 Abstract and Applied Analysis 4 Journal of Mathematical Biology 4 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 4 Applied Mathematics and Mechanics. (English Edition) 4 Applied Mathematical Modelling 4 Linear Algebra and its Applications 4 Advances in Difference Equations 4 International Journal of Systems Science. Principles and Applications of Systems and Integration 3 Journal of Optimization Theory and Applications 3 Applied Mathematics Letters 3 International Journal of Adaptive Control and Signal Processing 3 Automation and Remote Control 3 Chaos 2 International Journal of Modern Physics B 2 Journal of Fluid Mechanics 2 Mathematical Methods in the Applied Sciences 2 Theoretical Computer Science 2 Cybernetics and Systems 2 Dynamics and Stability of Systems 2 MCSS. Mathematics of Control, Signals, and Systems 2 International Journal of Computer Mathematics 2 Mathematical Programming. Series A. Series B 2 Journal of Difference Equations and Applications 2 Journal of Inequalities and Applications 2 Discrete Dynamics in Nature and Society 2 International Journal of Applied Mathematics and Computer Science 2 Discrete and Continuous Dynamical Systems. Series B 2 Journal of Systems Science and Complexity 2 Journal of Applied Mathematics and Computing 2 Journal of Control Theory and Applications 2 Nonlinear Analysis. Hybrid Systems 2 Arabian Journal for Science and Engineering 1 Modern Physics Letters B 1 Acta Mechanica 1 Biological Cybernetics 1 Computer Methods in Applied Mechanics and Engineering 1 Journal of Mathematical Physics 1 Mathematical Biosciences 1 Physics Letters. A 1 Acta Mathematica Vietnamica 1 Journal of Computer and System Sciences 1 Journal of Differential Equations 1 Meccanica 1 Numerical Functional Analysis and Optimization 1 SIAM Journal on Control and Optimization 1 Optimal Control Applications & Methods 1 Acta Applicandae Mathematicae 1 Signal Processing 1 Multidimensional Systems and Signal Processing 1 Japan Journal of Industrial and Applied Mathematics 1 Numerical Algorithms 1 SIAM Review 1 Journal of Dynamics and Differential Equations 1 Journal of Computer and Systems Sciences International 1 Applied Mathematics. Series B (English Edition) 1 Journal of Mathematical Sciences (New York) 1 Computational and Applied Mathematics 1 Science in China. Series E 1 Neural Processing Letters 1 Journal of Vibration and Control 1 Differential Equations and Dynamical Systems 1 Soft Computing 1 ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik 1 Acta Mathematica Sinica. English Series 1 International Journal of Nonlinear Sciences and Numerical Simulation 1 Nonlinear Analysis. Modelling and Control 1 Dynamical Systems 1 Sādhanā 1 Journal of Dynamical Systems and Geometric Theories 1 SIAM Journal on Applied Dynamical Systems 1 Structural and Multidisciplinary Optimization ...and 13 more Serials
all top 5

#### Cited in 35 Fields

 460 Systems theory; control (93-XX) 289 Ordinary differential equations (34-XX) 173 Biology and other natural sciences (92-XX) 147 Dynamical systems and ergodic theory (37-XX) 57 Computer science (68-XX) 41 Operations research, mathematical programming (90-XX) 30 Numerical analysis (65-XX) 18 Linear and multilinear algebra; matrix theory (15-XX) 18 Information and communication theory, circuits (94-XX) 16 Partial differential equations (35-XX) 16 Difference and functional equations (39-XX) 10 Calculus of variations and optimal control; optimization (49-XX) 8 Mechanics of particles and systems (70-XX) 7 Operator theory (47-XX) 7 Probability theory and stochastic processes (60-XX) 6 Mechanics of deformable solids (74-XX) 6 Fluid mechanics (76-XX) 4 Real functions (26-XX) 3 Functions of a complex variable (30-XX) 3 Integral equations (45-XX) 3 Optics, electromagnetic theory (78-XX) 2 General and overarching topics; collections (00-XX) 2 Field theory and polynomials (12-XX) 2 Statistics (62-XX) 2 Statistical mechanics, structure of matter (82-XX) 1 Mathematical logic and foundations (03-XX) 1 Number theory (11-XX) 1 Commutative algebra (13-XX) 1 Algebraic geometry (14-XX) 1 Several complex variables and analytic spaces (32-XX) 1 Special functions (33-XX) 1 General topology (54-XX) 1 Classical thermodynamics, heat transfer (80-XX) 1 Quantum theory (81-XX) 1 Game theory, economics, finance, and other social and behavioral sciences (91-XX)