text:
#### 题目列表

Select the sentence in which the author describes and counters an assessment of a particular artist.
The passage is primarily concerned with
Select the sentence that identifies a prevailing scholarly view.
The author of the passage would most likely agree with which of the following statements concerning Smith and Clayton?
Pueblo Bonito, the most impressive of the "great houses" at the prehistoric Chaco Canyon site in New Mexico, comprised over 600 rooms and 4 to 5 stories. Traditional interpretations have viewed the great houses as almost entirely residential, with some archaeologists estimating the population of Pueblo Bonito at 1,200. But Windes recently challenged this view by pointing out the paucity of hearths recorded during the excavation of Pueblo Bonito, which revealed only 3 upper-story hearths, in contrast to 59 ground-floor hearths: habituation rooms would have required hearths for cooking and heat. It is possible, however, that the collapse of upper-story floors disturbed evidence of upper-story hearths to such an extent that they were not revealed by early excavations such as those conducted by Peppers field crews in the 1890s and Judds in the 1920s. Additionally, reliance on room features for early population estimates is complicated by the Chacoans later remodeling, especially given Judds disinclination to destroy later structures and features to expose earlier ones. The failure of early excavations to strip off intact floors may have concealed evidence of hearths in upper-story rooms.
The author of the passage would be most likely to agree with which of the following statements about Windes argument?
It can be inferred from the passage that Windes would be most likely to agree with which of the following statements about room usage in Pueblo Bonito?
Which of the following can be inferred about the excavation work performed by Judds field crew?
According to the passage, archacologists have traditionally believed which of the following about the great houses of Chaco Canyon?
Since the 1970s, archaeological sites in Chinas Yangtze River region have yielded evidence of sophisticated rice-farming societies that predate signs of rice cultivation elsewhere in East Asia by a thousand years. Before this evidence was discovered, it had generally been assumed that rice farming began farther to the south. This scenario was based both on the geographic range of wild or free-living rice, which was not thought to extend as far north as the Yangtze, and on archaeological records of very early domestic rice from Southeast Asia and India (now known to be not so old as first reported). Proponents of the southern-origin theory point out that early rice-farming societies along the Yangtze were already highly developed and that evidence for the first stage of rice cultivation is missing. They argue that the first hunter-gatherers to develop rice agriculture must have done so in this southern zone, within the apparent present-day geographic range of wild rice. Yet while most strands of wild rice reported in a 1984 survey were concentrated to the south of the Yangtze drainage, two northern outlier populations were also discovered in provinces along the middle and lower Yangtze, evidence that the Yangtze wetlands may fall within both the present-day and the historical geographic ranges of rices wild ancestor.
Which of the following, if true, would most clearly undermine the conclusion that the author makes based on the 1984 survey?
Based on the passage, skeptics of the idea that rice cultivation began in the Yangtze River region pointed to which of the following for support?
Which of the following can be inferred from the passage about the "southern-origin theory?"
The disappearance of Stellers sea cow from the Bering and Copper islands by 1768 has long been blamed on intensive hunting. But its disappearance took only 28 years from the time Steller first described the species, a remarkably short time for hunting alone to depopulate the islands, especially given the large populations initially reported. However, by 1750, hunters had also targeted nearby sea otter populations. Fewer otters would have allowed sea urchin populations on which the otters preyed to expand and the urchins grazing pressure on kelp forests to increase. Sea cows were totally dependent on kelp for food, and within a decade of the onset of otter hunting, Steller noted that the islands sea cows appeared malnourished.
Which of the following can be inferred from the passage about kelp forests in the Bering and Copper is lands between 1750 and 1768?
According to the passage, it is likely that during the mid-1700s, sea urchin populations near the Bering and Copper islands
The conventional story of the American colonists revolt against Britain holds that the founders of the United States established a form of government that, although flawed by its leaders failure to recognize the rights of women and African Americans, was nevertheless unsurpassed in its promise of human equality. There is, however, a cynical counterstory, which details the founders lust for property and their crass manipulation of the colonial population, and characterizes leaders like Thomas Jefferson as having wielded promises of equality merely as deceptive tools, discarded once the Revolution was won, and as having deliberately allowed the United States to be governed by a small, powerful elite. Both of these stories assume that a homogeneous revolutionary leadership employed an equally homogeneous egalitarian discourse to justify its actions. Even a cursory examination of the public discourse from the period, however, makes it clear that the leadership was anything but homogeneous in its discourse, and that overall the relative emphasis placed on the words "liberty" and "property" was far greater than that placed on "equality".

25000 +道题目

8本备考书籍


text:
# Brokers - Simultaneous Co-simulations¶

This example shows how to configure a HELICS co-simulation so that multiple co-simulations can run simultaneously on one computer. Understanding this configuration is a pre-requisite to running the other advanced broker examples (which also involve multiple brokers running on one computer).

## Where is the code?¶

The code for the Advanced examples can be found in the HELICS-Examples repository on GitHub. This example on simultaneous co-simulations can be found here. If you have issues navigating the examples, visit the HELICS Gitter page or the user forum on GitHub.

## What is this co-simulation doing?¶

Using the exact same federates as in the Advanced Default example, the same co-simulation is run multiple times (simultaneously) with different random number generator seeds. The example both demonstrates how to run multiple HELICS co-simulations simultaneously on one computer without the messages between federates getting mixed up, but also shows a simple way to do sensitivity analysis. A better way is shown later in the orchestration example.

### Differences compared to the advanced default example¶

Two primary changes:

1. This example contains a set of co-simulations with each instance using a different random number generator seed in Battery.py

if __name__ == "__main__":
np.random.seed(2608)

and Charger.py

if __name__ == "__main__":
np.random.seed(1490)

The values shown above are from federation_1. Identical lines with alternative values can be found in federation_2 and federation_3.

1. The brokers are configured to ensure that messages from one federation do not get routed to federates in another federation.

#### HELICS differences¶

With no extra configuration, it is only possible to run one HELICS co-simulation on a given computer. In the most popular HELICS cores (ZMQ being the most common, by far), messages are sent between federates using the networking stack. (There are other ways, though. For example, the IPC core uses the Boost library inter-process communication.). If you want to run multiple co-simulations on one compute need, an extra step needs to be taken to keep the messages from each federation separate from each other and non-interfering. Since we’re using the network stack, this can be easily accomplished by assigning each broker a unique port to use. Looking at the federation launch config files, you can see this clearly expressed:

federation_1_runner.json

{
"federates": [
{
"directory": ".",
"exec": "helics_broker -f 3 --loglevel=1 --port=20100",
"host": "localhost",
"name": "broker"
},

federation_2_runner.json

{
"federates": [
{
"directory": ".",
"exec": "helics_broker -f 3 --loglevel=1 --port=20200",
"host": "localhost",
"name": "broker"
},

federation_3_runner.json

{
"federates": [
{
"directory": ".",
"exec": "helics_broker -f 3 --loglevel=1 --port=20300",
"host": "localhost",
"name": "broker"
},

#### Research question complexity differences¶

The Advanced Default example uses a random number generator to determine things like initial state-of-charge of the battery and charging power of the individual EV batteries. These factors have an impact on the charging duration for each EV battery and the peak charging power seen over the duration of the simulation. Since the later is the key metric of the simulation experiment, there is strong motivation to vary the seed value for the random number generator to expand the range of results, effectively increasing the sample size. These co-simulations could each be run serially but assuming the computer in question has the horsepower, there’s no reason not to run them in parallel.

## Execution and Results¶

To run the co-simulations simultaneously, all that is required is having the HELICS runner launch each individually. The trailing & in the shell commands below background the command and return another shell prompt to the user.

$helics run --path=./federation_1/federation_1_runner.json &$ helics run --path=./federation_2/federation_2_runner.json & \$ helics run --path=./federation_2/federation_3_runner.json &

The peak charging results are shown below. As can be seen, the peak power amplitude and the total time at peak power are impacted by the random number generator seed.

To do a more legitimate sensitivity analysis to the population of EVs that are being charged, a sample size larger than three is almost certainly necessary. We’ve put together another example to show how to orchestrate running larger sets of co-simulations to address exactly these kinds of needs.

## Questions and Help¶

Do you have questions about HELICS or need help?

1. Come to office hours!

2. Post on the gitter!

3. Place your question on the github forum!

format: plain


text:
Chapter 7

# To test the understanding of the lesson

## Question

• A Newtonian dynamic viscosity fluid, flowing with a velocity field of the type .

Give the expression of the shear stress exerted by the fluid located below the axis on the fluid are located above this axis.

• What is the volume expression of viscous forces acting on an incompressible viscous and Newtonian fluid ?

### Solution

• For unidirectional flow such that , the tangential surface force, known as shear force or viscous force, which is exerted through a area of surface normal to (it is the force exerted by the upper layer on the bottom layer) :

The viscosity effect in an unidirectional flow, accelerates the slow elements and slows down the fast elements.

It is therefore an internal transfer of momentum, which has the characteristics of diffusion of momentum.

• The volume force of viscosity is :

## Question

• Give the definition of the volume flow rate and mass flow rate of a flowing fluid.

• What is the relationship between them ?

### Solution

• The volume flow rate :

Called volume flow rate through a oriented surface ( ), the volume of fluid which crosses ( ) per unit of time, counted positively in the direction of the normal vector to the surface and negatively otherwise.

This flow rate is :

• Mass flow rate :

The mass flow rate corresponds to the mass of fluid passing through ( ) per unit of time, counted positively in the direction of the normal vector to the surface and negatively otherwise :

Where is the current density vector or vector density of flux of flow mass.

• If the fluid is incompressible and homogeneous, so if the density is constant and independent of the point, then :

It may be noted that these flow rates are equivalent to the electrical intensity, the heat flux and the particle flow rate (seen in diffusion).

## Question

• Give the definition and the expression of the Reynolds number. What is its dimension ?

• What is the relationship between the Reynolds number and the thickness of the boundary layer  ?

### Solution

• The Reynolds number is defined as the magnitude of the ratio between the convective term and the diffusion term :

Where is a characteristic size of the flow (the length of the obstacle, for example) and the velocity of the fluid away from the obstacle.

• The thickness of the boundary layer is expressed in terms of the Reynolds number :

## Question

• Give the definition of the vorticity vector of a flow fluid.

• What is the definition of a potential flow ?

### Solution

• The vorticity vector of a flow fluid is :

• If the vorticity vector is zero at any point in space :

We can then define a velocity potential, denoted , by :

The flow is said potential.

## Question

• Define an incompressible flow.

• For a potential flow, demonstrate the Laplace equation.

### Solution

• For an incompressible flow :

We can derive that the volume flow can be kept in a field tube and therefore evidently in pipeline.

In particular, if the speed is constant over a section of the pipeline :

Thus, when a flow field lines grow narrow, the norm of the vector speed increases.

• The connections between and leading to the Laplace equation (also seen in electrostatic) :

## Question

Give the expression of the particle acceleration in function of Eulerian field speeds.

## Question

Giving local expression of mass conservation law.

### Solution

This is a classic conservation equation :

With :

(current density vector or vector density of flux of flow mass)

## Question

• Write the Euler equation for a perfect flow in a Galilean frame of reference.

• Give the Navier - Stokes equation for viscous flow of an incompressible fluid in a Galilean frame of reference.

### Solution

• Euler equation for a perfect flow in a Galilean frame of reference :

Or again :

• Navier - Stokes equation for viscous flow of an incompressible fluid in a Galilean frame of reference :

## Question

• Give a Bernoulli equation on a current line in a perfect, homogeneous, stationary and incompressible flow.

• Give a Bernoulli equation in a perfect, homogeneous, stationary, incompressible and irrotational flow.

### Solution

• On a current line in a perfect, homogeneous, stationary and incompressible flow.

Between two points and of the same current line :

• For an irrotational flow :

Where is a constant (independent of the point in the fluid).

## Question

Emptying a container : give the Torricelli's formula . When does it apply ?

### Solution

Torricelli's formula is :

is the fluid velocity at the output of the hole and the height of liquid above the hole.

This formula rigorously applies when the height is constant.

It also applies when the flow is quasi-permanent ( varies slowly with time).

## Question

• What is called a Poiseuille flow ?

• Define the hydraulic resistance.

### Solution

• Poiseuille flow :

A incompressible viscous fluid of density flows through a cylindrical tube of length and radius .

The pressure at the input of the tube ( ) is . The pressure at the output of the tube is .

• The hydraulic resistance is :

Where is the volume flow rate of the Poiseuille flow.

Previous
Barrel drilled


text:
GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 16 Nov 2019, 19:13

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

### Show Tags

Updated on: 18 Jun 2015, 01:16
5
14
00:00

Difficulty:

5% (low)

Question Stats:

96% (00:36) correct 4% (00:48) wrong based on 2143 sessions

### HideShow timer Statistics

The price of a coat in a certain store is $500. If the price of the coat is to be reduced by$150, by what percent is the price to be reduced?

(A) 10%
(B) 15%
(C) 20%
(D) 25%
(E) 30%

2016 GMAT Official Guide, Question 1, Pg. 152

Originally posted by LithiumIon on 17 Jun 2015, 22:26.
Last edited by Bunuel on 18 Jun 2015, 01:16, edited 1 time in total.
Renamed the topic and edited the question.
EMPOWERgmat Instructor
Status: GMAT Assassin/Co-Founder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
Posts: 15461
Location: United States (CA)
GMAT 1: 800 Q51 V49
GRE 1: Q170 V170
Re: The price of a coat in a certain store is $500. If the price of the co [#permalink] ### Show Tags 18 Jun 2015, 17:42 3 Hi LithiumIon, This is one of the *new* questions from the OG2016. A PDF listing of all the new questions (by number) can be found here: empowergmat-blog-198415.html A certain percentage of questions in the Quant section of the GMAT are just 'math questions' - you'll use a formula, do some calculations and you'll have the answer. This is one of those types of questions. You still have to write everything down and stay organized, but the work involved is relatively straight-forward. Here, we're told that the price of a coat is$500 and that the price of the coat is to be reduced by $150. We're asked by what PERCENT is the price to be REDUCED? Original Price =$500
Reduction = $150 The percent reduction is relative to the original price of the coat: 150/500 = 15/50 = 3/10 = 30% Final Answer: GMAT assassins aren't born, they're made, Rich _________________ Contact Rich at: Rich.C@empowergmat.com The Course Used By GMAT Club Moderators To Earn 750+ souvik101990 Score: 760 Q50 V42 ★★★★★ ENGRTOMBA2018 Score: 750 Q49 V44 ★★★★★ CEO Status: GMATINSIGHT Tutor Joined: 08 Jul 2010 Posts: 2977 Location: India GMAT: INSIGHT Schools: Darden '21 WE: Education (Education) Re: The price of a coat in a certain store is$500. If the price of the co  [#permalink]

### Show Tags

17 Jun 2015, 23:53
1
2
LithiumIon wrote:
2016 GMAT Official Guide, Question 1, Pg. 152

The price of a coat in a certain store is $500. If the price of the coat is to be reduced by$150, by what percent is the price to be reduced?

(A) 10%
(B) 15%
(C) 20%
(D) 25%
(E) 30%

price of a coat in a certain store = $500 the price of the coat is to be reduced by$150

% change = (Final Value - Initial Value)*100 / Initial Value

% Reduction= (Reduction in Price)*100 / Initial Value

i.e. % Reduction= (150)*100 / 500 = 30%

_________________
Prosper!!!
GMATinsight
Bhoopendra Singh and Dr.Sushma Jha
e-mail: info@GMATinsight.com I Call us : +91-9999687183 / 9891333772
Online One-on-One Skype based classes and Classroom Coaching in South and West Delhi
http://www.GMATinsight.com/testimonials.html

ACCESS FREE GMAT TESTS HERE:22 ONLINE FREE (FULL LENGTH) GMAT CAT (PRACTICE TESTS) LINK COLLECTION
Manager
Joined: 20 Mar 2015
Posts: 54

### Show Tags

01 Nov 2018, 00:18
1
Agreed to all solutions; we can do it by quick math
100% of 500 = 500.0
10% of 500 = 50.0

Now we can apply either additive principle or multiplication one:

If we go with the additive principle:
10%+10%+10% = 30%
50+50+50 = 150

If we do it by the multiplicative principle:
10%*3 = 30%
50*3 = 150.0

Just two different approaches to get to the same answer.
GMAT Club Legend
Joined: 12 Sep 2015
Posts: 4064

### Show Tags

17 Apr 2016, 00:29
(150/500)*100 = 30%
Manager
Joined: 10 Apr 2016
Posts: 54
Concentration: Strategy, Entrepreneurship
GMAT 1: 520 Q29 V30
GPA: 3.01

### Show Tags

02 May 2016, 09:25
LithiumIon wrote:
The price of a coat in a certain store is $500. If the price of the coat is to be reduced by$150, by what percent is the price to be reduced?

(A) 10%
(B) 15%
(C) 20%
(D) 25%
(E) 30%

2016 GMAT Official Guide, Question 1, Pg. 152

Solution:

We are given that the price of a coat is $500 and that the price is reduced by$150. Thus, the price is reduced by:

150/500 x 100 = 3/10 x 100 = 30 percent

_________________

# Scott Woodbury-Stewart

Founder and CEO

Scott@TargetTestPrep.com
122 Reviews

5-star rated online GMAT quant
self study course

See why Target Test Prep is the top rated GMAT quant course on GMAT Club. Read Our Reviews

If you find one of my posts helpful, please take a moment to click on the "Kudos" button.


text:
# 온도의 변화가 수수, 수수X수단그라스교잡, 수단그라스 및 옥수수종자의 발아에 미치는 영향

• Published : 1986.09.30

#### Abstract

This experiment was carried out to study the influence of temperature treatments on germination of sorghum (cv. Pioneer 931), sorghum-sudangrass (cv. Pioneer 988), sudangrass (cv. Piper) and corn (cv. Suweon 19) in growth chamber. Each crop seeds were germinated under different day/night temperature of 10/5, 15/10, 20/15, 25/20, 30/25, 35/30 and $40/35^{\circ}C$ over 14/10 hour days. The results obtained are summarized as follows. 1. Germination of sorghum and sorghum-sudangrass were less affected by high temperature than those of corn and sudangrass. The optimum temperatures for germination were $20-40^{\circ}C$ for sorghum and $20-30^{\circ}C$ for corn. High temperature($40/35^{\circ}C$), however, resulted in great decrease of seed germination rates of seed germination rates of corn, while those of sorghum and sorghum-sudangrass shown high germination rates with the value of 95.8% and 89.9%, respectively. 2. Sorghum-sudangrass hybrid was shown to have a great tolerance to low temperature. Under low temperature($10/5^{\circ}C$) seed germination of sorghum-sudangrass was 80.0% but those of corn and sudangrass were 43.2% and 24.8% respectively. Germination rates of sorghum were decreased to about 68.8% at low temperature from a value of 98.1% at optimum temperature. 3. The days required to the seed germination of sorghum and sorghum-sudangrass hybrid were shortened by increasing of temperature. Under different temperature treatments, it was required 12 days ($15/10^{\circ}C$), 6 days ($25/20^{\circ}C$) and 3 days ($40/35^{\circ}C$) in sorghum but corn required 16, 7 and 3 days, respectively.


text:
# Is entropy violated inside black holes and worm holes?

Do the laws of thermodynamics hold true everywhere in universe ?

What about black holes and worm holes ?

• – Kyle Kanos Oct 5 '15 at 20:35
• As far as I know, there's a good chance wormholes don't and can't exist, which makes that part of the question rather Moot. It's a good question for black holes though. – userLTK Oct 5 '15 at 23:43

No, black holes do not violate the 2nd Law of Thermodynamics.

Imagine that we want to violate the 2nd Law of Thermodynamics by throwing some volume of ideal gas into a black hole. This would seem to violate the 2nd law because when it is outside the black hole the ideal gas contributes some calculable amount of entropy to the total entropy of the universe. Once the gas crosses the event horizon of the black hole, however, it is hidden from all observers who are outside the black hole. We conclude then that in this process $$dS_{universe} < 0.$$ That is, we have decreased the entropy of the universe. This is a big problem and we need a way to resolve it. The answer is to assume that black holes carry entropy.

An important mathematical theorem in General Relativity is the Area Theorem, which states that the surface area of a black hole's event horizon can never decrease (given reasonable physical assumptions). So $$dA \geq 0.$$ In the early 1970s, Jacob Bekenstein had the insight that he could relate the area of the event horizon of a black hole to another quantity that we know can never decrease: entropy. By dimension analysis, Bekenstein wrote down the equation for black hole entropy: $$S_{BH}=\alpha\frac{k_BAc^3}{G\hbar},$$ where $\alpha$ is some dimensionless constant (which Hawking proved is equal to exactly $1/4$).

Now, let's return to the problem of throwing an ideal gas into a black hole. We know that the mass of the black hole will increase, hence the area of the event horizon will increase. If, for example, our ideal gas also carries charge and angular momentum, then the charge and angular momentum of the black hole will change as well. We can relate the change in the area of the event horizon $dA$, the change in the mass of the black hole $dM$, the change in the charge of the black hole $dQ$, and the change in the angular momentum of the black hole $dJ$ via $$\frac{\kappa}{8\pi}dA=dM + \Phi dQ - \Omega_H dJ,$$ where $\kappa$ is the surface gravity of the black hole, $\Phi$ is the electrostatic potential of the black hole horizon, and $\Omega_H$ is the angular velocity of the event horizon. This equation should look familiar. It is the 1st Law of Black Hole Thermodynamics, and it is directly analogous to the ordinary 1st law.

We can write down a generalized 2nd Law of Thermodynamics:

$$S_{tot} = S_{universe} + S_{BH},$$ so that $dS_{tot} \geq 0$, even when we consider processes involving black holes.

This is far from the end of the story, and many interesting questions abound. For example, the entropy of a thermodynamic system is usually explained as characterizing the number of accessible microstates of the system. If black holes have a well defined entropy, then it should be possible to express this entropy as the logarithm of the number of accessible black hole microstates. In 1995, Andrew Strominger and Cumrun Vafa showed that by counting certain states of black holes in string theory, they could correctly reproduce the Bekenstein-Hawking entropy of those special types of black holes.


text:
spectrum$vects  Motif-based spectral clustering The overall aim of motifcluster is to use motifs for spectral clustering, so now we see how to extract clusters from the motif-based eigenvector embeddings. The run_motif_clustering function handles the entire process of building an MAM, restricting it to its largest connected component, performing eigenvector embedding, and extracting clusters. We therefore take the opportunity to showcase the ability of motifcluster to recover the blocks of a DSBM, demonstrating all of the methods outlined in this vignette. Let's use a DSBM with three blocks of 10 nodes each. block_sizes = rep(10, 3)  We use strong connections of 0.8 within the blocks, and weaker connections of 0.3 between the blocks. connection_matrix = matrix(c( 0.8, 0.2, 0.2, 0.2, 0.8, 0.2, 0.2, 0.2, 0.8 ), nrow = 3)  We also set the within-block edges to be Poisson-weighted with mean 20, and the between-block edges to be Poisson-weighted with smaller mean 10. weight_matrix = matrix(c( 20, 10, 10, 10, 20, 10, 10, 10, 20 ), nrow = 3) G4 = sample_dsbm(block_sizes, connection_matrix, weight_matrix, sample_weight_type = "poisson")  Now we can run the motif-based spectral clustering algorithm on this network with the 3-cycle motif$\mathcal{M}_1$. We build a functional MAM, weighting the instances by their mean edge weights, using the sparse formulation. We restrict this MAM to its largest connected component. Then we construct a random-walk Laplacian and embed it using the first four eigenvalues and eigenvectors. Finally we extract three clusters. motif_cluster = run_motif_clustering(G4, motif_name = "M1", motif_type = "func", mam_weight_type = "mean", mam_method = "sparse", type_lap = "rw", num_eigs = 4, num_clusts = 3 )  We can evaluate the performance by comparing it to the ground-truth labels using the adjusted Rand index from the mclust package: truth = c(rep(1, 10), rep(2, 10), rep(3, 10)) mclust::adjustedRandIndex(motif_cluster$clusts, truth)

A larger value indicates better recovery of the blocks, with a value of one indicating perfect agreement.

References

options(old_options)

Try the motifcluster package in your browser

Any scripts or data that you put into this service are public.

motifcluster documentation built on April 30, 2022, 1:06 a.m.


text:
# ACU-T: 3000 Enclosed Hot Cylinder: Natural Convection

This tutorial provides the instructions for setting up, solving and viewing results for a simulation of a hot cylinder contained within another air-filled cylinder. In this simulation, an internally heated cylinder is surrounded by air which heats up as it comes in contact with the surface of the inner cylinder. The localized heating near the surface induces a buoyancy driven flow in the air, generating convection currents. This tutorial is designed to introduce you to modeling concepts related to natural convection simulations.

The basic steps in any CFD simulation are shown in ACU-T: 2000 Turbulent Flow in a Mixing Elbow. The following additional capabilities of AcuSolve are introduced in this tutorial:
• Creating and specifying a new custom material in AcuConsole
• Specifying a volume group as a heat source
• Using the Boussinesq density model in buoyancy driven flows, such as cases involving natural convection
• Set up periodic boundary conditions
In this tutorial you will do the following:
• Analyze the problem
• Start AcuConsole and create a simulation database
• Set general problem parameters
• Set solution strategy parameters
• Create a new custom material model in AcuConsole and assign material properties to it
• Import the geometry for the simulation
• Create a volume group and apply volume parameters
• Create surface groups and apply surface parameters
• Set global and local meshing parameters
• Set periodic boundary conditions
• Generate the mesh
• Set the appropriate boundary conditions
• Run AcuSolve
• Monitor the solution with AcuProbe
• Post-processing the nodal output with AcuFieldView

## Prerequisites

You should have already run through the introductory tutorial, ACU-T: 2000 Turbulent Flow in a Mixing Elbow. It is assumed that you have some familiarity with AcuConsole, AcuSolve, and AcuFieldView. You will also need access to a licensed version of AcuSolve.

Prior to running through this tutorial, copy AcuConsole_tutorial_inputs.zip from <Altair_installation_directory>\hwcfdsolvers\acusolve\win64\model_files\tutorials\AcuSolve to a local directory. Extract twin_cylinder.x_t from AcuConsole_tutorial_inputs.zip.

The color of objects shown in the modeling window in this tutorial and those displayed on your screen may differ. The default color scheme in AcuConsole is "random," in which colors are randomly assigned to groups as they are created. In addition, this tutorial was developed on Windows. If you are running this tutorial on a different operating system, you may notice a slight difference between the images displayed on your screen and the images shown in the tutorial.

## Analyze the Problem

An important step in any CFD simulation is to examine the engineering problem at hand and determine the important parameters that need to be provided to AcuSolve. Parameters can be based on geometrical elements (such as inlets, outlets, or walls) and on flow conditions (such as fluid properties, velocity, or whether the flow should be modeled as turbulent or as laminar).

The system being simulated contains an internally-heated cylinder, which is surrounded by a cylindrical ring of a larger diameter. The annular volume between the two cylinders is filled with a fluid (air). The inner cylinder thus acts a heat source, and the fluid in contact with the surface of this heat source is heated up. This hot fluid, being lower in density than the cold fluid, then rises up to the upper part of the annulus due to buoyancy effects, and displaces the cold fluid at top. At the same time, the film of fluid which was in contact with the heating surface is replaced by the surrounding cold fluid. This new film of cold fluid goes through the same process until eventually a steady state convection current is achieved, or the inner cylinder ceases to generate heat and slowly the whole system achieves an equal temperature.

The system being simulated can be considered similar to a heat exchanger wherein the inner cylinder is akin to a tube through which a hot fluid passes by, and the air which surrounds this inner tube extracts heat from the inner tube. Another analogy can be of a wire carrying high current enclosed in an air cooled chamber. As the current heats up the wire due to resistance, the air around the wire keeps the wire temperature within control by extracting heat from the wire surface.

The schematics of the problem which will be addressed in this tutorial is shown in Figure 1. The inner cylinder is a solid volume with internal heat generation, and the outer cylinder is a fluid volume with air as the fluid. Both cylinders are assumed to be infinitely long and the system will be modeled using half symmetry and periodicity. The cylinders are infinite in z-direction and hence periodicity will be applied along this direction.

## Introduction to Theory

### Natural Convection

Convection is a heat transfer mechanism where the transfer of heat energy happens through the motion of matter. Since the definition of convection involves motion of matter a fluid state is usually present in convection. Usually this type of heat transfer takes place between a hot or a cold surface and a fluid. The film of fluid in contact with the surface absorbs heat from or transfers heat to the surface and is then replaced by a new film. This movement of fluid may either be governed by an external source, such as a fan or pump, or due to internal changes in the fluid properties. When no external sources are responsible for the fluid motion the heat transfer mechanism at work is called the Natural Convection. The driving force for motion of the fluid in a natural convection is density changes in the fluid due to temperature gradients induced in the fluid by heat transfer.

The natural convection mechanism works similarly as described above, whilst discussion of the problem. The fluid which is in contact with the surface absorbs or transfers heat from the surface and becomes hotter or colder than the surrounding fluid. Driven by buoyancy forces due to difference in densities caused by the temperature gradient, the fluid is displaced upwards or downwards. Surrounding fluid fills in the void created by the displaced fluid, which then undergoes the same process again. This gives rise to a convection current which drives the hot fluid to the top and cold fluid to the bottom of the convection cell. Buoyancy effects are driven by gravity, therefore natural convection requires presence of a gravitational force to work. It must be noted, however, that gravity is not the driving force behind the fluid movement. Presence of gravity only enables displacement of the fluid due to the density changes caused by temperature gradients.

Mathematical determination of the onset of natural convection is done through a dimensionless number called the Rayleigh number (Ra). The Rayleigh number is defined as:

where:
• x is the characteristic length (m)
• $R{a}_{x}$ is the Rayleigh number for characteristic length x
• $g$ is acceleration due to gravity (m/s2)
• ${T}_{s}$ is the surface temperature (K)
• ${T}_{\infty }$ is the quiescent temperature (fluid temperature far from the surface of the object) (K)
• $\nu$ is the kinematic viscosity (m2/s)
• α is the thermal diffusivity (m2/s)
• β is the thermal expansion coefficient (equals to $1/T$ for ideal gases where is absolute temperature).

The fluid properties $\nu$, α and β are evaluated at the film temperature, ${T}_{f}$, which is defined as:

${T}_{f}=\frac{{T}_{s}+{T}_{\infty }}{2}$

When the Rayleigh number is below a critical value for the fluid heat transfer is primarily in the form of conduction. When it exceeds this critical value the dominant heat transfer mechanism is convection.

### Boussinesq Density Model

The Boussinesq density model is an approximation method applied to buoyancy driven flows, such as natural convection flows. In the Boussinesq approximation, the density variation terms are neglected everywhere except when multiplied by acceleration due to gravity, $g$. The basis of this approximation is that since temperature changes are small, the resultant changes in density are small as well and thus can be neglected. However, when multiplied by $g$, the resultant term gives rise to forces which no longer are negligible. The Boussinesq approximation is:

where
• $\rho$ is the instantaneous density at temperature $T$ (kg/m3)
• ${\rho }_{0}$ is the density at reference temperature ${T}_{o}$ (kg/m3)
• $\text{Δ}T$ is change in temperature $T-{T}_{o}$ (K)

As stated in the approximation, the Boussinesq density model is only applicable when density variations are small. A general guideline is to check for the condition to be true. This indirectly puts a limitation on this model to be used to only for cases where expected temperature differences within the fluid are not large.

## Define the Simulation Parameters

### Start AcuConsole and Create the Simulation Database

In this tutorial, you will begin by creating a database, populating the geometry-independent settings, loading the geometry, creating volume and surface groups, setting group parameters, adding geometry components to groups, and assigning mesh controls and boundary conditions to the groups. Next, you will generate a mesh and run AcuSolve to solve for the number of time steps specified. Finally, you will visualize some characteristics of the results using AcuFieldView.

In the next steps you will start AcuConsole, and create the database for storage of the simulation settings.

1. Start AcuConsole from the Windows Start menu by clicking Start > Altair <version> > AcuConsole.
2. Click the File menu, then click New to open the New data base dialog.
Note: You can also open the New data base dialog by clicking on the toolbar.
3. Browse to the location that you would like to use as your working directory.
This directory is where all files related to the simulation will be stored. The AcuConsole database file (.acs) is stored in this directory. Once the mesh and solution are created, additional files and directories will be created within this directory.
4. Create a new directory in this location. Name it Natural_convection and navigate into this directory.
5. Enter NaturalConvection as the file name for the database, or choose any name of your preference.
Note: In order for other applications to be able to read the files written by AcuConsole, the database path and name should not include spaces.
6. Click Save to create the database.

### Set General Simulation Parameters

In next steps you will set parameters that apply globally to the simulation. To make this simple, the basic settings applicable for any simulation can be filtered using the BAS filter in the Data Tree Manager. This filter enables display of only a small subset of the available items in the data tree and makes navigation of the entries easier.

1. Click BAS in the Data Tree Manager to switch to basic view in the Data Tree.
2. Double-click the Global Data Tree item to expand it.
Tip: You can also expand a tree item by clicking next to the item name.
3. Double-click Problem Description to open the Problem Description detail panel.
Tip: You can also open a panel by right-clicking a tree item and clicking Open on the context menu.
4. Enter AcuSolve Tutorial as the Title.
5. Enter Natural Convection as the Sub title.
6. Change the Analysis type to Steady State.
7. Change the Temperature equation to Advective Diffusive.

### Set Solution Strategy Parameters

In the next steps you will set the parameters that control the behavior of AcuSolve as it progresses during the solution.

1. Double-click Auto Solution Strategy in the Data Tree to open the Auto Solution Strategy detail panel.
2. Check that Analysis type is set to Steady State.
3. Set the Max time steps to 100.
4. Change the Convergence tolerance to 0.0001.
5. Enter 0.25 for the Relaxation factor.
6. Check that Flow and Temperature are set to On.
7. Change the Temperature flow to On.
Changing the Temperature flow flag to On will instruct the solver to solve thermal-flow problems in fully coupled mode. Otherwise these problems are solved with a staggered strategy. In fully-coupled mode, the flow and temperature equations are solved simultaneously, while in the staggered approach, the flow equation will usually be solved first considering constant temperature, and then the temperature equation will be solved as the next step.

### Set Material Model Parameters

AcuConsole has three pre-defined materials, Air, Aluminum and Water, with standard parameters defined. In the next steps you will check and modify the material characteristics of the predefined Air model to match the desired properties for this problem. Since this a natural convection problem the density type for air will be set to use the Boussinesq approximation. Subsequently, you will create a new custom material and assign relevant material properties to it.
1. Double-click Material Model in the Data Tree to expand it.
2. Double-click Air in the Data Tree to open the Air detail panel.

The material type for air is Fluid. Fluid is the default material type for any new material created in AcuConsole.

3. Click the Density tab. Change the density type to Boussinesq.
4. Click the Viscosity tab. The viscosity of air is 1.781 x 10-5kg/m – sec.
5. Click the Specific Heat tab and make sure the Specific heat value is 1005.0 J/kg-K.
6. Similarly check the Conductivity tab and make sure the values are as follows:
1. Conductivity: 0.02521 W/m-K
2. Turbulent Prandtl number: 0.91
7. Save the database to create a backup of your settings. This can be achieved with any of the following methods.
• Click the File menu, then click Save.
• Click on the toolbar.
• Click Ctrl+S.
Note: Changes made in AcuConsole are saved into the database file (.acs) as they are made. A save operation copies the database to a backup file, which can be used to reload the database from that saved state in the event that you do not want to commit future changes.
8. Right-click Material Model in the Data Tree and select New from the context menu that appears.
A new entry, Material Model 1, will be created in the Data Tree under the Material Model branch.
9. Right-click Material Model 1 and select Rename in the context menu.
10. Type in Stainless Steel as the name and press Enter.
11. Double-click Stainless Steel in the Data Tree to open the Stainless Steel detail panel.
The Material type is listed as Fluid. This is the default type for any new material created in AcuConsole.
12. Change the Material type for Stainless Steel to Solid.
13. Set the material properties for Stainless Steel as follows by navigating through respective tabs in the detail panel:
1. Density: 8000 kg/m3.
2. Specific Heat: 500.0 J/kg-K
3. Conductivity: 16.2 W/m-K

## Import the Geometry and Define the Model

### Import the Geometry

You will import the geometry in the next part of this tutorial. You will need to know the location of twin_cylinder.x_t in order to complete these steps. This file contains information about the geometry in Parasolid ASCII format.
1. Click File > Import.
2. Browse to the directory containing twin_cylinder.x_t.
3. Change the file name filter to Parasolid File (*.x_t *.xmt *X_T …).
4. Select twin_cylinder.x_t and click Open to open the Import Geometry dialog.

For this tutorial, the default values for the Import Geometry dialog are used to load the geometry. If you have previously used AcuConsole, be sure that any settings that you might have altered are manually changed to match the default values shown in the figure. With the default settings, volumes from the CAD model are added to a default volume group. Surfaces from the CAD model are added to a default surface group. You will work with groups later in this tutorial to create new groups, set flow parameters, add geometric components, and set meshing parameters.

5. Click Ok to complete the geometry import.
6. Rotate the visualization to view the entire model.

### Set the Body Force

The body force commands add volumetric source terms to the governing conservation equations. Two types of body forces will be used in this tutorial.

The first one is the gravitational force on the fluid due to inertia of the fluid. As discussed in Analyze the Problem, gravity is an important aspect of the simulation. In fact, for thermal problems solved in AcuSolve with the Boussinesq approximation, the gravity is scaled by the product of the expansivity and the temperature minus reference temperature, while density remains constant. This variation in the gravitational force on fluid regions with different temperatures is what generated convection currents. For this tutorial gravity is defined as equal to standard gravity (g = 9.81 m/s2) along the negative Y-axis, which is the downward direction in the model.

The second body force which will be used in this model is the volumetric heat source, which specifies the heat energy source term per unit volume. This will be used to simulate the heat-generating inner cylinder in our model.

1. Double-click Body Force in the Data Tree to expand it.
2. Double-click Gravity to open the Gravity detail panel.

The medium for gravity is Fluid. This means that the gravity defined here is applicable only on material models whose material type is fluid.

3. Click Open Array.
4. In the Array Editor dialog, enter:
• X-component: 0.0
• Y-component: -9.81 m/s2
• Z-component: 0.0
5. Click OK to complete the definition of gravity.
Note: The definition of gravity here will have no effect on the simulation unless it is assigned to some volume set in the model.
6. Create a new body force by right-clicking on Body Force in the Data Tree and selecting New in the context menu that appears.
A new entry, Body Force 1, will be created under the Body Force branch.
7. Right-click on Body Force 1, select Rename in the context menu, and type in Heat Source as the entity name.
8. Double-click on Heat Source to open it in the detail panel.
9. Change the Medium to Solid.
10. Click on the drop-down selector next to first Type option and select Per unit volume.
This sets the type of heat source to volumetric heat source.
11. Click on the drop-down selector next to the second Type option and select Constant.
12. Set the Volumetric heat source value to 20000.0 W/m3

### Apply Volume Parameters

Volume groups are containers used for storing information about a volume region. This information includes solution and meshing parameters applied to the volume and the geometric regions that these settings are applied to.

When the geometry was imported into AcuConsole, all volumes were placed into the "default" volume container.

In the next steps you will create volume groups for each volume in the model, assign volumes to the respective volume groups, rename the default volume group container, and set the materials and other properties for each volume group.

1. Expand the Model Data Tree item.
2. Create a new volume group for the solid inner cylinder.
1. Right-click on Volumes.
2. Click New.
3. Rename the new volume group to solid.
4. Add the solid component in the geometry to this group.
1. Right-click solid under Volumes in the Data Tree.
3. Click the heating element portion of the geometry in the Visualization Area. Refer to the following figure to identify the correct portion.
Follow the instructions in the Add to dialog if you need to manipulate the display to select the correct portion of the geometry.
4. Click Done to add the selected volume to the solid volume group.
5. Set up the solid volume element set.
The material model for this volume will be set to Stainless Steel, which is the custom material model you created earlier in this tutorial, specifically for this solid volume. Also the solid volume is to be set up as the heat source
1. Expand the solid volume group in the tree.
2. Double-click Element Set to open the Element Set detail panel.
3. Change the Medium to Solid.
4. Change the Material model to Stainless Steel.
5. Change the Body force to Heat Source.
6. In the Data Tree, right-click on default and rename it to fluid.
7. Set up the Fluid volume element set.
1. Expand the fluid volume group in the tree.
2. Double click Element Set under fluid to open it in the detail panel.
3. Ensure that the Medium for the volume is set to Fluid. If not, change it to Fluid.
4. Change the Material model to Air.
5. Change the Body force to Gravity.

### Create Surface Groups and Apply Surface Parameters

Surface groups are containers used for storing information about a surface, including solution and meshing parameters, and the corresponding surface in the geometry that the parameters will apply to.

In the next steps you will define surface groups, assign the appropriate settings for the different characteristics of the problem, and add surfaces to the group containers.

In the process of setting up a simulation, you need to move into different panels for setting up the boundary conditions, mesh parameters, and so on, which can sometimes be cumbersome, especially for models with too many surfaces. To make it easier, less error prone, and to save time, two new dialogs are provided in AcuConsole. Use the Volume Manager and Surface Manager to verify and provide the information for all surface or volume entities at once. In this section some features of Surface Manager are exploited.

1. Turn-off display for Volumes by right-clicking on Volumes and selecting Display off .
2. Right-click on Surfaces in the Data Tree and select Surface Manager.
3. In the Surface Manager dialog, click New six times to create six new surface groups.

If you cannot see the Simple BC Active and Simple BC Type columns, click on Columns , select these two columns from the list and click Ok.

4. Turn off the display for all surfaces except for the default surface.
5. Rename the default surface to inner_wall.
6. Rename Surface 1 through Surface 6 according to the image below.
7. Set the Simple BC Active and Simple BC Type columns as per Figure 14.
8. Assign the periodic surfaces to the respective surface groups.
As mentioned earlier, the cylinders are assumed to be infinitely extended in z-direction. Hence periodicity will be applied in this direction.
1. In the solid_pos_z row in the Surface Manager, click Add to .
2. Select the planar symmetry surfaces as shown in Figure 15 and click Done.
3. Follow the procedure to assign all the surfaces that will extend in the z-direction to respective surface collectors.
9. Assign the outer wall of the geometry to the outer_wall surface group. Use Figure 16 as the reference for selecting the required surfaces.
10. Assign the surface for symmetry_plane.

When the geometry was loaded into AcuConsole, all geometry surfaces were placed in the default surface group container. This default surface group was renamed to inner_walls. In the previous steps, you assigned some surfaces to various other surface groups that you created. At this point, all that is left in the inner_walls surface group are the surfaces which make up the contact boundary between the inner cylinder and the fluid volume.

1. Close the Surface Manager.

### Assign Surface Parameters

The modeling for this simulation was done using half symmetry. The model is only a partial representation of the system, the complete geometry of which is a cylinder. Hence it is appropriate to set the surface that you chose as symmetry_plane with a symmetry boundary condition to simulate that effect.

This change was completed using the Surface Manager in the last section. The following steps are thus optional.

1. Update symmetry_plane.
1. Expand the symmetry_plane surface in the tree.
2. Double-click Simple Boundary Condition under symmetry_plane to open the Simple Boundary Condition detail panel.
3. Ensure that the Type is set to Symmetry.
2. Update outer_wall.
1. Expand the outer_wall surface group in the tree.
2. Double click Simple Boundary Condition under outer_wall to open the Simple Boundary Condition detail panel.
3. Ensure that the Type is set to Wall.
4. Verify that the Wall velocity type is set to Match Mesh Velocity.
5. Change Temperature BC type from Flux to Value.
6. Set the Temperature to 25° C.
The default unit for temperature input is K. You can change the unit for temperature by clicking on the unit button at the right of the input field, and selecting oC from the appearing menu.
3. Update inner_wall
The inner walls form the boundary surface of the inner cylinder volume, and enclose the fluid volume on the inside. Since the inner cylinder is a solid medium, this contact boundary will be a wall.
1. Expand the inner_wall surface group in the tree.
2. Double click Simple Boundary Condition under inner_wall to open the Simple Boundary Condition detail panel.
3. Ensure that the Type is set to Wall.
4. Verify Wall velocity type is set to Match Mesh Velocity.
4. Update the periodic surfaces solid_pos_z, solid_neg_z, fluid_pos_z, and fluid_neg_z
Physically the simulation domain is assumed to extend infinitely in the z-direction. However, only a small section of the cross section is being modelled and the solution is assumed to be consistent along the z-direction. Thus, these periodic surfaces are not physical boundaries but the solution on these surfaces is constrained to be equal by periodicity. This is achieved via a periodic boundary condition in AcuConsole, which links the corresponding pairs of nodes on the two surfaces which are to be constrained with a periodic boundary condition.

Periodicity can be defined before proceeding with mesh generation. With this workflow, when the mesh is generated, AcuMeshSim, which is the mesh generation engine for AcuSolve, will read the defined periodicity constraints and ensure a periodic mesh on the specified surface pairs.

1. Expand the Model Data Tree item, and right-click on Periodics.
2. Select New from the context menu to create a new entity, Periodic 1.
3. Repeat the above step to create a second entity Periodic 2.
4. Rename the two new entities as periodicity_fluid and periodicity_solid.
5. Right-click on periodicity_fluid and select Define from the context menu.
6. In the Periodic BC dialog, make the following settings.
• Use the drop-down arrow to select the surfaces for Side 1 and Side 2 as fluid_neg_z and fluid_pos_z, respectively
• Check that the Type is set to Translational.
• Set X, Y and Z-offset as 0.0, 0.0, 0.01 respectively.
Use the following figure for reference for setting up the periodic BC.
7. Click OK to close the dialog.
8. Using the same figure as reference, similarly define the periodic BC for the entity periodicity_solid, with only the following changes:
• Use the drop down arrows for Side 1 and Side 2 and select solid_neg_z and solid_pos_z, respectively.

### Create Time History Output Points

Time History Output commands enables you to extract the nodal solution at any point within the domain.

1. In the tree, double-click on Output, then right-click on Time History Output, and select New.
A new entry, Time History Output 1, will be created in the Data Tree under the Time History Output branch.
2. Right-click on Time History Output 1, select Rename, and type in Monitor points as the entity name.
3. Double click Monitor points to open the detail panel. In the detail panel,
1. Change the Type to Coordinates.
2. Click Open Array.
3. In the Array Editor, add a new row by clicking Add Row.
4. Fill in the values as follows:
4. Click OK.
5. Set Time step frequency to 1.
This will save the results for the defined time history points at every time step.
6. Save the database.

### Set the Initial Conditions

1. Double-click on Nodal Initial Condition in the Data Tree to open the detail panel.
2. Set the Temperature to 80° C.
1. The default unit for temperature input is K. You can change the unit for temperature by clicking on unit to the right of the input field, and selecting oC from the appearing menu.
2. Alternatively, enter 353.15 K in the temperature field.

## Assign Mesh Controls

### Set Global Mesh Parameters

Now that the flow characteristics have been set for the whole problem, a sufficiently refined mesh has to be generated.

Global mesh attributes are the meshing parameters applied to the model as a whole without reference to a specific geometric volume, surface, edge, or point. Local mesh attributes are used to create mesh generation controls for specific geometry components of the model.

In the next steps you will set the global mesh attributes.

1. Click MSH in the Data Tree Manager to filter the settings in the Data Tree to show only the controls related to meshing.
2. Double-click the Global Data Tree item to expand it.
3. Double-click Global Mesh Attributes to open the Global Mesh Attributes detail panel.
4. Change the Mesh size type to Absolute.
5. Enter 0.005 m for the Absolute mesh size.

### Set Surface Mesh Parameters

Surface mesh attributes are applied to a specific surface in the model. It is a type of local meshing parameter used to create targeted mesh controls for one or more specific surfaces.

Setting local mesh attributes, such as surface mesh attributes, is not mandatory. When a local mesh attribute is not found for a component, the global attributes are used as the mesh generation control for that component. If a local mesh attribute is present, it will take precedence over the global setting.

In the next steps you will set the surface meshing attributes.

1. Go to the Solver Browser, expand 01.Global, then click PROBLEM_DESCRIPTION.
2. Expand the Model Data Tree item.
3. Under the Model branch, expand the Surfaces. Under Surfaces, expand the inner_wall surface group.
4. If necessary, check the box next to Surface Mesh Attributes to activate it. Double-click it to open the Surface Mesh Attributes detail panel.
The detail panel should now be populated with options related to the local surface meshing controls.
5. Ensure that the Mesh size type is set to Absolute.
6. Enter 0.002 m for the Absolute mesh size.
7. Switch the Boundary layer flag to On.
Mesh controls related to boundary layer meshing become visible.
8. Check the Boundary layer type is set to Full Control.
9. Set Resolve to Total Layer Height.
This sets the total layer height based on the other settings you provide.
10. Set the remaining settings as follows:
Option Description
First element height 0.0001
Growth rate 1.2
Number of layers 8
Boundary layer elements type Tetrahedron

Instead of repeating the above steps for the outer_wall surface, you can choose to propagate the mesh attribute settings for inner_wall surface group to outer_wall surface group.

1. Under the inner_wall surface, right-click Surface Mesh Attributes and select Propagate.
2. In the Propagate dialog, select the surface outer_wall and click Propagate.

### Define Mesh Extrusion

The present simulation is equivalent to a 2D representation of the model, which actually extends infinitely in both sides along the z-direction. In AcuSolve, 2D models are simulated by having just one element across the faces of the cross section. Thus when these faces are set up with a similar boundary condition, it coerces the corresponding nodes across the faces to have same results. In this problem, these faces are the negative and positive z-surfaces. This kind of mesh is achieved in AcuSolve with mesh extrusion process. In the following steps, the process of extrusion of the mesh between these surfaces is defined.

1. Expand the Model Data Tree item.
2. Right-click Mesh Extrusions and select New from the context menu to create a new entity, Mesh Extrusion 1.
3. Repeat the above step to create a second entity, Mesh Extrusion 2.
4. Rename the two entities as extrusion_fluid and extrusion_solid.
5. Right-click extrusion_fluid and select Define from the context menu.
6. In the Mesh Extrusion dialog, make the following settings.
1. Check that the Geometry type is set to surface.
2. Use the drop down arrows to select the surfaces for Side 1 and Side 2 as fluid_neg_z and fluid_pos_z, respectively.
3. Check that the Extrusion type is set to Number of layers.
4. Set Number of layers equal to 1.
5. Set Extrusion options to All tets.
Use the following figure for reference for setting up the mesh extrusion for extrusion_fluid.
7. Click OK to close the dialog.
8. Using the same figure as reference, similarly define the mesh extrusion for the entity extrusion_solid, with only the following changes:
1. Use the drop down arrows to select the surfaces for Side 1 and Side 2 as solid_neg_z and solid_pos_z, respectively

### Generate the Mesh

In the next steps you will generate the mesh that will be used when computing a solution for the problem.

1. Click on the toolbar to open the Launch AcuMeshSim dialog.
For this case, the default settings will be used.
2. Click Ok to begin meshing.

During meshing an AcuTail window opens. Meshing progress is reported in this window. A summary of the meshing process indicates that the mesh has been generated.

Note: The actual number of nodes and elements, and memory usage may vary slightly from machine to machine.
3. Visualize the mesh in the modeling window. Turn on the display of surfaces and set the display type to solid and wire.
4. Rotate and zoom in the model to analyze the various mesh regions.

### Assign Reference Pressure

The present case does not have any inlet or outlet surfaces to define any boundary condition that sets the pressure level inside the domain. To make the solution more robust, you will set a pressure reference point using a nodal boundary condition. The following steps will show how to setup the reference pressure inside the CFD domain.

1. Click BAS in the Data Tree Manager to switch to basic view in the Data Tree.
2. Expand the Model Data Tree item.
3. Right-click on Nodes and select New to create a new entity, Node 1.
4. Rename Node 1 to Fixed Pressure Node.
5. Right-click Fixed Pressure Node and select Define.
6. In the Node Define Dialog Box, set Selection Type to Pressure Point and Volumes to fluid.
7. Click OK.
8. Expand Fixed Pressure Node and enable Pressure.
The single node will now act as the pressure reference point for the simulation. The default Type of Zero sets the nodes in this set to pressure = 0.0.
9. Examine the location of the reference pressure node and check that it is inside the domain.
1. Right-click on Fixed Pressure Node and select Display on.
2. Right-click on Surfaces and set Display type to outline.
3. Right-click Periodics and select Display off.
You should be able to see the fixed pressure node as a point, as shown in the figure below.

## Compute the Solution and Review the Results

### Run AcuSolve

In the next steps you will launch AcuSolve to compute the solution for this case.

1. Click on the toolbar to open the Launch AcuSolve dialog.
2. Click Ok to start the solution process.

While computing the solution, an AcuTail window opens. Solution progress is reported in this window. A summary of the solution process indicates that the run has been completed.

The information provided in the summary is based on the number of processors used by AcuSolve. If you use a different number of processors than indicated in this tutorial, the summary for your run may be slightly different than the summary shown.

3. Close the AcuTail window and save the database to create a backup of your settings.

### Post-Process with AcuProbe

AcuProbe can be used to monitor various variables over solution time.

Note: This solution was obtained by running AcuSolve
with four processors.
1. Open AcuProbe by clicking on the toolbar.
2. In the Data Tree on the left, expand Residual Ratio. Right-click on Final and select Plot All.
This will plot the residuals for the three variables, pressure, temperature and velocity, in the plot area.
Note: You might need to click on the toolbar in order to properly display the plot.
3. Right-click on Final under Residual Ratio and select Plot None.
4. Expand Time History > Monitor Points.
5. Expand node 1 and node 2.
6. One node at a time, right-click on temperature and select Plot.
Note: You might need to click on the toolbar in order to properly display the plot.

The node 1 lies in the bottom half of the model and the node 2 in the upper half. The temperature distribution in the above plot shows that in steady state upper half of the cylinder annulus is occupied by the hotter air and lower half has the colder air.

7. In the menu area, click the surfs collector and select all.
8. The time series data of the variables can also be exported as a text file for further post-processing.
1. Right-click on the variable that you want to export and click Export.
2. Enter a File name and choose .txt for the Save as type.
3. Click Save.

### View Results with AcuFieldView

The tutorial has been written with the assumption that you have become familiar with the AcuFieldView interface and basic operations. In general, it will be helpful to understand the following basics:
• How to find the data readers in the File menu and open up the desired reader panel for data input.
• How to find the visualization panels either from the toolbar or the Visualization panels from the main menu to create and modify surfaces in AcuFieldView.
• How to move the data around the modeling window using mouse actions to translate, rotate and zoom in to the data.
This tutorial shows you how to work with steady state analysis data.

#### Start AcuFieldView

1. Click on the AcuConsole toolbar to open the Launch AcuFieldView dialog.
2. Click Ok to start AcuFieldView.
You will see that the temperature contours have already been displayed on all the boundary surfaces with mesh.

#### Manipulate the Model View in AcuFieldView

1. Close the Boundary Surface dialog.
2. Click Viewer Options.
3. Turn off perspective view by deselecting the Perspective check box.
4. Disable axis markers by clicking on the Axis Markers button.
5. Close the Viewer Options dialog.
6. Click on the Colormap Specification icon on the toolbar.
7. Click on Background in the Scalar Colormap Specification dialog and select white from the color palette that opens.
8. Close both dialogs.
9. Click on the Toggle Outline icon on the toolbar to turn off the outline display.
Your AcuFieldView display should now look like this.

#### Create the Boundary Surface Showing Temperature for the Outer Surfaces with Mesh

1. Orient the geometry as shown in the figure below, so that the symmetry plane and periodic surfaces are visible.
2. Click to open the Boundary Surface dialog.
3. Click the Legend tab and check the Show Legend check box.
4. Change the color of labels to black from the color palette.
5. If desired, change the number of labels to show more labels.
6. Change the Annotation title color to black.
Note: You can move the legend using Shift + left click, and resize it using Shift + right click.

#### Coordinate the Surface Showing Temperature on the Mid-Coordinate Surface

1. In the Surface tab in the Boundary Surface dialog box, click Visibility to turn it off.
2. Click Create to create a new Boundary Surface set.
3. Check Visibility to turn it on.
4. Set the Display Type to Outlines.
5. Under Boundary Types, click Select All, and click Ok.
6. Click to open the Coordinate Surface dialog.
7. Click Create to create a new Coordinate Surface.
8. Set the Coord Plane to Z.
The coordinate surface created is the mid plane between the two periodic surfaces in the model.
9. Change the Coloring to Scalar.
10. Set the Display Type to Smooth.
11. In the Scalar Function list, select Temperature as the scalar function to be displayed.
12. In the Colormap tab, change Scalar Coloring to Local.
13. In the Legend tab, check the Show Legend check box to display the temperature values on the coordinate plane.
14. From the Defined Views, select viewing direction as +Z.

#### Coordinate the Surface Showing Vectors of Velocity on the Mid-Coordinate Surface

1. In the Surface tab in the Coordinate Surface dialog box, click Create to create a new Coordinate Surface set.
2. Set the Display Type to Vectors.
3. Change the Coloring to Scalar.
4. In the Scalar Function list, select Velocity Magnitude as the scalar function to be displayed.
5. Next to Vectors, click Options.
6. Activate Head Scaling and set it at 1.
7. Set the Length Scale to 4.
8. Activate the Skip option, and set the value to 50%.

## Summary

In this AcuSolve tutorial, you successfully set up and solved a natural convection problem. The problem simulated a hot cylinder placed in the center of another air-filled cylindrical volume. Air was modeled using a Boussinesq density approximation model, which is used for buoyancy driven flows, such as those involving natural convection. As the film of air in vicinity of the surface of the hot inner cylinder heated up, it generated convection currents within the annular volume.

You started the tutorial by creating a database in AcuConsole, importing and meshing the geometry and setting up the basic simulation parameters. The hot inner cylinder was represented by a solid volume also acting as a heat source. Once the case was setup, the solution was generated with AcuSolve.

Results were post-processed in AcuFieldView where you generated a temperature profile, and a velocity vector profile, on a cross-section of the model.

New features that were introduced in this tutorial include creating and specifying a new custom material in AcuConsole, specifying a volume group as a heat source using the Boussinesq density model and setting up periodic boundary conditions.


text:
This book is in Open Review. We want your feedback to make the book better for you and other students. You may annotate some text by selecting it with the cursor and then click the on the pop-up menu. You can also see the annotations of others: click the in the upper right hand corner of the page

## 4.6 Exercises

#### 1. Class Sizes and Test Scores

A researcher wants to analyze the relationship between class size (measured by the student-teacher ratio) and the average test score. Therefore he measures both variables in $10$ different classes and ends up with the following results.

 Class Size 23 19 30 22 23 29 35 36 33 25 Test Score 430 430 333 410 390 377 325 310 328 375

Instructions:

• Create the vectors cs (the class size) and ts (the test score), containing the observations above.

• Draw a scatterplot of the results using plot().

#### 2. Mean, Variance, Covariance and Correlation

The vectors cs and ts are available in the working environment (you can check this: type their names into the console and press enter).

Instructions:

• Compute the mean, the sample variance and the sample standard deviation of ts.

• Compute the covariance and the correlation coefficient for ts and cs.

Hint: Use the R functions presented in this chapter: mean(), sd(), cov(), cor() and var().

#### 3. Simple Linear Regression

The vectors cs and ts are available in the working environment.

Instructions:

• The function lm() is part of the package AER. Attach the package using library().

• Use lm() to estimate the regression model $TestScore_i = \beta_0 + \beta_1 STR_i + u_i.$ Assign the result to mod.

• Obtain a statistical summary of the model.

#### 4. The Model Object

Let us see how an object of class lm is structured.

The vectors cs and ts as well as the model object mod from the previous exercise are available in your workspace.

Instructions:

• Use class() to learn about the class of the object mod.
• mod is an object of type list with named entries. Check this using the function is.list().
• See what information you can obtain from mod using names().

#### 10. Regression Output: No Constant Case — Ctd.

In Exercises 8 and 9 you have dealt with a model without intercept. The estimated regression function was

$\widehat{TestScore_i} = \underset{(1.36)}{12.65} \times STR_i.$

The coefficient matrix coef from Exercise 9 contains the estimated coefficient on $STR$, its standard error, the $t$-statistic of the significance test and the corresponding $p$-value.

Instructions:

• Print the contents of coef to the console.
• Convince yourself that the reported $t$-statistic is correct: use the entries of coef to compute the $t$-statistic and save it to t_stat.

The matrix coef from the previous exercise is available in your working environment.

Hints:

• X[a,b] returns the [a,b] element of the matrix X.

• The $t$-statistic for a test of the hypothesis $H_0: \beta_1 = 0$ is computed as $t = \frac{\hat{\beta}_1}{SE(\hat{\beta}_1)}.$

#### 11. Two Regressions, One Plot

The two estimated regression models from the previous exercises are

$\widehat{TestScore_i} = \underset{(1.36)}{12.65} \times STR_i$

and

$\widehat{TestScore_i} = \underset{(23.96)}{567.4272} \underset{(0.85)}{-7.1501} \times STR_i.$

You are provided with the code line plot(cs, ts) which creates a scatterplot of ts and cs. Note that this line must be executed before calling abline()! You may color the regression lines by using, e.g., col = “red” or col = “blue” as an additional argument to abline() for better distinguishability.

The vectors cs and ts as well as the list objects mod and mod_ni from previous exercises are available in your working environment.

Instructions:

Generate a scatterplot of ts and cs and add the estimated regression lines of mod and mod_ni.

#### 12. $TSS$ and $SSR$

If graphical inspection does not help, researchers resort to analytic techniques in order to detect if a model fits the data at hand good or better than another model.

Let us go back to the simple regression model including an intercept. The estimated regression line for mod was

$\widehat{TestScore_i} = 567.43 - 7.15 \times STR_i, \, R^2 = 0.8976, \, SER=15.19.$

You can check this as mod and the vectors cs and ts are available in your working environment.

Instructions:

• Compute $SSR$, the sum of squared residuals, and save it to ssr.
• Compute $TSS$, the total sum of squares, and save it to tss.

#### 15. The Estimated Covariance Matrix

As has been discussed in Chapter 4.4, the OLS estimators $\widehat{\beta}_0$ and $\widehat{\beta}_1$ are functions of the random error term. Therefore, they are random variables themselves. For two or more random variables, their covariances and variances are summarized by a variance-covariance matrix (which is often simply called the covariance matrix). Taking the square root of the diagonal elements of the estimated covariance matrix obtains $SE(\widehat\beta_0)$ and $SE(\widehat\beta_1)$, the standard errors of $\widehat{\beta}_0$ and $\widehat{\beta}_1$.

summary() computes an estimate of this matrix. The respective entry in the output of summary (remember that summary() produces a list) is called cov.unscaled. The model object mod is available in your workspace.

Instructions:

• Use summary() to obtain the covariance matrix estimate for the regression of test scores on student-teacher ratios stored in the model object mod. Save the result to cov_matrix.

• Obtain the diagonal elements of cov_matrix, compute their square root and assign the result to the variable SEs.

Hint: diag(A) returns a vector containing the diagonal elements of the matrix A.


text:
# All Seminars

Show:
Title: Connections between Classical and Umbral Moonshine
Defense: Dissertation
Speaker: Sarah Trebat-Leder of Emory University
Contact: Sarah Trebat-Leder, sarah.trebat-leder@emory.edu
Date: 2018-04-02 at 2:00PM
Venue: W302
Abstract:
Both results of this dissertation involve finding unexpected connections between the classical theory of monstrous moonshine and the newer umbral moonshine. In our first result, we use generalized Borcherds products to associate to each pure A-type Niemeier lattice a conjugacy class g of the monster group and give rise to identities relating dimensions of representations from umbral moonshine to values of McKay-Thompson series. Our second result focuses on the Mathieu group M23. While it inherits a moonshine from being a subgroup of M24, we find a new and simpler moonshine for M23 such that the graded traces are, up to constant terms, identical to the monstrous moonshine Hauptmoduln.
Title: Maass forms and modular forms: applications to class numbers and partitions
Defense: Dissertation
Speaker: Olivia Beckwith of Emory University
Contact: Olivia Beckwith, olivia.dorothea.beckwith@emory.edu
Date: 2018-04-02 at 3:00PM
Venue: W302
Abstract:
This dissertation is about arithmetic information encoded by analytic characteristics (such as Fourier coefficients) of classical modular forms and a real-analytic generalization of modular forms called harmonic Maass forms. For example, I use the theory of harmonic Maass forms to extend and refine a theorem of Wiles on class number divisibility. I also prove asymptotic bounds for Rankin-Selberg shifted convolution L-functions in shift aspect. In partition theory, I use the circle method to describe the expected distribution of parts of integer partitions over residue classes, and I use effective estimates for partition functions to obtain simple formulas for functions arising in group theory.
Title: Patching and local-global principles for gerbes with an application to homogeneous spaces
Defense: Dissertation
Speaker: Bastian Haase of Emory University
Contact: Bastian Haase, bastian.haase@emory.edu
Date: 2018-04-02 at 4:00PM
Venue: W302
Abstract:
Starting in 2009, Harbater and Hartmann introduced a new patching setup for semi-global fields, establishing a patching framework for vector spaces, central simple algebras, quadratic forms and other algebraic structures. In subsequent work with Krashen, the patching framework was refined and extended to torsors and certain Galois cohomology groups. After describing this framework, we will discuss an extension of the patching equivalence to bitorsors and gerbes. Building up on these results, we then proceed to derive a characterisation of a local- global principle for gerbes and bitorsors in terms of factorization. These results can be expressed in the form of a Mayer-Vietoris sequence in non-abelian hypercohomology with values in the crossed-module $G->Aut(G)$. After proving the local-global principle for certain bitorsors and gerbes using the characterization mentioned above, we conclude with an application on rational points for homogeneous spaces.
Title: Truncated Singular Value Decomposition Approximation for Structured Matrices via Kronecker Product Summation Decomposition
Defense: Dissertation
Speaker: Clarissa Garvey of Emory University
Contact: James Nagy, jnagy@emory.edu
Date: 2018-04-02 at 4:30PM
Venue: W301
Abstract:
Singular value decompositions are a particularly attractive matrix factorization for ill-posed problems because singular value magnitudes reveal information about the relative importance of data in the matrix. However, computing a singular value decomposition is typically computationally infeasible for large problems, as the cost for traditional methods, such as Lanczos bidiagonalization-based approaches and randomized methods, scales linearly with the number of entries in the matrix times the number of singular values computed. In this work we present two new algorithms and one new hybrid approach for computing the singular value decomposition of matrices cheaply approximable as an ordered Kronecker summation decomposition. Unlike previous work using ordered Kronecker summation decompositions, the factorizations these methods produce are more accurate for certain classes of matrices and have nonnegative singular values. The three proposed methods are also faster, with lower computational and spatial complexity, although also lower accuracy, than traditional methods. Our Kronecker-based methods therefore enable singular value decomposition approximations on larger matrices than traditional methods, while providing more accurate results in many cases than previous Kronecker-based singular value decompositions. We demonstrate the efficacy of these methods on a variety of image deconvolution problems for which the image is modeled as a regular grid of data.
Title: Counting Problems for Elliptic Curves over a Fixed Finite Field
Seminar: Algebra
Speaker: Nathan Kaplan of UC Irvine
Contact: David Zureick-Brown, dzb@mathcs.emory.edu
Date: 2018-03-27 at 4:00PM
Venue: W304
Abstract:
Let $E$ be an elliptic curve defined over a finite field $\mathbb{F}_q$. Hasses theorem says that $\#E(\mathbb{F}_q) = q + 1 - t_E$ where $|t_E| \le 2\sqrt{q}$. Deuring uses the theory of complex multiplication to express the number of isomorphism classes of curves with a fixed value of $t_E$ in terms of sums of ideal class numbers of orders in quadratic imaginary fields. Birch shows that as $q$ goes to infinity the normalized values of these point counts converge to the Sato-Tate distribution by applying the Selberg Trace Formula. \\ In this talk we discuss finer counting questions for elliptic curves over $\mathbb{F}_q$. For example, what is the probability that the number of rational points is divisible by $5$? What is the probability that the group of rational points is cyclic? If we choose a curve at random, and then pick a random point on that curve, what is the probability that the order of the point is odd? We study the distribution of rational point counts for elliptic curves containing a specified subgroup, giving exact formulas for moments in terms of traces of Hecke operators. We will also discuss some open problems. This is joint with work Ian Petrow (ETH Zurich).
Title: On large multipartite subgraphs of H-free graphs
Seminar: Combinatorics
Speaker: Jan Volec of McGill Univeristy
Contact: Dwight Duffus, dwight@mathcs.emory.edu
Date: 2018-03-26 at 4:00PM
Venue: W303
Abstract:
A long-standing conjecture of Erd\H{o}s states that any n-vertex triangle-free graph can be made bipartite by deleting at most $n^2/25$ edges. In this talk, we study how many edges need to be removed from an H-free graph for a general graph H. By generalizing a result of Sudakov for 4-colorable graphs H, we show that if H is 6-colorable then G can be made bipartite by deleting at most $4n^2/25+O(n)$ edges. In the case of $H=K_6$, we actually prove the exact bound $4n^2/25$ and show that this amount is needed only in the case G is a complete 5-partite graph with balanced parts. As one of the steps in the proof, we use a strengthening of a result of $F\ddot{u}redi$ on stable version of $Tur\acute{a}n's$ theorem. This is a joint work with P. Hu, B. $Lidick\acute{y}$, T. Martins-Lopez and S. Norin.
Title: Estimating bilinear forms via extrapolation
Seminar: Numerical Analysis and Scientific Computing
Speaker: Marilena Mitrouli of National and Kapodistrian University of Athens
Contact: Michele Benzi, benzi@mathcs.emory.edu
Date: 2018-03-21 at 4:00PM
Venue: W301
Abstract:
A spectrum of applications arising from Statistics, Machine Learning, Network Analysis require computation of bilinear forms $x^Tf(A)y$, where $A$ is a diagonalizable matrix and $x$, $y$ are given vectors. In this work we are interested in efficiently computing bilinear forms primarily due to their importance in several contexts. For large scale computation problems it is preferable to achieve approximations of bilinear forms without exploiting the whole matrix function. For this purpose an extrapolation procedure has been developed, attaining the approximation of bilinear forms with one, two or three term estimates in a complexity of square order. The extrapolation procedure gives us the flexibility to define the moments of a matrix $A$ either directly or through the polarization identity. The presented approach is characterized by easy applicable formulae of low complexity that can be implemented in vectorized form.
Title: On Spanning Trees with few Branch Vertices
Defense: Dissertation
Speaker: Warren Shull of Emory University
Contact: Warren Shull, warren.edward.shull@emory.edu
Date: 2018-03-05 at 4:00PM
Venue: W301
Abstract:
Hamiltonian paths, which are a special kind of spanning tree, have long been of interest in graph theory and are notoriously hard to compute. One notable feature of a Hamiltonian path is that all its vertices have degree two in the path. In a tree, we call vertices of degree at least three \emph{branch vertices}. If a connected graph has no Hamiltonian path, we can still look for spanning trees that come "close," in particular by having few branch vertices (since a Hamiltonian path would have none). \bigskip A conjecture of Matsuda, Ozeki, and Yamashita posits that, for any positive integer $k$, a connected claw-free $n$-vertex graph $G$ must contain either a spanning tree with at most $k$ branch vertices or an independent set of $2k+3$ vertices whose degrees add up to at most $n-3$. In other words, $G$ has this spanning tree whenever $\sigma_{2k+3}(G)\geq n-2$. We prove this conjecture, which was known to be sharp.
Title: Numerical and Streaming Analyses of Centrality Measures on Graphs
Seminar: Numerical Analysis and Scientific Computing
Speaker: Eisha Nathan of Georgia Institute of Technology
Contact: Michele Benzi, mbenzi@emory.edu
Date: 2018-03-02 at 2:00PM
Venue: W301
Abstract:
Graphs are a natural representation for modeling relationships between entities across many different fields of research. In real-world networks today, new data is constantly being produced, leading to the notion of dynamic graphs. An important query when analyzing graphs is to identify the most important vertices in a graph. Vertex importance is termed as centrality, and centrality scores can be used to provide rankings on the vertices of a graph. Specifically, I focus on Katz Centrality, a linear algebra based metric that measures the affinity between vertices in a graph as a weighted sum of the number of walks between them. Calculating Katz scores involves approximating the solution to a linear system using iterative solvers. Currently, we do not have a sense of how the numerical error from the iterative method in the approximation to the centrality vector affects the identification of the highly ranked vertices. In the first part of the talk, I bridge the two research areas of numerical analysis and network analysis by providing insight into how bounding the error between the approximate and exact solution in the numerical problem affects the correct relative ranking of vertices in the data mining problem. Typically, Katz scores are calculated through linear algebra. In the second section of the talk, I present an new alternative, agglomerative method of calculating personalized Katz scores. The algorithm presented is several orders of magnitude faster than the typical linear algebra approach and is able to return good quality scores of vertices. Finally, updating centrality scores of the vertices in a dynamic graph quickly becomes expensive to recompute from scratch every time the underlying graph is changed, as is the case in evolving networks. I conclude the talk by presenting a new algorithm for updating the Katz scores of vertices in a dynamic graph that is faster than static recomputation while maintaining good quality of the scores returned.
Title: On Cycles, Chorded Cycles, and Degree Conditions
Defense: Dissertation
Speaker: Ariel Keller of Emory University
Contact: Ariel Keller, ariel.keller@emory.edu
Date: 2018-03-01 at 3:00PM
Venue: MSC N301
Abstract:
Sufficient conditions to imply the existence of certain substructures in a graph are of considerable interest in extremal graph theory, and conditions that guarantee a large set of cycles or chorded cycles are a recurring theme. This dissertation explores different degree sum conditions that are sufficient for finding a large set of vertex-disjoint cycles or a large set of vertex-disjoint chorded cycles in a graph. \vskip.1in For an integer $t\ge 1$, let $\sigma_t (G)$ be the smallest sum of degrees of $t$ independent vertices of $G$. We first prove that if a graph $G$ has order at least $7k+1$ and degree sum condition $\sigma_4(G)\ge 8k-3$, with $k\ge 2$, then $G$ contains $k$ vertex-disjoint cycles. Then, we consider an equivalent condition for chorded cycles, proving that if $G$ has order at least $11k+7$ and $\sigma_4(G)\ge 12k-3$, with $k\ge 2$, then $G$ contains $k$ vertex-disjoint chorded cycles. We prove that the degree sum condition in each result is sharp. Finally, we conjecture generalized degree sum conditions on $\sigma_t(G)$ for $t\ge 2$ sufficient to imply that $G$ contains $k$ vertex-disjoint cycles for $k \ge 2$ and $k$ vertex-disjoint chorded cycles for $k \ge 2$. This is joint work with Ronald J. Gould and Kazuhide Hirohata.


text:
# What is the perimeter and area of the polygon with the given vertices: W(5,-1), X(5, 6), Y(2,-1), Z(2, 6)?

Nov 13, 2016

Perimeter = 20 units
Area = 21 square units

#### Explanation:

Perimeter $= 2 \left(3\right) + 2 \left(7\right)$
$= 6 + 14$
$= 20$ units

$A r e a = 3 \cdot 7$
$= 21$ square units


text:
# Precise definition of relative consistency in Kunen's "Set Theory"

I'm reading Kunen's "Set Theory" (Revised edition 2013).

On page 108 he defines for axiomatic set theories $\Lambda, \Gamma$ which are strong enough to formalize finitistic arguments (e.g. ZFC, Z, BST...)

$$\Gamma \triangleleft \Lambda \quad\text{ :iff }\quad \Lambda \vdash Con(\Gamma)$$

and

$$\Gamma \leq \Lambda \quad\text{ :iff }\quad \text{we have a finitistic proof that } Con(\Lambda) \rightarrow Con(\Gamma)$$

Then he states

Lemma II.1.3

1. $\Gamma \triangleleft \Lambda \rightarrow \Gamma \leq \Lambda$
2. $\Gamma \leq \Lambda \ \wedge\ \Lambda \triangleleft \Theta \rightarrow \Gamma \triangleleft \Theta$
3. /4. ...

Proof. For (1): Assuming $\Lambda \vdash Con(\Gamma)$, we may argue finitistically that if we had a formal proof of contradiction from $\Gamma$, then $\Lambda \vdash \neg Con(\Gamma)$, which would give us a contradiction in $\Lambda$. So, $Con(\Lambda) \rightarrow Con(\Gamma)$.
For (2): Assuming $\Gamma \leq \Lambda \ \wedge\ \Lambda \triangleleft \Theta$, then working in $\Theta$, we can prove $Con(\Lambda)$ and $Con(\Lambda) \rightarrow Con(\Gamma)$, and hence $Con(\Gamma)$.

At the beginning of this chapter Kunen says:

The definitions and lemmas in this section are informal, and are intended to give an overview of the types of consistency results you will find in this book and in other works on set theory. The reader who is familiar with proof theory will see how to make them precise and formal. These remarks take place in the metatheory, using just finitistic reasoning.

My question: What is a precise formulation of 1. and 2.?

I see two possibilities.

Let $\vdash_{fin}$ denote the syntactic consequence relation of a finitistic system like PRA (Primitive Recursive Arithmetic). We write $\ulcorner \varphi \urcorner$ for the statement $\varphi$ encoded into PRA and let $\square \varphi$ be an abbreviation for $\ulcorner \vdash_{fin} \varphi \urcorner$.

Option a):

1. $\vdash_{fin} \ulcorner\Lambda \vdash \neg Incon(\Gamma) \urcorner \rightarrow \square( Incon(\Gamma) \rightarrow Incon(\Lambda) )$
2. $\vdash_{fin} \square( Incon(\Gamma) \rightarrow Incon(\Lambda) ) \ \wedge\ \ulcorner\Theta \vdash \neg Incon(\Lambda) \urcorner \rightarrow \ulcorner\Theta \vdash \neg Incon(\Gamma) \urcorner$

Option b):

1. $\vdash_{fin} \ulcorner\Lambda \vdash \neg Incon(\Gamma) \urcorner \rightarrow ( Incon(\Gamma) \rightarrow Incon(\Lambda) )$
2. $\vdash_{fin} ( Incon(\Gamma) \rightarrow Incon(\Lambda) ) \ \wedge\ \ulcorner\Theta \vdash \neg Incon(\Lambda) \urcorner \rightarrow \ulcorner\Theta \vdash \neg Incon(\Gamma) \urcorner$

In case of option a) I have a problem in proving 1, in b) in proving 2.

I use: Since we can assume that $Incon(\Gamma)$ is $\Sigma_1$, we have $\vdash_{fin} Incon(\Gamma) \rightarrow \square Incon(\Gamma)$. But for $Incon(\Gamma) \rightarrow Incon(\Lambda)$ this doesn't work.

Can anyone prove option a) or b)? Or am I completely missing the point here?

## 1 Answer

Let's look at $(1)$. The right way to interpret "$\Gamma\triangleleft\Lambda\implies\Gamma\le\Lambda$" is as $$(\Lambda\vdash Con(\Gamma))\implies [PRA\vdash (Con(\Lambda)\implies Con(\Gamma))].$$ This is just a direct translation of $(1)$: "If $\Lambda$ proves the consistency of $\Gamma$, then there is a finitistic proof that the consistency of $\Lambda$ implies the consistency of $\Gamma$."

Your versions have extraneous external $\vdash_{fin}$s; e.g. your version of $(1)$ says "$(1)$ is finitistically provable." Of course, this is a priori stronger than just saying that $(1)$ is true.

Now what about the proof?

Proving this formalization of $(1)$ is not trivial: we need to use $\Sigma_1$-completeness in a slightly clever way. Suppose $\Lambda\vdash Con(\Gamma)$. By $\Sigma_1$-completeness, $PRA$ proves "$\Lambda\vdash Con(\Gamma)$" (ignoring Godel-numbering issues). Now since $\Lambda$ is sufficiently strong, $\Lambda$ itself is $\Sigma_1$ complete, and this means that $Incon(\Gamma)\implies \Lambda\vdash Incon(\Gamma)$ is true.

Moreover, $PRA$ proves that $\Lambda$ is $\Sigma_1$ complete! This is the crucial trick. Put another way, we have $PRA\vdash Incon(\Gamma)\implies \Lambda\vdash Incon(\Gamma)$. To prove this, you have to go back to the proof that $\Lambda$ is $\Sigma_1$ complete and check carefully that it goes through in $PRA$. This is straightforward but tedious, so I'll leave it as an exercise to the reader. :P (Note that this whole mess is subsumed by Kunen's statement "we may argue finitistically.")

So we have that $PRA$ proves both of the following statements:

• $\Lambda\vdash Con(\Gamma)$.

• $Incon(\Gamma)\implies \Lambda\vdash Incon(\Gamma)$.

Putting these together, we get $$PRA\vdash Incon(\Gamma)\implies \Lambda\vdash Con(\Gamma)\wedge Incon(\Gamma),$$ and in particular $$PRA\vdash Incon(\Gamma)\implies Incon(\Lambda).$$ And this finishes the proof. $\quad\Box$.

Formalizing and proving $(2)$ is similar.

Incidentally, the same argument as above can also be used to prove the stronger statements you write down.

• Thanks! Isn't proving (2) trivial with your interpretation (or my option a)), since $\Lambda$ is strong enough to formalize finitistic arguments? The reason for my external $\vdash_{fin}$ was Kunen's comment: "These remarks take place in the metatheory, using just finitistic reasoning." Apr 2 '18 at 14:18
• @PopovFlorino Well, it depends what you mean by "trivial." :P What you need to do is show that $\Theta$ knows that $\Lambda$ would see any inconsistency in $\Gamma$, and hence (since $\Lambda$ proves that $\Gamma$ is consistent) would be inconsistent if $\Gamma$ were inconsistent. So again, you're not just using the strength of $\Lambda$, but also the more subtle strength of $\Theta$; namely, that $\Theta$ can prove that $\Lambda$ is $\Sigma_1$ complete. Apr 2 '18 at 14:33
• This subtlety - that we need provable $\Sigma_1$ completeness instead of just $\Sigma_1$ completeness - also crops up in this question which you might find interesting (see the statement $(*)$ in my answer). (Incidentally, in that question there is a separate subtlety which crops up, but which turns out to be unnecessary - see the followup question here.) Apr 2 '18 at 14:34
• I thought "$\Theta$ is strong enough to formalize finitistic arguments" means (ignoring Godel-numbering issues), that $(\vdash_{fin} \varphi) \Rightarrow (\Theta \vdash \varphi)$. (I meant "$\Theta$ is strong enough ..." in my first comment). With this we can argue: If $\vdash_{fin} Incon(\Gamma) \rightarrow Incon(\Lambda)$ and $\Theta \vdash \neg Incon(\Lambda)$, then $\Theta \vdash \neg Incon(\Lambda) \wedge Incon(\Gamma) \rightarrow Incon(\Lambda)$, i.e. $\Theta \vdash \neg Incon(\Gamma)$. What is wrong with this proof for (2)? Apr 2 '18 at 17:54
• @PopovFlorino That does work. I may have worked a bit too hard in my answer, I'll look at it when I have more time. Apr 2 '18 at 19:11