text:
The voltage ${\displaystyle {\mathcal {E}}}$ (in volts) induced in a wire of length ${\displaystyle l}$ (in meters) cutting a magnetic field of strength ${\displaystyle B}$ (in gauss) at the velocity ${\displaystyle v}$ (in m/s) is:
${\displaystyle {\mathcal {E}}=Bvl=-{\frac {d{\phi }}{d{t}}},}$
where ${\displaystyle d{\phi }/d{t}}$ is the rate of cutting of lines of magnetic flux in maxwell per second and the minus is inserted because of Lenz’s law (q.v.). In vector notation,
${\displaystyle {\mathcal {E}}=\int (\mathbf {v} \times \mathbf {B} )\cdot dl}$


text:
# The complexity of the envelope of line and plane arrangements Academic Article

•
• Overview
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• Research
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### abstract

• A facet of an hyperplane arrangement is called external if it belongs to exactly one bounded cell. The set of all external facets forms the envelope of the arrangement. The number of external facets of a simple arrangement defined by $n$ hyperplanes in dimension $d$ is hypothesized to be at least $d{n-2 \choose d-1}$. In this note we show that, for simple arrangements of 4 lines or more, the minimum number of external facets is equal to $2(n-1)$, and for simple arrangements of 5 planes or more, the minimum number of external facets is between $\frac{n(n-2)+6}{3}$ and $(n-4)(2n-3)+5$.

format: plain


text:
# What does this function do?

Hi,

So since I am a beginner, my question my be trivial, but still…

I was looking at this example and there is a function within the example, I do not quite understand:

def eps(v):
return as_tensor([[e[0, 0], e[0, 1], 0],
[e[0, 1], e[1, 1], 0],
[0, 0, 0]])

At first I thought eps(u) is the strain tensor but since it is later multiplied with strian

metadata = {"quadrature_degree": deg_stress, "quadrature_scheme": "default"}

a_Newton = inner(eps(v), sigma_tang(eps(u_)))*dxm

the eps can’t be the strain tensor but it must do something else. I think that simply because I don’t think multiplying strain tensor with stress tensor has any meaning. Or does it?

# question

Is there a mathematical interpretation of what the eps does? I am trying to rewrite the example in a different framework.

Is this the strain-displacement matrix, usually denoted by B?

You might want to take a step back from plasticity, and first make sure you understand @bleyerj’s 2D elasticity tutorial here. It seems like the root of the misunderstanding may be with the concept of weak forms of partial differential equations (PDEs), in which case you might even start with the Poisson demo.

I’ll answer your questions directly in the context of the 2D elasticity tutorial:

Is there a mathematical interpretation of what the eps does?

eps(v) returns the symmetric gradient of its argument, v, i.e.,

\boldsymbol{\varepsilon}(\mathbf{v}) = \operatorname{sym}\nabla\mathbf{v} = \frac{1}{2}(\nabla\mathbf{v} + \nabla\mathbf{v}^T)\text{ .}

If v is the displacement, then this is the strain, but notice that the Cauchy stress is contracted with eps(v), where v is a test function. (In the plasticity demo, the 2D symmetric gradient is then padded with zeros to fit into a 3D formulation.)

I don’t think multiplying strain tensor with stress tensor has any meaning. Or does it?

The contraction \frac{1}{2}\boldsymbol{\sigma}(\boldsymbol{\varepsilon}(\mathbf{u})):\boldsymbol{\varepsilon}(\mathbf{u}) = \frac{1}{2}\boldsymbol{\varepsilon}(\mathbf{u}):\mathbf{c}:\boldsymbol{\varepsilon}(\mathbf{u}) is the elastic potential energy density (where \mathbf{c} is the rank-4 stiffness tensor). One way to get the weak problem is to take a variation of the sum of elastic potential energy and change in external potential energy (associated with conservative force field \mathbf{f}):

\delta\int_\Omega\left(\frac{1}{2}\boldsymbol{\varepsilon}(\mathbf{u}):\mathbf{c}:\boldsymbol{\varepsilon}(\mathbf{u}) - \mathbf{f}\cdot\mathbf{u}\right)\,d\Omega = \int_\Omega\left(\boldsymbol{\sigma}(\boldsymbol{\varepsilon}(\mathbf{u})):\boldsymbol{\varepsilon}(\delta\mathbf{u}) - \mathbf{f}\cdot\delta\mathbf{u}\right)\,d\Omega = 0\text{ ,}

where the arbitrary “virtual displacement” \delta\mathbf{u} is the test function, denoted in \mathbf{v} in the tutorial.

Is this the strain-displacement matrix, usually denoted by B ?

The strain–displacement matrix in your example is essentially a discrete representation of the symmetric gradient operator. However, when using the FEniCS Unified Form Language (UFL), you simply state the weak PDE system, and lower-level details of the discretization beyond that point are all automated.

2 Likes

Stress components expressed symbolically in UFL (e.g., sigma(u)[0,0] in the elasticity demo) can be projected to the space P0 and written to files, similar to the way in which p is projected to P0, assigned to p_avg, and written to file_results in the plasticity demo.


text:
Accelerating Dual Momentum: 150 Year Backtest

Many of you may have heard of Gary Antonacci's strategy called Global Equities Momentum (GEM) in which he uses a "dual momentum" signal to compare US stocks to global stocks to bonds over the trailing 12 months. The strategy has done quite well historically but I felt that a 12-month signal was too long and too rigid. I wanted to pick up not only on the direction of an asset but also the rate of change of that direction.

In a strategy I'm calling "Accelerating Dual Momentum" we look at this "acceleration" of an asset by simply adding the 6 month, 3 month, and 1 month returns. Whichever asset is higher between US stocks and global small cap stocks we buy. If both are negative than we buy bonds. Hold for the next month and repeat a month later.

The algorithm below provides a backtest over the last 14 years. I also wrote an extensive article on my blog further explaining the strategy and providing performance data going back to 1871. Here's the link: Accelerating Dual Momentum.

1290
Total Returns
--
Alpha
--
Beta
--
Sharpe
--
Sortino
--
Max Drawdown
--
Benchmark Returns
--
Volatility
--
 Returns 1 Month 3 Month 6 Month 12 Month
 Alpha 1 Month 3 Month 6 Month 12 Month
 Beta 1 Month 3 Month 6 Month 12 Month
 Sharpe 1 Month 3 Month 6 Month 12 Month
 Sortino 1 Month 3 Month 6 Month 12 Month
 Volatility 1 Month 3 Month 6 Month 12 Month
 Max Drawdown 1 Month 3 Month 6 Month 12 Month
'''
Even weight 1, 3, 6 month
Default to Long Treasuries

https://engineeredportfolio.com/2018/05/02/accelerating-dual-momentum-investing/
'''
import pandas as pd
import math
import numpy as np
import datetime

MAX_ASSETS = 1
ROBINHOOD_GOLD = 0

def initialize(context):
"""
Called once at the start of the algorithm.
"""

schedule_function(set_allocation, date_rules.month_start(), time_rules.market_open())
schedule_function(my_rebalance, date_rules.month_start(days_offset=0), time_rules.market_open(hours=1))

context.sp500 =         sid(8554)    #S&P 500                             SPY
context.midvalue =      sid(21770)   #US Midcap Value                     IJJ, VOE after  7/1/2008
context.world =         sid(33486)   #All World ex-US Stocks              EFA, VEU after  1/1/2010
context.world_small =   sid(22972)   #All World ex-US Small Cap Stocks    EFA, SCZ after  7/1/2008, VSS after  1/1/2012
context.outofmarket =   sid(23921)   #Long Term Treasuries                TLT
context.tips =          sid(25801)   #Inflation Protected (TIPS)          TIP

context.started = 0

def set_allocation(context, data):
"""
Get our portfolio allocation
"""
if get_datetime('US/Eastern').date() >= datetime.date(2008, 7, 1):
context.world_small =   sid(35248)   #SCZ
context.world =         sid(22972)   #EFA
context.midvalue =      sid(32521)   #VOE
if get_datetime('US/Eastern').date() >= datetime.date(2010, 4, 1):
context.world_small =   sid(38272)   #VSS
context.world =         sid(33486)   #VEU

assets = [context.sp500, context.midvalue, context.world, context.world_small, context.outofmarket, context.tips]

#Get best asset class within each subcategory
df = pd.DataFrame(columns=['Weight','Score1','Score2','1Year','6Mon','3Mon','1Mon'])

#Calculate Momentum Ratios
for stock in assets:
his = data.history(stock, "price", 252, frequency="1d")
df.loc[stock, '1Mon'] = his[-1] / his[-21] - 1
df.loc[stock, '3Mon'] = his[-1] / his[-63] - 1
df.loc[stock, '6Mon'] = his[-1] / his[-126] - 1
df.loc[stock, '1Year'] = his[-1] / his[0] - 1

#Check Term Trend is Positive
df = df.astype(float)
df['Score1'] = df['1Mon'] + df['3Mon'] + df['6Mon']
df['Score2'] = df['3Mon'] + df['6Mon'] + df['1Year']
df['Weight'] = df['Score1']
df.loc[df['Weight'] < 0, 'Weight'] = 0.0

#Set outofmarket, and allworld to 0
df.loc[context.world, 'Weight'] = 0.0
df.loc[context.outofmarket, 'Weight'] = 0.0
df.loc[context.midvalue, 'Weight'] = 0.0
df.loc[context.tips, 'Weight'] = 0.0

#Get only top assets
df.loc[~df.index.isin(df['Weight'].nlargest(MAX_ASSETS).index.tolist()),'Weight'] = 0.0

#Add Safe if none others are positive
if len(df[df.Weight > 0]) == 0:
if df.loc[context.tips, '1Mon'] > df.loc[context.outofmarket, '1Mon']:
df.loc[context.tips, 'Weight'] = 1.0
else:
df.loc[context.outofmarket, 'Weight'] = 1.0

#Determine Weights
sum_weight = sum(df['Weight'])
df['Weight'] = df['Weight']/sum_weight

log.info(df.round(4))
context.good = df

record(sp500 = df.Weight.loc[context.sp500],
world_small = df.Weight.loc[context.world_small]*0.75,
treasuries = df.Weight.loc[context.outofmarket]*0.5,
tips = df.Weight.loc[context.tips]*0.25,
leverage = context.account.leverage)

"""
Determine how much of each asset to buy and place orders, making sure no extra cash is used
"""
stocks = context.good.index.tolist()
weight = context.good['Weight'].values.tolist()
n = len(weight)
if n < 1:
return

#Determine necessary contribution
for x in range(0, n):
desired_balance = context.good.loc[stocks[x], 'Weight']*context.portfolio.portfolio_value
curr_price = data.current(stocks[x],'price')
current_balance = context.portfolio.positions[stocks[x]].amount*curr_price
context.good.loc[stocks[x], 'Need'] = desired_balance-current_balance
context.good.loc[stocks[x], 'Price'] = curr_price*1.005

#Determine how much to get of each (truncate by share price)
context.good['Get'] = context.good['Need']
context.good.loc[context.good.Get < 0,'Get'] = 0 #set all gets less than 0 to 0
get_sum = context.good['Get'].sum()
if get_sum == 0:
get_sum = 1
cash = context.portfolio.cash + ROBINHOOD_GOLD
context.good['Get'] = context.good['Get']*cash/get_sum #scale gets by available cash
context.good.loc[context.good.Get < MIN_BUY,'Get'] = 0 #set all gets less than 0 to 0
context.good['Shares'] = np.trunc(context.good['Get']/context.good['Price']) #determine number of shares to buy
context.good['Get'] = context.good['Shares'] * context.good['Price'] #recalculate how much will be bought from truncated shares

#Figure out remaining cash and buy more of the stock that needs it most
cash = cash - context.good['Get'].sum()
context.good.loc[context.good['Need'].idxmax(),'Get'] += cash #use up all cash
context.good['Shares'] = np.trunc(context.good['Get']/context.good['Price']) #recalculate number of shares after adding left over cash back in
context.good['Get'] = context.good['Shares'] * context.good['Price'] #recalculate how much will be bought from truncated shares

#place orders for each asset
for x in range(0, n):
order(stocks[x], context.good.loc[stocks[x], 'Shares'], style=LimitOrder(context.good.loc[stocks[x], 'Price']))
log.info(context.good[['Weight','Need','Get']].sort_values(by='Need', ascending=False))

def my_rebalance(context,data):
"""
Scale down stocks held that are in portfolio, sell any that aren't
"""
context.started = 1

context.long_turnover = 0
good_stocks = context.good.index.tolist()
print_out = ''

#Sell stocks that are not in our lists
for security in context.portfolio.positions:
cost = context.portfolio.positions[security].cost_basis
price = context.portfolio.positions[security].last_sale_price
amount = context.portfolio.positions[security].amount
gain = (price-cost)*amount
if security not in good_stocks and data.can_trade(security):
print_out += '\nSell: ' + security.symbol + ' | Gains: \$' + '{:06.2f}'.format(gain) + ' | Gain: ' + '{:04.2f}'.format((price/cost-1)*100) + '%'
order_target_percent(security,0)
context.long_turnover += 1

#Determine weights and trim good stocks
n = len(good_stocks)
curr_weights = np.zeros(n)
weight = context.good['Weight'].values.tolist()
for x in range(0, n):
security = good_stocks[x]
curr_weights[x] = context.portfolio.positions[security].amount * context.portfolio.positions[security].last_sale_price / context.portfolio.portfolio_value
if curr_weights[x] >  weight[x]:
print_out += '\nTrim: ' + security.symbol
order_target_percent(good_stocks[x],weight[x])
log.info(print_out)
There was a runtime error.
18 responses

Attached is a notebook that has some more research information. It also includes a cell which can be used to calculate the signal now or at any day defined.

314

Hi Stephen!

When you said "compute 1+3+6 month momemtum signal" did you mean we must add these values?

Cool post! Haven't read it 100% yet, but I think this is kind of what I'm looking for nowadays strategy wise.

Yep, just add the signals. The backtest and the attached notebook show the calculation in more detail but it's super simple!

Hi @Stephen, very nice > 100 year LONG-term perspective & expansion on Gary's (excellent) work. I love it!!

Wow great work! What do you think of adding leveraged ETFs instead (SPXL, TMF, etc.)? They have a much higher expense ratio but the dynamic nature of this algo might make up for it. I am a newcomer so please relieve me of my ignorance if I am way off with this idea. Thank you!

Govind, thanks for sharing, I did see this! I'm not entirely sure Gary read my full blog (don't blame him if he didn't, it was long!) because I directly addressed his concerns before he voiced them - because I shared them! I admit to first doing some guess and checking on the ~20 year period I had at first in portfolio visualizer. But I recognized how it could have been a total fluke what happened in the past 20 years, and before posting the strategy I wanted to test it on data I hadn't seen before. So I went and compiled and shared data going back to 1871. And this "accelerating dual momentum" strategy I shared worked better than the base GEM in that period too, without any tweaking or data mining done by me there. I fully discussed it here, although I probably need to write a more focused and shorter blog on the topic.

I also computed the rolling 30 year returns to show how the ADM strategy is beating or at least tying GEM in virtually every 30-year period. I can submit to the fact that data mining might have been done in the more recent history; but I'm not sure how one can objectively look at the over 100-year backtest and say it's the result of data mining.

I'd also like to remind Gary and others about the fact that Gary admited in his book about the viability of looking at accelerating momentum singles. Here's his quote:

Accelerating momentum as either curvature or trend salience might be effective with stock indexes and other assets, in addition to its use with individual stocks.

So I'm not sure why he's knocking the strategy on merits that aren't entirely true, and when he as admitted in the past to the potential that accelerating momentum offers. I suppose in his book he only gave it a few paragraphs and was dismissive there. But again, the strategy has worked well in all time periods better than regular GEM and the strategy makes a decent amount of intuitive sense to me (and hopefully others!).

BTW Andrew, avoid leveraged ETFs like the plague. Because of compounding they don't perform as you'd initially expect beyond a single day's return. I need to probably write my own post on the subject, but some googling will come up with good articles on the pitfalls of leveraged ETFs for investment horizons longer than a day.

Stephen - While you give reasons to justify their use based on the last 20 years of data, I think Gary's point is that your use of shorter look-back periods, long-term bonds, and small-cap international stocks may all come from data mining. He shows they do not hold up so well on longer-term data. Haven't both strategies performed the same over a trailing 30 year period since the 1990s according to your chart? Prior to 1970, isn't your strategy no longer dual momentum, since you have no international stocks to compare to the S&P 500? In other words, you are just looking at the trend of the S&P 500. I'm also unclear why you call your strategy Accelerating Dual Momentum. In Gary's book, he refers to 2 studies of accelerating momentum. The first uses a linear regression of daily returns against the square of time to see if momentum has been accelerating. The second uses the slope of performance relative to the 12-month geometric average rate of return. It looks to me like you have just dual momentum with shorter look-back periods from 1970 forward and simple trend following prior to 1970.

This algo run in live account ?

Hi Stephen

I noticed that you are using price momentum vs total returns momentum (including dividends) in your algorithm. I believe Gary's algorithm works with total returns. Can you please confirm that my understanding is correct about how your algorithm is calculating momentum?

update - Stephen - never mind - looks like quantopian price data is already dividend adjusted.

A second question: Gary's algo compares total return of each stock to that of the risk free asset total return before rank ordering them. The algo in your pipeline simply looks for positive momentum vs comparison to a risk free asset, per my understanding. Am I getting this correct?

Mike,

The attached Quantopian algorithm did not explicitly include dividends, I always had a tough time understanding how Quantopian handled dividends in the price data.

I don't use the risk-free asset total return, just look to see if the average 1,3,6 month return is positive. My testing showed that the absolute return relative to the risk-free asset just confused the matter and marginally reduced returns (it's another way of trying to keep you out of the market, and in the long run, it's better to be in than out!).

I've actually gone away from using the Quantopian algorithm and just go directly to Portfolio Visualizer (go to the "Timing Periods" tab). I found that my Quantopian algorithm is doing a fixed number of day look back when the calendar month look back can actually differ quite a bit. I've just found it simpler to use the Portfolio Visualizer tool, but this Quantopian algorithm should agree 99% of the time (except during highly volatile times and inconsistent lengthed months... like the beginning of 2019).

After portfolio visualizer introduced paid plans and stopped giving forward looking signal results Quantopian is back to being the only free option left (besides manually calculating the signal). Attached is a super simple notebook to provide that calculation.

102

It looks great! Also, you can easily run this strategies on you own computer by IBridgePy. www.IBridgePy.com because it does not need Q's pipeline.
IBridgePy is an easy-to-use python platform to run Quantopian's strategies on your own computer. And there are a lot of tutorials on YouTube.
For example, Intro of IBridgePy: backtest and live trading https://youtu.be/xWMzTgGWv48

Disclaimer: This is Dr. Hui Liu, who created IBridgePy

I cloned your algorithm and backtested it but got drastically lower results. Not sure where the discrepancy lies.

Hi

Has this strategy been tested with limit price entering and exit, and possibly with target sell price (stop gain rule)?

Since the strategy uses so few ETFS it is probably well suited to investigate whether it is possible to get in and out of positions to better levels than market close price end of month

For example:
Entering 0.5% above yesterday's low
Exit 0.5% below yesterday's high
otherwise, a simple position stop-gain rule with a threshold around 10% -15%

Has anyone tested the strategy with trading costs?

I don't know exactly why, but I can't produce the same signals as you, for SPY VSS and TLT.

As you can see from the attached note, when I use the sum of 1 month, 3 months and 6 months (from Barchart), it is VSS, which is the purchase signal for July, but if you look at the print screen from your script it is SPY which is the signal for July.

What's right here? Is there anything about the strategy I haven't understood?


text:
#### Category : mongodb

$model =$user->find(1); $model->where(‘test’,"like",’%23%’); Sourc.. Read more i am NOT getting result thought the data is there in DB$conditions = collect($this->getColumns())->map(function ($column, $key) use ($keyword, $or_conditions) { return$or_conditions[] = [ $column => [ ‘$regex’ => "$keyword", ‘$options’ => ‘i’ ], ]; }); $final_or_condition = [ ‘$or’ => $conditions->toArray() ];$this->collection = $this->collection->whereRaw($final_or_condition); when i search in Datatable input field on ..

In my project, I use Laravel 8 with Mongodb as my database, and jenssegers/laravel-mongodb as its driver. Let say I have a collection named products which has relationship embedsOne to vendors. when I create a new product, how can I insert existing vendor to this new product? I tried the following $vendor = Vendor::find($request->vendor); $product .. Read more I try to query my collection with only one query and 3 potentials search method : fulltext search, classic search and search regex. This 3 matches can be executed at the same time or just one of them. The fulltext search is the first stage pipeline as we know. Does this fulltext search can be .. Read more i am having collection of employee assosiated with company use IlluminateDatabaseEloquentModel; use JenssegersMongodbEloquentModel as Eloquent; class Employee extends Eloquent { protected$casts = [ ‘has_pp’ => ‘Boolean’, ‘has_eal’ => ‘Boolean’, ‘support’ => ‘String’, ‘has_cin’ => ‘Boolean’, ‘has_lac’ => ‘Boolean’ ]; protected $fillable = [ ‘first_name’, ‘last_name’, ‘has_pp’, ‘has_eal’, ‘support’, ‘has_cin’, ‘has_lac’ ]; public function company(){ .. Read more I was using "jenssegers/mongodb": "2.1" and switched to "jenssegers/mongodb": "3.0.x" while upgrading from Laravel5.0 to Laravel 5.1 with php 5.6.x I modified my codes for the upgraded version, but I’m no longer able to get the results count as used to before with this code:$results = DB::connection($connection) ->collection("listing") ->timeout(-1) ->raw(function ($collection) use ($filters,$cols) ..

I am having multiple forms as collection and needs to validate their ids in one request for which i am taking from*name as name of forms and form id wrt to it. but i am not able to validate form_id wrt to formname provided. any idea to achive this would be very grateful. $v = .. Read more I am new to mongodb and using the aggregate function to fetch some records Query I used is:$records = $this->raw()->aggregate([ [‘$match’ => ["ts" => [‘$gte’=>$startTime,’$lt’=>$endTime]]], [‘$group’ => ["_id" =>["config" => ‘$conf’, "status" => ‘$s’ ], "count" =>[‘$sum’ =>1]], "ts" => [ ‘$dateToString’ => [ ‘format’ => ‘%Y-%m-%d’, ‘date’ =>$startTime, ‘timezone’ => ..

It must be a very stupid question but I am unable to fetch a date key in the mongo aggregate function in laravel framework I used this query to fetch the records but I am not getting how to store the "ts" key in the records too. $records =$this->raw()->aggregate([ [‘$match’ => ["ts" => [‘$gte’=> ..

I’m trying to crear a seeder on laravel 6 using mongodb this is database.php use IlluminateSupportStr; return [ ‘default’ => env(‘DB_CONNECTION’), ‘connections’ => [ ‘mongodb’ => [ ‘driver’ => ‘mongodb’, ‘host’ => env(‘DB_HOST’), ‘port’ => env(‘DB_PORT’), ‘database’ => env(‘DB_DATABASE’), ‘username’ => env(‘DB_USERNAME’), ‘password’ => env(‘DB_PASSWORD’), ‘options’ => [ ‘database’ => ‘mydb’ ] ], ], and ..

I am working on a project using Laravel and MongoDB from jenssegers/laravel-mongodb package and I am struggling with crud operations on cart. Before changing the database to a NoSQL one, my functions were working, it was searching the product in my products table by id and adding, removing or updating as the case may be. ..

hello so this is my first laravel project and i’m lost.. i have a mongodb database that has a users collection and a posts collection this is my Post model: <?php namespace AppModels; use IlluminateDatabaseEloquentFactoriesHasFactory; use IlluminateDatabaseEloquentModel; use JenssegersMongodbEloquentSoftDeletes; /** * @method static create(array $array) * @method static firstWhere(string$string, int $id) */ class Post .. Read more I tried to create a mongodb logging channel ‘mongo’ => [ ‘driver’ => ‘monolog’, ‘handler’ => MonologHandlerMongoDBHandler::class, ‘formatter’ => MonologFormatterMongoDBFormatter::class, ‘handler_with’ => [ ‘mongo’ => new MongoDBClient("mongodb://localhost:27017"), ‘database’ => ‘testdb’, ‘collection’ => ‘testcoll’ ] ], However, im getting error: IlluminateContractsContainerBindingResolutionException(code: 0): Unresolvable dependency resolving [Parameter #0 [ <required>$mongodb ]] in class MonologHandlerMongoDBHandler The error ..

this code working fine when i not change j.acc_no value, If we change j.acc_no value it does not update the document. any one suggest any idea to fix this $updateData = [ ‘bank.$[i].details.$[j].acc_name’ => ‘raja K’, ‘bank.$[i].details.$[j].acc_no’ => ‘35252555’, ‘bank.$[i].details.$[j].min_deposit’ => ‘10000’, ‘bank.$[i].details.$[j].max_deposit’ => ‘1000000’, ‘bank.$[i].details.$[j].status’ => 1, ];$filters = [ [‘i.bankname_slug’ => [‘$eq’ .. Read more I have an array(or collection) like this for my notifications. Each notification has an array field for all users who has seen the notification. array:2 [ 0 => array:8 [ "_id" => "604485ddf92e0000050079c6" "title" => "Notification title" "message" => "Notification message" "type" => "success" "expire_date" => "2021-05-09T19:30:00.000000Z" "read_by_users" => array:2 [ 0 => array:2 [ .. Read more Let’s say I declared a variable$a = "users" and I know there is a collection named users in my mongodb. How can I fetch the collection’s instance in folders like commands? I used Schema facade but it only returns the collection info or it can return if collection exists. Sourc..
When using jenssegers/laravel-mongodb, the belongsTo relationship properly always returns null, despite, when dumping, the value appearing. How can I get the belongsTo relationship? I have two models, a post and a user. Where the post has a belongs to relationship with user. When I use $post->user, I always get null. Despite the fact that when .. Read more I’m trying to setup MongoDB in a Laravel Jetstream + Sail. What I’m doing is trying to install the following packages: mongodb/mongodb jenssegers/mongodb but I get the following errors : Problem 1 – jenssegers/mongodb[v3.8.0, …, v3.8.2] require mongodb/mongodb ^1.6 -> satisfiable by mongodb/mongodb[1.6.0, …, 1.9.x-dev (alias of dev-master)]. – mongodb/mongodb 1.9.x-dev is an alias of .. Read more i have a lumen project that is using mongo db now i want to access the data and do some logic on them my object in mongo is like below : _id: ObjectId(‘602cfb30bc865100073f0e56’), serviceType: ‘normal’, segment: ‘Basic’, steps: { ‘step1’: 1, ‘step2’: 2 } nnow in my laravel application i do it like below : .. Read more I would like to save the following object to a field on a mongodb collection via eloquent "project_settings" : { "resource" : true, "zone" : true, "contractor" : true, "responsible_person" : false } this is my php code: return ProjectModel::where($project_id)->first()->update($project_data); dont worry about the model it works fine i need to make a query that .. Read more I am running following query in mongodb collection and I am using this package https://github.com/jenssegers/laravel-mongodb to communicate with mongodb in laravel.$ibQuery = $this->model->project([‘message_count’ => [ ‘$size’ => [ ‘$ifNull’ => [‘$messages’, []] ] ] ]) ->get(); return $ibQuery; and I am getting this error. exception: "MongoDBDriverExceptionCommandException" file: "/opt/lampp/htdocs/my-project/vendor/mongodb/mongodb/src/Operation/Find.php" line: 337 message: "Unsupported projection option: .. Read more So I’ve managed to do a clean install of Laravel in Windows using WSL2. One of the requirements of my project is to have two databases running at the same time mysql and mongodb. I added an image of MongoDB in my docker-compose.yml as the dock-hub tells me, but when I try to add the .. Read more i have data in mongodb it is a road object that has a property and an array of points that it consists of: my model in laravel <?php namespace AppModels; use AppTraitsUuids; use GraphQLTypeDefinitionType; use RebingGraphQLSupportFacadesGraphQL; use JenssegersMongodbEloquentModel as Eloquent; class Road extends Eloquent { //use HasFactory; use Uuids; protected$connection = ‘mongodb’; protected $collection .. Read more I get wrong date from MongoDB using Eloquent in Laravel. My record in database looks like this "created_at" : ISODate("2020-11-17T15:30:42.131+01:00") Code to get records from MongoDB$taskObj = TaskComments::where(‘task_id’, \$task_id)->get()->toArray(); Result date for created_at is 1970-01-25 20:31:23 which is wrong, I would like to get in this format 2020-11-17 15:30:42 I would like to create ..


text:
## Algebra 1: Common Core (15th Edition)

Published by Prentice Hall

# Chapter 3 - Solving Inequalities - 3-1 Inequalities and Their Graphs - Practice and Problem-Solving Exercises - Page 168: 11

#### Answer

$\frac{k}{9}\gt \frac{1}{3}$

#### Work Step by Step

To find the inequality, the quotient of $k$ and $9$ is greater than $\frac{1}{3}$, arrange your inequality in the order stated: $\frac{k}{9}$ greater than $\frac{1}{3}$ The greater than sign is $\gt$. Add this in between $\frac{k}{9}$ and $\frac{1}{3}$. $\frac{k}{9}\gt \frac{1}{3}$

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.


text:
My watch list
my.chemeurope.com

# Kolchuga passive sensor

The Kolchuga passive sensor is an ESM system developed in Ukraine. Its detection range is limited by line-of-sight but may be up to 800 km (500 miles) for very high altitude, very powerful emitters. Frequently referred to as Kolchuga Radar, the system is not really a radar, but an ESM system comprising three or four receivers, deployed tens of kilometres apart, which detect and track aircraft by triangulation and multilateration of their RF emissions.

## History

This was developed by the Special Radio Device Design Bureau public holding, the Topaz holding, the Donetsk National Technical University, the Ukrspetsexport state company, and the Investment and Technologies Company. It took them eight years (1993 - 2000) to conduct research, develop algorithms, test solutions on experimental specimens, and launch serial production. The new product dramatically changed the balance in the constant competition between offensive and defensive means. The relatively cheap Ukrainian Kolchuga radar station is able to detect and identify practically all known active radio devices mounted on ground, airborne, or marine objects.

## Mode of operation

Kolchuga is an electronic support measures system that employs two or more sites to locate emitters by triangulation. The system is vehicle mounted and comprises a large vertical meshed reflector, with two smaller circular parabolic dishes beneath and a pair of VHF-to-microwave log periodic antennas above. The dishes may exploit amplitude monopulse techniques for improved direction finding, whilst the angled spacing of the log-periodic antenna suggests that they may use phase interferometry to improve angle measurements. Various smaller antennas, presumably for inter-site communications are to the side and rear of the dish.

The detection range is one of the best in its class, but it is highly dependent on the emitted power of the transmitter being tracked, and requires satisfaction of the line of sight condition to at least two receiving sites for triangulation (compared with three sites for a multilateration system such as the VERA passive sensor). A Kolchuga complex can detect and locate air and surface targets and trace their movement to a range generally limited only by the common line-of-sight of the stations. Assuming no terrain masking, the line-of-sight range of a single Kolchuga station (in km) is approximately (Skolnik 1981):

$d(km) =130(\sqrt{hr(km)}+\sqrt{ht(km)})$

where hr(km) is the height of the radar in km, and ht(km) is the height of the target in kilometres, and assuming standard atmospheric radio refraction. Thus, for a Kolchuga at 100 m altitude (above local terrain) and a target at 10 km (30 kft), the range of the system would be approximately 450 km. For targets at altitudes of 20 km (60 kft) the line of sight limitation would be 620 km—but few targets fly at such altitudes. However, being line-of-sight limited, the system is an effective early warning air defense system.

### System parameters

According to the manufacturer's brochure (from AIDEX 1997), the upgraded Kolchuga-M is equipped [corrected for English grammatical mistakes]:

• With four antennas in the VHF, UHF and SHF radio bands with narrow and wide beams
• With parallel receivers allowing the instant discovery and analysis of signals of radio technical equipment (RTE) in the range 100 MHz -18 GHz and continual tracking across the entire band. [Other sources claim from 130MHz to 18GHz, and a 36 channel preset receiver].
• With sensitivity of approximately -145 dBW
• With equipment for the analysis, processing and continual calculation and tracking of coordinates by triangulation and chronometrical-hyperbolic method.
• With equipment for the identification, operative displaying of information, long term memory and recording of the results of processing.

The brochure also claims that the system provides [corrected for English grammatical mistakes]:

• The detection, analysis of signals with pulsed and continuous waves and resolution of practically all known radio technical equipment (RTE) deployed on ground, sea and air platforms, including radar units of all classes, identification systems, air traffic control system and navigation systems
• The composition, character of operation and flight paths of RTE in a zone 600 km deep with complete exception of blind zones and continual high accuracy
• The detection of take-off of aerial targets and their formation in groups at a distance exceeding that of modern radars.
• The direction of arrival of solitary and group targets and target indication to active facilities of anti-aircraft systems

Special inhibitory sorters omit up to 24 interfering signals, and tracking sorters make it possible to synchronously sort out and track signals from 32 targets;

### Target identification

Kolchuga is able to detect and identify many types of radio devices mounted on ground, airborne, or marine objects. Target detection relies only on an emitter having sufficient power and being within Kolchuga's frequency range. Target identification, however, is more complex and is based on the measurement of different parameters of the transmitted signal—such as its frequency, bandwidth, pulse width, pulse repetition interval, etc. Kolchuga has been reported to use around forty different parameters when identifying a target. These parameters are compared to a database in order to identify both the type of emitter and, in some cases, even the specific piece of equipment (by identifying the unique signature or "fingerprint" that most transmitters have, due to the variations and tolerances in individual components). The database within Kolchuga is said to have the capacity to store around three hundred different types of emitter and up to five hundred specific signatures for each type.

## Exports

In 2002 the U.S. State Department accused Ukraine of selling Kolchuga to Iraq, based on recordings of the then Ukrainian president Leonid Kuchma supposedly made by Mykola Mel'nychenko. This was followed by political steps from United Kingdom and the United States. No material confirmation has been found in Iraq. See Cassette Scandal for further information.

It has been exported to Russia where additional development has been performed and the system is marketed in conjunction with the new S-400 surface to air missile (SAM) system. [1]

Unconfirmed reports in September 2006 suggested a sale was made to Iran [2][3] although this was denied by the Ukrainian government [4].

## Rumours and speculation of performance

Since becoming publicly known following the Cassette Scandal, the capabilities of Kolchuga have been the source of many rumours and uninformed speculation in forums such as [5]. Many observers have a tendency to credit it with magical powers of detection. Many of these do not stand up to detailed engineering analysis, or have not been confirmed, but are recorded in this section for completeness, together with reasons for doubting the claim. Note that the material in this section should not be regarded as accurate. Claims include:

• That Kolchuga has a range of 800 km. It will certainly have the sensitivity to see high power sources at such a range. However, basic line-of-sight arguments outlined in the sections above show that this would require aircraft to be flying at impractically high altitudes and/or for the Kolchuga stations to be deployed on very high mountains. The claim is thus technically possible but operationally very unlikely.
• That Kolchuga can detect US stealth aircraft by their radio and radar emissions. Whilst technically possible, no stealth aircraft radiate when in battle, so this mode of operation is militarily irrelevant.
• Kolchuga is sufficiently sensitive to detect US stealth aircraft from unconventional sources of RF emissions, including radiation from exhaust trails and electromagnetic interference from the engine. (Technically the power levels of these sources are likely to be so small, if at all, that there would be insufficient energy for Kolchuga to measure these effects at one site, let alone the two or more required for triangulation. They would also be almost impossible to distinguish for normal background RF noise and would not appear like the conventional emissions types Kolchuga is designed to receive and analyse).
• Kolchuga is sufficiently sensitive to operate as a bistatic radar receiver and hence exploit other transmitters in the environment. (This is likely to be true for the specific case of forward scatter in which the radar cross section of even a stealth aircraft can become very large for a few moments. However, achieving forward scatter simultaneously in two or more receivers is geometrically impossible, so triangulation would not work. Secondly, unless Kolchuga has been specifically designed to work as a bistatic radar receiver with a certain radar type—and there's no evidence of Kolchuga being deployed in conjunction with specific radar types—then it will lack the essential matched filter required to reliably detect reflected pulses in the presence of noise. Therefore, if this effect works at all, it is likely to be unreliable and fleeting).
• Ukrainian sources often make claims such as "[Kolchuga] is head and shoulders above all American, Russian, French, Czech, or Brazilian developments in this field"[6]. Such claims are unsubstantiated and pure speculation, as the performance and even existence of many ESM systems are highly classified and not deducible by ELINT. It may or may not be true, but must be regarded with caution, particularly as those making the claims are often either journalists or those with an interest in exporting the system.

• Military of Ukraine
• ELINT\ESM
• VERA – a similar system using multilateration developed in the Czech Republic
• BORAP – a similar system using triangulation developed in the Czech Republic
• Ukrainian Ground Forces
• Multinational force in Iraq

## References

• Skolnik, Merrill (1981). Introduction to Radar Systems. McGraw-Hill. ISBN 0-07-057909-1.


text:
Article

# Search for B -> phi pi decays

• ##### V. Zhulanov
06/2012;
Source: arXiv

ABSTRACT We report on a search for the charmless decays $B^{+} \to\phi\pi^{+}$ and
$B^{0} \to\phi \pi^{0}$ that are strongly suppressed in the Standard Model. The
analysis is based on a data sample of $657 \times 10^6$ $B \bar{B}$ pairs
collected at the $\Upsilon(4S)$ resonance with the Belle detector at the KEKB
asymmetric-energy $e^+ e^-$ collider. We find no significant signal and set
upper limits of $3.3 \times 10^{-7}$ for $B^{+} \to \phi \pi^{+}$ and $1.5 \times 10^{-7}$ for $B^0 \to \phi \pi^0$ at the 90% confidence level.

0 Bookmarks
·
197 Views
• Source
01/2001;


text:
# Finding sum of divisors of numbers between the range (1 to 1e16)

The exact link to the question : https://www.spoj.com/problems/DIVSUM2/

Input: An integer stating the number of test cases (equal to 500), and that many lines follow, each containing one integer between $$\1\$$ and $$\10^{16}\$$ inclusive.

Output: Print one integer, the sum of the proper divisors of the number.

An easier version of this question exists in which the constraints are relaxed.

My code works for the easier version , but exceeds the time limit for the harder one.

Example of approach: $$\36=2^2 \times 3^2\$$. So sum of proper factors = $$(2^0+2^1+2^2)\times(3^0+3^1+3^2)-36=55$$

## My code:

#include<iostream>
long long sum_proper_divisors(long long n)
{
long long sum=1,x=n;
for(long long i=2;i*i<=n;i++)
{
long long current_sum=1,current_term=1;
while(n%i==0)
{
n/=i;
current_term*=i;
current_sum+=current_term;
}
sum*=current_sum;
}
if(n>=2){
sum*=(n+1);
}
return (sum-x);
}
int main()
{
long long q;
std::cin>>q;
while(q--)
{
long long n;
std::cin>>n;
std::cout<<sum_proper_divisors(n)<<std::endl;
}
}

• Questions should be self-contained: i.e. the spec shouldn't be hidden behind a link. By all means leave the link for detail, but please summarise the requirements directly. Jul 2, 2019 at 15:14
• @PeterTaylor Done . Jul 2, 2019 at 15:22

## 1 Answer

What's going on with the indentation?

long long

No. We're not in the 1970s. Use <cstdint>. Here you probably want std::uint_fast64_t.

IMO n should be const and the copy (x) should be the variable which is modified in the loop.

    for(long long i=2;i*i<=n;i++)

The first step to optimising prime divisor searches is to use a wheel. If you special-case i=2 then you can search for(i=3;i*i<=x;i+=2), which is already a speedup by about a factor of 2.

On some architectures, multiplication is expensive and it's faster to maintain a variable ii or iSquared which is updated using the identity $$\(i+a)^2 - i^2 = 2ai + a^2\$$: e.g. (untested, may be buggy): for(i=3,ii=9;ii<=x;ii+=(i+1)<<2,i+=2)

          long long current_sum=1,current_term=1;
while(n%i==0)
{
n/=i;
current_term*=i;
current_sum+=current_term;
}

This can be simplified:

           std::uint_fast64_t current_sum=1;
while(x%i==0)
{
x/=i;
current_sum=current_sum*i+1;
}

The first big step to improving the algorithm would be to exploit the fact that up to 500 test cases are passed at a time. If you read them all and find the largest, you can do a variant of Eratosthenes' sieve once to set up an array which allows you to factor all numbers up to that largest one very fast.

If that's not enough, it's probably necessary to switch factorisation approach, starting with Pollard's rho algorithm.

• << 2 should give the same exact code as * 4
– alx
Jul 2, 2019 at 18:51


text:
• ### 3D Placement of an Unmanned Aerial Vehicle Base Station for Maximum Coverage of Users with Different QoS Requirements(1709.05235)

Sept. 15, 2017 cs.IT, math.IT
The need for a rapid-to-deploy solution for providing wireless cellular services can be realized by unmanned aerial vehicle base stations (UAV-BSs). To the best of our knowledge, this letter is the first in literature that studies a novel 3D UAV-BS placement that maximizes the number of covered users with different Quality-of-Service requirements. We model the placement problem as a multiple circles placement problem and propose an optimal placement algorithm that utilizes an exhaustive search (ES) over a one-dimensional parameter in a closed region. We also propose a low-complexity algorithm, namely, maximal weighted area (MWA) algorithm to tackle the placement problem. Numerical simulations are presented showing that the MWA algorithm performs very close to the ES algorithm with a significant complexity reduction.
• ### 3D Placement of an Unmanned Aerial Vehicle Base Station (UAV-BS) for Energy-Efficient Maximal Coverage(1705.03415)

May 9, 2017 cs.IT, math.IT
Unmanned Aerial Vehicle mounted base stations (UAV-BSs) can provide wireless services in a variety of scenarios. In this letter, we propose an optimal placement algorithm for UAV-BSs that maximizes the number of covered users using the minimum transmit power. We decouple the UAV-BS deployment problem in the vertical and horizontal dimensions without any loss of optimality. Furthermore, we model the UAV-BS deployment in the horizontal dimension as a circle placement problem and a smallest enclosing circle problem. Simulations are conducted to evaluate the performance of the proposed method for different spatial distributions of the users.
• ### Degrees of Freedom in Cached MIMO Relay Networks With Multiple Base Stations(1702.05528)

Feb. 17, 2017 cs.IT, math.IT
The ability of physical layer relay caching to increase the degrees of freedom (DoF) of a single cell was recently illustrated. In this paper, we extend this result to the case of multiple cells in which a caching relay is shared among multiple non-cooperative base stations (BSs). In particular, we show that a large DoF gain can be achieved by exploiting the benefits of having a shared relay that cooperates with the BSs. We first propose a cache-assisted relaying protocol that improves the cooperation opportunity between the BSs and the relay. Next, we consider the cache content placement problem that aims to design the cache content at the relay such that the DoF gain is maximized. We propose an optimal algorithm and a near-optimal low-complexity algorithm for the cache content placement problem. Simulation results show significant improvement in the DoF gain using the proposed relay-caching protocol.
• ### Optimal Beamforming for MIMO Shared Relaying in Downlink Cellular Networks with ARQ(1610.03089)

Oct. 10, 2016 cs.IT, math.IT
In this paper, we study the performance of the downlink of a cellular network with automatic repeat-request (ARQ) and a half duplex decode-and-forward shared relay. In this system, two multiple-input-multiple-output (MIMO) base stations serve two single antenna users. A MIMO shared relay retransmits the lost packets to the target users. First, we study the system with direct retransmission from the base station and derive a closed form expression for the outage probability of the system.We show that the direct retransmission can overcome the fading, however, it cannot overcome the interference. After that, we invoke the shared relay and design the relay beamforming matrices such that the signal-to-interference-and-noise ratio (SINR) is improved at the users subject to power constraints on the relay. In the case when the transmission of only one user fails, we derive a closed form solution for the relay beamformers. On the other hand when both transmissions fail, we pose the beamforming problem as a sequence of non-convex feasibility problems. We use semidefinite relaxation (SDR) to convert each feasibility problem into a convex optimization problem. We ensure a rank one solution, and hence, there is no loss of optimality in SDR. Simulation results are presented showing the superior performance of the proposed relay beamforming strategy compared to direct ARQ system in terms of the outage probability.
• ### Optimal Energy Allocation For Delay-Constrained Traffic Over Fading Multiple Access Channels(1607.00711)

Sept. 13, 2016 cs.IT, math.IT
In this paper, we consider a multiple-access fading channel where $N$ users transmit to a single base station (BS) within a limited number of time slots. We assume that each user has a fixed amount of energy available to be consumed over the transmission window. We derive the optimal energy allocation policy for each user that maximizes the total system throughput under two different assumptions on the channel state information. First, we consider the offline allocation problem where the channel states are known a priori before transmission. We solve a convex optimization problem to maximize the sum-throughput under energy and delay constraints. Next, we consider the online allocation problem, where the channels are causally known to the BS and obtain the optimal energy allocation via dynamic programming when the number of users is small. We also develop a suboptimal resource allocation algorithm whose performance is close to the optimal one. Numerical results are presented showing the superiority of the proposed algorithms over baseline algorithms in various scenarios.
• ### Asymmetric Degrees of Freedom of the Full-Duplex MIMO 3-Way Channel with Unicast and Broadcast Messages(1608.00209)

July 31, 2016 cs.IT, math.IT
In this paper, we characterize the asymmetric total degrees of freedom (DoF) of a multiple-input multiple-output (MIMO) 3-way channel. Each node has a separate-antenna full-duplex MIMO transceiver with a different number of antennas, where each antenna can be configured for either signal transmission or reception. We study this system under two message configurations; the first configuration is when each node has two unicast messages to be delivered to the two other nodes, while the second configuration is when each node has two unicast messages as well as one broadcast message to be delivered to the two other nodes. For each configuration, we first derive upper bounds on the total DoF of the system. Cut-set bounds in conjunction with genie-aided bounds are derived to characterize the achievable total DoF. Afterwards, we analytically derive the optimal number of transmit and receive antennas at each node to maximize the total DoF of the system, subject to the total number of antennas at each node. Finally, the achievable schemes for each configuration are constructed. The proposed schemes are mainly based on zero-forcing and null-space transmit beamforming.
• ### Proactive Location-Based Scheduling of Delay-Constrained Traffic Over Fading Channels(1607.00329)

July 1, 2016 cs.IT, math.IT
In this paper, proactive resource allocation based on user location for point-to-point communication over fading channels is introduced, whereby the source must transmit a packet when the user requests it within a deadline of a single time slot. We introduce a prediction model in which the source predicts the request arrival $T_p$ slots ahead, where $T_p$ denotes the prediction window (PW) size. The source allocates energy to transmit some bits proactively for each time slot of the PW with the objective of reducing the transmission energy over the non-predictive case. The requests are predicted based on the user location utilizing the prior statistics about the user requests at each location. We also assume that the prediction is not perfect. We propose proactive scheduling policies to minimize the expected energy consumption required to transmit the requested packets under two different assumptions on the channel state information at the source. In the first scenario, offline scheduling, we assume the channel states are known a-priori at the source at the beginning of the PW. In the second scenario, online scheduling, it is assumed that the source has causal knowledge of the channel state. Numerical results are presented showing the gains achieved by using proactive scheduling policies compared with classical (reactive) networks. Simulation results also show that increasing the PW size leads to a significant reduction in the consumed transmission energy even with imperfect prediction.
• ### On the Degrees of Freedom of SISO X-networks with Synergistic Alternating Channel State Information at transmitters(1605.07069)

May 23, 2016 cs.IT, math.IT
In this paper, we consider the two-user single-input single-output (SISO) X-channel and $K$-user SISO X-network in fast fading environment. It is assumed that the transmitters have access to synergistic alternating channel state information (CSI). Specifically, the CSIT alternates between three states, namely, perfect, delayed and no-CSIT, in a certain manner to enable these states to work together cooperatively. These states are associated with fractions of time denoted by $\lambda_P$, $\lambda_D$ \text{and} $\lambda_N$, respectively. For the two-user $X$-channel, simple upper bound is developed to prove the tightness of the achievability result of $4/3$ DoF under a certain distribution of the availability of three CSIT states for $\Lambda(\lambda_P=1/3, \lambda_D= 1/3, \lambda_N=1/3)$. For the $K$-user $X$-network, it is shown that the sum Degrees of freedom (DoF) is at least $2K/(K + 1)$, using two-phase transmission schemes over finite symbols channel extension and under the same distribution of the availability of $\Lambda(\lambda_P=1/3, \lambda_D= 1/3, \lambda_N=1/3)$.This achievability result, can be considered as a tight lower bound, coincides with the best lower bound known for the same network but with partial output feedback in stead of alternating CSIT. Hence, we show that the role of synergistic alternating CSIT with distribution $\Lambda(1/3,1/3,1/3)$ is equivalent to the partial output feedback. Also, this lower bound is strictly better than the best lower bound known for the case of delayed CSI assumption for all values of $K$. All the proposed transmission schemes are based on two phases transmission strategy, namely, interference creation and interference resurrection, which exploit the synergy of instantaneous CSI and delay CSIT to retrospectively align interference in the subsequent channel uses.
• ### Achievable Degrees of Freedom of the K-user MISO Broadcast Channel with Alternating CSIT via Interference Creation-Resurrection(1605.02928)

May 10, 2016 cs.IT, math.IT
Channel state information at the transmitter affects the degrees of freedom of the wireless networks. In this paper, we analyze the DoF for the K-user multiple-input single-output (MISO) broadcast channel (BC) with synergistic alternating channel state information at the transmitter (CSIT). Specifically, the CSIT of each user alternates between three states, namely, perfect CSIT (P), delayed CSIT (D) and no CSIT (N) among different time slots. For the K-user MISO BC, we show that the total achievable degrees of freedom (DoF) are given by $\frac{K^{2}}{2K-1}$ through utilizing the synergistic benefits of CSIT patterns. We compare the achievable DoF with results reported previously in the literature in the case of delayed CSIT and hybrid CSIT models.
• ### On Optimizing Cooperative Cognitive User Performance under Primary QoS Constraints(1603.09530)

March 31, 2016 cs.NI
We study the problem of optimizing the performance of cognitive radio users with opportunistic real-time applications subject to primary users quality-of-service (QoS) constraints. Two constrained optimization problems are formulated; the first problem is maximizing the secondary user throughput while the second problem is minimizing the secondary user average delay, subject to a common constraint on the primary user average delay. In spite of the complexity of the optimization problems, due to their non-convexity, we transform the first problem into a set of linear programs and the second problem into a set of quasiconvex optimization problems. We prove that both problems are equivalent with identical feasible sets and optimal solutions. We show, through numerical results, that the proposed cooperation policy represents the best compromise between enhancing the secondary users QoS and satisfying the primary users QoS requirements.
• ### Efficient 3-D Placement of an Aerial Base Station in Next Generation Cellular Networks(1603.00300)

Feb. 26, 2016 cs.NI, math.OC
Agility and resilience requirements of future cellular networks may not be fully satisfied by terrestrial base stations in cases of unexpected or temporary events. A promising solution is assisting the cellular network via low-altitude unmanned aerial vehicles equipped with base stations, i.e., drone-cells. Although drone-cells provide a quick deployment opportunity as aerial base stations, efficient placement becomes one of the key issues. In addition to mobility of the drone-cells in the vertical dimension as well as the horizontal dimension, the differences between the air-to-ground and terrestrial channels cause the placement of the drone-cells to diverge from placement of terrestrial base stations. In this paper, we first highlight the properties of the dronecell placement problem, and formulate it as a 3-D placement problem with the objective of maximizing the revenue of the network. After some mathematical manipulations, we formulate an equivalent quadratically-constrained mixed integer non-linear optimization problem and propose a computationally efficient numerical solution for this problem. We verify our analytical derivations with numerical simulations and enrich them with discussions which could serve as guidelines for researchers, mobile network operators, and policy makers.
• ### Interactive Multiple Model Estimation of Doubly-Selective Channels for OFDM systems(1602.01038)

Feb. 2, 2016 cs.IT, math.IT
In this paper, we propose an algorithm for channel estimation, acquisition and tracking, for orthogonal frequency division multiplexing (OFDM) systems. The proposed algorithm is suitable for vehicular communications that encounter very high mobility. A preamble sequence is used to derive an initial estimate of the channel using least squares (LS). The temporal variation of the channel within one OFDM symbol is approximated by two complex exponential basis expansion models (CE-BEM). One of the Fourier-based BEMs is intended to capture the low frequencies in the channel (slow variations corresponding to low Doppler), while the other is destined to capture high frequencies (fast variations corresponding to high Doppler). Kalman filtering is employed to track the BEM coefficients iteratively on an OFDM symbol-by-symbol basis. An interactive multiple model (IMM) estimator is implemented to dynamically mix the estimates obtained by the two Kalman filters, each of which matched to one of the BEMs. Extensive numerical simulations are conducted to signify the gain obtained by the proposed combining technique.
• ### Achievable Degrees of Freedom on K-user MIMO Multi-way Relay Channel with Common and Private Messages(1510.00359)

Oct. 15, 2015 cs.IT, math.IT
This paper investigates the achievable total degrees of freedom (DoF) of the MIMO multi-way relay channel that consists of K users, where each user is equipped with M antennas, and a decode-and-forward relay equipped with N antennas. In this channel, each user wants to convey K-1 private messages to the other users in addition to a common message to all of them. Due to the absence of direct links between the users, communication occurs through the relay in two phases; a multiple access channel phase (MAC) and a broadcast (BC) phase. We drive cut-set bounds on the total DoF of the network, and show that the network has DoF less than or equal to K min(N,M). Achievability of the upper bound is shown by using signal space alignment for network coding in the MAC phase, and zero-forcing precoding in the BC phase. We show that introducing the common messages besides the private messages leads to achieving higher total DoF than using the private messages only.
• ### Sensing/Decision-Based Cooperative Relaying Schemes With Multi-Access Transmission: Stability Region And Average Delay Characterization(1506.08895)

June 29, 2015 cs.IT, math.IT
We consider a cooperative relaying system which consists of a number of source terminals, one shared relay, and a common destination with multi-packet reception (MPR) capability. In this paper, we study the stability and delay analysis for two cooperative relaying schemes; the sensing-based cooperative (SBC) scheme and the decision-based cooperative (DBC) scheme. In the SBC scheme, the relay senses the channel at the beginning of each time slot. In the idle time slots, the relay transmits the packet at the head of its queue, while in the busy one, the relay decides either to transmit simultaneously with the source terminal or to listen to the source transmission. The SBC scheme is a novel paradigm that utilizes the spectrum more efficiently than the other cooperative schemes because the relay not only exploits the idle time slots, but also has the capability to mildly interfere with the source terminal. On the other hand, in the DBC scheme, the relay does not sense the channel and it decides either to transmit or to listen according to certain probabilities. Numerical results reveal that the two proposed schemes outperform existing cooperative schemes that restrict the relay to send only in the idle time slots. Moreover, we show how the MPR capability at the destination can compensate for the sensing need at the relay, i.e., the DBC scheme achieves almost the same stability region as that of the SBC scheme. Furthermore, we derive the condition under which the two proposed schemes achieve the same maximum stable throughput.
• ### A Probabilistic MAC for Cognitive Radio Systems with Energy Harvesting Nodes(1408.2779)

Oct. 29, 2014 cs.IT, math.IT, cs.NI
In this paper, we consider a cognitive radio (CR) system where the secondary user (SU) harvests energy from both the nature resources and the primary user (PU) radio frequency(RF) signal. We propose an energy-based probabilistic access scheme in which SU probabilistically accesses and senses the primary channel. The decision is based on the available energy and the PU's activity. We investigate the problem of maximizing the SU's success rate provided that the PU average quality of service (QoS) constraint is satisfied. We also assume multi-packet reception (MPR) capability and sensing errors under a Rayleigh fading channel. Numerical results show the effectiveness of the proposed probabilistic access scheme.
• ### On the Stable Throughput of Cooperative Cognitive Radio Networks with Finite Relaying Buffer(1410.2419)

Oct. 9, 2014 cs.NI
In this paper, we study the problem of cooperative communications in cognitive radio systems where the secondary user has limited relaying room for the overheard primary packets. More specifically, we characterize the stable throughput region of a cognitive radio network with a finite relaying buffer at the secondary user. Towards this objective, we formulate a constrained optimization problem for maximizing the secondary user throughput while guaranteeing the stability of the primary user queue. We consider a general cooperation policy where the packet admission and queue selection probabilities, at the secondary user, are both dependent on the state (length) of the finite relaying buffer. Despite the sheer complexity of the optimization problem, attributed to its non-convexity, we transform it to a linear program. Our numerical results reveal a number of valuable insights, e.g., it is always mutually beneficial to cooperate in delivering the primary packets in terms of expanding the stable throughput region. In addition, the stable throughput region of the system, compared to the case of infinite relaying queue capacity, marginally shrinks for limited relaying queue capacity.
• ### On the Degrees of Freedom of SISO X-Channel with Alternating CSIT(1407.8348)

July 31, 2014 cs.IT, math.IT
In this paper, we establish the degrees of freedom (DoF) of the two-user single input single output (SISO) X-channel with alternating channel state information at the transmitters (CSIT). Three cases are considered for the availability of CSIT; perfect, delayed and no-CSIT. Each state is associated with a fraction of time denoted by $\lambda_P, \lambda_D$ and $\lambda_N$, respectively. We provide new results for DoF of the two-user SISO X-channel when the available CSIT alternates between these three cases under a certain distribution $\Lambda(\lambda_P, \lambda_D, \lambda_N )$. Specifically, we show that the X-channel with alternating CSIT for $\Lambda(1/8, 3/8, 1/2)$ can achieve $5/4$ DoF. The interesting thing about $5/4$ is that it represents a position of compromise or a middle ground between the channel knowledge that transmitters need to steer interference and the degrees of freedom that the network can achieve. Moreover, $5/4$ is strictly greater than $6/5$ which is the upper bound for the X-channel with \textit{fixed} delayed CSIT.
• ### On the Coexistence of a Primary User with an Energy Harvesting Secondary User: A Case of Cognitive Cooperation(1307.4744)

July 28, 2014 cs.IT, math.IT, cs.NI
In this paper, we consider a cognitive scenario where an energy harvesting secondary user (SU) shares the spectrum with a primary user (PU). The secondary source helps the primary source in delivering its undelivered packets during periods of silence of the primary source. The primary source has a queue for storing its data packets, whereas the secondary source has two data queues; a queue for storing its own packets and the other for storing the fraction of the undelivered primary packets accepted for relaying. The secondary source is assumed to be a battery-based node which harvests energy packets from the environment. In addition to its data queues, the SU has an energy queue to store the harvested energy packets. The secondary energy packets are used for primary packets decoding and data packets transmission. More specifically, if the secondary energy queue is empty, the secondary source can neither help the primary source nor transmit a packet from the data queues. The energy queue is modeled as a discrete time queue with Markov arrival and service processes. Due to the interaction of the queues, we provide inner and outer bounds on the stability region of the proposed system. We investigate the impact of the energy arrival rate on the stability region. Numerical results show the significant gain of cooperation.
• ### The asymmetric DoF Region for the 3-user MxN Interference Channel(1407.2981)

July 10, 2014 cs.IT, math.IT
In this paper, the 3-user Gaussian MIMO interference channel is considered. The asymmetric distribution of the DoF, where different users have different number of DoF, is studied. Two cases are presented, the first is when all transmitters and receivers have equal number of antennas $M$, the other when the transmitters have $M$ antennas each, while the receivers have $N$ antennas each. It is assumed that the channel coefficients are constant and known to all transmitters and receivers. The region of the achievable DoF tuple $(d_1; d_2; d_3)$ is presented.
• ### Energy-Efficient Cooperative Cognitive Relaying Protocols for Full-Duplex Cognitive Radio Users and Delay-Aware Primary Users(1406.2255)

July 8, 2014 cs.IT, math.IT, cs.NI
This paper considers a network in which a primary user (PU) may cooperate with a cognitive radio (CR) user for transmission of its data packets. The PU is assumed to be a buffered terminal operating in a time-slotted fashion. We develop two protocols which involve cooperation between primary and secondary users. To satisfy certain quality of service requirements, users share time slot duration and frequency bandwidth. Moreover, the secondary user (SU) may leverage the primary feedback signal. The proposed protocols are designed such that the secondary rate is maximized and the primary queueing delay is maintained less than the queueing delay in case of non-cooperative PU. In addition, the proposed protocols guarantee the stability of the primary queue and maintain the average energy emitted by the CR user below a certain value. The proposed protocols also provide more robust and potentially continuous service for SUs compared to the conventional practice in cognitive networks where SUs transmit in the spectrum holes and silence sessions of the PUs. We include primary source burstiness, sensing errors, and feedback reception errors to the analysis of the proposed cooperative cognitive protocols. Numerical results show the beneficial gains of the cooperative protocols in terms of secondary rate and primary throughput, queueing delay, and average energy savings.
• ### Maximum Secondary Stable Throughput of a Cooperative Secondary Transmitter-Receiver Pair: Protocol Design and Stability Analysis(1206.6302)

June 9, 2014 cs.IT, math.IT, cs.NI
In this paper, we investigate the impact of cooperation between a secondary transmitter-receiver pair and a primary transmitter (PT) on the maximum stable throughput of the primary-secondary network. Each transmitter, primary or secondary, has a buffer for storing its own traffic. In addition to its own buffer, the secondary transmitter (ST) has a buffer for storing a fraction of the undelivered primary packets due to channel impairments. Moreover, the secondary destination has a relaying queue for storing a fraction of the undelivered primary packets. In the proposed cooperative system, the ST and the secondary destination increase the spectrum availability for the secondary packets by relaying the unsuccessfully transmitted packets of the PT. We consider two multiple access strategies to be used by the ST and the secondary destination to utilize the silence sessions of the PT. Numerical results demonstrate the gains of the proposed cooperative system over the non-cooperation case.
• ### A DoF-Optimal Scheme for the two-user X-channel with Synergistic Alternating CSIT(1404.6348)

April 25, 2014 cs.IT, math.IT
In this paper, the degrees of freedom (DoF) of the two-user single input single output (SISO) X-channel are investigated. Three cases are considered for the availability of channel state information at the transmitters (CSIT); perfect, delayed, and no-CSIT. A new achievable scheme is proposed to elucidate the potency of interference creation-resurrection (IRC) when the available CSIT alternates between these three cases. For some patterns of alternating CSIT, the proposed scheme achieves $4/3$ DoF, and hence, coincides with the information theoretic upper bound on the DoF of the two-user X-channel with perfect and instantaneous CSIT. The CSIT alternation patterns are investigated where the patterns that provide extraordinary synergistic gain and dissociative ones are identified.
• ### Coverage Probability Analysis for Wireless Networks Using Repulsive Point Processes(1309.3597)

Sept. 13, 2013 cs.NI
The recent witnessed evolution of cellular networks from a carefully planned deployment to more irregular, heterogeneous deployments of Macro, Pico and Femto-BSs motivates new analysis and design approaches. In this paper, we analyze the coverage probability in cellular networks assuming repulsive point processes for the base station deployment. In particular, we characterize, analytically using stochastic geometry, the downlink probability of coverage under a Matern hardcore point process to ensure minimum distance between the randomly located base stations. Assuming a mobile user connects to the nearest base station and Rayleigh fading, we derive two lower bounds expressions on the downlink probability of coverage that is within 4% from the simulated scenario. To validate our model, we compare the probability of coverage of the Matern hardcore topology against an actual base station deployment obtained from a public database. The comparison shows that the actual base station deployment can be fitted by setting the appropriate Matern point process density.
• ### A New Achievable DoF Region for the 3-user MxN Symmetric Interference Channel(1105.4026)

May 20, 2011 cs.IT, math.IT
In this paper, the 3-user multiple-input multiple-output Gaussian interference channel with M antennas at each transmitter and N antennas at each receiver is considered. It is assumed that the channel coefficients are constant and known to all transmitters and receivers. A novel scheme is presented that spans a new achievable degrees of freedom region. For some values of M and N, the proposed scheme achieve higher number of DoF than are currently achievable, while for other values it meets the best known upperbound. Simulation results are presented showing the superior performance of the proposed schemes to earlier approaches.
• ### Power Control for Maximum Throughput in Spectrum Underlay Cognitive Radio Networks(1002.1584)

Feb. 8, 2010 cs.IT, math.IT, cs.NI, math.OC
We investigate power allocation for users in a spectrum underlay cognitive network. Our objective is to find a power control scheme that allocates transmit power for both primary and secondary users so that the overall network throughput is maximized while maintaining the quality of service (QoS) of the primary users greater than a certain minimum limit. Since an optimum solution to our problem is computationally intractable, as the optimization problem is non-convex, we propose an iterative algorithm based on sequential geometric programming, that is proved to converge to at least a local optimum solution. We use the proposed algorithm to show how a spectrum underlay network would achieve higher throughput with secondary users operation than with primary users operating alone. Also, we show via simulations that the loss in primary throughput due to the admission of the secondary users is accompanied by a reduction in the total primary transmit power.


text:
# Integrate the Functions in ((X+1)(X + Logx)^2)/X - Mathematics

Integrate the functions in ((x+1)(x + logx)^2)/x

#### Solution

Is there an error in this question or solution?

#### APPEARS IN

NCERT Class 12 Maths
Chapter 7 Integrals
Q 36 | Page 305


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# Cartier dual of $\alpha_p$

How can I prove that the Cartier dual of $\alpha_p$ is again $\alpha_p$ (using the Yoneda lemma)? It should be something like $\alpha_p(R) \to (\alpha_p(R) \to \mu_p(R)), x \mapsto (y \mapsto \exp_{p-1}(x + y))$, where $\exp_{p-1}$ is the truncated exponential sequence. My problem is that this isn't a homomorphism.

Definition 1 Let $A$ be a finite commutative group scheme over $k$. Let $\hat{A}$ be the dual vector space to $A$. Define $\Delta : \hat{A} \to \hat{A} \otimes \hat{A}$ to be dual to the multiplication map $A \otimes A \to A$, and define a multiplication on $\hat{A}$ by the dual of the comultiplication of $A$. Then $\hat{A}$ is the Cartier dual.
Application to $\alpha_p$: As a ring, $\alpha_p$ is $k[t]/t^p$. Set $e_i = t^i$, for $0 \leq i \leq p-1$. In this basis, multiplication is given by $e_i e_j = \sum \delta_{i+j}^k e_k$ and comultiplication is given by $\Delta(e_k) = \Delta(t)^k = (t \otimes 1 + 1 \otimes t)^k = \sum \frac{k!}{i! j!} \delta_{i+j}^k e_i \otimes e_j$. (Here $\delta_a^b$ is $1$ if $a=b$ and $0$ otherwise.) Taking $f_i$ to be the basis of $\hat{A}$ with $\langle e_i, f_j\rangle = \delta_i^j i!$, we see that $A \cong \hat{A}$.
Definition 2: For any $k$-scheme $S$, the $S$-points of $\hat{A}$ are $\mathrm{Hom}(A \times S, \mathbb{G}_m \times S)$, with group structure given by multiplication of characters.
Application to $\alpha_p$: By a happy coincidence, I computed $\mathrm{Hom}(\mathbb{G}_a \times S, \mathbb{G}_m \times S)$ in an earlier answer. Every homorphism is of the form $t \mapsto e^{tN}$ for $N$ a nilpotent of $\mathcal{O}(S)$. If we want that morphism to factor through $\alpha_p$, a simple modification of that argument gives a bijection $$\mathrm{Hom}(\mathbb{G}_a \times S, \mathbb{G}_m \times S) \to \{ N \in \mathcal{O}(S) : N^p=0 \}.$$ with the bijection between the two sides again being that $N$ corresponds to the map $N \mapsto e^{tN}$.
Obviously, there is also a bijection between $\{ N \in \mathcal{O}(S) : N^p=0 \}$ and $\alpha_p(S)$. I leave it to you to check the necessary compatibilities for the full Yoneda proof.


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# Two-way Anova Worksheet

Two-way Anova Worksheet
• Page 1
1.
The means of each group of a two-way analysis of variance for two independent variables are plotted and shown in the three graphs. Which of these represents a disordinal interaction? Consider the interaction to be significant.

 a. Both graph 1 and graph 2 b. Graph 1 c. Graph 3 d. Graph 2

#### Solution:

In a disordinal interaction, the lines cross each other and an intersection occurs and the interaction is significant.

In the above 3 graphs, graph 1 satisfies this condition.

So, graph 1 represents a disordinal interaction.

2.
In a two-way ANOVA, variable A has three levels and variable B has two levels. The number of data values in each group is five. Find the degrees of freedom for factor A × B and for the within (error) factor.
 a. 2, 24 b. 4, 3 c. 2, 1 d. 3, 2

#### Solution:

The number of levels of factor A and B are 3 and 2 respectively.

The number of data values in each group is 5.

The degrees of freedom are as follows:
Interaction (A × B): d.f.N = (3 - 1)(2 - 1) = 2
Within(error): d.f.D = 3 · 2 · (5 - 1) = 24
[Interaction (A × B): d.f.N = (a - 1)(b - 1)
Within(error): d.f.D = ab(n - 1)
where a, b are levels of factors A, B respectively.]

3.
In a two-way ANOVA, factor A has six levels and factor B has five levels. The number of data values in each group is seven. Find the degrees of freedom for factor A and factor B.
 a. 7, 6 b. 5, 4 c. 6, 5 d. 20, 180

#### Solution:

The degrees of freedom for factor A and factor B having 6 and 5 levels respectively, are as follows:

Factor A: d.f.N = a - 1 = 6 - 1 = 5

Factor B: d.f.N = b - 1 = 5 - 1 = 4

4.
The means of each group of a two-way analysis of variance for two independent variables are plotted as shown in the graph. What does the graph represent about the interaction of the variables?

 a. Disordinal interaction b. Ordinal interaction c. No interaction

#### Solution:

When there is no significant interaction effect among the variables, the lines in the graph will be parallel or approximately parallel.

The given two lines are parallel to each other, so they represent 'no interaction'.

5.
In a two-way ANOVA, factor A has $a$ levels and factor B has $b$ levels. The number of data values in each group is $n$. Find the degrees of freedom for factor A and factor B.
 a. ($a$ - 1)($b$ - 1), $a$$b$($n$ - 1) b. $a$ - 1, $b$ - 1 c. $a$ + 1, $b$ + 1 d. $a$, $b$

#### Solution:

The degrees of freedom for factor A and factor B having a and b levels respectively, are as follows:
Factor A: d.f.N = a - 1
Factor B: d.f.N = b - 1

6.
In a two-way ANOVA, factor A has $a$ levels and factor B has $b$ levels. The number of data values in each group is $n$. Find the degrees of freedom for factor A × B (Interaction) and for the Within (error) factor.
 a. $a$ - 1, $b$ - 1 b. $a$ + 1, $b$ + 1 c. $a$, $b$ d. ($a$ - 1)($b$ - 1), $a$$b$($n$ - 1)

#### Solution:

The number of levels of factor A and B are a and b respectively.

The number of data values in each group is n.

The degrees of freedom are as follows:
Interaction (A × B): d.f.N = (a - 1)(b - 1)
Within(error): d.f.D = ab(n - 1)

7.
The means of each group of a two-way analysis of variance for two independent variables are plotted and shown in the three graphs. Which of these represents an ordinal interaction? Consider the interaction to be significant.

 a. Graph 2 b. Both graph 1 and graph 2 c. Graph 3 d. Graph 1

#### Solution:

In an ordinal interaction, the lines do not cross each other, nor are they parallel.

In the above 3 graphs, graph 2 satisfies this condition.

So, graph 2 represents an ordinal interaction.

8.
Which of the following are the assumptions for two-way ANOVA?
I. All samples are drawn from normally distributed populations.
II. All populations have a common variance.
III. The groups must be equal in sample size.
IV. Within each group, the observations were sampled randomly and independently of each other.
 a. I and II b. I, II, III, and IV c. I, II, and III d. Only I

#### Solution:

The following are the assumptions for the two-way ANOVA

The populations from which the samples were obtained must be normally distributed or approximately normally distributed.

The samples must be independent.

The variances of the populations from which the samples were selected must be equal.

The groups must be equal in sample size.

I, II, III, and IV are all the assumptions for the two-way ANOVA.

9.
In a medical school a new method of teaching in which professional actors played the roles of patients was introduced. The test scores of male and female students who were taught by either the conventional method or by a new form of training using role-play are shown in the table. Find FA , FB. Is there any difference in the mean test score under (i) Gender (ii) Teaching Method, using a two-way ANOVA at $\alpha$ = 0.05. [A : Gender and B : Teaching Method ]

 a. 57.45, 0.771, (i) no, (ii)yes b. 21.04, 57.45, (i) no, (ii) no c. 21.04, 0.771, (i) yes, (ii) no d. 21.04, 57.45, (i) yes, (ii) yes

#### Solution:

H0: There is no difference in the mean test score depeding on the gender of the student.
H1: There is a difference in the mean test score depeding on the gender of the student.
[Hypotheses for the variable Gender.]

H0: There is no difference in the mean test score depending on the teaching methods.
H1: There is a difference in the mean test score depending on the teaching methods.
[Hypotheses for the variable Teaching Method.]

a = 2, b = 2, and n = 5. The degrees of freedom(d.f.) are
Factor A: d.f.N = a - 1 = 2 - 1 = 1
Factor B: d.f.N = b - 1 = 2 - 1 = 1
Interaction (A × B): d.f.N = (a - 1)(b - 1) = (2 - 1)(2 - 1) = 1
Within(error): d.f.D = ab(n - 1) = 2·2 (5 - 1) = 16

[a, b are levels of factors A, B respectively and n is the number of data values in each group.]

The critical value FA corresponding to α = 0.05, d.f.N = 1, and d.f.D = 16 is 4.49

The critical value FB corresponding to α = 0.05, d.f.N = 1, and d.f.D = 16 is 4.49

The critical value FA × B corresponding to α = 0.05, d.f.N = 1, and d.f.D = 16 is 4.49

SSA = 994.05, SSB = 2714.5, SSA × B = 36.45 and SSW = 756

The mean squares are as follows:
MSA = SS / Aa-1 = 994.05
MSB = SS / Bb-1 = 2714.5
MSA × B = SS / A×B(a-1)(b-1) = 36.45
MSW = SS / Wab(n-1) = 47.25

[Substitute and simplify.]

MSA = 994.05, MSB = 2714.5, MSA × B = 36.45 and MSW = 47.25

The F values are computed as follows:
FA = MS / AMS / W = 21.04
FB = MS / BMS / W = 57.45
FA × B = MS / A×BMS / W = 0.771
[Substitute and simplify.]

Since FA = 21.04 and FB = 57.45 are greater than the critical value 4.49, the null hypotheses concerning the gender of the student and teaching methods should be rejected.

So, it can be concluded that there is a difference in the mean test score of students under gender and also the teaching method.

The ANOVA summary table is shown below.

10.
There are three groups of plants that were exposed to 8 hours, 12 hours and 16 hours of sunlight per day during a given growing period and two kinds of fertilizers (P and Q) were used. The heights of the plants(in feet) after six month are tabulated as follows.
At $\alpha$ = 0.05, find the test values FA , FB. Is there any difference in the mean height of plants depending on the (i) type of fertilizer (ii) duration of exposure to sun using a two-way ANOVA. [ A : Fertilizer and B : Exposure to sun ]

 a. 4.7, 3.9, (i) no, (ii) no b. 30.08, 64.03, (i) no, (ii) no c. 30.08, 64.03, (i) yes, (ii) yes d. 4.7, 3.9, (i) yes, (ii) yes

#### Solution:

H0: There is no difference in the mean height depending on the type of fertilizer used.
H1: There is a difference in the mean height depending on the type of fertilizer used.
[Hypotheses for the variable fertilizer.]

H0: There is no difference in the mean height depending on the duration of exposure to sun.
H1: There is a difference in the mean height depending on the duration of exposure to sun.
[Hypotheses for the variable exposure to sun.]

a = 2, b = 3, and n = 3. The degrees of freedom(d.f.) are
Factor A: d.f.N = a - 1 = 2 - 1 = 1
Factor B: d.f.N = b - 1 = 3 - 1 = 2
Interaction (A × B): d.f.N = (a - 1)(b - 1) = (2 - 1)(3 - 1) = 2
Within(error): d.f.D = ab(n - 1) = 2·3 (3 - 1) = 12
[a, b are levels of factors A, B respectively and n is the number of data values in each group.]

The critical value FA corresponding to α = 0.05, d.f.N = 1, and d.f.D = 12 is 4.7

The critical value FB corresponding to α = 0.05, d.f.N = 2, and d.f.D = 12 is 3.9

The critical value FA × B corresponding to α = 0.05, d.f.N = 2, and d.f.D = 12 is 3.9

SSA = 20.1, SSB = 85.8, SSA × B = 0.444 and SSW = 8

The mean squares are as follows:
MSA = SS / Aa-1 = 20.1
MSB = SS / Bb-1 = 42.89
MSA × B = SS / A×B(a-1)(b-1) = 0.222
MSW = SS / Wab(n-1) = 0.667
[Substitute and simplify.]

MSA = 20.1, MSB = 42.89, MSA × B = 0.222 and MSW = 0.667

The F values are computed as follows:
FA = MS / AMS / W = 30.08
FB = MS / BMS / W = 64.33
FA × B = MS / A×BMS / W = 0.333
[Substitute and simplify.]

Since FA = 30.08 > 4.7, FB = 64.33 > 3.9, the null hypothesis concerning the type of fertilizer, and the duration of exposure to sun should be rejected.

So, it can be concluded that the type of fertilizer does affect the mean height of the plants and also the duration of exposure to sun does affect the mean height of the plants.

The ANOVA summary table is shown below.