text:
# Kulraj Suri

Software Engineer / Quant Dev

# Portfolio Optimisation

1. What is Portfolio Optimisation?

Given:

• A set of equities
• Target return

Find:

• Allocation to each equity that minimizes risk

Given a set of equities that you want to invest in and a target return that you want to achieve, a portfolio optimiser will calculate the best weights of those equities to give you the lowest risk for that portfolio.

2. Risk

• Stock A and Stock B both rise 10% over a given period of time
• Stock A is much more volatile than Stock B – large peaks and troughs

Stock B is less risk than Stock A given the same 10% return:

We can measure the risk of a stock through its volatility:

Risk = Volatility = Standard Deviation of Returns

3. Portfolios

A portfolio is a weighted blend of assets

We can consider a portfolio containing technology stocks. We can view the stocks using a Risk vs Return graph. As is usually the case, the highest return stocks tend to have the highest volatility (risk).

When we build a portfolio by allocating a weighted amount of cash to each one of these equities, we get a portfolio that is a weighted combination of these equities.

Now the portfolio itself is somewhere on this graph. It has a certain amount of risk and a certain amount of expected return and you would guess that it would be somewhere in the middle of all the equities.

Portfolio optimisation allows us to do better than this ‘middle ground’. We can find portfolios that for a given target return may have lower risk than any of the individual equities.

4. Portfolio Optimisation

What goes into a portfolio optimiser?

Inputs:

• Expected return for each equity
• Volatility (risk) for each equity
• Covariance matrix
• Portfolio target return

Output:

• Portfolio weights for each equity that minimise risk for target return

These relationships are found through a technique called optimisation which is an iterative process.

5. Covariance Matrix

What is the covariance matrix?

• The covariance is the correlation (similarity) between different objects
• Covariance measures the degree to which two variables are linearly associated
• Correlation is a scaled version of covariance that takes on values in [1,1] with a correlation of ±indicating perfect linear association and 0 indicating no linear relationship.

Let’s take a look at covariance in the context of three different example stocks:

• Correlation = Covariance
• A high positive correlation means the two stocks move together (e.g. Apple and Intel or Z and Y)
• A low negative correlation means the two stocks move opposite to one another (e.g. Tesla and General Motors or X and Z/Y)
• Stock Y and Stock Z move closely together and are closely correlated
• Correlation between Stocks Y and Z = 0.9
• Stock X shows the same return of 10% over a given time period but moves in an opposite direction to Stocks Y and Z
• Correlation between X and Y = -0.9
• Correlation between X and Z = -0.7

The purpose of portfolio optimisation is to combine all three stocks to reduce volatility but give the same return of 10%.

For example, the optimum weights for a portfolio containing Stocks X, Y and Z with a target return of 10% could be:

• X = 50%
• Y = 25%
• Z = 25%

Now the portfolio is diversified because Stocks Y & Z are anti-correlated to Stock X. As we can see below, we have smoother less volatile performance:

6. Efficient Frontier

For each target return there is a unique portfolio – that is the minimum risk portfolio for that amount of return.

The line represents an infinite number of different portfolios with each one giving a different target return.

1. Minimum risk portfolio: lowest volatility but has lowest return
2. Balanced risk-reward: this is the maximum Sharpe Ratio – maximises the ratio of return to volatility (Sharpe Ratio)
3. Maximum return portfolio: highest return but often highest risk


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# Invariance under diffeomorphisms of the Sobolev $H^s$ spaces

Let $\Omega, \Omega' \subset \mathbb{R}^n$ be two open subsets and let $\psi : \Omega \rightarrow \Omega'$ be a $C^k$ diffeomorphism. Then, $\psi$ induces by pullback a linear isomorphism $$u \mapsto u \circ \psi$$ between the Sobolev spaces $W^{k,p}_{\text{loc}}(\Omega')$ and $W^{k,p}_{\text{loc}}(\Omega)$. I assume that there is an analogous result for the Hilbert-Sobolev spaces $H^s_{\text{loc}}(\Omega)$ with real exponent $s$. Can someone please provide me a reference where this is discussed?

-
I guess the result should be obvious once you write down the definition for $H^s_{loc}(\Omega)$. But if you insist on a reference, I would recommend Friedlander's Introduction to the theory of distributions. –  Hui Yu Nov 11 '12 at 3:32
Not for me, and the reference you provide doesn't help at all... –  levap Nov 12 '12 at 17:10
Really? Can you tell me what did you get and where are you stuck? –  Hui Yu Nov 12 '12 at 17:13
Say I work with the Fourier transform definition. So $u \in H^s(\Omega)$ if for all $\phi \in C^{\infty}_c(\Omega)$, we have $u\phi \in H^s(\mathbb{R}^n)$, where $H^s(\mathbb{R}^n)$ is defined using Fourier transform conditions. I have no idea how to estimate the Fourier transform of the composition $u \circ \psi$ of $u$ with a diffeomorphism in terms of the Fourier transform of $u$. –  levap Nov 12 '12 at 17:38
I think it's better to consider $H^s$ as the Besov space $B^s_{2,2}$ and use their local characterization. The invariance under diffeomorphisms ought to be someone in the 1st or 2nd volume of Triebel's Function Spaces but this is only a guess, and I don't have access to the book now. –  user53153 Dec 22 '12 at 2:40
The idea is as follows. Let $\Psi$ be the linear operator defined by $\Psi u = u \circ \psi$. Prove the following: $$\Psi \in \mathcal{L}(H^k(\Omega),H^k(\Omega)) \text{ and } \Psi \in \mathcal{L}(H^{k-1}(\Omega),H^{k-1}(\Omega))$$ by using the chain rule and change of variables. Then interpolation theory provides the estimate $$\Psi \in \mathcal{L}(H^s(\Omega),H^s(\Omega)) \text{ for } s \in (k-1,k)$$ since $H^s$ is the interpolation between $H^{k-1}$ and $H^{k}$. Moreover, we get an estimate $$\lVert \Psi \rVert_{H^s \to H^s} \le C \lVert \Psi \rVert_{H^{k-1} \to H^{k-1}}^\theta \lVert \Psi \rVert_{H^k \to H^k}^{1-\theta}$$ where $\theta \in (0,1)$ is determined by $s$ and $k$.

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Journal topic
SOIL, 6, 89–94, 2020
https://doi.org/10.5194/soil-6-89-2020
SOIL, 6, 89–94, 2020
https://doi.org/10.5194/soil-6-89-2020

Short communication 04 Mar 2020

Short communication | 04 Mar 2020

A new model for intra- and inter-institutional soil data sharing

A new model for intra- and inter-institutional soil data sharing
José Padarian and Alex B. McBratney José Padarian and Alex B. McBratney
• Sydney Institute of Agriculture & School of Life and Environmental Sciences, University of Sydney, Sydney, New South Wales, Australia

Abstract

Data sharing and collaboration are critical to solving large-scale problems. The prevailing soil data-sharing model is based on different groups sending their data to a lead party. This model is of a centralised nature and, consequently, results in the participants ceding control and governance over their data to the lead party. Here we explore the use of a distributed ledger (blockchain) to solve the aforementioned issues. We explain what a blockchain is and some of its characteristics to then describe some features of a blockchain that make it an interesting candidate for an inter-institutional database. Finally, we describe the potential use case of developing a global soil spectral library with multiple, independent international institutions constituting the network.

1 Introduction

Soil is a key component of ecosystems, and the need for soil information to monitor its condition is increasing. A large amount of soil data has been collected in the last century, with a special increment during the 70s to 80s, and many organisations are performing the exhaustive task of “rescuing” and organising that data in more accessible formats . Additionally, in many countries, a large amount of new soil data is being generated, partially aided by the advancements in methods such as soil spectroscopy, which makes the acquisition of soil data faster and cheaper compared with traditional wet chemistry methods .

Most collected soil data are useful for solving problems locally, but they are too fragmented to tackle more general issues. This applies at various levels of granularity, including different teams within an institution, a single institution in different regional locations, and multiple institutions either within a country or internationally. In these cases, collaboration and data sharing become paramount. The soil community recognises this collaboration need and has responded by creating different data-sharing initiatives. For instance, compiled a global soil spectral library for soil mapping, modelling, and monitoring with datasets from 92 countries (mainly data from the United States, Australia, and Europe). Another global spectral library has been promoted by the FAO's Global Soil Partnership via the Global Soil Laboratory Network (GLOSOLAN). The FAO also promoted different initiatives to establish collaboration networks to share soil profile information, including between Latin American countries (SISLAC) or the Global Soil Information System (GLOSIS). All these initiatives are designed as centralised systems where, in order to collaborate, different parties (either individuals or organisations) should send their data to the lead organisation (Fig. 2a). A centralised information system has a series of disadvantages that we explore in this work. Two of the most important disadvantages of using a centralised network, especially in the context of a collaborative network of multiple independent parties, are that the control and data governance (norms, principles, and rules) are completely ceded to the initiating party. Usually, these aspects could be defined in a data-sharing agreement prior to the establishment of the collaboration but, in practice, there are no controls to avoid unilateral decisions.

A potential solution for the data control and governance issues derived from the implementation of a centralised data-sharing system is the use of a distributed ledger or blockchain. The aim of this paper is to delineate the requirements for a functional, decentralised, inter-institutional database (IIDB) to share soil information in a distributed ledger or blockchain. We mainly focus on the technical considerations of data sharing instead of its social, political, and organisational aspects, keeping in mind that the latter are important for any data-sharing system, decentralised or not. First, we introduce some terms that will be used throughout this paper to then explain what a blockchain is and what some of its characteristics are. Second, we describe some features of a blockchain that make it an interesting candidate for an IIDB. Finally, we present a use case of collaborative effort that could be a good fit for using the proposed model.

2 Blockchain

Before defining what a blockchain is, we introduce a list of definitions that are used throughout this paper.

Point of failure:

a potential risk caused by a poor system design where a single fault at that point can affect the correct functioning of the system.

Hash:

alpha-numeric string generated by mapping the data of an arbitrary size onto data of a fixed size (Dworkin2015). When the original data are unknown, it is very difficult to reconstruct them from the hash value, which makes it a good candidate to ensure the integrity of a transaction.

In simple terms, a blockchain is a linked sequence of records of the transactions of digital assets (Fig. 1). These transactions can be of different types, including data creation (adding new data to the blockchain) or transfer (transferring the ownership of the data to another party or to the same owner to edit data). The best known assets are crypto-currencies (e.g. Bitcoin), but in practice can be anything that can be represented by data. Each new transaction is cryptographically signed using the party's private key which is verified against the public key (included in the asset). The transaction also includes a hash that is generated using its public–private key pair and the hash of the previous block. Any attempts to modify a block that has already been incorporated into the blockchain would change the signature and the hash of that transaction, which can be detected.

Figure 1Diagram of three consecutive blocks (two transactions) within a blockchain.

It is worth detailing what a key pair is and how it operates in the context of signing transactions. In asymmetric cryptography, two keys are used – private and public keys . The private key is used to generate a signature based on the data included in the transaction and the public key is used to verify the signature. As the names suggest, the private key is only known by the signing party, and the public key is available to everyone to verify the signature. Generally, the signature and the public–private key pair are long sequences of alpha-numeric characters which are algorithmically generated. There are many implementations of these algorithms but, in general, the algorithm to generate the signature s can be rationalised as a function $s=f\left(h,{k}_{\text{priv}}\right)$, where h is the hash of the data to sign and kpriv is the private key. In order to verify the signature, the verifier should compute the hash h using the signature s and the public key kpub such as ${h}^{\prime }=g\left(s,{k}_{\text{pub}}\right)$ and also independently compute the hash of the data h using the same hashing algorithm (publicly known). If $h={h}^{\prime }$, the signature is valid.

By design, a blockchain usually operates within a network of interconnected nodes (Fig. 2b). Each node keeps a copy of the chain (public ledger) and acts as a validator, ensuring the validity of new transactions. After enough nodes have reached consensus about the validity of the transaction, the new data block is appended to the chain.

Figure 2Data flows in two different soil information system infrastructures: (a) centralised; (b) decentralised.

Blockchain technology is a diverse ecosystem with many implementations that differ in their characteristics and efficiency. For instance, popular implementations such as Bitcoin, Ethereum, Litecoin, and Monero are computation-intensive and require large energy input due to their consensus algorithm (proof-of-work), consuming more energy than mineral mining (copper, gold, platinum, and rare earth oxides) to produce an equivalent market value . Of course, other consensus algorithms (e.g. proof-of-stake) do not require intensive computations. Given the diversity of implementations, it is completely possible to design a system that is secure, reliable, and efficient to serve as a soil data-sharing platform.

3 A soil data-sharing platform based on blockchain

Besides providing a solution to the aforementioned problems, namely centralised data control and governance, a blockchain has other characteristics that make it an interesting candidate for a IIDB. Some of these solutions and characteristics are described in this section.

3.1 Decentralisation

As mentioned before, the main characteristic of a blockchain is the decentralised nature of the system. Each node of the network keeps a copy of the blockchain, which is synchronised after every new transaction (creation or transfer). Assuming that each node of the network is controlled by a different party, there is no centralised data storage, and hence no single point of failure or control. Normally, in a well-designed, diverse network, a significant number of the nodes can be compromised without affecting its integrity.

Because all the nodes have a copy of the blockchain and act as validators, malicious modifications to the data are very difficult (see immutability section). The only possible way of tampering with the data is if most of the nodes are colluded, which can be avoided by ensuring a diverse network.

For intra-institutional data sharing, a blockchain system can also be implemented to replace a traditional, permissioned database. The advantages are similar to the inter-institutional case, including each team leader having “ownership” of their data, data traceability, data access logging and potentially preventing unauthorised access, and preventing malicious modifications or deletions. Data are one of the most valuable assets of any company, and adding this extra layer of security to ensure its integrity should be a priority, and even mandatory for publicly funded institutions.

3.2 Data governance

Data governance defines the norms, principles, and rules under which the activities of a consortium should be conducted. It might include important details such as data release and rights to publish with consortium data first, research output rules (e.g. authorship sequence in consortium publications), if the data should be shared with non-consortium members , and the addition of new members. In a data-sharing network, data governance is stipulated on an agreement and any modifications can be agreed between the members. In practice, control and governance over the data are ceded to the central node, and the system has no way to prevent unilateral changes from being made.

Using a technology such as blockchain does not replace the initial process of negotiations or the effort of setting rules, but it can help reduce some of the friction points. Many of the clauses included in a data-sharing agreement can be programmatically enforced and, since the network is collectively governed, changed over time via a democratic process. Usually, any node of the network can propose an election process where the rest of the nodes cast a vote transaction, which is also appended to the chain. If the “super-majority” (usually a large proportion such as at least 2∕3) of nodes approves the changes, the suggested changes are incorporated into the system.

3.3 Data ownership

When a new asset is created, it is cryptographically signed and assigned to one or more users' public key(s). If the data need to be transferred (either to make corrections or include new information, or to another user), only the owners are capable of doing so by using the corresponding private keys, even if all the blockchain data are available at every node. This process is automatically validated by all the nodes by ensuring that the signatures match with the owner's(s') public key(s) before proceeding with the transfer.

Here we refer to data ownership as the link between a user and a digital asset, without any legal implication. Like in any database, decentralised or not, we are assuming that the user has legal rights to upload the data, which should be properly acknowledged, following the rules defined by the consortium. All this information can be included within each asset, permanently linking data and metadata, where any change can be recorded in case of ownership changes. If required, the network can perform basic checks to ensure that the metadata are included or even just provide access to encrypted data to authorised users.

3.4 Immutability

Since the blocks of the chain are linked (Fig. 1), in practice, it is not possible to remove or edit a transaction. When a party needs to change the content of some asset that they own, either to correct some error or add new information to an asset, a transfer transaction needs to be performed, transferring the asset to itself. When the transaction is approved by the network, the new version is appended to the blockchain. This design feature produces data redundancy, but also makes it possible to keep the history of every asset (versioning), which is key for auditability. Thanks to this immutability, the parties within the network can always trust that the data have not been tampered with.

Similarly to the data ownership case, here we assume that the asset contains data that are legitimate and error-free. In any system, decentralised or not, it is difficult to control what happens to the data before their ingestion into the system. Although it could be possible to implement pre-ingestion solutions, probably it would always be possible to “cheat the system”. It is important to consider that there are implicit incentives for the parties to provide legitimate data, especially considering the transparency of a decentralised system (ownership and immutability), such as maintaining their credibility.

4 Potential use case: global soil spectral library

Although a blockchain data-sharing model has applications at many levels of granularity (inter- and intra-institutional, and international), we would like to focus on the use case of creating a multi-party (e.g. multi-institutional, multi-national, global) soil spectral library. Spectral soil data can be compared to the digital fingerprint of a particular soil sample which encodes information about its physical, chemical, and biological properties (Grunwald2016). In pedometrics, a discipline that applies quantitative methods to study the variation of soils, the use of spectral data in conjunction with statistical or machine learning models to predict soil properties is already broadly implemented . Nevertheless, the development and application of models derived from spectral data still present a series of challenges. For instance, models derived from local data, despite showing a good performance, have a limited applicability to other areas since they might lose their validity . A potential solution is to develop models trained on data obtained from larger extents, which can be then “localised”, taking advantage of the global knowledge to make predictions at the local scale . This approach has shown significant improvements in local predictions for multiple soil properties. To take full advantage of these advanced models, and since they are considered “data-hungry” methods, it is recommended to train them on a large soil spectral library. Of course, collating a large spectral library that spans a large extent is not a trivial task. This is when collaboration and data sharing become important. Multiple, independent organisations can join efforts to reach a solution to a problem that is very difficult to solve independently, which yields institutional (local) benefits greater than what it is possible when working in isolation.

After all the efforts from different institutions to collaborate in a common initiative, it is only fair that the data-sharing infrastructure is carefully designed to ensure a democratic access and control and governance over the data. We believe that, in general, a decentralised system can guard those interests for all parties involved. Particularly in the case of a global soil spectral library, the use of a decentralised database is of critical importance since the resulting database could be used by national reference centres for soil analysis. The level of transparency and security that a distributed ledger offers ensures that the reference data have not been tampered with and also, given its decentralised model, will maximise accessibility. In the following sections we explore certain implementation aspects of a decentralised data-sharing system in the context of a global soil spectral library.

4.1 Consortium initiation

The potential members of the consortium would have enough analytical capacity to measure the spectral response of soil samples and also to perform laboratory analyses to measure the corresponding physical, chemical, and biological soil properties. This includes universities and commercial soil laboratories from different countries.

Each member should have available the computational infrastructure to become a node of the network. The requirements are not prohibitive and include enough capacity to store all the data and Internet connection. Each node should generate their public–private key pair, securely store a copy of the private key, and distribute the public key to the rest of the members. To start the network, all the public keys should be known by all the members. Once the network is functional, more nodes can be added with the approval of most of the current members via an election process.

In terms of the network users, it is possible to have multiple users per node (e.g. different researchers from a single university). Ideally, all the users should have their own public–private key pair to sign their transactions, and their public keys should be known to all the users. This information can also be stored in the blockchain as a public ledger of who can access the data.

As mentioned in Sect. 3.2, here we do not consider the legal/organisational aspects of creating a consortium, which, arguably, are more complex than the technical challenges. What voting power should each party have (affecting governance)? Should it be proportional to the data they provide? How should the data be referenced? Should all the contributors be co-authors of the publications derived from the database? These are some of the questions that should be resolved before implementing a technical solution such as a database, decentralised or not.

4.2 Providing data

After the network is functional, any member can create new transactions to add data that will be synchronised between all the nodes, ensuring immediate accessibility to the data to all the members. The structure of what constitutes an “asset” should be defined during the consortium initiation period. For instance, the asset could be a single soil sample with its corresponding analytical data (Snippet 1). The system should support the use of numerical and text data to store all the necessary soil properties and metadata. Complex data structures such as soil spectral data can be stored as comma-separated numbers or compressed.

Snippet 1. Example Sample asset. The soil property codes are just for illustration purposes.

{
 "date": "2019-05-27",
 "user": "University1",
 "top": 0,
 "bottom": 10,
 "oc": 5.2,
 "sand": 10,
 "silt": 20,
 "clay": 70,
 "bd": 1.1,
 "spectra": "EOhM2lSd6j/o19ZP/9nqP+..."
 # This spectra is encoded as base64
}

The new transaction should be signed with the user's private key and the asset ownership set to a user's public key. This provides a way of authenticating the origin of the data and allows the user, and only that user, to create updated versions of that asset if needed (e.g. when new properties are measured or to correct potential errors). Before a new transaction is appended to the blockchain, a “super-majority” of the voting power must agree on the validity of that transaction. The most basic validation is to ensure that the owner(s) are signing the transaction, but in practice it is possible to set any logical rules. This provides the opportunity to give certain groups of users the control over an asset, define minimum number of owners, perform basic data integrity checks (plausible values, names encoding), etc. Of course, as mentioned in Sect. 3.4, the legitimacy of the data is hard to prove, which should be considered when designing the system.

4.3 Retrieving data

Since every node keeps a copy of the blockchain locally, it is possible to retrieve data from any node from the network, providing extra redundancy and hence assuring accessibility in case of malfunction of some of the nodes. Advanced users can query their local copy of the database directly. A friendlier way of providing access to read the data is via an application programming interface (API) that connects any user with a node. That API can perform tasks such as querying the blockchain to retrieve specific data, provide the history of any asset, and potentially process data using pipelines approved by the consortium.

4.4 User interaction

Most of the specific blockchain operations (i.e. signing and verifying transaction) are performed in the background. There is no extra overhead for the users besides keeping their respective private keys safe. A user interface can be built on top of an API so users can access the system as if it were a traditional data management system (DMS), with capabilities to query and retrieve the data from the network.

In terms of the types of users with access to the system, any person with access to a node has complete reading access to the blockchain. If public access is required to allow non-consortium members to connect to the database, multiple solutions are available, including single or multiple nodes acting as a web server. Using multiple nodes as web servers might reduce latency, especially when the consortium spans different countries (i.e. an external user can connect to the closest node). Again, a platform can be built to ensure the public experience is identical to a normal DMS.

5 Summary

The prevailing soil data-sharing model is centralised, with users ceding control and governance over their data to a lead party. We propose the use of a public ledger (blockchain) to create a decentralised soil data-sharing network. This network provides a series of advantages to the participant institutions, including

• allowing institutions to preserve the ownership and control over their data,

• ensuring that once the data are appended to the blockchain, they cannot be tampered with, and

• actively participating in governance decisions such as adding new members through elections facilitated by the system.

Ultimately, any consortium data-sharing agreement is based on trust between the participants. By using a blockchain network, the need for trust is removed since rules can be programmatically enforced and the data become tamper-resistant. This protects the already existing trust bond between the consortium members and, potentially, allows the consortium to expand its reach by working with new parties that are not fully trusted.

For intra-institutional data sharing, a blockchain system can also be implemented to replace a traditional, permissioned database. The advantages include each team leader having “ownership” of their data, data traceability, data access logging and potentially preventing unauthorised access, and preventing malicious modifications or deletions.

Author contributions
Author contributions.

JP conceived the concept and wrote the first draft. Both authors contributed to generating and reviewing the subsequent versions of the manuscript.

Competing interests
Competing interests.

The authors declare that they have no conflict of interest.

Review statement
Review statement.

This paper was edited by Jan Vanderborght and reviewed by Dominique Arrouays and two anonymous referees.

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Viscarra-Rossel, R., Behrens, T., Ben-Dor, E., Brown, D., Demattê, J., Shepherd, K., Shi, Z., Stenberg, B., Stevens, A., Adamchuk, V., Aïchi, H., Barthès, B., Bartholomeus, H., Bayer, A., Bernoux, M., Böttcher, K., Brodský, L., Du, C., Chappell, A., Fouad, Y., Genot, V., Gomez, C., Grunwald, S., Gubler, A., Guerrero, C., Hedley, C., Knadel, M., Morrás, H., Nocita, M., Ramirez-Lopez, L., Roudier, P., Campos, E., Sanborn, P., Sellitto, V., Sudduth, K., Rawlins, B., Walter, C., Winowiecki, L., Hong, S., and Ji, W.: A global spectral library to characterize the world's soil, Earth-Sci. Rev., 155, 198–230, 2016. a

Singh, J. A. and Daar, A. S.: Intra-consortium data sharing in multi-national, multi-institutional genomic studies: gaps and guidance, The HUGO Journal, 3, 11–4, 2009. a


text:
# Terminal Velocity of a Car

When explaining how a car reaches its terminal velocity, is it solely down to how aerodynamic the car is? My thoughts are that if you have a fixed mass $m$, then the contact frictional force will be constant,$F=\mu{R}$, and because the thrust due to the engine will be much greater than this value, then surely only air resistance will increase the total value of drag? And this is why as the speed increase the acceleration begins to tend to 0 as Drag ∝$v^2$.

• There is more than just air friction and surface contact friction that acts to resist acceleration in a car. – JMac May 8 '17 at 13:09
• Rolling resistance is not constant, but it depends on many factors. The formula $F = \mu R$ is only valid for sliding dry friction (with slipping). – ja72 May 8 '17 at 13:40
• @ja72 does it follow though that if the mass decreases, is the rolling resistance decreases too? – DLB May 8 '17 at 14:04
• Not necessarily. Resistance of internal drivetrain parts for example may depend on engine torque more than vehicle mass. Many rotating parts follow the stribeck curve where resistance is high at low speeds, drops and the rises again. I think tires follow a similar pattern. – ja72 May 8 '17 at 14:58
• Note that $F=\mu R$ is not happening here. This is the formula for kinetic friction, while a car is influenced by static friction. Static friction is a variable force unlike kinetic friction, and does not stay at a fixed value according to that formula. – Steeven Feb 5 '18 at 14:51

It is just a simplification to say that total drag is $\propto v^2$ and to ignore all other forms of friction. This applies because the power to overcome aerodynamic drag is $\gg$ power loss on the drivetrain and rolling resistance. If you included all the other terms the final speed will decrease only very slightly.

So a simplified model of acceleration (as a function of speed $v$) is

$$a = \frac{P(v)}{m v} - \beta v^2$$

Where $P(v)$ is the engine power at speed $v$, $m$ is the mass and $\beta$ is some drag constant.

Top speed occurs when $a=0$ and hence $$v_{top} \approx \sqrt[3]{ \frac{P_{max}}{m \beta} }$$

This applies only the the car is drag limits (and not gear limited) and that it is designed with getting peak power at the rpm corresponding to top speed. Some cars with 6 speeds, may only achieve top speed in 5th gear, and 6th gear is just an "overdrive".

The car reaches terminal velocity when power in = power out. The power that an engine can deliver depends on the gear ratio and the rpm.

The car has several sources of power dissipation - rolling friction and air drag are the two most important ones. The power these take is a function of velocity - so more power (but a smaller fraction) goes into rolling friction as the car goes faster.

Simple way of formulating this:

\begin{align}P_{engine}&=f(v,...)\\ F_{air}&=\frac12\rho v^2 A\cdot C_D\\ F_{roll}&=\mu_r F_n\\ P_{resistance}&=(F_{air}+F_{roll})v\end{align}

• Floris, rolling resistance is not just a frictional thing. At normal inflation pressure, the bottom surface of the tires of an automobile (the part that contacts the road) deform as they rotate. This deformation is continuous while the car is moving, it heats the tires, and it is a very substantial portion of total power consumption (approximately 50%) at normal highway speeds (e.g., 60 miles per hour). – David White Jan 1 '18 at 20:41
• @DavidWhite - I have no argument with your statement. Would you prefer that I change "friction" to "resistance"? Or are you suggesting that I formulate a more complex set of equations to describe this? I was trying to keep it simple - maybe I overdid it. – Floris Jan 1 '18 at 20:43
• Floris - no, I'm not advocating that you change your statement. I merely want the OP to realize that there is more involved than the rolling resistance typically associated with a static friction coefficient. – David White Jan 5 '18 at 2:41


text:
# When exponents meet trigonometry...

Calculus Level 3

The integration of $$e^{3x}$$ cos x with respect to x can be expressed as $$\frac{p}{q}$$ $$e^{3x}$$ sin x + $$\frac{r}{s}$$ $$e^{3x}$$ cos x + c where c is the arbitrary constant. Assuming gcd (p, q) = 1 and gcd (r, s) = 1, what is p+q+r+s?

×


text:
# Solve a transcendental equation in Mathematica

I believe it is called a "transcendental equation", and I would like to find the best possible solution.

I tried to use Solve:

Solve[Cos[ϕ0]/ϕ0 == h/(2 s0), ϕ0]

During evaluation of In[5]:= Solve::nsmet: This system cannot be solved with the methods available to Solve. >>

Solve[Cos[ϕ0]/ϕ0 == h/(2 s0), ϕ0]

And it didn't work. Since this is the first time I am working with this kind of equation, could somebody help me? I know I can use Taylor expansion for h/s0 << 1 or h/s0 >> 1, but I don't want to do that. The reason why not is because this would not be a realistic model. In my case h is approximately the same as s0.

### Edit

BTW, I would be satisfied with numerical solutions as well -- but only in case that is the only option.

• Numerics is your only option, yes… – J. M. will be back soon Jul 8 '15 at 8:33
• Ok. If so.. How? I though NSolve would do the job, but it doesn't. Problem: "This system cannot be solved with the methods available to NSolve. " – skrat Jul 8 '15 at 8:43
• Use FindRoot[], and have a good initial guess ready. – J. M. will be back soon Jul 8 '15 at 8:44
• For any numerical value of the right-hand side, Solve does a nice job when you add the domain Reals, such as Solve[Cos[ϕ0]/ϕ0 == 1/4, ϕ0, Reals] – Fred Simons Jul 8 '15 at 10:13
• This message is not an error message. It is information for the user how the solution is obtained: first the inexact (machine) numbers are replaced with exact numbers, then Solve finds the exact solution of the resulting equation and then returns the numerical approximation of the exact result. Keep in mind that the solution set of an equation can be highly senitive for small changes in the coefficients. – Fred Simons Jul 15 '15 at 11:26

You're in luck if you're only interested in a domain where x = h/(2 s0) is approximately 1/2. That domain is contained by the domain of InverseFunction applied to Cos[φ0]/φ0:

ϕ = InverseFunction[Cos[#]/# &]

InverseFunction can only work on a domain on which the function is one-to-one. In this case, it's from zero to the first minimum around ϕ0 -> 2.79839:

Plot[Cos[ϕ0]/ϕ0, {ϕ0, 0, 10}]

Now the equation has a unique solution for h/(2 s0) greater than the first maximum, which about h/(2 s0) == 0.161228 or for 0 < ϕ0 < 1.35119:

{minhs, foo} = FindMaximum[Cos[ϕ0]/ϕ0, {ϕ0, 6}]
{maxϕ} = NSolve[Cos[ϕ0]/ϕ0 == minhs && 0 < ϕ0 < 2, {ϕ0}]
(*
{0.161228, {ϕ0 -> 6.12125}}
{{ϕ0 -> 1.35119}}
*)

If these limitations are acceptable, then the inverse function is your solution:

ϕ0 -> ϕ[h/(2 s0)]

It is a symbolic solution and computable to arbitrary precision.

ϕ[0.5]
(*  1.02987  *)

N[ϕ[1/2], 20]
(*  1.0298665293222588276  *)

Plot[ϕ[x], {x, 0, 1},
GridLines -> {{minhs}, {ϕ0 /. maxϕ}}, Mesh -> {{minhs}},

The lower right region shows the valid solution.

Alexei Boulbitch's answer is quite nice, but here are another couple of useful techniques for situations like this. First, if all you need is a graph of the function, you can use ParametricPlot to get that. You want $\phi(x)$, where $\cos \phi/\phi = x$; so you can use $\phi$ as a parameter and plot the curve $(\cos \phi/\phi, \phi)$:

ParametricPlot[{Cos[ph]/ph, ph}, {ph, 0, Pi/2}, PlotRange -> {{0, 1}, Automatic}, AspectRatio -> 1/GoldenRatio]

Second, you can use Interpolation to return an approximate inverse function. Generate a list of points you want your graph to go through; this will be a table of points of the form $(\cos \phi/\phi, \phi)$, as above. Then feed it into Interpolation:

pts = Table[{Cos[ph]/ph, ph}, {ph, Pi/40, Pi/2, Pi/40}];
phiroot[x_] = Interpolation[pts][x];

phiroot now returns the function $\phi(x)$ that you were looking for:

N[phiroot[1/2]]

(* 1.02986 *)

For comparison, the value found by FindRoot differs from this value by about one part in $10^5$:

FindRoot[Cos[x]/x == 1/2, {x, 1}]

(* {x -> 1.02987} *)

The weakness of this approach is that if you try to find phiroot for a value of x outside the range of pts, then you'll be extrapolating rather than interpolating and your results will go pear-shaped rather quickly. Still, if you're only concerned with a fixed range of $x$, this approach works rather well.

Looking at the plot:

Manipulate[Plot[Cos[x]/x - y, {x, 0, 2 \[Pi]}], {y, 0, 1}]

that is, here:

and varying the parameter y=h0/(2s0), one finds that the first root is close to x==1. Let us look for this root only, than it is easy to find a list of values with the structure {y,x0}, where y is the value of the parameter and x0 is the root:

    lst = Table[{y, FindRoot[Cos[x]/x - y == 0, {x, 1}][[1, 2]]}, {y, 0,
1, 0.05}]

(*  {{0., 1.5708}, {0.05, 1.49593}, {0.1, 1.42755}, {0.15, 1.36464}, {0.2,
1.30644}, {0.25, 1.25235}, {0.3, 1.20191}, {0.35, 1.15474}, {0.4,
1.11051}, {0.45, 1.06896}, {0.5, 1.02987}, {0.55, 0.993021}, {0.6,
0.958252}, {0.65, 0.925404}, {0.7, 0.894337}, {0.75,
0.864927}, {0.8, 0.837061}, {0.85, 0.810634}, {0.9,
0.785553}, {0.95, 0.76173}, {1., 0.739085}}  *)

Then this:

    model = a/(1 + b*y);
ff = FindFit[lst, model, {a, b}, y]
Show[{
ListPlot[lst],
Plot[model /. ff, {y, 0, 1}, PlotStyle -> Red]
}]

returns the fitting parameters for the model:

(* {a -> 1.58663, b -> 1.10311}  *)

and a plot for the visual inspection of the fitting quality:

Though it is approximate, the obtained solution:

x=1.59/(1+1.1*y)

is analytic, simple and rather accurate. The other solution can be found similarly.

Have fun!

• I have to ask you one question though... Where did you get that model? Based on what? Why didn't you for example go for a third degree polynomial or something..? – skrat Jul 8 '15 at 11:17
• Also, please help me understand what this does. I will walk you through and let me know if my understanding is wrong. Basically what you did is you firstly created a list based on numerical solution of the original equation for different parameter values y. Than you fitted a function which fits the best and the end result is a nice analytical solution (which is not exact but good) to the original equation? Right? – skrat Jul 8 '15 at 11:31
• @skrat Right. "Where did you get that model?" Just proposed one that I expected to behave like what I saw in the output of the ListPlot. You could also propose another one. An infinite number of models will fit this list. In this situation, and since the model is anyway an approximation, the simplest one is the best. – Alexei Boulbitch Jul 8 '15 at 12:29


text:
# POUMM 2.1.6 2019-12-05

• Fixed an S3-class related problem on r-devel.
• Disabled OPENMP compilation flags due to potential test failures on some platforms, e.g. Solaris.

# POUMM 2.1.5 2019-03-27

• Improved documentation for some functions (specifyPOUMM).
• Fixed a bug: duplicated rows for “g0” in the summary of a POUMM object.
• Compatibility with v1.12.2 of package data.table - using rep instead of recycling in columns with more than 1 but fewer than the max-number of rows (see https://github.com/venelin/POUMM/pull/2).

# POUMM 2.1.3 Unreleased

• Improved man page for gPOUMM function.
• Testing minor changes in the source package - store RData files in lfs.
• Testing github release and zanodo doi-issuing.

# POUMM 2.1.2 2018-11-15

• Temporarily removed bivariate posterior density plotting functionality; univariate posterior density plotting still available.
• Fixed a bug in the C++ likelihood calculation for the case when both sigmae = 0 and se = 0.
• Refactored unit tests.
• Enabled travis continuous integration with a status badge in the README.Rmd.
• Some bugs were fixed.

# POUMM 2.1.1 Unreleased

• Release available from github only

# POUMM 2.1.0 Unreleased

• SPLiTTree dependency is renamed to SPLITT
• Release available from github only

# POUMM 2.0.0 Unreleased

• Using SPLiTTree for the pruning likelihood calculation.
• Automatic web-site generation using the R-package pkgdown.
• Project source-code published on github: https://github.com/venelin/POUMM.git.
• Written a new vignette “Interpreting the POUMM”.

# POUMM 1.3.2 2017-07-02

• Fixing test-errors on r-patched-solaris-x86 and notes on r-devel-linux-x86_64-fedora-clang.
• Fixed memory errors reported by ASAN and valgrind.
• Improved default values for the parameter sigmae.

# POUMM 1.3 Unreleased

• Parallel likelihood calculation: (tested on Linux using Intel complier v16.0.0).
• Added possibility to specify known measurement standard error for each tip of the tree. (see ?POUMM).
• Added a bivariate densplot of the posterior samples displaying a pairs-plot, i.e. a matrix of plots with marginal densities at the diagonal, scatter plots on the lower triangle and correlation on the upper (see ?plot.summary.POUMM); possibility to change the color-palette used in the plot with color-blindness friendly default colors.
• New default parameter settings: now the ML-limits (parUpper and parLower) as well as the priors for the parameters alpha and sigma are scaled according to the length of the tree (tMean) and the empirical variance of the trait (zVar) (see ?specifyPOUMM). Please note that the default parameter settings may change in future releases.
• Improved ML-fit: now the optimization is run 50 times starting from different points in the search region. The ML-fit is still kept deterministic, i.e. consecutive runs should find the same optimum.
• Fixed a dependency on the adaptMCMC package (now added as an import).
• Update references.

# POUMM 1.2.1 2017-03-11

• First version of the package released on CRAN.


text:
# Definition:Real Vector Space

Jump to navigation Jump to search

## Definition

Let $\R$ be the set of real numbers.

Then the $\R$-module $\R^n$ is called the real ($n$-dimensional) vector space.

## Also known as

A real vector space is also known as a real linear space.

## Also see

This object is proved to be a vector space in Real Vector Space is Vector Space.

The definition is also expanded upon in Real Numbers form Vector Space.

The real vector spaces have direct applications to the real world. In fact, it could be suggested that they are the interface between mathematics and physical reality, as follows:

• The $\R$-vector space $\R^3$ can be shown (given appropriate assumptions about the nature of the universe) to be isomorphic to the spatial universe.


text:
HomeEnglishClass 11MathsChapterRelations

Let A="{"1,2"]"a n d\ B={3,4} ...

Let A="{"1,2"]"a n d\ B={3,4} . Find the total number of relations from A into B.

Updated On: 17-04-2022

Get Answer to any question, just click a photo and upload the photo
and get the answer completely free,

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Step by step solution by experts to help you in doubt clearance & scoring excellent marks in exams.

Transcript

this question we have a which is equal to 1 number of patients so cal we have to and B which is further and offer which is the number of elements in the formula for the total number of iterations number of relations from a to distil power number of elements of a number of elements of B so that number of

relation to be 22 and 24 which is the total number of relation relation to


text:
# All Questions

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### How PCA is performed in the paper “Markov Models…”

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### Cross Currency Swap pricing

I have seen two methods for calculating the value of a xccy swap - 1) Convert the future foreign payments to the base currency using forward FX rates, net with the base currency payments and ...
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### Trinomial model for stock options with deterministic interest curve

I am implementing a basic trinomial model with constant volatility right now. I want to do an extension that does not take a constant riskfree rate as input, but interpolates between different given ...
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Anyone know how to build worldquant alpha 101 using pyalgotrade? Could someone give me the sample code? Or how could I start this project? Or there is another tool can simulate the worldquant alpha ...
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### Data of Credit Migration Matrices

Please advise that how to get the data of credit migration matrices There is a paper of credit migration matrices, I would import the data to Matlab or R for credit analysis. https://www....
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### How to get all securities in an asset class from IBPy (Interactive Brokers python API)

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### How to create a synthetic put?

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### Question regarding volatility forecasting using High Frequency Data

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I know that Bloomberg will give me the swap rates for Treasury 30's-5's, but I don't have a Bloomberg. Can anyone direct me to a source?
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Can anyone give me a detailed explanation of how below equations (3) and (4) are derived from (1) and (2)? \begin{align*} \frac{dF_{t,T}}{F_{t,T}} &=\sigma e^{-\lambda(T-t)}dB_t, \tag{1}\\ \ln(F_{...
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### What is the benefit of having proximity to the Bloomberg datacenter?

I own and operate a datacenter adjacent to Bloombergs Datacenter in Orangeburg NY. We have had a couple of trading firms come to us due to our proximity to Bloomberg to receive "data" from them ...
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### Levered beta with changing equity/debt ratios

I know how to calculate a bottom up levered beta for a privately held and not publicly traded company with Hamada (Proof of Hamada's Formula (Relationship between levered and unlevered beta)) and ...
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### Calibration of intensity model

I could use some advice on calibration of stochastic intensity models. I am thinking that the CIR model is most suitable, as it can not take negative values (when feller condition is satisfied). I ...
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### Stock market cash flow

I want to understand better cash flow of stock market and it's participants, but could not find any reasonable information online, hope more experienced people here could help. Money IN flow: (1)...
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### Understanding Vega calculation in black Scholes model

I am attempting to calculate the Greeks, and I understand their derivation. However when it comes to actually implementing Vega I am a little lost. Vega is defined analytically as:  SN'(d_1)\sqrt{T-...
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### Importance Sampling for pricing options with longstaff and schwartz

I have been asking this similar question before. However, I really want to be concrete and get and concrete explanation. I have been reading the paper by Moreni and try to implement the same ...
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### Using CME DV01 to predict Futures price at 0.00% Yield

DV01 is published at CME Group for the cheapest-to-deliver bond here: http://www.cmegroup.com/trading/interest-rates/invoice-spread-calculator.html. If my goal is to get an approximation where the ...
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### What is “swap rate differentials” in relation to currency exchange?

So I'm working on a excel spreadsheet that relates relevant drivers for the FX market. In particular, I was told to look into swap rate differentials. I've read that the interest rate differential is ...
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### Sovereign credit default risk

so I'm tasked with trying to calculate the sovereign credit risk based on a 1 year default probability, and I know that Bloomberg already has a model for the 1 year default probability under the ...


text:
# Trouble understanding the partial differentiation used in reinforcement learning

I am studying deterministic actor-critic algorithms in reinforcement learning.

I try to give a brief explanation of actor-critic algorithms before jumping into the mathematics. The actor takes in state $$s$$ and outputs a deterministic action $$a$$ based on the distribution policy $$u$$.

The state and action are fed into the critic. The critic sees how good it is to take a particular action from a given state using the action-value function $$Q(s,a,w)$$.

The critic is then updated via temporal difference (TD) learning and the actor updated in the direction of the critic

Thus it can be seen that the actor's goal is to try and maximise the state action value function $$Q(s,a,w)$$ by picking the best actions in the given state.

I am having trouble understanding the mathematics behind the updating of the actor.

The below equation gives how the actor is updated.

$$$$\frac{\partial l}{\partial u} = \frac{\partial Q(s, a, w)}{\partial a} \frac{\partial a}{\partial u}$$$$

What I understand is that we are taking the partial derivative of $$l$$ with respect to $$u$$, and we are backpropogating the critic gradient to the actor.

It seems that $$l$$ is a differentiable function of the variable $$a$$, but I am confused when it comes to describing what is happening in the equation above as it seems to consist of two functions multiplied together.

Can someone kindly explain what is really happening in the mathematics above?

Your understanding of what's going on seems to be correct, just one little clarification: $$u$$ should be the model parameters of the deterministic policy $$\mu(s,u)$$ and not a distribution itself, same as $$w$$ are the model parameters of $$Q(s,a,w)$$, but that's probably what you meant (or I might be unfamiliar with the formulation).

Concerning your actual question, the update step implied by $$\frac{\partial l}{\partial u}$$ is supposed to make the deterministic policy $$\mu(s,u)$$ to get closer to the optimal $$a$$, which maximizes $$Q(s,a,w)$$. As $$a = \mu(s,u)$$, we have a composite function on our hands

$$Q(s,a,w) = Q(s, \mu(s, u), w)$$

When updating the actor paramters $$u$$ such that $$Q$$ is maximized, we need to make a step in the direction of the gradient of $$Q$$ with respect to $$u$$ which, since it is a composite function, is computed using the chain rule

$$\frac{\partial Q}{\partial u} = \frac{\partial Q}{\partial a}\frac{\partial a}{\partial u}$$

The notation is a bit sloppy, replacing $$\mu$$ and $$a$$ and so on, but that also seems to be the case in the literature. So what is going on intuitively consits of two parts:

1. moving $$a$$ in the direction of $$\frac{\partial Q}{\partial a}$$ will increase $$Q$$, e.g. in 1D if $$\frac{\partial Q}{\partial a} > 0$$ increasing $$a$$ would increase $$Q$$ and if $$\frac{\partial Q}{\partial a} < 0$$ increasing $$a$$ would decrease $$Q$$

2. moving $$u$$ in the direction of $$\frac{\partial a}{\partial u}$$ will increase $$a$$, in 1D the example would be the same as above

If you multiply these together and update $$u$$ according to the product you end up moving $$u$$ such that $$Q$$ increases by either increasing or decreasing $$a$$, which is exactly what you want to do.


text:
# Energy and Work (pV diagram problem)

#### Zach00

Hi there! So I've never taken physics until now (first year in uni) so I'm struggling to say the least. I'm trying to do this problem of a pV diagram and I'm stuck on b). I've done a) and c). I'm just confused how to find Vc in an isothermal process because if it was adiabatic then I could use the adiabatic law with gamma. See the attached files for the question and my work

#### Attachments

• 153.4 KB Views: 2
• 52.4 KB Views: 4

#### benit13

For isothermal expansion:

$$\displaystyle P_1 V_1 = P_2 V_2$$

We know that $$\displaystyle P_1 = 3 P_2$$, so

$$\displaystyle V_2 = 3 V_1 = 3 \times 0.004 = 0.012 m^3$$

Last edited:


text:
# Using doc-view with auto-revert to view LaTeX PDF output in Emacs

October 3, 2009
By

(This article was first published on Blogistic Reflections, and kindly contributed to R-bloggers)

When I am authoring a $LaTeX$ document in Emacs, such as a report or my CV, it is useful for me to compile the $LaTeX$ source file periodically to see what the resulting file PDF looks like. I used to run a separate PDF viewer to look at the output, but I now have a complete Emacs solution.

### The editing environment

When writing a $LaTeX$ document, I usually want the output to be a PDF file. Accordingly, I put the following in my .emacs file.

```(setq TeX-PDF-mode t)
```

I then split my Emacs frame into two buffers vertically, using C-x 3 (see screencast below). After compiling my $LaTeX$ file with C-c C-c, I visit the resulting PDF file in the other Emacs window. The Emacs doc-view package will display the PDF file.

### Including auto-revert functionality

The final piece to the puzzle is to set files visited in doc-view-mode to auto-revert when changed on disk. That way, then I update my $LaTeX$ file and recompile with C-c C-c, the PDF in the other window will automatically update.

This is achieved by placing the following line in my .emacs.

```(add-hook 'doc-view-mode-hook 'auto-revert-mode)
```

### Screencast Example

Here is a screencast of this process in action.

This is a simple setup that I use to author reports, edit them, and see immediate updates to my output file without leaving Emacs.

R-bloggers.com offers daily e-mail updates about R news and tutorials on topics such as: Data science, Big Data, R jobs, visualization (ggplot2, Boxplots, maps, animation), programming (RStudio, Sweave, LaTeX, SQL, Eclipse, git, hadoop, Web Scraping) statistics (regression, PCA, time series, trading) and more...

Tags: , ,


text:
906 views

Consider an array $A[1...n]$. It consists of a permutation of numbers $1....n$. Now compute another array $B[1...n]$ as follows: $B[A[i]]:= i$ for all $i$. Which of the following is true?

1. $B$ will be a sorted array.
2. $B$ is a permutation of array $A$.
3. Doing the same transformation twice will not give the same array.
4. $B$ is not a permutation of array $A$.
5. None of the above.
in DS
recategorized | 906 views
0
Good question. ?
0
It consists of a permutation of numbers 1....n1....n wht does it mean

@Bikram sir
+2

It consists of a permutation of numbers 1....n1....n wht does it mean

It means all the different permutations of n numbers.

see the best answer, example is given.

0

For anybody who's still curious as to what is 'permutation of elements, its just one of the random arrangements of the n elements.

https://gateoverflow.in/260210/permutations-tifr2011-b-30

Option (b) B is a permutation of array A.

In fact, $B$ gives the reverse index of all the elements of array $A$. Since the array $A$ contains numbers $[1 .. n]$ mapped to the locations $[1 .. n]$ and $A$ is a permutation of the numbers $[1 .. n]$, the array $B$ will also be a permutation of the numbers $[1 .. n]$.

For example: $$\begin{array}{l|cccccccc} \text{index} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\[1em] \hline A & 5 & 1 & 3 & 7 & 6 & 2 & 8 & 4\\ B & 2 & 6 & 3 & 8 & 1 & 5 & 4 & 7 \end{array}$$

To see that option c is incorrect, let array $C$ be the array attained from doing the same transformation twice, that is, $C[B[i]] = i , \forall i \in [1 .. n]$. We get, $$\begin{array}{l|cccccccc} \text{index} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\[1em] \hline A & 5 & 1 & 3 & 7 & 6 & 2 & 8 & 4\\ B & 2 & 6 & 3 & 8 & 1 & 5 & 4 & 7\\ C & 5 & 1 & 3 & 7 & 6 & 2 & 8 & 4 \end{array}$$

We can see that $C = A$, which makes option c incorrect.

by Boss (22.4k points)
edited by
0
Hi,

Could you please elaborate on How Array A contains 5,1,2 ... sequence ? Is it random sequence of numbers which we got after permutations of 8 elements ?

Thanks
0
nice sagar sir
0
+1 vote

let n=2
A[1,2] and it consists of a permutation of numbers 1,2 which are
case 1: (1,2)
case 2: (2,1)
B[A[i]]:=i  for all i (GIVEN)
case 1: B[A[1]]:=1 B[1]:=1
B[A[2]]:=2 B[2]:=2 so B=(1,2)
case 2: B[A[1]]:=1 B[2]:=1
B[A[2]]:=2 B[1]:=2 so B=(2,1)
Hence array B have permutation of 1,2

Ans is B

by Active (2.6k points)
+1 vote

Let n=4 ,so indexes from 1 to 4.

Ex 1 - A[3,2,1,4] then B is B[3,2,1,4]

Ex 2 - A[4,2,1,3] then B is B[3,2,4,1]

From the examples above

Option C - Doing the same transformation twice will not give the same array. is proven false by Ex 1. As A=B.

Option A - B will be a sorted array. is proven false by Ex 2.

Option B - B is a permutation of array A.

From google - meaning of permutation - "each of several possible ways in which a set or number of things can be ordered or arranged.". As we see both Ex 1 and Ex 2 B's are permutations of their A's.

Option D - is false if option B is True.

by (31 points)
edited

–1 vote
1


text:
# Thread: Contour Integral where f(z) is not continuous

1. ## Contour Integral where f(z) is not continuous

$\int_C \frac{z+i}{z^3 + 2z^2} dz$ , where C is the circle |z|=1 traversed once counterclockwise.

ok so the parameterisation of the contour gives $e^{\pi i t}, 0 \leq t \leq 2$

but i'm not sure how to continue becuase the $f(z)= \frac{z+i}{z^3+2z^2}$ is undefined at z=0 and at z=-2

2. ## Re: Contour Integral where f(z) is not continuous

Originally Posted by linalg123
$\int_C \frac{z+i}{z^3 + 2z^2} dz$ , where C is the circle |z|=1 traversed once counterclockwise.

ok so the parameterisation of the contour gives $e^{\pi i t}, 0 \leq t \leq 2$

but i'm not sure how to continue becuase the $f(z)= \frac{z+i}{z^3+2z^2}$ is undefined at z=0 and at z=-2
You probably need to use the residue theorem.


text:
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# Max Born German-Jewish physicist and mathematician

German-Jewish physicist and mathematician
The basics
Quick Facts
Intro German-Jewish physicist and mathematician
A.K.A. ম্যাক্স বর্ন
Was Scientist Physicist Mathematician Academic Professor Educator Writer Non-fiction writer Theoretical physicist
From Germany United Kingdom
Type Academia Education Literature Mathematics Science
Gender male
Birth 11 December 1882, Wrocław, People's Republic of Poland
Death 5 January 1970, Göttingen, Germany (aged 87 years)
Star sign Sagittarius
Residence Göttingen, Germany; Edinburgh, United Kingdom; Wrocław, People's Republic of Poland; Cambridge, United Kingdom; Bad Pyrmont, German Reich
Family
 Father: Gustav Jacob Born Children: Gustav Victor Rudolf Born
The details
Biography

Max Born ([ˈmaks ˈbɔɐ̯n]; 11 December 1882 – 5 January 1970) was a German-Jewish physicist and mathematician who was instrumental in the development of quantum mechanics. He also made contributions to solid-state physics and optics and supervised the work of a number of notable physicists in the 1920s and 1930s. Born won the 1954 Nobel Prize in Physics for his "fundamental research in quantum mechanics, especially in the statistical interpretation of the wave function".

Born entered the University of Göttingen in 1904, where he found the three renowned mathematicians Felix Klein, David Hilbert, and Hermann Minkowski. He wrote his Ph.D. thesis on the subject of "Stability of Elastica in a Plane and Space", winning the University's Philosophy Faculty Prize. In 1905, he began researching special relativity with Minkowski, and subsequently wrote his habilitation thesis on the Thomson model of the atom. A chance meeting with Fritz Haber in Berlin in 1918 led to discussion of the manner in which an ionic compound is formed when a metal reacts with a halogen, which is today known as the Born–Haber cycle.

In the First World War, after originally being placed as a radio operator, he was moved to research duties regarding sound ranging due to his specialist knowledge. In 1921, Born returned to Göttingen, arranging another chair for his long-time friend and colleague James Franck. Under Born, Göttingen became one of the world's foremost centres for physics. In 1925, Born and Werner Heisenberg formulated the matrix mechanics representation of quantum mechanics. The following year, he formulated the now-standard interpretation of the probability density function for ψ*ψ in the Schrödinger equation, for which he was awarded the Nobel Prize in 1954. His influence extended far beyond his own research. Max Delbrück, Siegfried Flügge, Friedrich Hund, Pascual Jordan, Maria Goeppert-Mayer, Lothar Wolfgang Nordheim, Robert Oppenheimer, and Victor Weisskopf all received their Ph.D. degrees under Born at Göttingen, and his assistants included Enrico Fermi, Werner Heisenberg, Gerhard Herzberg, Friedrich Hund, Pascual Jordan, Wolfgang Pauli, Léon Rosenfeld, Edward Teller, and Eugene Wigner.

In January 1933, the Nazi Party came to power in Germany, and Born, who was Jewish, was suspended from his professorship at the University of Göttingen. He emigrated to the United Kingdom, where he took a job at St John's College, Cambridge, and wrote a popular science book, The Restless Universe, as well as Atomic Physics, which soon became a standard textbook. In October 1936, he became the Tait Professor of Natural Philosophy at the University of Edinburgh, where, working with German-born assistants E. Walter Kellermann and Klaus Fuchs, he continued his research into physics. Born became a naturalised British subject on 31 August 1939, one day before World War II broke out in Europe. He remained at Edinburgh until 1952. He retired to Bad Pyrmont, in West Germany, and died in hospital in Göttingen on 5 January 1970.

## Early life

Max Born was born on 11 December 1882 in Breslau, which at the time of Born's birth was part of the Prussian Province of Silesia in the German Empire, to a family of Jewish descent. He was one of two children born to Gustav Born, an anatomist and embryologist, who was a professor of embryology at the University of Breslau, and his wife Margarethe (Gretchen) née Kauffmann, from a Silesian family of industrialists. She died when Max was four years old, on 29 August 1886. Max had a sister, Käthe, who was born in 1884, and a half-brother, Wolfgang, from his father's second marriage, to Bertha Lipstein. Wolfgang later became Professor of Art History at the City College of New York.

Initially educated at the König-Wilhelm-Gymnasium in Breslau, Born entered the University of Breslau in 1901. The German university system allowed students to move easily from one university to another, so he spent summer semesters at Heidelberg University in 1902 and the University of Zurich in 1903. Fellow students at Breslau, Otto Toeplitz and Ernst Hellinger, told Born about the University of Göttingen, and Born went there in April 1904. At Göttingen he found three renowned mathematicians: Felix Klein, David Hilbert and Hermann Minkowski. Very soon after his arrival, Born formed close ties to the latter two men. From the first class he took with Hilbert, Hilbert identified Born as having exceptional abilities and selected him as the lecture scribe, whose function was to write up the class notes for the students' mathematics reading room at the University of Göttingen. Being class scribe put Born into regular, invaluable contact with Hilbert. Hilbert became Born's mentor after selecting him to be the first to hold the unpaid, semi-official position of assistant. Born's introduction to Minkowski came through Born's stepmother, Bertha, as she knew Minkowski from dancing classes in Königsberg. The introduction netted Born invitations to the Minkowski household for Sunday dinners. In addition, while performing his duties as scribe and assistant, Born often saw Minkowski at Hilbert's house.

Born's relationship with Klein was more problematic. Born attended a seminar conducted by Klein and professors of applied mathematics, Carl Runge and Ludwig Prandtl, on the subject of elasticity. Although not particularly interested in the subject, Born was obliged to present a paper. Using Hilbert's calculus of variations, he presented one in which, using a curved configuration of a wire with both ends fixed, he demonstrated would be the most stable. Klein was impressed, and invited Born to submit a thesis on the subject of "Stability of Elastica in a Plane and Space" – a subject near and dear to Klein – which Klein had arranged to be the subject for the prestigious annual Philosophy Faculty Prize offered by the University. Entries could also qualify as doctoral dissertations. Born responded by turning down the offer, as applied mathematics was not his preferred area of study. Klein was greatly offended.

Klein had the power to make or break academic careers, so Born felt compelled to atone by submitting an entry for the prize. Because Klein refused to supervise him, Born arranged for Carl Runge to be his supervisor. Woldemar Voigt and Karl Schwarzschild became his other examiners. Starting from his paper, Born developed the equations for the stability conditions. As he became more interested in the topic, he had an apparatus constructed that could test his predictions experimentally. On 13 June 1906, the rector announced that Born had won the prize. A month later, he passed his oral examination and was awarded his PhD in mathematics magna cum laude.

On graduation, Born was obliged to perform his military service, which he had deferred while a student. He found himself drafted into the German army, and posted to the 2nd Guards Dragoons "Empress Alexandra of Russia", which was stationed in Berlin. His service was brief, as he was discharged early after an asthma attack in January 1907. He then travelled to England, where he was admitted to Gonville and Caius College, Cambridge, and studied physics for six months at the Cavendish Laboratory under J. J. Thomson, George Searle and Joseph Larmor. After Born returned to Germany, the Army re-inducted him, and he served with the elite 1st (Silesian) Life Cuirassiers "Great Elector" until he was again medically discharged after just six weeks' service. He then returned to Breslau, where he worked under the supervision of Otto Lummer and Ernst Pringsheim, hoping to do his habilitation in physics. A minor accident involving Born's black body experiment, a ruptured cooling water hose, and a flooded laboratory, led to Lummer telling him that he would never become a physicist.

In 1905, Albert Einstein published his paper On the Electrodynamics of Moving Bodies about special relativity. Born was intrigued, and began researching the subject. He was devastated to discover that Minkowski was also researching special relativity along the same lines, but when he wrote to Minkowski about his results, Minkowski asked him to return to Göttingen and do his habilitation there. Born accepted. Toeplitz helped Born brush up on his matrix algebra so he could work with the four-dimensional Minkowski space matrices used in the latter's project to reconcile relativity with electrodynamics. Born and Minkowski got along well, and their work made good progress, but Minkowski died suddenly of appendicitis on 12 January 1909. The mathematics students had Born speak on their behalf at the funeral.

A few weeks later, Born attempted to present their results at a meeting of the Göttingen Mathematics Society. He did not get far before he was publicly challenged by Klein and Max Abraham, who rejected relativity, forcing him to terminate the lecture. However, Hilbert and Runge were interested in Born's work, and, after some discussion with Born, they became convinced of the veracity of his results and persuaded him to give the lecture again. This time he was not interrupted, and Voigt offered to sponsor Born's habilitation thesis. Born subsequently published his talk as an article on "The Theory of Rigid Bodies in the Kinematics of the Relativity Principle" (German: Die Theorie des starren Elektrons in der Kinematik des Relativitätsprinzips), which introduced the concept of Born rigidity. On 23 October Born presented his habilitation lecture on the Thomson model of the atom.

## Career

### Berlin and Frankfurt

Born settled in as a young academic at Göttingen as a privatdozent. In Göttingen, Born stayed at a boarding house run by Sister Annie at Dahlmannstraße 17, known as El BoKaReBo. The name was derived from the first letters of the last names of its boarders: "El" for Ella Philipson (a medical student), "Bo" for Born and Hans Bolza (a physics student), "Ka" for Theodore von Kármán (a Privatdozent), and "Re" for Albrecht Renner (another medical student). A frequent visitor to the boarding house was Paul Peter Ewald, a doctoral student of Arnold Sommerfeld on loan to Hilbert at Göttingen as a special assistant for physics. Richard Courant, a mathematician and Privatdozent, called these people the "in group."

In 1912, Born met Hedwig (Hedi) Ehrenberg, the daughter of a Leipzig University law professor, and a friend of Carl Runge's daughter Iris. She was of Jewish background on her father's side, although he had become a practising Lutheran when he got married, as did Max's sister Käthe. Despite never practising his religion, Born refused to convert, and his wedding on 2 August 1913 was a garden ceremony. However, he was baptised as a Lutheran in March 1914 by the same pastor who had performed his wedding ceremony. Born regarded "religious professions and churches as a matter of no importance". His decision to be baptised was made partly in deference to his wife, and partly due to his desire to assimilate into German society. The marriage produced three children: two daughters, Irene, born in 1914, and Margarethe (Gritli), born in 1915, and a son, Gustav, born in 1921. Through marriage, Born is related to jurists Victor Ehrenberg, his father-in-law, and Rudolf von Jhering, his wife's maternal grandfather, as well as Hans Ehrenberg, and is a great uncle of British comedian Ben Elton.

By the end of 1913, Born had published 27 papers, including important work on relativity and the dynamics of crystal lattices (3 with Theodore von Karman), which became a book. In 1914, received a letter from Max Planck explaining that a new professor extraordinarius chair of theoretical physics had been created at the University of Berlin. The chair had been offered to Max von Laue, but he had turned it down. Born accepted. The First World War was now raging. Soon after arriving in Berlin in 1915, he enlisted in an Army signals unit. In October, he joined the Artillerie-Prüfungs-Kommission, the Army's Berlin-based artillery research and development organisation, under Rudolf Ladenburg, who had established a special unit dedicated to the new technology of sound ranging. In Berlin, Born formed a lifelong friendship with Einstein, who became a frequent visitor to Born's home. Within days of the armistice in November 1918, Planck had the Army release Born. A chance meeting with Fritz Haber that month led to discussion of the manner in which an ionic compound is formed when a metal reacts with a halogen, which is today known as the Born–Haber cycle.

Even before Born had taken up the chair in Berlin, von Laue had changed his mind, and decided that he wanted it after all. He arranged with Born and the faculties concerned for them to exchange jobs. In April 1919, Born became professor ordinarius and Director of the Institute of Theoretical Physics on the science faculty at the University of Frankfurt am Main. While there, he was approached by the University of Göttingen, which was looking for a replacement for Peter Debye as Director of the Physical Institute. "Theoretical physics," Einstein advised him, "will flourish wherever you happen to be; there is no other Born to be found in Germany today." In negotiating for the position with the education ministry, Born arranged for another chair, of experimental physics, at Göttingen for his long-time friend and colleague James Franck.

### Göttingen

For the 12 years Born and Franck were at the University of Göttingen (1921 to 1933), Born had a collaborator with shared views on basic scientific concepts — a benefit for teaching and research. Born's collaborative approach with experimental physicists was similar to that of Arnold Sommerfeld at the University of Munich, who was ordinarius professor of theoretical physics and Director of the Institute of Theoretical Physics — also a prime mover in the development of quantum theory. Born and Sommerfeld collaborated with experimental physicists to test and advance their theories. In 1922, when lecturing in the United States at the University of Wisconsin–Madison, Sommerfeld sent his student Werner Heisenberg to be Born's assistant. Heisenberg returned to Göttingen in 1923, where he completed his habilitation under Born in 1924, and became a privatdozent at Göttingen.

In 1919 and 1920 Max Born became displeased about the large number of objections against Einstein's relativity, and gave speeches in the winter of 1919 in support of Einstein. Born received pay for his relativity speeches which helped with expenses through the year of rapid inflation. The speeches in German language became a book published in 1920 of which Einstein received the proofs before publication. A third edition was published in 1922 and an English translation was published in 1924. Born represented light speed as a function of curvature, "the velocity of light is much greater for some directions of the light ray than its ordinary value c, and other bodies can also attain much greater velocities."

In 1925, Born and Heisenberg formulated the matrix mechanics representation of quantum mechanics. On 9 July, Heisenberg gave Born a paper entitled Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen ("Quantum-Theoretical Re-interpretation of Kinematic and Mechanical Relations") to review, and submit for publication. In the paper, Heisenberg formulated quantum theory, avoiding the concrete, but unobservable, representations of electron orbits by using parameters such as transition probabilities for quantum jumps, which necessitated using two indexes corresponding to the initial and final states. When Born read the paper, he recognized the formulation as one which could be transcribed and extended to the systematic language of matrices, which he had learned from his study under Jakob Rosanes at Breslau University.

Up until this time, matrices were seldom used by physicists; they were considered to belong to the realm of pure mathematics. Gustav Mie had used them in a paper on electrodynamics in 1912, and Born had used them in his work on the lattices theory of crystals in 1921. While matrices were used in these cases, the algebra of matrices with their multiplication did not enter the picture as they did in the matrix formulation of quantum mechanics. With the help of his assistant and former student Pascual Jordan, Born began immediately to make a transcription and extension, and they submitted their results for publication; the paper was received for publication just 60 days after Heisenberg's paper. A follow-on paper was submitted for publication before the end of the year by all three authors. The result was a surprising formulation:

${\displaystyle pq-qp={h \over 2\pi i}I}$

where p and q were matrices for location and momentum, and I is the identity matrix. Note that the left hand side of the equation is not zero because matrix multiplication is not commutative. This formulation was entirely attributable to Born, who also established that all the elements not on the diagonal of the matrix were zero. Born considered that his paper with Jordan contained "the most important principles of quantum mechanics including its extension to electrodynamics." The paper put Heisenberg's approach on a solid mathematical basis.

Born was surprised to discover that Paul Dirac had been thinking along the same lines as Heisenberg. Soon, Wolfgang Pauli used the matrix method to calculate the energy values of the hydrogen atom and found that they agreed with the Bohr model. Another important contribution was made by Erwin Schrödinger, who looked at the problem using wave mechanics. This had a great deal of appeal to many at the time, as it offered the possibility of returning to deterministic classical physics. Born would have none of this, as it ran counter to facts determined by experiment. He formulated the now-standard interpretation of the probability density function for ψ*ψ in the Schrödinger equation, which he published in July 1926.

In a letter to Born on 4 December 1926, Einstein made his famous remark regarding quantum mechanics:

Quantum mechanics is certainly imposing. But an inner voice tells me that it is not yet the real thing. The theory says a lot, but does not really bring us any closer to the secret of the 'old one'. I, at any rate, am convinced that He is not playing at dice.

This quotation is often paraphrased as 'God does not play dice'.

In 1928, Einstein nominated Heisenberg, Born, and Jordan for the Nobel Prize in Physics, but Heisenberg alone won the 1932 Prize "for the creation of quantum mechanics, the application of which has led to the discovery of the allotropic forms of hydrogen", while Schrödinger and Dirac shared the 1933 Prize "for the discovery of new productive forms of atomic theory". On 25 November 1933, Born received a letter from Heisenberg in which he said he had been delayed in writing due to a "bad conscience" that he alone had received the Prize "for work done in Göttingen in collaboration — you, Jordan and I." Heisenberg went on to say that Born and Jordan's contribution to quantum mechanics cannot be changed by "a wrong decision from the outside." In 1954, Heisenberg wrote an article honouring Planck for his insight in 1900, in which he credited Born and Jordan for the final mathematical formulation of matrix mechanics and Heisenberg went on to stress how great their contributions were to quantum mechanics, which were not "adequately acknowledged in the public eye."

Those who received their Ph.D. degrees under Born at Göttingen included Max Delbrück, Siegfried Flügge, Friedrich Hund, Pascual Jordan, Maria Goeppert-Mayer, Lothar Wolfgang Nordheim, Robert Oppenheimer, and Victor Weisskopf. Born's assistants at the University of Göttingen's Institute for Theoretical Physics included Enrico Fermi, Werner Heisenberg, Gerhard Herzberg, Friedrich Hund, Pascual Jordan, Wolfgang Pauli, Léon Rosenfeld, Edward Teller, and Eugene Wigner. Walter Heitler became an assistant to Born in 1928, and completed his habilitation under him in 1929. Born not only recognised talent to work with him, but he "let his superstars stretch past him; to those less gifted, he patiently handed out respectable but doable assignments." Delbrück, and Goeppert-Mayer went on to win Nobel Prizes.

## Later life

In January 1933, the Nazi Party came to power in Germany. In May, Born became one of six Jewish professors at Göttingen who were suspended with pay; Franck had already resigned. In twelve years they had built Göttingen into one of the world's foremost centres for physics. Born began looking for a new job, writing to Maria Göppert-Mayer at Johns Hopkins University and Rudi Ladenburg at Princeton University. He accepted an offer from St John's College, Cambridge. At Cambridge, he wrote a popular science book, The Restless Universe, and a textbook, Atomic Physics, that soon became a standard text, going through seven editions. His family soon settled into life in England, with his daughters Irene and Gritli becoming engaged to Welshman Brinley (Bryn) Newton-John (Olivia Newton-John's parents; Born is Olivia's grandfather and Irene is her mother) and Englishman Maurice Pryce respectively.

Born's position at Cambridge was only a temporary one, and his tenure at Göttingen was terminated in May 1935. He therefore accepted an offer from C. V. Raman to go to Bangalore in 1935. Born considered taking a permanent position there, but the Indian Institute of Science did not create an additional chair for him. In November 1935, the Born family had their German citizenship revoked, rendering them stateless. A few weeks later Göttingen cancelled Born's doctorate. Born considered an offer from Pyotr Kapitsa in Moscow, and started taking Russian lessons from Rudolf Peierls's Russian-born wife Genia. But then Charles Galton Darwin asked Born if he would consider becoming his successor as Tait Professor of Natural Philosophy at the University of Edinburgh, an offer that Born promptly accepted, assuming the chair in October 1936.

In Edinburgh, Born promoted the teaching of mathematical physics. He had two German assistants, E. Walter Kellermann and Klaus Fuchs, and one Scottish assistant, Robert Schlapp, and together they continued to investigate the mysterious behaviour of electrons. Born became a Fellow of the Royal Society of Edinburgh in 1937, and of the Royal Society of London in March 1939. During 1939, he got as many of his remaining friends and relatives still in Germany as he could out of the country, including his sister Käthe, in-laws Kurt and Marga, and the daughters of his friend Heinrich Rausch von Traubenberg. Hedi ran a domestic bureau, placing young Jewish women in jobs. Born received his certificate of naturalisation as a British subject on 31 August 1939, one day before the Second World War broke out in Europe.

Born remained at Edinburgh until he reached the retirement age of 70 in 1952. He retired to Bad Pyrmont, in West Germany, in 1954. In October, he received word that he was being awarded the Nobel Prize. His fellow physicists had never stopped nominating him. Franck and Fermi had nominated him in 1947 and 1948 for his work on crystal lattices, and over the years, he had also been nominated for his work on solid state physics, quantum mechanics and other topics. In 1954, he received the prize for "fundamental research in Quantum Mechanics, especially in the statistical interpretation of the wave function" — something that he had worked on alone. In his Nobel lecture he reflected on the philosophical implications of his work:

I believe that ideas such as absolute certitude, absolute exactness, final truth, etc. are figments of the imagination which should not be admissible in any field of science. On the other hand, any assertion of probability is either right or wrong from the standpoint of the theory on which it is based. This loosening of thinking (Lockerung des Denkens) seems to me to be the greatest blessing which modern science has given to us. For the belief in a single truth and in being the possessor thereof is the root cause of all evil in the world.

In retirement, he continued scientific work, and produced new editions of his books. In 1955 he became one of signatories to the Russell-Einstein Manifesto. He died at age 87 in hospital in Göttingen on 5 January 1970, and is buried in the Stadtfriedhof there, in the same cemetery as Walther Nernst, Wilhelm Weber, Max von Laue, Otto Hahn, Max Planck, and David Hilbert.

## Personal life

Born's wife Hedwig (Hedi) Martha Ehrenberg (1891–1972) was a daughter of the jurist Victor Ehrenberg and Elise von Jhering (a daughter of the jurist Rudolf von Jhering). Born was survived by his wife Hedi and their children Irene, Gritli and Gustav. Singer Olivia Newton-John is a daughter of Irene (1914-2003). The actor Max Born (Fellini Satyricon) is his grandson. His great-grandchildren include songwriter Brett Goldsmith, singer Tottie Goldsmith, racing car driver Emerson Newton-John, and singer Chloe Rose Lattanzi.

## Awards and honors

• 1934 – Stokes Medal of Cambridge
• 1939 – Fellow of the Royal Society
• 1945 – Makdougall–Brisbane Prize of the Royal Society of Edinburgh
• 1945 – Gunning Victoria Jubilee Prize of the Royal Society of Edinburgh
• 1948 – Max Planck Medaille der Deutschen Physikalischen Gesellschaft
• 1950 – Hughes Medal of the Royal Society of London
• 1953 – Honorary citizen of the town of Göttingen
• 1954 – Nobel Prize in Physics The award was for Born's fundamental research in quantum mechanics, especially for his statistical interpretation of the wavefunction.
• 1954 – Nobel Prize Banquet Speech
• 1954 – Born Nobel Prize Lecture
• 1956 – Hugo Grotius Medal for International Law, Munich
• 1959 – Grand Cross of Merit with Star of the Order of Merit of the German Federal Republic
• 1972 – Max Born Medal and Prize was created by the German Physical Society and the British Institute of Physics. It is awarded annually.
• 1982 – Ceremony at the University of Göttingen in the 100th Birth Year of Max Born and James Franck, Institute Directors 1921–1933.
• 1991 – Max-Born-Institut für Nichtlineare Optik und Kurzzeitspektroskopie [de] – Institute named in his honor.
• 2017 – On 11 December 2017, Google showed a Google doodle, designed by Kati Szilagyi, in honouring the 135th birth anniversary of Born.
The contents of this page are sourced from Wikipedia article on 08 Mar 2020. The contents are available under the CC BY-SA 4.0 license.


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# spaCy - Text Preprocessing - Keeping "Pronouns" in text

I am fairly new to machine learning and NLP in general. I am trying to wrap my head around how to do proper text pre-processing (cleaning the text).

I have built a custom text classification model. I have below method that I run on all input text, before serving it to my model. (both in training and testing).

The method will remove stopwords, punctuations and lemmatize the text.

import spacy
from spacy.lang.en.stop_words import STOP_WORDS
import string

def normalize(text, lowercase, remove_stopwords, remove_punctuation):
nlp = spacy.load("en_core_web_sm", disable=['parser', 'tagger', 'ner'])
stops = spacy.lang.en.stop_words.STOP_WORDS

if lowercase:
text = text.lower()
text = nlp(text)
if remove_punctuation:
text = [t for t in text if t.text not in string.punctuation]
lemmatized = list()
for word in text:
lemma = word.lemma_.strip()
if lemma:
if not remove_stopwords or (remove_stopwords and lemma not in stops):
lemmatized.append(lemma)

return " ".join(lemmatized)

Consider below input string:

Input: You're very beautiful!

If I clean that text, using my method:

test_text = "You're very beautiful!"
test_text = normalize(test_text, lowercase=True, remove_stopwords=True, remove_punctuation=True)

It will return: -PRON- beautiful

Is this a "good" way to clean the text? I notice that -PRON- is kept and therefore also used when training the model (same goes for when testing the model, as I use the same normalize method).

Should the -PRON- also be removed from the text? And should I complete even more text preprocessing?

• Thank you for your reply! I'm glad I am on the right path. I have a few counter questions. - Why is it faster to load the nlp() object outside the function call? - How can I prepare two models to figure out what model works the best? (One with pre processed data and one without). Does spaCy offer some insights into the models? May 22 '20 at 13:03


text:
# rmLB_sm: Sample from the matrix Langevin-Bingham on the Stiefel... In SMFilter: Filtering Algorithms for the State Space Models on the Stiefel Manifold

## Description

This function draws a sample from the matrix Langevin-Bingham on the Stiefel manifold.

## Usage

 1 rmLB_sm(num, mJ, mH, mC, mX, ir)

## Arguments

 num number of observations or sample size. mJ symmetric ip*ip matrix mH symmetric ir*ir matrix mC ip*ir matrix mX ip*ir matrix, the initial value ir ir

## Details

The matrix Langevin-Bingham distribution on the Stiefel manifold has the density kernel:

f(X) \propto \mathrm{etr}\{ H X' J X + C' X \}

where X satisfies X'X = I_r, and H and J are symmetric matrices.

## Value

an array containing a sample of draws from the matrix Langevin-Bingham on the Stiefel manifold.

## Author(s)

Yukai Yang, yukai.yang@statistik.uu.se

#' @section References: Hoff, P. D. (2009) "Simulation of the Matrix Bingham<e2><80><94>von Mises<e2><80><94>Fisher Distribution, With Applications to Multivariate and Relational Data", Journal of Computational and Graphical Statistics, Vol. 18, pp. 438-456.

SMFilter documentation built on May 1, 2019, 8:01 p.m.


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## A powerful symmetry formula for definite integrals calculation

Let $f(x)$ be an even function, integrable in $\left [ 0,\alpha \right ]$ and $g(x)$ another function such that $g(-x)=\frac{1}{g(x)}$, $g(x)\neq0$, $g(x)\neq-1$ in $\left [ -\alpha,\alpha \right ]$.

It can be proved that:

$$\int_{-\alpha }^{\alpha }\frac{f(x)}{1+g(x)}dx=\int_{0}^{\alpha}f(x)dx (1)$$

It turns out to be an interesting formula in the sense that it allows to simplify and speed up some integrals calculations.

Let’s see some examples.

1. Let $f(x)=x^2$ and $g(x)=e^x$. In this case (1) becomes $$\int_{-1}^{1}\frac{x^2}{1+e^x}dx=\int_{0}^{1}x^2dx=\left [ \frac{1}{3}x^3 \right ]_{0}^{1}=\frac{1}{3}$$

2. Let now $f(x)=x^2+cosx$ and $g(x)=e^{xcosx}$. We have$$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{x^2+cosx}{1+e^{xcosx}}dx=\int_{0}^{\pi/2}(x^2+cosx)dx=\left [ \frac{1}{3}x^3 +sinx\right ]_{0}^{\pi/2}=\frac{\pi}{24}+1$$

3. Finally, let $f(x)=x^2$ and $g(x)=(1+x^2)^x$. Formula (1) becomes$$\int_{-1}^{1}\frac{x^2}{1+(1+x^2)^{x}}dx=\int_{0}^{1}x^2dx=\left [ \frac{1}{3}x^3\right ]_{0}^{1}=\frac{1}{3}$$

Proof

It’s a straightforward calculation.

$$\int_{-\alpha }^{\alpha }\frac{f(x)}{1+g(x)}dx=\int_{-\alpha }^{0}\frac{f(x)}{1+g(x)}dx+\int_{0}^{\alpha }\frac{f(x)}{1+g(x)}dx=$$

$$=\int_{-\alpha }^{0}\frac{f(x)\frac{1}{g(x)}}{1+\frac{1}{g(x)}}dx+\int_{0}^{\alpha}\frac{f(x)}{1+g(x)}dx=\int_{-\alpha }^{0}\frac{f(x)(\frac{1}{g(x)}+1-1)}{1+\frac{1}{g(x)}}dx+\int_{0}^{\alpha}\frac{f(x)}{1+g(x)}dx=$$

$$=\int_{-\alpha }^{0}\left [f(x)-\frac{f(x)}{1+\frac{1}{g(x)}} \right ]dx+\int_{0}^{\alpha}\frac{f(x)}{1+g(x)}dx=\int_{-\alpha }^{0}f(x)dx-\int_{-\alpha }^{0}\frac{f(x)}{1+\frac{1}{g(x)}}dx+\int_{0}^{\alpha}\frac{f(x)}{1+g(x)}dx=$$

applying the variable change $t=-x$ ($dt=-dx$) in the second integral and using the fact that $f(x)$ is even in the first integral,

$$=\int_{0}^{\alpha}f(x)dx-\int_{0}^{\alpha}\frac{f(t)}{1+g(t)}dt+\int_{0}^{\alpha}\frac{f(x)}{1+g(x)}dx=\int_{0}^{\alpha}f(x)dx$$

and the formula is proven.