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# IPMAT Question Paper 2019 | IPM Indore Quants

###### IPMAT Sample Paper | IPMAT Question Paper | Question 27

IPMAT 2019 Question Paper IPM Indore Quantitative Ability. Solve questions from IPMAT 2019 Question Paper from IPM Indore and check the solutions to get adequate practice. The best way to ace IPMAT is by solving IPMAT Question Paper. To solve other IPMAT Sample papers, go here: IPM Sample Paper

Question 27 : If x ∈ (a, b) satisfies the inequality $$frac{x - 3}{x^{2} + 3x + 2} \geq 1,\\$ then the largest possible value of b - a is 1. 3 2. 1 3. 2 4. No real valued x satisfies the inequality ## Best CAT Online Coaching Try upto 40 hours for free Learn from the best! #### 2IIM : Best Online CAT Coaching. ### Video Explanation ## Best CAT Coaching in Chennai #### CAT Coaching in Chennai - CAT 2022Limited Seats Available - Register Now! The question is "If x ∈$a, b) satisfies the inequality $$frac{x - 3}{x^{2} + 3x + 2} \geq 1,\\$ then the largest possible value of b - a is" ##### Hence, the answer is 1 Choice B is the correct answer ###### Best Indore IPM & Rohtak IPM CoachingSignup and sample 9 full classes for free. Register now! ###### Already have an Account? ##### Where is 2IIM located? 2IIM Online CAT Coaching A Fermat Education Initiative, 58/16, Indira Gandhi Street, Kaveri Rangan Nagar, Saligramam, Chennai 600 093 ##### How to reach 2IIM? Mobile:$91) 99626 48484 / 94459 38484
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Simple beam support reaction

The calculator gives support reactions of the simple beam under various vertical load.

Anton

Created: 2021-07-28 17:55:14, Last updated: 2021-07-28 18:45:36

This online calculator finds support reactions of the simple beam under transverse loads. A simple beam is a rectangular beam, fixed on two supports: one - pinned ('A' support), the other - roller ('B' support). The calculator displays support reactions VA and VB, moment equilibrium equations, and draws the beam system. Note: The distance to 'A' support value must be negative for the force acting to the left of the 'A' support. The theory and calculation formulas are below the calculator.

Simple beam support reactions

Distance to the support ALoadValueDirectionMoment directionRangeDifference
Items per page:

Digits after the decimal point: 2
Support reaction A, kN

Support reaction B, kN

Moment equilibrium equation

The file is very large. Browser slowdown may occur during loading and creation.
The file is very large. Browser slowdown may occur during loading and creation.
The file is very large. Browser slowdown may occur during loading and creation.

Support reactions

Under the influence of loads, opposing forces occur in the beam supports, called support reactions. These forces depend on the type of load and the type of supports.

The roller support (in our model designated as "B") allows the beam to move freely in the horizontal plane and prevents vertical movement. Therefore, it has only a vertical reaction VB.

A pinned support is attached to the beam. It prevents its horizontal and vertical movement. In the presence of forces acting on the beam in the horizontal plane, this support additionally produces a horizontal reaction. Our model allows only transverse forces acting on the beam, so the horizontal reaction of the 'A' support will always be zero. We denote the vertical reaction of the support A by VA .

Equilibrium Equations

As we know from statics, all forces and moments of forces in a fixed system are balanced. Therefore, the sums of forces and moments at any point of this system are equal to zero.
All forces under a transverse load on a simple beam act parallel to the Y-axis. Therefore, one can draw only two independent equilibrium equations for the projection of forces on the Y-axis. It is quite enough to find two unknown support reactions: VA and VB.
When writing equilibrium equations, we have a choice:

• compose one equation of equilibrium of the projection of forces and one equation of the moments balance at some point
• compose two equations of equilibrium of moments at two points.

Let's use the second method, and leave the first one to check the result.
It is most convenient to draw up equations for points of supports A and B:
$\sum M_A = 0 \\ \sum M_B = 0$

Recall that the moment of force at a certain point is the product of the force F by the shortest distance from this point to the line of action of the force (lever) l: $M = F l$

Based on this, the equilibrium equations for the moments at points A and B for the system of transferse forces F1 ... Fn acting on the beam take the form:
$\sum_ {i = 0}^n {s_{i_A} F_i l_{i_A}} = 0 \\ \sum_ {i = 0}^n {s_{i_B} F_i l_{i_B}} = 0$
Where Fi is the magnitude of the force or support reaction in Newtons. liA and liB - the lever in meters (the shortest distance from the point of application of force i to the support A or B respectively). siA and siB are the sign of the moment of force i at points A or B, respectively.
The rule for choosing the signs of the moments: the sign is positive (+1) for the moment bending the beam around the selected point clockwise ↻ and negative (-1) for the opposite direction ↺. You can also choose opposite values, then the equations will take a slightly different form, but the final result will not change.

For example, for the system of forces shown by the figure above, the equilibrium equations can be written as follows:
$-F_ {1} -4.5 F_ {2} +7 F_ {3} -6 V_B = 0 \\ -7F_ {1} +1.5 F_ {2} + F_ {3} +6 V_A = 0$
Calculating, we get the values of the support reactions: V A = 15.42 and V B = 14.58. Let's check that the sum of all forces is equal to zero (for forces acting downwards - the sign is positive, for those acting upwards - negative):
$F_1-F_2 + F_3 - V_A - V_B = 20-15 + 25-15.42-14.58 = 0$
When composing the equations, we assume the reactions of both supports are directed upward. Calculating, it may turn out that the reaction of a support turns out to be negative. This means that the reaction of such a support must point opposite direction. (The sum of the moments of forces acting on the beam tries to tear it away from the support.)

Sometimes it is required to specify a load distributed in a certain way over a segment of length a. For the purpose of calculating the support reactions, such a load can be replaced by its resultant force. The point of application of such a force is located at the center of mass of the distributed load shape. The modulus is calculated by integrating the load distribution function in a given range. For simple functions, the module can be easily expressed in terms of a given load rate.

The table below shows the formulas for the modulus of the concentrated resultant force and the points of its application for all types of distributed loads supported by the calculator:

Uniform $q a$ 1/2 a
Linearly decreasing $\frac{q a}2$ 1/3 a
Linearly increasing $\frac{q a}2$ 2/3 a

In the formulas above, q is the load intensity in N / m, a is the range of the distributed load. Note: the point of application of the force is measured from the beginning of the range of the distributed load. The intensity of a linearly distributed load is set for the section of maximum load (we assume that at the point of minimum, intensity = 0).
After calculating the modulus and lever of the resultant distributed load, they can be substituted into the equations of moments, just as we did with concentrated forces.

Concentrated moment

Another way to set the load in the calculator is by using a moment in Nm applied to a point. The value of the concentrated moment is added to the equilibrium equations with a sign determined by the direction of the moment according to the rule of signs. The point of application of the concentrated moment is irrelevant for calculating the support reactions.

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PLANETCALC, Simple beam support reaction

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# DSLA v2.0

Written by DSLA Core Team on

## GRC Automation Middleware

DSLA enables stakeholders on different sides of a service or event to trade risk using Decentralized Service Level Agreements (SLA), and with the roll out of DSLA v2.0 SLA stakeholders have more flexibility, control and a better overall experience than ever before.

We’re excited to cover the improvements and changes in v2.0 that make it more streamlined and intuitive for stakeholders, and make DSLA GRC Automation Middleware more attractive for developers and integrators.

## A Better Way to Manage Risk

Today DSLA v2.0 is live on four networks: Ethereum, Avalanche, Arbitrum and Polygon, with work ongoing to port DSLA to Solana. The DSLA token is also available on Fantom, Harmony and Binance Smart Chain, though v2.0 of the protocol is not being deployed on these chains.

The release of v2.0 also enables a number of new flagship use cases, like NFT Floor Price Protection, that broaden the general audience of end-users that could make use of a DSLA risk management product.

It’s safe to say that with v2.0, swapping risk with DSLA is more accessible than ever.

Let’s look at some key structural changes and improvements DSLA v2.0 incorporates:

• SLA Owners, SLA Liquidity Providers and SLA Users, three distinct SLA stakeholders with different roles, responsibilities and benefits;
• Claim Fees, improving the overall tokenomics of DSLA and offsetting the up front verification fee deposit required of SLA Owners;
• Multi-provider SLA Support, enabling many-to-many SLA’s where anyone can sign on to back an SLA on the Provider side;
• Deviation Based Rewards, making compensations and rewards proportional to performance, and incentivizing providers to outperform;
• Severity Levels and Penalties, allowing the setup of SLA’s that support fine tuned compensations according to the severity of failure, using pre-agreed severity levels
• Introduction of Branded Service Credits, modifying the previous DSLA-SP and DSLA-LP position tokens to be Branded Service Credits specific to each use case;
• Instant Claim Availability, claim as you go for compensation or rewards you are entitled to;
• SLA’s Don’t Terminate on Breach, providing a more intuitive user experience, SLA’s no longer terminate on the first breach, but stay active through the whole term set when the SLA contract was created;
• More Flexible SLA Policies, SLA’s can be setup to measure performance using any of the following operators: >, >=, <, <=, = and !=;
• Integration Tests and Unit Tests, improved integration and unit tests, ensuring DSLA Protocol performs as expected and making the Developer Toolkit (DTK) more robust and developer friendly;
• Refactoring, Gas Optimizations and Bug Fixes, improving basic protocol operation;
• Audits by Hacken and Astra

## Risk Prediction Markets and SLA Stakeholders

Prediction Markets let you trade the outcome of real world events. We now have three distinct stakeholders within DSLA Protocol who play a role in trading risk using DSLA GRC Automation Middleware and the DSLA.network dApp.

With DSLA, service stakeholders can trade third-party risk with each other, by staking on the outcome of periodic events, called SLA verifications.

SLA Owners: Create an SLA contract that SLA Liquidity Providers and SLA Users can sign onto to trade risk. The SLA Owner sets the terms of the SLA contract, and is also required to deposit DSLA tokens to pay for the SLA verifications. SLA Owners in v2.0 are entitled to a claim fee of 0.3% for all claims paid from the SLA contract. Usually an SLA Owner will also create the compensation pool and be an SLA Liquidity Provider.

SLA Liquidity Providers: Contribute to the compensation pool, backing that the performance targets set in an SLA contract will be met. SLA Liquidity Providers trade risk with SLA Users. When the SLA targets are met, SLA Liquidity Providers may claim a reward from the user pool.

SLA Users: Contribute to the user pool, hedging risk that the targets set in an SLA contract are not met. SLA Users trade risk with SLA Liquidity Providers. If the SLA targets are not met, SLA Users may claim a compensation.

## A Deeper Look at v2.0 Enhancements

Okay, so let’s take a deeper look at some of the major changes, improvements and enhancements that v2.0 is shipping. 👇

### What’s That About Claim Fees?

The introduction of Claim Fees improves overall protocol tokenomics and incentivizes stakeholders to create SLAs that liquidity providers and users will want to sign onto. With the new claim fee structure, all compensation and reward claims are subject to the following claim fees:

• 99.55% sent to the SLA Stakeholder (the SLA User or SLA Liquidity Provider) making the claim
• 0.3% claim fee sent to the SLA Owner
• 0.15% claim fee sent to DSLA Core Team

The claim fee incentivizes the creation of SLA’s that other SLA Liquidity Providers and SLA Users will want to sign onto. The more stakeholders that sign onto and use an SLA, the more fees will be generated.

All SLA Owners have to stake DSLA tokens to pay for periodic verifications when they create an SLA. The 0.3% claim fee is meant to offset this upfront cost, with upside that claim fees can easily exceed upfront verification fees.

Verification fees are handled the same in v2.0 as they have been previously with 25% of the fee going to the user who performs the verification, 25% going to the developer who created the use case, 25% to the DSLA Core Team and 25% burned in the DSLA Bonfire Burn.

### Many-to-Many SLA’s

Allowing multiple SLA Liquidity Providers to sign onto the same SLA makes the protocol more efficient and easier to scale. Now SLA Liquidity Providers who just want to back a certain SLA can sign onto it and provide liquidity, without having to necessarily create their own SLA.

This makes Decentralized SLA’s function more like typical DeFi liquidity pools that many liquidity providers would be familiar with. By allowing anyone to assess current SLA’s in the marketplace and provide liquidity to make a return, without having to issue a new SLA and attract SLA Users, we reduce friction and overhead for SLA Liquidity Providers.

### Deviation Based Rewards

DSLA v2.0 modifies how rewards and compensations are calculated so that they scale according to performance. What does this mean?

When a Service Level Indicator (SLI) verification is done, DSLA checks actual performance against the Service Level Objective (SLO), which is the target set in the SLA. The difference between actual and target, as a percentage, is the deviation, which is then used in the rewards or compensations calculation.

So when actual performance very close to the target performance, rewards or compensations are smaller. If there is a large deviation then rewards or compensations are also larger.

In practice this means that service providers and SLA Liquidity Providers are incentivized to outperform the SLO target, because the better they perform the more rewards they earn (and the worse they perform the more compensations are paid out).

For SLA Users compensations and premiums paid will also scale according to deviation. The worse a provider performs, the larger the compensation for the SLA User. On the flip side, smaller premiums will be deducted from the funds the SLA User staked in the user pool if the provider just meets SLA targets.

In practice SLA Users will often be able to claim back a portion of their staked funds at the end of the SLA term, if the provider met, but did not really exceed, SLA targets.

## Severity Levels and Penalties

Allows. For example a Third Party Logistics (3PL) provider and one of their customers may want to set a SLA with severity levels that have structured penalties if a shipment is One Hour Late, 12 Hours+ Late, One Day+ Late, or One Week+ Late. With Severity Levels this could easily be accomplished.

DSLA is utilizing this new Severity Level feature in our new Parametric Staking product, where rewards are adjusted according to DSLA token price performance as follows:

• Severity 1: Any increase under 100% results in a 1% deviation
• Severity 2: Any increase of 100%+ results in a 10% deviation
• Severity 3: Any increase of 250%+ results in a 25% deviation

Severity Levels are an excellent and necessary feature to make DSLA more flexible and broadly appealing.

### Branded Service Credits

The tokens to represent SLA User and SLA Liquidity Provider stakes in each SLA have been updated to be more reflective of the SLA contract type and how risk is being swapped using DSLA contracts.

Service credit tokens will now be specific to the use case, retiring the standard DSLA-SP and DSLA-LP tokens used previously. New use cases and SLA contract types will be able to assign the name for their specific Branded Service Credits.

For example, if we have a staking performance SLA for guaranteed APR/APY returns, instead of SLA Users getting issued a generic DSLA-SP token to represent their stake, they could receive APRMISS tokens. APRMISS tokens would be entitled to a claim if the APR target in the SLA is missed. It’s more practical and intuitive than “short” tokens, when regular users aren’t shorting anything, they are just swapping risk to receive protection.

In the future, Branded Service Credit names will be one of a number of protocol specifications that will be approved via governance. Governance rights will be granted based on NFTs earned in the DSLA Metaverse and by using DSLA to hedge risk.

### Instant Claim Availability

SLA Liquidity Providers and SLA Users both stake tokens through the term of the SLA when they sign onto an SLA contract. With DSLA v2.0, however, if a stakeholder is entitled to compensations or rewards after periodic verifications, they may claim those funds.

### SLA’s Don’t Terminate on Breach

SLA contracts no longer terminate on the first SLO breach. This simple change makes the way SLA contracts work much more intuitive and will make managing SLA contracts much more streamlined.

Prior to DSLA v2.0, if a periodic verification was performed and the SLA was found to have not met targets, compensations would be made available to SLA Users and the SLA contract would terminate.

A new SLA would then need to be created and signed onto again by all parties to hedge and trade risk. Now stakeholders can choose an SLA, sign on for the entire term, and claim compensation or rewards as they go, giving a much smoother user experience for all involved.

### More Flexible SLA Policies

DSLA now supports SLO targets to be set using any of the following operators: >, >=, <, <=, = and !=. This gives SLA Owners more flexibility in how they want to create SLAs and measure performance in comparing SLI and SLO.

### Enhanced Integration Tests and Unit Tests

The core team put a lot of effort developing comprehensive integration tests and unit tests to ensure the protocol works as expected with the v2.0 upgrades.

Including these tests in the DTK provides a robust framework for other developers who create a use case or integrate DSLA GRC Automation Middleware into their application. Developers using the DTK now have a great testing toolkit to ensure everything works as expected.

The enhanced integration tests and unit tests will make developing and deploying new use cases much more streamlined.

## Security Audits & Bug Bounty

As always, security is paramount and we continue to put it first. With the significant structural changes and additions to DSLA in v2.0 we have engaged external auditors to review the code for any vulnerabilities.

Our security process for DSLA v2.0 includes:

• An audit from leading smart contract auditor Hacken
• An audit from leading security firm Astra
• A security bug bounty campaign live on Immunefi
• Enhanced integration test and unit test framework developerd for v2.0

# 💡DSLA Protocol

### GRC Automation Middleware

DSLA’s approach to integrated risk management enables the downside-protected delivery of goods, services and digital assets without intermediaries.

👉 Get started at dsla.network

👉 Check our blog at blog.stacktical.com

👉 Build a use case at readme.stacktical.com


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# Teacher Notes

### Why use this resource?

This resource gives students a chance to develop a method for differentiating a product. By looking at factorised polynomial functions, which they can already differentiate by first expanding, students’ natural desire to find a way of jumping a few steps is being harnessed here. The Chain Rule could also be developed from this resource. The motivation for doing so could come from considering how to tackle this review question.

This resource can be used to reinforce the idea of finding a generalisation by building complexity up in steps.

The final tab A certain perspective explains how this simple version of the Product Rule can be used to justify why it works for any pair of functions.

### Preparation

Students need to be able to differentiate polynomials, they do not need any other prior knowledge.

Sets of printouts of the Things you might have noticed might be useful for groups to study. Note that this section refers to the Chain Rule.

### Possible approach

This works well with students tackling the initial two functions in the Warm-up from the board and trying to spot a rule alone. These ideas can then be shared via a plenary as a large group and finally rules can be checked as a class using the toggle. Alternatively, rather than use the Warm-up examples you could ask each student to come up with their own expression with two brackets.

Students could then be asked to work in pairs or small groups as they investigate the functions in the main problem.

You might next like students to work through How does your rectangle grow? which offers a visual cue to why the Product Rule works.

### Key questions

• What in the original expression affects the coefficients in the derivative?
• What is similar and what has changed in each new function?
• How can this help us with harder functions?

### Possible support

• Encourage students struggling to find connections, to invent more of their own functions to check, or extra functions can be suggested.
• It’s a good idea to follow a building process doing two or three of each type until connections are made rather than random examples. \begin{align*} y&=(x+1)(ax+b) \\ y&=(x+c)(ax+b) \\ y&=(1-x)(ax+b) \\ y&=(cx+d)(ax+b) \\ \end{align*}

### Possible extension

Students could unpack ways of generalising, thinking about how they might extend to higher degree polynomials, or might start thinking about other functions (see the final tab).

If the Chain Rule has not yet been covered, students can consider functions of the form $y=(ax+b)^c$ then $y=(ax^2+bx+c)^d$ or $(x^3+2)^2$. Students can conjecture a general rule in function notation.

Students could look at some review questions such as this one.


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# Does the MixColumns step come before or after AddRoundKey in AES decryption?

I found these images depicting the AES decryption process:

In the first image, the MixColumns step comes before the AddRoundKey step, while in the second image, the AddRoundKey will come before the MixColumns.

Which order is the correct one?

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Have a look at my answer to »Why is MixColumns omitted from the last round of AES?«. Both are correct, but with a different expanded key. –  Paŭlo Ebermann May 25 '12 at 17:54
what do you mean by different expanded key? –  goldroger May 25 '12 at 18:02
Did you actually read the linked answer? You'll see that the third code block uses InvMixColumns(ExpandedKey[i]) instead of ExpandedKey[i] (in the last block, this is shown as EqExpandedKey[i]). –  Paŭlo Ebermann May 25 '12 at 18:06
@JohnPaulParreño: I'd correct my answer (I originally talked about the ShiftRows and SubBytes operation), however, I can't really say it better than that Paŭlo said in his answer that he linked to; look at that. –  poncho May 25 '12 at 23:18
possible duplicate of Why is MixColumns omitted from the last round of AES? –  mikeazo Jun 4 '12 at 19:21

Not sure if this is answered fully, but if you check wikipedia, you'll see that AddRoundKey occurs 3 times in total with 1 time before and 1 time after the MixColumns operation.

However, if you are just considering the middle rounds then the MixColumns comes before

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After the AddRoundKey step, except for the last round, where it does not happen

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InvMixColumn(x+RoundKey) = InvMixColumn(x) + InvMixColumn(RoundKey)


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# Burnside problem for hyperbolic groups?

Let $\Gamma$ be a nonelementary hyperbolic group. Then is it true that $\Gamma^{p}:=<\gamma^{p}| \gamma \in \Gamma>$ is a finite index subgroup of $\Gamma$? Here $p$ is a prime number. What is known about this problem? What can we say when $p$ is just an odd number(not necessarily prime) or just a postive integer?

• Since free groups are hyperbolic, surely the index can be infinite for sufficiently large $p$? The Burnside group $B(2,n)$ is known to be infinite for $n \ge 8000$ - I don't know if that is still the best known. – Derek Holt Nov 29 '11 at 9:03
• What if p=2 or 3 or some small numbers? is there known results? – user9552 Nov 29 '11 at 10:26
• For $p=2,3,4$ the group $G/G^p$ is finite for any finitely generated group $G$, because it is a quotient of the free Burnside group of exponent $p$, which is known to be finite for such small values of $p$. The smallest non-trivial value for which this is not yet known is $p=5$. – Ashot Minasyan Nov 29 '11 at 10:30
• To Ashot Minasyan:Is there a reference for that for p=2,3,4, the group $G/G^{p}$ is the quotient of the free Burnside group of exponent $p$? Thanks. – user9552 Nov 29 '11 at 12:11
• This web-page contains an overview of the results together with the references: www-gap.dcs.st-and.ac.uk/~history/HistTopics/… – Ashot Minasyan Nov 29 '11 at 13:07

A. Yu. Olshanskii in the paper "Periodic quotient groups of hyperbolic groups." ((Russian) Mat. Sb. 182 (1991), no. 4, 543--567; translation in Math. USSR-Sb. 72 (1992), no. 2, 519–541) proved that for every torsion-free non-elementary hyperbolic group $G$ there is a number $N \in \mathbb{N}$ such that for any odd $n \ge N$ the quotient $G/G^n$ is infinite.
In a more recent article, Ivanov and Olshanskii ("Hyperbolic groups and their quotients of bounded exponents". Trans. Amer. Math. Soc. 348 (1996), no. 6, 2091–2138) proved a similar statement for an arbitrary non-elementary hyperbolic group $G$ (torsion is allowed): there is $n=n(G) \in \mathbb{N}$ such that $G/G^n$ is infinite. In this case one cannot say that $G/G^k$ is infinite for any sufficiently large odd $k$, because if the group $G$ is generated by elements of, say, order $3$, then for any $k$ not divisible by $3$, $G^k=G$.
• @Mark, surely the second result (of Ivanov-Olshanskii) cannot be improved? In the torsion-free case you are probably thinking about the paper by Delzant and Gromov? But Ivanov-Olshanskii already prove this for all $n$ diisible by $2^5$. – Ashot Minasyan Nov 29 '11 at 11:32
• I meant a paper by Ivanov and Olshanskii. In your answer you always put $n$ odd. For the OP, it does not matter probably. – Mark Sapir Nov 29 '11 at 13:47
• I would like to use this opportunity to advertise the work of a young colleague: more than infinite, these quotients $G/G^n$ ($G$ hyperbolic, $n$ odd large enough) have exponential growth, arbitrarily close to the growth of $G$. See math.vanderbilt.edu/~coulonrb/papiers/growth.pdf – Benoît Kloeckner Nov 29 '11 at 17:24


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# Resolution III Design Example

This example appears in the Experiment Design and Analysis Reference book.

A baker wants to investigate the factors that most affect the taste of the cakes made in his bakery. He chooses to investigate seven factors, each at two levels: flour type (factor $A\,\!$), conditioner type (factor $B\,\!$), sugar quantity (factor $C\,\!$), egg quantity (factor $D\,\!$), preservative type (factor $E\,\!$), bake time (factor $F\,\!$) and bake temperature (factor $G\,\!$). The baker expects most of these factors and all higher order interactions to be inactive. On the basis of this, he decides to run a screening experiment using a 2 $_{\text{III}}^{7-4}\,\!$ design that requires just 8 runs. The cakes are rated on a scale of 1 to 10. The design properties for the 2 $_{\text{III}}^{7-4}\,\!$ design (with generators $D=AB\,\!$, $E=AC\,\!$, $F=BC\,\!$ and $G=ABC\,\!$) are shown in the following figure.

The resulting design along with the rating of the cakes corresponding to each run is shown in the following figure.

The normal probability plot of effects for the unreplicated design shows main effects $B\,\!$, $C\,\!$, $D\,\!$ and $G\,\!$ to be significant, as shown in the next figure.

However, for this design, the following alias relations exist for the main effects:

\begin{align} & A= & A+BD+CE+FG \\ & B= & B+AD+CF+EG \\ & C= & C+AE+BF+DG \\ & D= & D+AB+CG+EF \\ & E= & E+AC+BG+DF \\ & F= & F+BC+AG+DE \\ & G= & G+CD+BE+AF \end{align}\,\!

Based on the alias structure, three separate possible conclusions can be drawn. It can be concluded that effect $CD\,\!$ is active instead of $G\,\!$ so that effects $C\,\!$, $D\,\!$ and their interaction, $CD\,\!$, are the significant effects. Another conclusion can be that effect $DG\,\!$ is active instead of $C\,\!$ so that effects $D\,\!$, $G\,\!$ and their interaction, $DG\,\!$, are significant. Yet another conclusion can be that effects $C\,\!$, $G\,\!$ and their interaction, $CG\,\!$, are significant. To accurately discover the active effects, the baker decides to a run a fold-over of the present design and base his conclusions on the effect values calculated once results from both the designs are available.

The present design is shown next.

Using the alias relations, the effects obtained from the DOE folio for the present design can be expressed as:

\begin{align} & Effec{{t}_{A}}= & 0.025=A+BD+CE+FG \\ & Effec{{t}_{B}}= & -0.225=B+AD+CF+EG \\ & Effec{{t}_{C}}= & 2.075=C+AE+BF+DG \\ & Effec{{t}_{D}}= & 2.875=D+AB+CG+EF \\ & Effec{{t}_{E}}= & -0.025=E+AC+BG+DF \\ & Effec{{t}_{F}}= & -0.075=F+BC+AG+DE \\ & Effec{{t}_{G}}= & 3.825=G+CD+BE+AF \end{align}\,\!

The fold-over design for the experiment is obtained by reversing the signs of the columns $D\,\!$, $E\,\!$, and $F\,\!$. In a DOE folio, you can fold over a design using the following window.

The resulting design and the corresponding response values obtained are shown in the following figures.

Comparing the absolute values of the effects, the active effects are $B\,\!$, $C\,\!$, $D\,\!$ and the interaction $CD\,\!$. Therefore, the most important factors affecting the taste of the cakes in the present case are sugar quantity, egg quantity and their interaction.


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## How to calculate the moment of inertia of the rigid body?

I don't know how to calculate the following rigid bodies with different geometries, can anybody help me?

Thin spherical shell: I=(2/3)MR^2

Solid sphere: I=(2/5)MR^2

 Recognitions: Homework Help Science Advisor Use the definition of moment of inertia: $$I = \int r^2 dm$$ In the case of the shell the element of mass is $dm = M {dA} /{4 \pi R^2}$ where $dA = R^2 \sin \theta d\theta d\phi[/tex]. The distance to a point on the shell from the z-axis is [itex]R^2 \sin^2 \theta$ so $$I = \frac {M}{4 \pi R^2} \int_{0}^{2 \pi} d\phi \int_{-\pi /2}^{\pi /2}R^4 \sin^3 \theta d\theta$$ from which the desired result follows. In the case of the solid sphere you will work with a volume integral.


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## FANDOM

A yellow hypergiant (also known as a post-red supergiant/hypergiant), formerly a late S Doradus variable, is a star that has a spectral classification between A0 to K2. This gives them temperatures between $4,000$ and $8,000$ K. Yellow hypergiants are very luminous with $200,000$ to $600,000-750,000$ times greater than the Sun. It is a star that was once a red supergiant, but is now getting hotter and shrinking down to a luminous blue variable and then a Wolf-Rayet star. All yellow hypergiants stars are in an extremely active phase of their evolution.

This article will cover a few examples of yellow hypergiants.

## Rho Cassiopeiae

Shortcut:
Rho Cassiopeiae, also known as 7 Cassiopeiae, is the best known yellow hypergiant. It is located in the constellation Cassiopeia 8,200 light years away from Earth. It has a diameter of 450 (= 627 million km) times that of our sun, a mass of 14 and 30 times that of our sun and a temperature between $5,777$ and $7,200$ degrees Kelvin. It is visible to the naked eye as it has a luminosity approximately 500,000 times brighter than the Sun.

## V382 Carinae

Shortcut:

V382 Carinae, also known as x Carinae, is the second best known yellow hypergiant. It is easily visible with the naked eye. It is located in the constellation Carina $8,900$ light years away from Earth. It has a diameter of $747 (= 1.04$ billion km) times that of our sun and a mass of $20$ times that of our sun. It has a temperature approximately $5,866$ degrees Kelvin. It is visible to the naked eye as it has a luminosity approximately $316,000$ times brighter than the Sun.

## HR 5171

HR 5171, also known as the "Peanut star" or V766 Centauri, is a yellow hypergiant located in the Centaurus constellation $11,700$ light years away from Earth.

The star system was originally discovered by French astronomer Olivier Chesneau. However, it was later revealed to be a binary star in $2014$. Both are so close to each other that the two stars are in contact.

### HR 5171 A

The main component is a star named HR 5171 A. It has a diameter of $1,315 (= 1.83$ billion km) times greater than the Sun, based on a distance of $11,700$ light years and an angular diameter of $3.39$ milliarcseconds, making it the largest known yellow hypergiant star. It has a mass $27 to 36$ times that of our sun, as well as a luminosity of 630,000 to a million times that of the sun. Its temperature vary between $4,290 to 5,050$ degrees Kelvin.

### HR 5171 Ab

HR 5171 Ab is a small yellow hypergiant with a diameter between $401$ and $650$ times greater than the sun $(= 558$ to $905$ million km), with a temperature of $4,800 - 5,200$ degress Kelvin, and a small mass of only 5 times greater than the sun.

### HR 5171 B

HR 5171 B, is a blue supergiant with a temperature of $26,000$ Kelvin and a luminosity of $316,000$ times brighter than our sun. This star is not in contact.

### Gallery

<< 8. KY Cygni 9. HR 5171 10. Mu Cephei >>

## V509 Cassiopeiae

Shortcut:
V509 Cassiopeiae, also known as HR 8752, is also located in the constellation Cassiopeia 4,500 light years away from Earth. It has a diameter between 400 and 900 (= 557 to 1,253 million km) times that of our sun. It has a mass of 11 times that of our sun. It has a temperature between 4,000 and 8,000 degrees Kelvin. Just like Rho Cassiopeiae, it is visible to the naked eye as it has a luminosity between 200,000 and 400,000 times brighter than the Sun.

## HD 179821

Shortcut:
HD 179821 is a yellow hypergiant star in the Aquila constellation, surrounded by a detached dust shell. It is a semi-regular variable and either a post-AGB star or what it is more likely to be, a yellow hypergiant.

## Yellow Evolutionary Void

The Yellow Evolutionary Void separates yellow hypergiants from luminous blue variables, although yellow hypergiants are the hottest and luminous blue variables are the coolest, meaning they can have approximately the same temperature (between $7,500$ and $10,000$ K). They are sometimes considered as stars in a Pre-Luminous Blue Variable stage. This section will cover two yellow hypergiants that are near in the bounds of the Yellow Evolutionary Void.

### HD 33579

Shortcut:
HD 33579, also known as R76, is a white/yellow hypergiant located in the Large Magellanic Cloud and in the constellation Dorado $165,000$ light years away from Earth. It is amoung the brightest stars in the Large Magellanic Cloud. It has a diameter $380$ R $(= 530$ million km), a mass between $20$ and $30$ M. It has a temperature $7,980$ degrees Kelvin. It is visible to the naked eye as it has a luminosity approximately $525,000$ times brighter than the Sun.

### IRC +10420

Shortcut:

IRC+10420, also known as V1302 Aquilae, is a yellow hypergiant located in the Aquila constellation at a distance of $5,000$ parsecs $(= 16,300$ light years) from Earth on average. Its temperature is $6,000$ to $8,000$ degrees Kelvin, which mean that the star has increased its own temperature into the LBV range. It has a mass $10$ times that of our sun and a diameter varying between $357$ and $1,342$ times that of the sun $(= 497$ to $1,869$ million km).

### IRAS 17163-3907

Shortcut:

IRAS 17163-3907 (or Hen 3-1379), also known as Fried Egg Nebula, is a possible yellow hypergiant located in the constellation Scorpius at $13,000$ light years away from Earth. It has a diameter of $300 to 400$ R $(= 417$ to $557$ million km), with a temperature between $7,500$ to $10,000$ degrees Kelvin, which mean that the star is in a pre-LBV stage. Assuming the higher temperature is true, IRAS $17163-3907$ would be the hottest yellow hypergiant star. It is visible to the naked eye as it has a luminosity approximately $500,000$ times brighter than the Sun.[6]

IRAS 17163-3907 has also a double dust envelope, and like VY Canis Majoris and IRC +10420, this star has ejected much of its mass off of its atmosphere, it may explode into a supernova.


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Volume 12, Issue 4
Analysis of a Special $Q_1$-Finite Volume Element Scheme for Anisotropic Diffusion Problems

Numer. Math. Theor. Meth. Appl., 12 (2019), pp. 1141-1167.

Published online: 2019-06

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• Abstract

In this paper, we  analyze a special $Q_1$-finite volume element   scheme which is obtained by using the midpoint rule to approximate the line integrals in the standard $Q_1$-finite volume element method. A necessary and sufficient condition  for the positive definiteness of  the element stiffness matrix is  obtained. Based on  this result, a sufficient condition  for the coercivity of the scheme is  proposed. This sufficient condition has an explicit form  involving the information of the diffusion tensor and the mesh. In particular, this condition can reduce to a pure geometric one  that covers some special meshes, including  the parallelogram meshes,   the $h^{1+\gamma}$-parallelogram meshes and  some trapezoidal meshes. Moreover, the $H^1$ error estimate is proved rigorously without the $h^{1+\gamma}$-parallelogram assumption required by existing works. Numerical results are also presented to validate the   theoretical analysis.

• Keywords

$Q_1$-finite volume element scheme, midpoint rule, coercivity, $H^1$ error estimates.

65M10, 78A48

fangf512@nenu.edu.cn (Fang Fang)

hq1162377655@163.com (Qi Hong)

wu_jiming@iapcm.ac.cn (Jiming Wu)

• BibTex
• RIS
• TXT
@Article{NMTMA-12-1141, author = {Fang , Fang and Hong , Qi and Wu , Jiming }, title = {Analysis of a Special $Q_1$-Finite Volume Element Scheme for Anisotropic Diffusion Problems}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2019}, volume = {12}, number = {4}, pages = {1141--1167}, abstract = {

In this paper, we  analyze a special $Q_1$-finite volume element   scheme which is obtained by using the midpoint rule to approximate the line integrals in the standard $Q_1$-finite volume element method. A necessary and sufficient condition  for the positive definiteness of  the element stiffness matrix is  obtained. Based on  this result, a sufficient condition  for the coercivity of the scheme is  proposed. This sufficient condition has an explicit form  involving the information of the diffusion tensor and the mesh. In particular, this condition can reduce to a pure geometric one  that covers some special meshes, including  the parallelogram meshes,   the $h^{1+\gamma}$-parallelogram meshes and  some trapezoidal meshes. Moreover, the $H^1$ error estimate is proved rigorously without the $h^{1+\gamma}$-parallelogram assumption required by existing works. Numerical results are also presented to validate the   theoretical analysis.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2018-0080}, url = {http://global-sci.org/intro/article_detail/nmtma/13218.html} }
TY - JOUR T1 - Analysis of a Special $Q_1$-Finite Volume Element Scheme for Anisotropic Diffusion Problems AU - Fang , Fang AU - Hong , Qi AU - Wu , Jiming JO - Numerical Mathematics: Theory, Methods and Applications VL - 4 SP - 1141 EP - 1167 PY - 2019 DA - 2019/06 SN - 12 DO - http://dor.org/10.4208/nmtma.OA-2018-0080 UR - https://global-sci.org/intro/nmtma/13218.html KW - $Q_1$-finite volume element scheme, midpoint rule, coercivity, $H^1$ error estimates. AB -

In this paper, we  analyze a special $Q_1$-finite volume element   scheme which is obtained by using the midpoint rule to approximate the line integrals in the standard $Q_1$-finite volume element method. A necessary and sufficient condition  for the positive definiteness of  the element stiffness matrix is  obtained. Based on  this result, a sufficient condition  for the coercivity of the scheme is  proposed. This sufficient condition has an explicit form  involving the information of the diffusion tensor and the mesh. In particular, this condition can reduce to a pure geometric one  that covers some special meshes, including  the parallelogram meshes,   the $h^{1+\gamma}$-parallelogram meshes and  some trapezoidal meshes. Moreover, the $H^1$ error estimate is proved rigorously without the $h^{1+\gamma}$-parallelogram assumption required by existing works. Numerical results are also presented to validate the   theoretical analysis.

Fang Fang, Qi Hong & Jiming Wu. (2019). Analysis of a Special $Q_1$-Finite Volume Element Scheme for Anisotropic Diffusion Problems. Numerical Mathematics: Theory, Methods and Applications. 12 (4). 1141-1167. doi:10.4208/nmtma.OA-2018-0080
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# A new model for Auckland commercial property yields

Moshe Szweizer (Department of Research, CBRE Group Inc., Auckland, New Zealand)

ISSN: 1463-578X

Article publication date: 29 November 2018

Issue publication date: 24 January 2019

## Abstract

### Purpose

The purpose of this paper is to extend the studies of commercial property yields by providing a cross-field approach through the implementation of methods used in physics.

### Design/methodology/approach

Based on the equations used to describe real gases in physics, the commercial property yields are expressed through a model, as a product of two terms. The first term estimates the influence of the income change and investment on yields. The second estimates the yield variation as a function of property size. Additionally, the model combines the macroeconomic and microeconomic components influencing yield adjustment. Calculation of each component involves procedures developed in physics, with the investment volume being linked to the amount of gas and the microeconomic yield being linked to the gas compressibility.

### Findings

The model was applied to the Auckland office and industrial markets, both to the historic and current cycle. At the macro-level, it was found that the use of accumulation of investment over a relevant cycle, results in a high data to model correlation. When modelling the yields at the micro-level, a relationship between the outlying properties and the yield softening was observed.

### Practical implications

The paper provides an enhanced modelling power through association of the cyclic and investment activity with the yield change. Moreover, the model may be used to decouple the local and the international investment components and the extent of their influence on the local property market. Furthermore, it may be used to estimate the influence of the property size on the yield.

### Originality/value

This research provides a new cross-field application of modelling techniques and enhances the understanding of factors influencing yield adjustments.

## Citation

Szweizer, M. (2019), "A new model for Auckland commercial property yields", Journal of Property Investment & Finance, Vol. 37 No. 1, pp. 42-57. https://doi.org/10.1108/JPIF-06-2018-0043

## Publisher

:

Emerald Publishing Limited

## 1. Introduction

The paper presents a comprehensive model of commercial yields, and applies its findings to the Auckland prime and secondary office and industrial markets. The approach builds upon that proposed in Szweizer (2018), where a rental model was presented. Both approaches apply the methodology developed in physics to commercial property markets. Szweizer states that statistical methods developed in physics, which deal with complex multibody interactions, may be re-applied to property markets, providing a statistical toolset which may be used universally. In the work presented here, we employ a similar approach when developing the yield model.

The model expands the equation of state developed in thermodynamics, which is used in the context of real gases. This equation is transformed here to express property related qualities. Two major parts are arrived at. The first part models yield behaviour in the general economic context, and the second describes modifications to yield due to individual buildings’ competition. Therefore, a macroeconomic approach is provided by the first component, and the microeconomic approach is contributed through the second.

The paper is organised as follows. First, the general formula is discussed, which splits the yield into macro- and micro-level components. Subsequently, we discuss the macro-level component as Part I of the paper and the micro-level component as Part II. At the completion of Part II, we arrive at a composite formula for yield modelling. Each part contains a discussion and examples of the formulae application to the Auckland office and industrial markets.

### Data sources

The rents, vacancy, yields and transactions data are sourced from CBRE. Interest rates are sourced from Reserve Bank of New Zealand and International Investment Position data series are sourced from Statistics New Zealand.

## 2. Literature review – property

The standard model is exposed in numerous publications. These include Sivitanides et al. (2001), Hendershott and MacGregor (2005), Chen et al. (2004), Chichernea et al. (2008), Sivitanidou and Sivitanides (1999) and Shilling and Sing (2007). The standard model assumes cap rates to be expressed through an adjustment around equilibrium values, which are determined through real estate fundamentals such as rent levels, rent growth and risk-free interest rates.

Chervachidze et al. (2009) revisit the standard model and extend it by incorporating three additional parameters. These are economy-wide risk premium over the risk-free rate, the overall availability of debt in the economy and the investor sentiment. In the extended model, it is assumed that the macroeconomic capital flows and the availability of debt significantly impact capital pricing. It is noted that large capital flows from developing countries to developed ones have become manifest since the early 2000s. These capital flows increased debt availability in the developed countries, and have bid up prices of real estate assets.

In the model lagged capitalisation rate, real rent index and real T-Bond yield (calculated as nominal yield minus inflation rate) are used as standard variables. These are extended by economy-wide risk premium over the risk-free rate, calculated as the difference between Moody’s AAA Corporate Bond Index and the 10-year T Bond, and the overall availability of debt in the economy, calculated as a ratio of total net borrowing and lending to the nominal GDP level.

The final version of the model incorporates investor sentiment variable which is to address structural changes to the model performance, especially in the period preceding GFC. It is argued that the real estate investor sentiment, together with the dramatic increase in availability of liquidity, which happened in the early 2000s can be used as an explanation of the “great cap rate compression” observed before 2007.

McAllister and Nanda (2016) investigate the relation between the activity of foreign investors and capitalisation rates in major European office markets. The study involves 28 major cities over the period between 1999 and 2013. The modelling revealed a significant effect of foreign investment on market capitalisation rates. The strategy used employs the base capitalisation rates model as described by Archer and Ling (1997), Chervachidze et al. (2009), Chervachidze and Wheaton (2013). The main hypothesis being tested is: all else equal, an increase in the proportion of transactions involving foreign investors will lower capitalisation rates. The model involves: lagged capitalisation rate, real risk-free rate, the real rent ratio as a ratio of the real rent in a year and the same period average of real rent and the availability ratio. All are applied to a specific city and country. Country risk premium and debt availability are also included in the model.

The results suggest a wide spread of foreign investment in major European cities, ranging from as low as 7 per cent in Dublin to as high as 99 per cent in Bucharest and 94 per cent in Warsaw. London is estimated at 50 per cent, Paris at 60 per cent, Amsterdam at 63 per cent and Berlin at 29 per cent. It is important to note that the above study assumed foreign investment relating to entities or persons of foreign nationality investing in a local office market. This definition differs somewhat from the one used in the paper presented here, where we associate the origin of the investment with the country from which the investment funds are derived from.

For example, in the model presented here, if a New Zealand citizen borrows funds from a Chinese bank and invests in the New Zealand property market, we would treat the transaction as a foreign investment, as the funds are obtained from overseas, even though the person executing the purchase is a local citizen. Conversely, a foreigner may borrow funds from a local bank and invest in the New Zealand market. Such a case would be understood as a local investment.

Chaney and Hoesli (2015) extend the standard model through the incorporation of up to 30 variables and compare results obtained with the transaction-based and appraisal-based data sets. The data comprise 3,500 transactions and 8,700 appraisals spanning the period between 1995 and 2010 in Switzerland. The objective of the study is to establish if the appraisal-based yields are smoothed with respect to the transaction-based yields, but also to determine the statistical strength with which each type of variable enters the modelling equation.

The variables are grouped into eight economically meaningful categories. These include refurbishment risk, illiquidity risk, tenant risk, land leverage, commercial tenancy commitment, location, vacancy rate and performance of the economy. The variables are also divided into two groupings with respect to their influence either at the macroeconomic or microeconomic level. The modelling equation is composed of additive parts, which clearly distinguish the variable influence as being either at the macro or the micro level.

The illiquidity risk is associated with a natural logarithm of the property net lettable area (NLA), which in the paper is denoted as the “volume”. The inclusion of the property area as a modelling variable is taken after Saderion et al. (1994), who used property size, age and location when modelling 500 transactions of apartment complexes in Houston between 1978 and 1988.

In the model, the natural logarithm of the NLA is taken to address the problem of its distribution being very strongly skewed.

As the model is applied to two data sets, the results obtained in each case are compared to the other. The first set of modelling parameters is obtained for investors’ data, and the second for appraisers’ data and the relevance of each variable for each tested group is shown against the other. It is established that appraisers find macro location and building condition to be of the dominant importance, while investors are highly motivated by the property age, rent to median rent ratio, risk-free rate and the building size.

Szweizer (2018) introduces equations borrowed from physics to property market modelling. The equation of an ideal gas is compared to the formula linking income, capital value and initial yield as the first step. Subsequently, a general treatment of rental cycles is developed. The main feature of the model is the identification of four turning points, labelled as market shocks, during a property cycle. A cycle is divided into four distinct phases responsible for the compression of vacancy, overheated market, market deflation and the market bottom. The approach assumes yields to be constant throughout each cycle, although a variation of yields between each cycle is expected. In the paper presented here, we expand this model by undertaking a detailed analysis of intra-cyclical variations in yields.

## 3. Literature review – physics

In physics, the gas law may be stated in terms of the compressibility factor Z. When the compressibility is assumed to be equal to unity, the gas is called ideal. For real gases, the compressibility represents intermolecular interactions of gas constituents. The equation of state with compressibility expressed as an infinite series of terms, called virial expansion, was first proposed by Onnes (1902). In the paper, he proposes seven different forms for expansion of the compressibility factor and applies them so to model the behaviour of hydrogen, oxygen, nitrogen and carbon dioxide. The virial expansion is expressed in powers of gas density, but it could also be expressed in terms of temperature.

Schreiber et al. (2011) provide an educative paper with a description of how the coefficients are calculated and with some examples of the virial expansion applications. The virial coefficients are shown to be expressible in terms of integrals of Mayer f-functions, and the Mayer expansion is discussed. Mayer (1942) proposes that the virial expansion may be formulated in terms of specific integrals, all depending on the same base function and representing interactions between pairs, triplets and higher combinatoric arrangements of molecules. The integrals may be represented through a collection of graphs, each relating to interactions among increasingly more complex molecule arrangements.

The advantage of the Mayer approach comes through the observation that simultaneous interactions among multiple molecules are of much lower probability than pairwise interactions. Therefore, a rapidly converging series is expected when the virial expansion is expressed in this way.

The Schreiber paper provides examples of second virial coefficient calculation over simple potential models. First, an infinite barrier is discussed, which is used to model a hard sphere interaction. (Two molecules bouncing from each other in a similar way as two snooker balls when colliding). Second, a finite square well potential is discussed to illustrate interactive forces and the second virial coefficient is recalculated over it.

## 4. The equation of state for real gases

Real gases are described by an equation of state employing compressibility Z, as in Formula 1:

(1) P V T = n Z R

Here, P stands for pressure, V stands for volume, T stands for temperature, n stands for the amount of gas, Z stands for compressibility factor, and R is the universal gas constant.

When the Formula 1 is translated into property markets one arrives at the following relationship (Formula 2):

(2) R e n t × O c c u p a n c y C a p i t a l V a l u e = y 0 × Z

Here, y0 stands for yield arrived at through macroeconomic analysis and Z stands for yield correction due to microeconomic considerations. In the following section, we construct a model of y0, and in the subsequent section, the compressibility Z correction is discussed.

## 5. Part I – macroeconomic component

In Formula 2, the y0 replaces the amount of gas “n” as used in physics (Formula 1). In property, there are three contributing factors influencing this quantity. The first relates to the cyclic position, and the impact the general economy has on the property market. The second relates to the local availability of liquidity and is modelled through the local interest rates. The third factor represents the international willingness to invest in the local property.

Figure 1 shows the Otto process as employed by Szweizer (2018) when modelling property rental cycle. During the property cycle, the value of the market is influenced by two factors. The first is the change in value due to rental and vacancy changes. In Figure 1 this contribution is represented through vertical arrows, related to the “physical work” input or output, or in property income changes. Thus, the contraction of vacancy and increasing rents result in income growth and sharpening of yields. When the process is reversed between points 3 and 4, yields soften as a result.

The market value is also changed due to the rental changes when the occupancy remains stable (phase between points 2 and 3). This contribution denoted as “heat flow” in physics or investment flow in property is shown with horizontally pointing arrows. The heat flow represents the strength of investment in property, both local and international. The flow in, represented by the left arrow, relates to the increased investment activity, and the flow out, represented by the right arrow, corresponds to disinvestment, which sometimes happens in the Auckland market. The effective change in value due to the cyclic activity corresponds to the area enclosed by the points 1–4.

The presence and magnitude of the disinvestment phase depends on the market and the cycle. In some markets, the disinvestment may be negligible, while in the others quite pronounced. As an example, Auckland experienced substantial outflow of foreign investment prior and following GFC. At that time, the Australian investors sold most of their properties in Auckland and did not return to the New Zealand market for several years.

When developing the model, it was observed that the yields do not respond directly to the individual transactions which happen at any given time, but rather, yields respond to the long-term accumulation of investment in the market. Thus, if we look at Figure 1, yields at point 2 are not influenced by the transactions happening at that time, but are a result of investment accumulation along the curve from point 1 to point 2.

The above observation resulted in the model being developed as relying on the integration of both “work – income growth” and “heat – value growth” over the cyclic progression.

The work – or the income growth component, is integrated with respect to vacancy over the cycle starting at point 1. In the Otto cycle, the rental growth is expressed through the adiabatic function with elasticity gamma. When this is integrated over a period from point 1 to some subsequent point P, the following formula is arrived at:

(3) Δ y W = A V 1 V P R 1 V γ d V = A R 1 γ + 1 ( V P γ + 1 V 1 γ + 1 ) ,

where “A” is a constant weight with which this contribution enters the yield model, VP is the vacancy at some future point P, at which the yield is being evaluated, R1 is the rent at point 1, and γ is the elasticity as calculated through the rental model.

The heat – or the investment component, is integrated with respect to rent over the cycle, and is expressed through the following formula:

(4) Δ y Q = B R 1 R P m C v C a p d R = B R 1 R P m O c c γ 1 R d R = B m O c c 2 ( γ 1 ) ( R P 2 R 1 2 ) ,
where Cv is the specific heat at constant vacancy, Cap stands for capital value, Occ is the occupancy rate, and m is the amount of funds being invested. The constant B denotes the weight with which the investment component enters the model. In the above formula, the amount of funds invested between points 1 and P is assumed to be constant, which limits the applicability of Formula 4 to short time intervals.

The data were provided as a time granulated series. To calculate the accumulation of investment between periods 1 and k the Formula 4 becomes Formula 5:

(5) Δ % V a l Q = i = 1 k B Δ % m i + 1 O c c i + 1 2 ( γ 1 ) ( R i + 1 2 R i 2 ) ,
where Δ%mi represents the percentage change of the investment measuring parameter of the iteration, and the Δ%Val is the percentage change in property value between partitions 1 and k.

Formula 5 was used in two forms. First, by setting Δ%mi to represent the local interest rates, the contribution of the local investment market is captured. Secondly, by setting Δ%mi to the percentage change in New Zealand International Investment Position the attractiveness of the local investment against the overseas investment is captured. While this last term does not directly measure the foreign investment in the local property market, it does represent the relative attractiveness of New Zealand investment against the international one.

There is no precise data series available that would capture the amount of foreign investment in the New Zealand commercial property market. Therefore, we could not use such a direct approach when modelling. However, the model only requires the knowledge of the period-by-period percentage change in the relative attractiveness of the New Zealand market to the foreign investment. To estimate this, we used period-by-period percentage change in the International Investment Position as provided by Statistics New Zealand.

By combining Formulae 3 and 5 we arrive at the following modelling expression for market yields:

(6) y k = y 1 ( 1 + A i = 1 k Δ y W i + i = 1 k B % I i + 1 O c c i + 1 2 ( γ 1 ) ( R i + 1 2 R i 2 ) + i = 1 k C Δ % I I P i + 1 O c c i + 1 2 ( γ 1 ) ( R i + 1 2 R i 2 ) ) ,
where %I stand for the five-year bond rate, and Δ%IIP is the international investment position percentage change over the surveying iteration.

When modelling, we used five-year bond rate as the interest rate driving variable. It is customary when modelling property markets, to use the 10-year bond rate, and we tested the model against such setup as well. For completeness, the model was also tested against one-year and two-year bond rates as well as against short-term interest rates in the form of 30-day, 60-day and 90-day bank bill yields. It was found that the five-year bond rate gave the best data fit and therefore this is what is used here. One may wish to observe, that the specific choice of the interest rate measure, as used in the model here, may be related to the New Zealand property market only.

Figures 2–5 show the modelling results when parameters y1, A, B and C are numerically adjusted to provide for the best fit. The figures represent two cycles which have been fitted separately, but are shown on the charts as a continuous sequence.

The model coefficients and their relative strength as contributors to Formula 6 are shown in Table I. All coefficients are shown as absolute values and are normalised with respect to their total, so to show the proportion with which each component influences yield changes.

The historic cycle ended in the period between 2009 and 2011 depending on the sector. The current cycle has just passed the peak or point 3, as shown in Figure 1.

It is clear from Table I that in most cases the variations in income drive the yield changes. The notable exceptions are historic prime office cycle and current secondary industrial. The current prime industrial yield is also influenced strongly by investment, even though to a lesser degree. When looking at the investment (Q) part of the table it is important to recall that yield changes, as treated by the model, are not influenced by individual sales, but by the accumulation of net investment throughout the entire cycle.

Prime office historic yield shows a substantial influence by local investors, as well as a strong overseas involvement. During the pre-GFC market rush, the local investment in New Zealand was highly influenced by numerous investment funds being created at that time. These investment funds, together with already existing property vehicles, were aggressively buying prime office space which is reflected in Table I. Overseas participation was mostly represented by Australian investors, who entered the Auckland office market in the period between 2003 and 2007. During that time, Australian investors represented a share between 14 and 26 per cent of Auckland prime office purchases by volume. The Australian disinvestment started in 2006 and lasted until 2014 influencing both the historic and the current cycle.

The current prime office cycle, while peaking already, does not show such highly heated state as the one prior to the pre-GFC, which is reflected in much smaller investment contributions shown in Table I. There were large international prime office transactions in the current cycle, but these were between overseas vendors and purchasers, and therefore do not contribute to the net accumulation of the overseas investment.

In recent years, Chinese and Indian banks increased their presence in New Zealand. During the 2013–2014 period, three Chinese banks opened operations in New Zealand. These include Industrial and Commercial Bank of China (ICBC), China Construction Bank and Bank of China. The locally established overseas banks provide funds to investors of local origin and those who have recently re-settled in New Zealand from other countries. The effect is growth in overseas-funded investment, especially in the industrial space.

During the current cycle, Auckland industrial space constitutes one of the most attractive investment options. This is due to persistently low vacancies, especially in the prime sector. Therefore, there are three main drivers behind the current surge in industrial investment.

First, very low prime vacancy, even against persistently large supply of industrial space, drives the investment with locally and internationally borrowed money. Second, the strong presence of overseas banks in New Zealand, which lend to the local investors, results in increased overseas investment accumulation. Third, the migrant population, which is interested in conducting secondary retail, purchases secondary industrial space, which is used as supportive warehousing to their retail activity. These factors may be seen in Table I through increased values of current prime and secondary industrial investment components.

## 6. Part II – microeconomic component

The real gas consists of molecules. These molecules interact through a combination of attractive and repulsive forces. When two molecules are further apart the attractive force is dominant. When they are close together the repulsive force pushes them away resulting in a collision and change of relative motion. To take these interactions into account the compressibility factor Z is added to the ideal gas equation.

When properties are valued, the yield achieved by one property influences the valuation of another. Like in the case of molecules, the properties interact with each other within the market context.

Compressibility factor Z is evaluated through the virial expansion. In the realm of property markets this is expressed as:

(7) Z = 1 + B 2 ( T ) ρ + B 3 ( T ) ρ 2 + = 1 + B 2 ( T ) ρ ( 1 + B 3 ( T ) B 2 ( T ) ρ + B 4 ( T ) B 2 ( T ) ρ 2 ) ,
where T stands for the capital value and ρ stands for density.

In the case of gases, the density ρ is the number of molecules per unit volume. In the case of the property market, we define ρ as the number of properties in a NLA- related basket.

### The density

All statistical distributions used in the model depend on exponential functions of the NLA. Therefore, to obtain a linear distribution of data points when calculating densities, we transform the set of NLA’s into a set consisting of natural logarithms of these. Thus, the density is defined as the number of properties in a basket partition, which is based on equally split hyperspace of natural logarithms of properties’ NLAs. The basket width is defined as two standard errors of the such transformed NLA data set.

We define density space as:

D e n s i t y s p a c e : l n N L A = { 0 , ln ( max ( N L A ) ) } ,
B a s k e t s i z e : b a s k e t = 2 Standard Error ( ln N L A ) .

### Convergence of virial expansion

The infinite series in Formula 7 involves constants B which are evaluated through multidimensional integrals of the Mayer f-functions. These integrals progress in complexity with each interaction. In general, it is not feasible to perform a summation of the series in any analytical manner.

We postulate that the sale events occur continuously, that the probability of a sale taking place is independent of other sales, and that the sale events occur at a constant average rate. Therefore, we postulate without proof, that the infinite series in Formula 7, converges to an exponential distribution, providing for the following formula:

(8) Z = 1 + B 2 ( T ) ρ e ϵ a k T ρ ,
where T stands for the capital value, ϵ stands for the energy (potential) of building to building interaction, ρ stands for the density, k originally stands for the Boltzmann constant which is set to unity here and “a” stands for the extent of the density space and is equal to the natural logarithm of the NLA of the largest property.

### The potential

The strength of the interaction between properties is described through the potential ϵ(r), where r accounts for the difference between any two properties. When evaluating the model, the function ϵ(r) is chosen in such a way as to first, replicate modelled reality in a reasonably consistent manner, and second is simple enough so that the integrals depending on it may be integrated readily. Square wells and/or barriers are a common choice as these compare well with the modelled reality and can be integrated easily.

In the model presented here a square barrier was selected as follows:

(9) ϵ ( r ) = { ϵ 0 r a ln ( max ( NLA ) ) 0 r > a ln ( max ( NLA ) ) ,
where ϵ is a constant positive number.

Therefore, we assume that properties repel each other with a constant force given by the potential ϵ. Such definition may be understood as representing competition among similar properties.

### B2 coefficient

Assuming a constant barrier potential, as given above, the coefficient B2 may be evaluated through integration with respect to the distance over one-dimensional space:

(10) B 2 = 12 0 a = ln ( max ( N L A ) ) ( e ϵ k T 1 ) d r = 12 ( 1 e ϵ k T ) a ,

In Formula 10, the integrand is the Mayer f-function.

Combining Formulae 8 and 10 the formula for compressibility over a constant barrier potential becomes:

(11) Z = 1 + 12 ( 1 e ϵ k T ) a ρ e ϵ a k T ρ .

## 7. Application of the model

The model was applied to the data comprising both office and industrial prime sales spanning the period from 2012 to 2017. When fitting the model two parameters were varied: the macroeconomic yield and the potential ϵ. The fitting parameters are shown in Table II and the fit quality shown in Figures 7 and 9. The distribution of the data points over the natural logarithm of the NLA is shown in Figures 6 and 8.

It was found that the best fit was achieved when the term ϵ/kT was set to the weighted average yield as obtained through the macroeconomic part of the model (described in part I). The weighting was done with respect to the number of sales, but also with the age of the sale through an exponential moving average. Therefore, the sales which were older were contributing less to the weighted yield average than the new sales.

The best-fit parameters are shown in Table II. The fitting process was done in two steps. While fitting the data the ϵ/kT and y0 parameters were varied. The parameter “a” is the logarithm of the NLA of the largest property and thus stayed constant. It was of interest to estimate the value of interaction potential ϵ. To do so, the capital value (T) was taken as the average of capital values of the entire sample data for each sector, and the constant k was set to unity. The potential ϵ was calculated, resulting in values, agreeing closely with the average net effective rent of the data set.

The similarity of ϵ/kT and y0 suggests that the average effective rent may be used as the potential ϵ. In such a case, the model reduces to the following formula:

(12) y i e l d = y 0 ( 1 + 12 ( 1 e y 0 ) a ρ e y 0 a ρ ) ,
where y0 is the macroeconomic yield, as obtained in the first part of the paper, “a” is the ln(max(NLA)) and ρ is the number of properties in a basket as defined above.

Formula 12 represents the combined macro- and micro-economic model as presented here. The value of y0 is obtained through the macroeconomic expression shown in Formula 6, and the term in brackets, represents the microeconomic correction. This may be compared to Formula 2, where the RHS term consists of the macroeconomic yield, which is multiplied by the microeconomic correction (the Z factor). Thus, the microeconomic correction depends on all: the building size, the density but also the macroeconomic yield itself.

It is clear from Figures 7 and 9 that the yield stabilizes in the region where the number of properties per basket is large, but is softer when the properties are distinct by their size. Thus, the very large and the very small properties which differ in this way from the bulk of properties attain softer yields.

The model was tested against individual sales data points. Formula 12 was used with y0 set to the macroeconomic yield for each year, as obtained through the macroeconomic model described in the first part of the paper. The other two variables “a” and “ρ” were applied as specified in Table II, that is the overall data sample size and the individual basket association spanning throughout the entire period were used.

Subsequently, the difference between factual yield and the modelled was calculated. The standard error shown in Table III represents, averaged over each year, the discrepancy between the actual yield achieved through the sales and the yield estimated through the model.

## 8. Conclusion and discussion

The technique presented in the paper attempts to provide for a comprehensive commercial yield model. It is assumed, based on the real gas equation, that yield may be expressed as a product of two terms. The first term represents the macro-level approach and the second term the micro-level approach.

When compared to the standard model several observations may be entertained. The standard model is an adjustment model with an expectation that the yields may be modified with respect to some underlying equilibrium value. Here, the yields are expected to evolve through subsequent cycles with no reference to any fixed base value. The standard model represents yields as proportionally dependent on both the rents and the risk-free interest rates, with the rents scaled through an elasticity factor, and the risk-free rate through an exponential contribution. Here, the rents contribute through a quadratic dependence, which is a result of a different approach to the understanding of the market dynamics.

In the standard model, yields are assumed to be responding to the current market conditions. Therefore, the current rental changes and the current availability of debt is factored into the model. In the approach presented here, we expect the yields to respond to the accumulation of both the income generated by the property and the investment in the market, both taken over the period of the entire cycle. This approach results in an evolution of yields, with yields becoming increasingly sharper in subsequent cycles, just because the amount of the total investment in the property markets accumulates over longer periods of time. In this way, yields correspond to the total accumulated value of the market, and if a substantial disinvestment was to happen, the yields would soften as a result. To calculate the accumulation of income and investment, these need to be integrated over the entire period considered. The process of integration results in the quadratic dependence on rents.

The most important observation utilised in the first part of this paper is the assumption that yields do not respond to individual sales data, but rather correlate with the long-term accumulation of investment in the property market. Three main components contributed to the macro-level model. It was assumed that yield changes are due to the accumulation of income generated by the property, the accumulation of local investment and the accumulation of foreign investment. All three components were calculated over the duration of a property cycle.

The macro-level model allows for further splitting of the relative strength with which the three components contribute to the yield changes. This allowed for the identification of the overseas derived funds contribution. In New Zealand, the direct identification of the strength of such contribution is very difficult if approached directly and the model provides for a clear advantage in this area.

When analysing micro-level yield variation, the buildings’ sizes were used to assess yield distribution. It showed that properties which differ substantially from the bulk of properties, acquire softer yields. Therefore, very large or very small buildings, which differ substantially from the mainstream constructions are penalised through softer yields.

It should be noted that the choice of the NLA as the density factor in compressibility expansion, was used due to the density used in the original physics approach to compressibility. Replacement of density with other factors pertaining to property markets would seem to be of potential interest. For example, as noted in the literature review, building age could constitute a promising research avenue leading to the expansion of this study.

The model presented here has a potential to be used as a forecasting tool. This depends on the ability to forecast the main independent variables used by the model. Therefore, the interest rates, IIP changes, or any other variable which could be used in its place as a measure of international attractiveness of the market, would need to be forecasted first. Also, the ability to construct a forecast of the rental model, which is used as a basis for the income generated component of the model, would be required.

The other difficulty is associated with the establishment of the location of the market turning points. When forecasting, these need to be predicted, which constitutes a considerable challenge. On the other hand, the ability to locate future turning points, may be a prime objective of forecasting.

## Figures

#### Figure 1

Rent vacancy curve Income and Investment Flow

#### Figure 2

Auckland Prime Office Yield 1999-2017

#### Figure 3

Auckland Secondary Office Yield 1999

#### Figure 4

Auckland Prime Industrial Yield 2001-2017

#### Figure 5

Auckland Secondary Industrial Yield 2001-2017

#### Figure 6

Auckland prime office histogram

#### Figure 7

Micro-level fit to Auckland office yield data

#### Figure 8

Auckland prime industrial histogram

#### Figure 9

Micro-level fit to Auckland industrial yield data

## Table I

The macro-level model coefficients

Cycle Yield (P1) (%) Income (%) Q-Local (%) Q-Overseas (%) R2 (%)
Prime Office Historic 9.00 19.9 72.6 7.5 54.4
Secondary Office Historic 12.04 99.2 0.4 0.4 66.3
Prime Industrial Historic 9.41 91.5 3.9 4.7 75.8
Secondary Industrial Historic 11.07 94.9 4.8 0.3 85.0
Prime Office Current 8.36 92.7 6.2 1.1 97.9
Secondary Office Current 9.33 95.9 3.8 0.3 93.8
Prime Industrial Current 8.49 64.2 24.1 11.6 95.5
Secondary Industrial Current 9.65 0.7 73.1 26.2 97.3

## Table II

Micro-level model best fit parameters

Office Industrial
ϵ/kT (%) 6.56 6.31
a 10.68 11.09
y0 (%) 6.56 6.31
Cap value (T) ($psm) 5,979 2,287 ϵ ($psm) 392 144
Average rent (\$psm) 387 140
Error (%) −1.5 −3.0

## Table III

The model applied to the Auckland prime stock

Prime office Prime industrial
Year No. of sales SE (%) No. of sales SE (%)
2012 7 0.254 79 0.127
2013 17 0.240 73 0.116
2014 34 0.158 102 0.121
2015 31 0.224 133 0.119
2016 22 0.180 137 0.094
2017 24 0.259 118 0.098
Total 135 0.093 642 0.048

## References

Archer, W.A. and Ling, D.C. (1997), “The three dimensions of real estate markets: linking space, capital, and property markets”, Real Estate Finance, Vol. 14 No. 3, pp. 5-11.

Chaney, A. and Hoesli, M. (2015), “Transaction-based and appraisal-based capitalization rate determinants”, International Real Estate Review, Asian Real Estate Society, Vol. 18 No. 1, pp. 1-43.

Chen, J., Hudson-Wilson, S. and Nordby, H. (2004), “Real estate pricing: spreads and sensibilities: why real estate pricing is rational”, Journal of Real Estate Portfolio Management, Vol. 10 No. 1, pp. 1-21.

Chervachidze, S. and Wheaton, W. (2013), “What determined the great cap rate compression of 2000–2007, and the dramatic reversal during the 2008–2009 financial crisis?”, The Journal of Real Estate Finance and Economics, Vol. 46 No. 2, pp. 208-231.

Chervachidze, S., Costello, J. and Wheaton, W.C. (2009), “The secular and cyclic determinants of capitalization rates: the role of property fundamentals, macroeconomic factors, and ‘structural changes’”, The Journal of Portfolio Management, Vol. 35 No. 5, pp. 50-69.

Chichernea, D., Miller, N., Fisher, J., Sklarz, M. and White, R. (2008), “A cross sectional analysis of cap rates by MSA”, Journal of Real Estate Research, Vol. 30 No. 3, pp. 249-292.

Hendershott, P.H. and MacGregor, B.D. (2005), “Investor rationality: evidence from UK property capitalization rates”, Real Estate Economics, Vol. 33 No. 2, pp. 299-322.

McAllister, P. and Nanda, A. (2016), “Do foreign buyers compress office real estate cap rates?”, Journal of Real Estate Research, Vol. 38 No. 4, pp. 569-594.

Mayer, J.E. (1942), “Contribution to statistical mechanics”, The Journal of Chemical Physics, Vol. 10 No. 10, pp. 629-643.

Onnes, K.H. (1902), “Expression of state of gases and liquids by means of series”, KNAW Proceedings, Vol. 4, pp. 125-147.

Saderion, Z., Smith, B. and Smith, C. (1994), “An integrated approach to the evaluation of commercial real estate”, Journal of Real Estate Research, Vol. 9 No. 2, pp. 151-167.

Schreiber, F., Zanini, F. and Roosen-Runge, F. (2011), “Virial expansion a brief introduction”, available at: www.soft-matter.uni-tuebingen.de/teaching/Tutorial_Virial_Expansion.pdf (accessed 5 June 2018).

Shilling, J.D. and Sing, T.F. (2007), “Do institutional real estate investors have rational expectations?”, Asian Real Estate Society (AsRES) Conference, Macau, 9-12 July.

Sivitanides, P., Southard, J., Torto, R. and Wheaton, W. (2001), “The determinants of appraisal-based capitalization rates”, Real Estate Finance, Vol. 18 No. 2, pp. 27-37.

Sivitanidou, R. and Sivitanides, P. (1999), “Office capitalization rates: real estate and capital market influences”, Journal of Real Estate Finance and Economics, Vol. 18 No. 3, pp. 297-322.

Szweizer, M. (2018), “Thermodynamics and London office property cycles”, Journal of Property Investment & Finance, Vol. 36 No. 3, pp. 273-294.

## Corresponding author

Moshe Szweizer can be contacted at: moshe.szweizer@cbre.co.nz


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sou:CGS

Type
Description

The centimetre-gram-second system (abbreviated CGS or cgs) is a variant of the metric system of physical units based on centimetre as the unit of length, gram as a unit of mass, and second as a unit of time. All CGS mechanical units are unambiguously derived from these three base units, but there are several different ways of extending the CGS system to cover electromagnetism. The CGS system has been largely supplanted by the MKS system, based on metre, kilogram, and second. Note that the term cgs is ambiguous, since there are several variants with conflicting definitions of electromagnetic quantities and units. The unqualified term is generally associated with the Gaussian system of units, so this more precise URI is preferred.

Properties

The centimetre-gram-second system (abbreviated CGS or cgs) is a variant of the metric system of physical units based on centimetre as the unit of length, gram as a unit of mass, and second as a unit of time. All CGS mechanical units are unambiguously derived from these three base units, but there are several different ways of extending the CGS system to cover electromagnetism. The CGS system has been largely supplanted by the MKS system, based on metre, kilogram, and second. Note that the term cgs is ambiguous, since there are several variants with conflicting definitions of electromagnetic quantities and units. The unqualified term is generally associated with the Gaussian system of units, so this more precise URI is preferred.

Annotations
CGS System of Units(en)

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# How do you find the length of the curve y=3x-2, 0<=x<=4?

May 15, 2018

$L = {\int}_{0}^{4} \sqrt{1 + 9} \cdot \mathrm{dx} = {\left[\sqrt{10} x\right]}_{0}^{4} = 4 \sqrt{10}$

#### Explanation:

show below

$y = 3 x - 2$

$y ' = 3$

$L = {\int}_{a}^{b} \sqrt{1 + {\left(y '\right)}^{2}} \cdot \mathrm{dx} = {\int}_{0}^{4} \sqrt{1 + 9} \cdot \mathrm{dx}$

$= {\left[\sqrt{10} x\right]}_{0}^{4} = 4 \sqrt{10}$


text:
# Quantum mechanics/Atomic spectra

 Subject classification: this is a chemistry resource.
 Type classification: this is a lesson resource.

## Combination of two orbital angular momenta (different orbitals)

In classical physics, the total angular momentum is the (vector) sum of the angular momenta of the particles present in the system; the resulting total depends on the orientation of the components. In quantum mechanics, if a system has two particles with angular momentum L1 and L2 and angular momentum quantum numbers l1 and l2, we can define a total angular momentum L=L1+L2. L behaves as any angular momentum and the eigenvalues of ${\displaystyle {\hat {\mathbf {L} }}^{2}}$  are ħ²L(L+1), i.e.

${\displaystyle {\hat {\mathbf {L} }}^{2}\psi =\hbar ^{2}L(L+1)\psi }$  (Eq. 1)

L is the total angular momentum quantum number and can take values |l1 - l2|, |l1 - l2|+1, ..., l1 + l2.

Example 1. Consider an atom with one electron in orbital 2p (l1=1) and one electron in orbital 3d (l2=2). The total angular momentum quantum number for this system can be L=1,2,3. States with different total angular momentum have different energy so that there are (at least) 3 different states deriving from a configuration 2p3d.

Example 2. Consider the excited He atom with configuration 1s2p. The allowed quantum number for this configuration is just L=1.

The component along the z axis of the angular momentum is ${\displaystyle {\hat {L}}_{z}}$  and its eigenvalues can take the values ħmL with mL = -L, -L+1, ..., L.

Wavefunctions with L=0, 1, 2, 3,... are indicated with the capital letters S, P, D, F,...

## Combination of two spin angular momenta (different orbitals)

In a system with two electrons in different orbitals, you can define that total spin as the sum of the individual spins: ${\displaystyle {\hat {\mathbf {S} }}={\hat {\mathbf {S} }}_{1}+{\hat {\mathbf {S} }}_{2}}$ . The total spin behaves as any angular momentum, and the eigenvalues of ${\displaystyle {\hat {\mathbf {S} }}^{2}}$  are ħ²S(S+1), i.e.

${\displaystyle {\hat {\mathbf {S} }}^{2}\psi =\hbar ^{2}S(S+1)\psi }$  (Eq. 2)

S is the total spin quantum number and takes the value between |s1 - s2| and s1 + s2, i.e. between 0 and 1 in the case of 2 electrons, because the individual spin quantum numbers are each equal to ½.

The component along the z axis of the total spin is ${\displaystyle {\hat {S}}_{z}}$  and its eigenvalues can take the values ħmS with mS = -S, -S+1, ..., S. There are in total 2S+1 eigenvalues. 2S+1 is called spin multiplicity. States with multiplicity 1, 2, 3, 4,... are called singlet, doublet, triple, quartet, etc.

In the two examples in the section above, the total spin can take the value of 0 and 1 and so the configuration gives rise to a singlet and a triplet. The energy of a wavefunction depends on the spin multiplicity (or S).

## Spectroscopic term

The electronic configuration is not enough to describe the wavefunction, because for each configuration there are different ways to combine the orbital and spin angular momenta of the electrons (different combinations have different energies). The labels of a wavefunction which indicate the total angular momentum (S, P, D,...) and the spin multiplicity (singlet, doublet, triplet) are called spectroscopic terms.

The spectroscopic terms originating from the configuration 2p3d of the example 1 above are 1P, 1D, 1F, 3P, 3D, 3F.

The spectroscopic terms originating from the configuration 1s2p of the excited helium atom are 1P and 3P.

You can neglect the electrons in filled sub-shells, so that the spectroscopic term of the ground state of lithium 1s22s is 2S. The spectroscopic terms of the excited configuration of beryllium 1s22s2p are 1P and 3P.

## Electrons in equivalent orbitals

In sections 1 and 2 we considered electrons in different orbitals. If they are in the same orbital the calculation of the allowed spectroscopic terms is slightly more laborious. Consider atomic carbon with 2 electrons in 2p. You have to arrange the electrons in all possible ways allowed by the Pauli principle and then identify by inspection the spectroscopic terms on the basis of the values of mS and mL The spectroscopic terms of atomic carbon are 1D, 3P, 1S.

## Hund's empirical rule

It is useful to know what is the lowest energy spectroscopic term originating from a given configuration. Hund's rule states that:

 The level with the largest multiplicity has the lowest energy. For a given multiplicity, the term with the largest L has the lowest energy.

The ground state of atomic carbon is 3P (which has the largest multiplicity). This is why in elementary textbooks, electrons in partially filled shells have parallel spin.

## The last complication: spin-orbit coupling

Everything said so far implied that the spin does not appear in the Hamiltonian and its only effect is to change the structure of the spatial wavefunction. It is true that this is the main role of spin, but there is a very small term in the Hamiltonian that depends on the spin and this is called the spin-orbit coupling (which will not be written explicitly). Spin orbit coupling is important only for heavy atoms or when very accurate energy levels are needed (spectroscopy). The effect of spin orbit coupling is to split the spectroscopic terms in energy levels. The number of energy levels originating from a spectroscopic term is given by the possible values of the total angular momentum J = L + S (angular plus spin) with eigenfunction

${\displaystyle {\hat {\mathbf {J} }}^{2}\psi =\hbar ^{2}J(J+1)\psi }$

The quantum number J can take the values |L - S|, |L - S|+1, ..., L + S.

The spectroscopic term 3P (L=1, S=1) can have J=0,1,2 and therefore 3 finely spaced energy levels indicated as 3P0, 3P1, 3P2.

## Summary

Lessons 12-13 give an example of how quantum mechanics is used when the problem is too difficult to be solved exactly. You have to proceed by successive approximations, each time more refined. The aim is to understand the experiment (in this case, how many energy levels are present in an atom) and not to predict the exact value of the energy levels.

1. The first level of approximation is the independent electron approximation. Each electron is in an orbital and the energy is determined only by the orbital occupation or electronic configuration.
2. The second level of approximation is considering that the energy is different if the total orbital angular momentum and spin angular momentum are different. We differentiate the energy in spectroscopic terms depending on the values of L and S.
3. The most accurate level takes into account the spin orbit coupling. The energy levels are identified by the total angular momentum J.

Exercise
1. Find the spectroscopic terms originating from the following configurations (two electrons in two non-equivalent orbitals):
• 1s2s
• 1s3s
• 2p3p
• 2s3d
2. Find the spectroscopic term originating from the ground states configuration of the sodium atom and the boron atom.
3. Find the spin-orbit energy levels of hydrogen with electron in 2p and 3d.

Next: Lesson 13 - Introduction to molecular orbital theory


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# If $\vec{a},\vec{b},\vec{c}$ are coplanar vectors, the prove required result.

If $\vec{a},\vec{b},\vec{c}$ are coplanar vectors and $\vec{a}$ is not parallel to $\vec{b}$, then prove that $\{{(\vec{c} \times \vec{b}) \cdot(\vec{a} \times \vec{b})} \}\vec{a}+\{{(\vec{a} \times \vec{c}) \cdot(\vec{a} \times \vec{b})} \}\vec{b}$ is equal to $\{{(\vec{a} \times \vec{b}) \cdot(\vec{a} \times \vec{b})} \}\vec{c}$

I am trying to use the fact that three coplanar vectors are linearly dependent but not able to get the desired result. Could someone please help me with this.

• If $\vec a$ and $\vec b$ are not parallel, then $\vec c=x\vec a+y\vec b$. You could try inserting that and distribute as much as possible. – Arthur Feb 2 '17 at 21:51

Hint: that's a good idea. Let $\vec c=\lambda \vec a+\mu \vec b$ then distributing and using $\vec u \cdot \vec u = |\vec u|^2\,$, $\vec u \times \vec u = 0\,$:
$$\lambda |\vec a\times \vec b|^2 \;\vec a + \mu |\vec a \times \vec b|^2 \;\vec b = |\vec a \times \vec b|^2\,(\lambda \vec a + \mu \vec b) = |\vec a \times \vec b|^2\;\vec c$$


text:
## Differential and Integral Equations

### Global existence and blow up results for a heat equation with nonlinear nonlocal term

Miguel Loayza

#### Abstract

We study the nonlocal parabolic equation $$u_t-\Delta u=\int_0^t k(t,s)|u|^{p-1}u(s) \, ds$$ with $p>1$. We assume that $k$ is continuous and there exists $\gamma \in {\mathbb{R}}$ such that $k(\lambda t, \lambda s)=\lambda^{-\gamma}k(t,s)$ for all $\lambda>0$, $0<s<t$. We consider the problem in ${\mathbb{R}}^N$ and a Dirichlet problem in a bounded smooth domain $\Omega$. We analyze the conditions for either blow up or global existence of solutions.

#### Article information

Source
Differential Integral Equations, Volume 25, Number 7/8 (2012), 665-683.

Dates
First available in Project Euclid: 20 December 2012

https://projecteuclid.org/euclid.die/1356012657

Mathematical Reviews number (MathSciNet)
MR2975689

Zentralblatt MATH identifier
1265.35122

Subjects
Primary: 35K154 35B33: Critical exponents 35B44: Blow-up

#### Citation

Loayza, Miguel. Global existence and blow up results for a heat equation with nonlinear nonlocal term. Differential Integral Equations 25 (2012), no. 7/8, 665--683. https://projecteuclid.org/euclid.die/1356012657


text:
Math Help - Please help me with this Chain Rule and Cosine theorem problem

1. Please help me with this Chain Rule and Cosine theorem problem

Hi All,

I have the next problem to solve and I am trying to solve it for 2 days with no correct result. I used Chain Rule and I got:

but the answer is still not correct.

Please someone if can see let me know where is my mistake

Thank you

2. Re: Please help me with this Chain Rule and Cosine theorem problem

note that $\frac{df}{dt} < 0$ because the distance between the two is decreasing.


text:
# Find a matrix $P$ that orthogonally diagonalizes $I-uu^T$ if $u=\pmatrix{1\\-1\\1}$.

Find a matrix $P$ that orthogonally diagonalizes $I-uu^T$ if $u=\pmatrix{1\\-1\\1}$.

Let matrix $A=I-uu^T=\pmatrix{0&1&-1\\1&0&1\\-1&1&0}$.

I have followed the usual procedure and obtained matrix $P=\pmatrix{\frac{1}{\sqrt3}&-\frac{1}{\sqrt6}&\frac{1}{\sqrt2}\\-\frac{1}{\sqrt3}&\frac{1}{\sqrt6}&\frac{1}{\sqrt2}\\\frac{1}{\sqrt3}&\frac{2}{\sqrt6}&0}$.

In the midst of solving for $P^TAP$, I realized it got really tedious and long-winded. Is there is a better method to go about obtaining the diagonal matrix $D$ instead of computing it directly like so? I can't help but feel like I have missed something.

• Did you actually want the orthonormal eigenbasis (i.e., the diagonalizing matrix $P$), as you stated right at the top and in the question’s title, or did you simply way to find the diagonal matrix $D$? There are some short cuts for finding $P$ as well. – amd Apr 24 '17 at 23:35

Note that the diagonal entries of $D$ are simply the eigenvalues of $A$. Once you confirm that the eigenvalues of $A$ are $-2,1,1$, you can immediately conclude that we have $$D = \pmatrix{-2&0&0\\0&1&0\\0&0&1}$$ no computation of eigenvectors (or of $P$) required.

• Oh, I feel really silly now, thank you so much! Another question, if you don't mind: does the sequence of the column vector obtained for matrix $P$ matter? Sometimes I obtain a matrix $P$ with the same column entries but arranged differently from the solution. – iamaweed Apr 24 '17 at 22:07
• The order of the columns in $P$ just needs to match up with the order of the eigenvalues in $D$. In this case, my version of $D$ would be the correct version since your first column is the eigenvector for $\lambda = -2$. – Ben Grossmann Apr 24 '17 at 22:11
• Ah, I get it now. Thanks so much for your help! (: – iamaweed Apr 24 '17 at 22:12

Assuming that you actually wanted the matrix $P$ as stated in the title and first sentence of your question, here are some short cuts.

Every eigenvector $v$ of a matrix $A$ is also an eigenvector of $I-A$: $$(I-A)v=v-A v=v-\lambda v=(1-\lambda)v.$$ We can also see from this that if $\lambda$ is an eigenvalue of $A$, then $1-\lambda$ is an eigenvalue of $I-A$.

The matrix $uu^T$ has rank one—every column is a scalar multiple of $u$— and its kernel consists of all vectors $v$ for which $u^Tv=0$, i.e., the orthogonal complement of $u$. Thus, an orthonormal basis of $u^\perp$ together with $u/\|u\|$ is an orthonormal basis of $\mathbb R^3$ that consists of eigenvector of $I-uu^T$, so you can take as $P$ the matrix with these vectors as its columns. The eigenvalues of $uu^T$, incidentally, are $0$ and $u^Tu$. The former is obvious because the nullity of $uu^T$ is $2$, while the latter is easily found by rearranging $(uu^T)u$.

For this problem, you can take another short cut since you’re working in $\mathbb R^3$: if you take any vector $v$ that’s orthogonal to $u$ and then compute $u\times v$, the three vectors form an orthogonal basis and all that’s left to do is to normalize them. You can find by inspection that $(1,1,0)^T$ is orthogonal to $u$, and $(1,-1,1)^T\times(1,1,0)^T=(-1,1,2)^T$, from which a quick computation produces $$P=\left[\begin{array}{rcr}\frac1{\sqrt3}&\frac1{\sqrt2}&-\frac1{\sqrt6}\\-\frac1{\sqrt3}&\frac1{\sqrt2}&\frac1{\sqrt6}\\\frac1{\sqrt3}&0&\frac2{\sqrt6}\end{array}\right].$$ The matrix $D=P^{-1}(I-uu^T)P$ has the corresponding eigenvalues along its main diagonal and $u^Tu=3$, so $D=\operatorname{diag}(-2,1,1)$.