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# Phased array ultrasonics

Animation showing the principle of an ultrasonic scanner used in medical ultrasonic imaging. It consists of a beamforming oscillator (TX) that produces an electronic signal consisting of pulses of sine waves oscillating at an ultrasonic frequency, which is applied to an array of ultrasonic transducers (T) in contact with the skin surface that convert the electric signal into ultrasonic waves traveling through the tissue. The timing of the pulses emitted by each transducer is controlled by programmable delay units (φ) that are controlled by a microprocessor control system (C). The moving red lines are the wavefronts of the ultrasonic waves from each transducer. The wavefronts are spherical, but they combine (superpose) to form plane waves, creating a beam of sound traveling in a specific direction. Since the pulse from each transducer is progressively delayed going up the line, each transducer emits its pulse after the one below it. This results in a beam of sound waves emitted at an angle (θ) to the array. By changing the pulse delays, the computer can scan the beam of ultrasound in a raster pattern across the tissue. Echoes reflected by different density tissue, received by the transducers, build up an image of the underlying structures.
Weld examination by phased array. TOP: The phased array probe emits a series of beams to flood the weld with sound. BOTTOM: The flaw in the weld appears as a red indication on the instrument screen.

Phased array ultrasonics (PA) is an advanced method of ultrasonic testing that has applications in medical imaging and industrial nondestructive testing. Common applications are to noninvasively examine the heart or to find flaws in manufactured materials such as welds. Single-element (non-phased array) probes, known technically as monolithic probes, emit a beam in a fixed direction. To test or interrogate a large volume of material, a conventional probe must be physically scanned (moved or turned) to sweep the beam through the area of interest. In contrast, the beam from a phased array probe can be focused and swept electronically without moving the probe. The beam is controllable because a phased array probe is made up of multiple small elements, each of which can be pulsed individually at a computer-calculated timing. The term phased refers to the timing, and the term array refers to the multiple elements. Phased array ultrasonic testing is based on principles of wave physics, which also have applications in fields such as optics and electromagnetic antennae.

## Principle of operation

The PA probe consists of many small ultrasonic transducers, each of which can be pulsed independently. By varying the timing, for instance by pulsing the elements one by one in sequence along a row, a pattern of constructive interference is set up that results in a beam at a set angle. In other words, the beam can be focused and steered electronically. The beam is swept like a search-light through the tissue or object being examined, and the data from multiple beams are put together to make a visual image showing a slice through the object.

## Phased array use in the industry

Phased array is widely used for nondestructive testing (NDT) in several industrial sectors, such as construction, pipelines, and power generation. This method is an advanced NDT method that is used to detect discontinuities i.e. cracks or flaws and thereby determine component quality. Due to the possibility to control parameters such as beam angle and focal distance, this method is very efficient regarding the defect detection and speed of testing.[1] Apart from detecting flaws in components, phased array can also be used for wall thickness measurements in conjunction with corrosion testing.[2][3] Phased array can be used for the following industrial purposes:

• Inspection of welds[4]
• Thickness measurements
• Corrosion inspection
• Flaw detection
• Rolling stock inspection (wheels and axles)

## Features of phased array

At a construction site, a technician tests a pipeline weld for defects using an ultrasonic phased array instrument. The scanner, which consists of a frame with magnetic wheels, holds the probe in contact with the pipe by a spring. The wet area is the ultrasonic couplant that allows the sound to pass into the pipe wall.
• The method most commonly used for medical ultrasonography.
• Multiple probe elements produce a steerable and focused beam.[5]
• Focal spot size depends on probe active aperture (A), wavelength (γ) and focal length (F).[6] Focusing is limited to the near field of the phased array probe.
• ${\displaystyle {\text{Focal spot size}}=F\lambda /A}$
• ${\displaystyle {\text{Near Field}}=A^{2}/4\lambda }$
• Produces an image that shows a slice through the object.
• Compared to conventional, single-element ultrasonic testing systems, PA instruments and probes are more complex and expensive.
• In industry, PA technicians require more experience and training than conventional UT technicians.

## Standards

European Committee for Standardization (CEN)
• prEN 16018, Non destructive testing - Terminology - Terms used in ultrasonic testing with phased arrays
• ISO/WD 13588, Non-destructive testing of welds – Ultrasonic testing – Use of (semi-) automated phased array technology

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# Chapter R - Review of Basic Concepts - R.1 Sets - R.1 Exercises: 97

$(M\cap N) \cup R = R$

#### Work Step by Step

RECALL: (1) The symbol $\cup$ represents union of sets which lists down in a set the combined elements of the given sets. (2) The symbol $\cap$ represents intersection of sets which lists down in a set the elements common to the given sets. Use (2) to obtain: $M \cap N = \emptyset$ Use (1) above, and the fact that the empty set has no elements, to obtain: $(M\cap N) \cup R = \emptyset \cup R = R$

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.


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# 0.3 Modelling corruption  (Page 10/11)

 Page 10 / 11

Thus, the impulse response describes how a system behaves directly in time, while the frequency responsedescribes how it behaves in frequency. The two descriptions are intimately related because the frequency response is theFourier transform of the impulse response. This will be used repeatedly in [link] to design filters for the manipulation (augmentation or removal)of specified frequencies.

In Exercise  [link] , a system was defined to have an impulse response that is a sinc function.The Fourier transform of a sinc function in time is a rect function in frequency [link] . Hence, the frequency response of the system is a rectangle that passes all frequencies below ${f}_{c}=1/T$ and removes all frequencies above (i.e., the system is a lowpass filter).

M atlab can help to visualize the relationship between the impulse response and the frequency response.For instance, the system in convolex.m is defined via its impulse response, which is a decaying exponential. [link] shows its output when the input is a simple sum of delta functions, andExercise  [link] explores the output when the input is a white noise. In freqresp.m , the behavior of this system is explained by looking at its frequency response.

Ts=1/100; time=10;             % sampling interval and total time t=0:Ts:time;                   % create time vectorh=exp(-t);                     % define impulse response plotspec(h,Ts)                 % find and plot frequency response freqresp.m numerical example of impulse and frequency response (download file)

The output of freqresp.m is shown in [link] . The frequency response of the system (which is just the magnitude spectrum ofthe impulse response) is found using plotspec.m . In this case, the frequency response amplifies low frequenciesand attenuates other frequencies more as the frequency increases. This explains, for instance, why the output of theconvolution in Exercise  [link] contained (primarily) lower frequencies, as evidenced by the slower undulations in time.

Suppose a system has an impulse response that is a $\text{sinc}$ function. Using freqresp.m , find the frequency response of the system. What kind of filter does this represent?Hint: center the $\text{sinc}$ in time; for instance, use h=sinc(10*(t-time/2)) .

Suppose a system has an impulse response that is a $sin$ function. Using freqresp.m , find the frequency response of the system. What kind of filter does this represent?Can you predict the relationship between the frequency of the sine wave and the location of the peaks in the spectrum?Hint: try h=sin(25*t) .

Create a simulation (analogous to convolex.m ) that inputs white noise into a system with impulse responsethat is a $\text{sinc}$ function (as in Exercise  [link] ). Calculate the spectra of the input and outputusing plotspec.m . Verify that the system behaves as suggested by thefrequency response in Exercise  [link] .

Create a simulation (analogous to convolex.m ) that inputs white noise into a system with impulse responsethat is a $\text{sin}$ function (as in Exercise  [link] ). Calculate the spectra of the input and outputusing plotspec.m . Verify that the system behaves as suggested by thefrequency response in Exercise  [link] .

can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
a perfect square v²+2v+_
kkk nice
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
rolling four fair dice and getting an even number an all four dice
Kristine 2*2*2=8
Differences Between Laspeyres and Paasche Indices
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
is it 3×y ?
J, combine like terms 7x-4y
im not good at math so would this help me
yes
Asali
I'm not good at math so would you help me
Samantha
what is the problem that i will help you to self with?
Asali
how do you translate this in Algebraic Expressions
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
what is nanomaterials and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
Got questions? Join the online conversation and get instant answers!


text:
2011-03-31 11:03

# 从Zend Studio 8运行phpUnit测试套件时出现奇怪的错误

Zend Studio 8 has phpUnit integration and a dedicated view, which is great, so I would like to use it... but:

When I run my phpunit.xml on the command line, the tests are executed just fine, when I try to run phpunit.xml in Zend Studio as unit test, I get the following fatal error:

Fatal error: Uncaught exception 'PHPUnit_Framework_Error' with message 'Assigning the return value of new by reference is deprecated' in C:\Program Files (x86)\Zend\ZendServer\bin\PEAR\PEAR\Config.php:650

Stack trace:

#0 C:\Program Files (x86)\Zend\ZendServer\bin\PEAR\PEAR\RunTest.php(22): ZendPHPUnitErrorHandler->handle(8192, 'Assigning the r...', 'C:\Program File...', 650, Array)
#1 C:\Program Files (x86)\Zend\ZendServer\bin\PEAR\PEAR\RunTest.php(22): require_once()
#2 C:\Program Files (x86)\Zend\ZendServer\bin\PEAR\PHPUnit\Extensions\PhptTestCase.php(49): require_once('C:\Program File...')
#3 C:\Program Files (x86)\Zend\Zend Studio - 8.0.0\plugins\com.zend.php.phpunit_8.0.0.v20101001-0100esources\ZendPHPUnit.php(103): require_once('C:\Program File...')
#4 {main} thrown in C:\Program Files (x86)\Zend\ZendServer\bin\PEAR\PEAR\Config.php on line 650

I really don't have a clue what that means... why should some deprecated code in the PEAR config keep my tests from running?

My colleague who has neither PEAR nor phpunit installed, can run unit tests from Zend Studio 8 just fine but he can't run any from the CLI. So it seems that Zend Studio not only integrates with phpUnit but comes bundled with a version of it. I wanted the newest version and that's why I installed PEAR and then via PEAR I installed phpUnit. As a consequence, the phpUnit library of Zend Studio was outdated and I had to replace the file 'ZendPHPUnit.php' in the Zend Studio plugin folder with a version of a Zend Forum user. This lead to correct inclusion of the phpUnit classes and the current problem. [/edit]

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#### 1条回答

• dongtuoleng8624 10年前

Line 650 in PEAR\Config.php does

``````\$this->_registry['default'] = &new PEAR_Registry(
\$this->configuration['default']['php_dir']
);
``````

This will raise an `E_DEPRECATED` message depending on whether you have enabled error_reporting to include that level. The error handlers of `PHPUnit` and `ZendPHPUnit` will convert those to `PHPUnit_Framework_Exceptions`.

Because PHP will raise `E_DEPRECATED` already at compile time, the only way to suppress these errors is in php.ini. My assumption is, that your PHPUnit you run from CLI is using a different php.ini than that your Zend Studio and in that php.ini the error level is not low enough to raise `E_DEPRECATED`.

If you can provide a reproducable example, we might be able to make more sense of it. In any case, try to update your PEAR installation to the latest version.

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# Тренировочный CES тест по Integrated Survey Program (ISP)

Добро пожаловать на страницу, где вы можете пройти тест CES онлайн по теме “Integrated Survey Program”. Данная практика поможет вам, как морскому специалисту, подготовиться к сдаче экзаменов, получению сертификата компетентности или аттестата.

Выберите режим, в котором хотите пройти CES тест:

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Coating is applied in ballast tanks for protection of steel against corrosion. Which type of coating is not considered by the class to be an efficient corrosion protection system?
Semi-hard coating.
Soft coating.
Hard coating.
Принять ответ
During inspection of the aft peak, where are cracks unlikely to be found?
Vibration crack at connection between floor and hull plating.
Indentation due to bottom slamming.
Fatigue cracks at the top of tanks.
Принять ответ
For a simply supported beam with a concentrated load at middle of the span, the maximum bending moment is higher:
At middle of span.
At the end supports.
At the quarter length of span.
Принять ответ
For efficient cathodic protection of ballast tanks which of the following does not apply?
Anodes are to be properly attached.
Anodes are to be submerged in water.
Anodes are to be painted.
Consumed anodes are to be replaced.
Принять ответ
How does the class notify deficiencies?
Memo to the owner (MO) covering the deficiency.
Condition of Class (CC) for owner in case of minor deficiency.
Memo to the surveyor.
Write a report only.
Принять ответ
In case securing equipment’s on board a Container ship is worn beyond use and new equipment has to be ordered. In that connection, please mark which of the below statement that is correct:
The equipment should be as specified in the container securing manual.
The equipment should be certified by the classification society.
The equipment should be of similar type and size.
Принять ответ
In connection with loading of containers, the correct information about the weight and location of containers are crucial for ships stability and strength, as well as the stowing and securing of them. Where would you look for information on how to provide proper securing of the containers?
In the ISM system.
In the Container securing manual.
Принять ответ
In the bottom plating in a ballast tank, the thickness diminution may be due to, general corrosion, pitting corrosion, grooving corrosion or erosion corrosion. Where would you look for pitting corrosion?
On horizontal flanges.
Under suction heads for ballast line.
In way of cut-outs for drainage.
Принять ответ
It is very important to be familiar with the technical terminology when dealing with container ships. Is a floor a part of the:
Ship.
Deck structure.
Double bottom structure.
Принять ответ
Maintenance of loose stowing and securing equipment is very important for the safety of the cargo during transportation overseas, such as (please indicate which is incorrect):
Semi-automatic twist locks.
Manual twist locks.
Sliding base sockets.
Cell guides.
Принять ответ
On a container ship, which part of the ship represents the web of hull girder?
The longitudinal bulkheads and the ship sides.
Принять ответ
Pitting corrosion is most commonly found on horizontal surface in:
Ballast tanks.
Cargo holds.
Heavy fuel oil tanks.
Принять ответ
Regarding inspection of the cargo holds, where is the least likely location for finding cracks?
At the cell guides.
At corner of stringer decks.
At hatch corner.
Принять ответ
Reporting in PIMS of the results from the inspection is very important Checklist and Narrative reports. Consider you have carried out an inspection in a ballast tank. The tank was found in order except for an indention in the ship side, which has caused a slight buckling of two side longitudinal. How would you report this?
Fill in the checklist, take some photo and make a narrative report where you describe the indent and location.
Fill in the checklist and take some photos.
Fill in the checklist.
Принять ответ
Similar to Classification Society, the Owner also has acceptance criteria to be used when deficiencies are found. These acceptance criteria should normally be somewhat stricter than the class. Where do you find Owner’s acceptance criteria?
In the working guidance by chief officer.
In the PIMS manual on board.
In the ISM system Reg. 10.
Принять ответ
Some areas of minimal attention with respect to coating, during inspection of ballast tanks are:
Inside edge of cut-outs.
Edge part of stiffeners.
On the web of stiffeners.
Принять ответ
The class society has acceptance criteria for different types of deficiencies, such as: which two alternatives are not correct?
Coating breakdown.
Corrosion.
Cracks.
Buckling.
Принять ответ
The coating condition in the ballast tanks may be rated as: indicate which of these coating conditions require annual follow up of the ballast tank in question by the class surveyor?
Good.
Fair.
Poor.
Принять ответ
The fore ship may suffer damage due to impact from the sea. Which structural element will then suffer the most significant damage?
The shell plating.
The stiffeners from the shell plating.
The web frames and girders.
Принять ответ
The function of a loaded beam is to transfer loads towards the end supports (in shear). Which part of beam has this function?
The flange.
The web.
Both web and flange.
Принять ответ
The hull girder of a container ship is as other ships subject to tension and compression due to the loads from the lightweight and cargo, and the counteracting buoyancy loads. Since the container ship normally operates in a hogging loading condition, where would you expect tension stresses in the hull girder?
In the deck area.
In the bottom area.
Принять ответ
The pressure testing to prove strength and tightness is not required for which of the following items?
Piping systems.
Tank boundaries.
Cargo holds.
Принять ответ
This is another example of good condition but what do we call the deficiency you can see on the coating? Click on the right option:
Delamination.
Cracking.
Blistering.
Принять ответ
What is assumed by the class for retention of class?
Owner/manager to ensure adequate handling and manning of the ship.
Owner/manager to ensure adequate maintenance, handling and manning of the ship.
Owner to ensure that the vessel is presented for surveys.
Принять ответ
What is the Integrated Survey Program (ISP)?
A maintenance program for cargo ships.
A hull inspection and maintenance program for container ships.
A maintenance system for machinery on container ships.
A survey program, which is integrated in the maintenance system on board.
Принять ответ
What is the classification in the SiO phase?
To verify that the standard of ship in operation (SiO) acceptable after conversion.
Verify that the standard of the ship is acceptable after damage repairs.
To verify that the standard of the ship is acceptable through a system periodical surveys.
To make sure that the vessel is well maintained.
Принять ответ
What is the condition for the 5-year validity of the class Certificate?
That surveys are carried out every 2,5 years and with satisfactory result.
That survey is carried out every 5 years with satisfactory result.
That annual surveys are carried out with satisfactory result.
That all condition of class (CC) are carried out.
Принять ответ
When doing inspections of the upper deck area, where would you not find cracks immediately?
At the hatch coaming stays.
At the lashing bridges connection to the deck.
At the middle of hatch end coaming.
At the hatch end coaming brackets.
Принять ответ
When looking for cracks in the ship structure like for instance the bottom structure, would you then first focus on:
The middle of the plate.
At the connection between plate and stiffeners.
At the connection between stiffeners and girders.
Принять ответ
Where would you look for cracks in the side ballast tanks in the first place?
At the tank top longitudinal connection to the web frames.
At ship side longitudinal connection to the web frames below waterline.
At the longitudinal bulkhead longitudinal connection to the web frames.
Принять ответ
Which is outside the scope of Planned Inspection and Maintenance System (PIMS)?
Dry docking/bottom survey.
Survey of ballast tanks with Fair coating.
Survey of container fittings.
Piping in ballast tanks.
Принять ответ
Which is required from the crew on board for practising of ISP?
Carry out inspections in accordance with the maintenance system on board, which has been accepted by the class.
Carry out inspections when deemed necessary.
Carry out inspections and report in accordance with the maintenance system, which has been accepted by the class.
Carry out inspections and submit report to the class.
Принять ответ
Which of the following items is not mandatory in the PIMS?
Cargo holds.
Deck and hatches.
Safety Equipment.
Tanks.
Piping system outside machinery space.
Принять ответ
Which of the following precautions while pressure testing of a ballast tank is most important?
Filling should be to the top of tank.
Venting done through the air pipes.
Hatch covers and/or manhole covers should be kept open during pressure testing.
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# How to prove this identity of summation of ramanujan?

In the first book of ramanujan him find this identity:

Let k , n $\in\mathbb{N}$* and define $A_k=3^k\left(n+\frac{1}{2}\right)-\frac{1}{2}$. the if r is a positive interger

$$\sum_{k=n+1}^{A_r}\frac{1}{k}=r+2\sum_{k=0}^{r-1}\left(r-k\right)\sum_{j=A_{k-1}+1}^{A_k}\frac{1}{\left(3j\right)^3-3j}$$ where $A_{-1}=0$

I want to know a demonstration for that identity, because he uses it to demonstrate his very popular affirmation:

$$\sum_{n=1}^{1000}\frac{1}{n}=7\frac{1}{2} "very\ nearly"$$

First we need the result from entry $2$ of the reference. Lets start with the parial fractions \begin{eqnarray*} \frac{1}{(3j)^3-3j} =\frac{1/2}{3j-1}-\frac{1}{3j}+\frac{1/2}{3j+1}. \end{eqnarray*} So \begin{eqnarray*} 1+2 \sum_{j=1}^{n} \frac{1}{(3j)^3-3j} =1+ \sum_{j=1}^{n} \left( \frac{1}{3j-1}\color{red}{+\frac{1}{3j}}+\frac{1}{3j+1} \color{red}{-\frac{3}{3j}} \right) \\ = \sum_{j=1}^{3n+1} \frac{1}{j} -\sum_{j=1}^{n} \frac{1}{j} = \sum_{j=n+1}^{3n+1} \frac{1}{j} . \end{eqnarray*} Now with $A_k=3^k n + \frac{1}{2} (3^k-1)$ then $A_{k+1}=3A_k+1$, the above result gives \begin{eqnarray*} \sum_{j=A_k +1}^{A_{k+1}=3A_k+1} \frac{1}{j} =1+ 2 \sum_{j=1}^{A_k} \frac{1}{(3j)^3-3j}. \end{eqnarray*} Now sum the above formula from $k=0,1,\cdots, r-1$ and we have \begin{eqnarray*} \sum_{j=n +1}^{A_{r}} \frac{1}{j} =r+ 2 \sum_{k=0}^{r-1} \sum_{j=1}^{A_k} \frac{1}{(3j)^3-3j}. \end{eqnarray*} Note that the inner sums on the RHS are contained within next sum, so this can be rearranged to the result \begin{eqnarray*} \sum_{j=n +1}^{A_{r}} \frac{1}{j} =r+ 2 \sum_{k=0}^{r-1} (r-k) \sum_{j=A_{k-1}+1}^{A_k} \frac{1}{(3j)^3-3j}. \end{eqnarray*}
• RHS $=$ right hand side. – Donald Splutterwit Oct 29 '17 at 21:55


text:
# Math Help - Latex

1. ## Latex

Hello people!

How do I insert mathematical symbols in a forum? ie what tool to use? Latex install touch me?

For example to other forums is a tool called TexAide in which one writes as in Mathtype and then paste the formulas in the forum

the glue comes out is the code is placed between [Text] and are the images

how is here?

2. Originally Posted by Dogod11
Hello people!

How do I insert mathematical symbols in a forum? ie what tool to use? Latex install touch me?

For example to other forums is a tool called TexAide in which one writes as in Mathtype and then paste the formulas in the forum

the glue comes out is the code is placed between [Text] and are the images

how is here?
We use  tags here.

3. Hey, but that does not answer my question because I was there and I could not come up with the solution,

TexAide served or not served.

Or is there another program?

Thanks and greetings!

4. Originally Posted by Dogod11
Hey, but that does not answer my question because I was there and I could not come up with the solution,

TexAide served or not served.

Or is there another program?

Thanks and greetings!
I did answer your question. you basically asked how LaTeX works here. And I told you. you need no tool from which to paste code. it is built into the forum. just type your code between  tags and the symbols will show up. e.g. If you type $$\int x^2 ~dx$$ you get $\int x^2 ~dx$

5. O.K.


text:
• # question_answer Earth is orbiting the sun continuously and rotating around its axis. What is the earth's axial tilt? A) $50{}^\circ$               B) $23.4{}^\circ$ C) $25.3{}^\circ$     D)        $100{}^\circ$ E) None of these

Correct Answer: B

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# Set of real numbers | TOMATO B.Stat Objective 714

Try this problem from I.S.I. B.Stat Entrance Objective Problem based on Set of real numbers.

## Set of real Numbers (B.Stat Objective Question )

The set of all real numbers x satisfying the inequality $x^{3}(x+1)(x-2) \geq 0$ can be written as

• [-1,infinity)
• none of these
• [2,infinity)
• [0,infinity)

### Key Concepts

Equation

Roots

Algebra

B.Stat Objective Problem 714

Challenges and Thrills of Pre-College Mathematics by University Press

## Try with Hints

First hint

$x^{3}(x+1)(x-2) \geq 0$

case I $x^{3}(x+1)(x-2) \geq 0$

or, $0 \leq x, -1 \leq x, 2 \leq x$ which is first inequation

case II $x^{3} \geq 0, (x+1) \leq 0, (x-2) \leq 0$

or, $x \geq 0, x \leq -1, x \leq 2$ which is second equation

Second Hint

case III $x^{3} \leq 0, (x+1) \leq 0, (x-2) \geq 0$

or, $x \leq 0 x \leq -1, 2 \leq x$ which is third equation

case IV $x^{3} \leq 0, (x+1) \geq 0, (x-2) \leq 0$

or, $x \leq0, x \geq -1, x \leq 2$ which is fourth equation

Final Step

Combining we get $x^{3}(x+1)(x-2) \geq 0$ satisfy if $x\in$ $[-1,0] \bigcup [2,infinity)$

or, answer option none of these

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Projector of $\mathbb{R}^{n}$ of rank $p$

Let $f \in \mathcal{L}(\mathbb{R}^{n},\mathbb{R}^{p})$ and $g \in \mathcal{L}(\mathbb{R}^{p},\mathbb{R}^{n})$. We assume that $g \circ f$ is a projector ($g \circ f$ satisfies : $\big( g \circ f \big)^{2} = g \circ f$) of $\mathbb{R}^{n}$ of rank $p$. I would like to prove that $f \circ g = \mathrm{Id}_{\mathbb{R}^{p}}$ .

I have noticed that $\mathrm{Im}(g \circ f) \subset \mathrm{Im}(g)$, which ensures that $\mathrm{rg}(g) \geq \mathrm{rg}(g \circ f) = p$. By the rank-nullity theorem for $g$ : $p = \dim \mathrm{ker}(g) + \mathrm{rg}(g)$. It follows that $\dim \mathrm{ker}(g) = 0$. Therefore, $\mathrm{ker}(g) = \lbrace 0 \rbrace$. But from that, I don't see how I could conclude that $f \circ g = \mathrm{Id}_{\mathbb{R}^{p}}$. I would like to prove that $\mathrm{ker}(f \circ g) = \lbrace 0 \rbrace$, but I can't make a link with $g \circ f$.

If you prove $Ker(f\circ g)=\{0\}$. Then by rank-nullity thm $f\circ g$ is an isomorphism. Now $(f\circ g)^3=(f\circ g)^2$ hence $f\circ g =id$, As$[g \circ f \circ g \circ f=g \circ f \Rightarrow f\circ g \circ f\circ g \circ f\circ g = f\circ g \circ f\circ g]$

You have established that $g$ is injective, and has as image the subspace $V=\operatorname{Im}(g\circ f)$. The fact that $(g\circ f)^2=g\circ f$ means that $(g\circ f)(v)=v$ for all $v\in V$, so the restriction $(g\circ f)|_V$ equals $\def\I{\operatorname{Id}}\I_V$. But then $g$ gives an isomorphism $\Bbb R^p\to V$ since $\dim(V)=p$, and $f|_V:V\to\Bbb R^p$ is its inverse. It follows that $f\circ g=f|_V\circ g=\I_{\Bbb R^p}$, QED.


text:
# American Institute of Mathematical Sciences

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## Chaotic oscillations of linear hyperbolic PDE with variable coefficients and implicit boundary conditions

 1 Department of Mathematics, South China University of Technology, Guangzhou, 510640, China 2 Department of Mathematics and Big Data, Foshan University, Foshan, 528000, China

* Corresponding author: Qigui Yang

Received  March 2019 Revised  November 2019 Published  April 2020

In this paper, the chaotic oscillations of the initial-boundary value problem of linear hyperbolic partial differential equation (PDE) with variable coefficients are investigated, where both ends of boundary conditions are nonlinear implicit boundary conditions (IBCs). It separately considers that IBCs can be expressed by general nonlinear boundary conditions (NBCs) and cannot be expressed by explicit boundary conditions (EBCs). Finally, numerical examples verify the effectiveness of theoretical prediction.

Citation: Qigui Yang, Qiaomin Xiang. Chaotic oscillations of linear hyperbolic PDE with variable coefficients and implicit boundary conditions. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020335
##### References:
 [1] G. Chen, S.-B. Hsu and J. X. Zhou, Chaotic vibrations of the one-dimensional wave equation due to a self-excitation boundary condition. Ⅰ: Controlled hysteresis, Trans. Amer. Math. Soc., 350 (1998), 4265-4311.  doi: 10.1090/S0002-9947-98-02022-4.  Google Scholar [2] G. Chen, S.-B. Hsu and J. X. Zhou, Chaotic vibrations of the one-dimensional wave equation due to a self-excitation boundary condition. Ⅱ: Energy injection, period doubling and homoclinic orbits, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 8 (1998), 423-445.  doi: 10.1142/S0218127498000280.  Google Scholar [3] G. Chen, S.-B. Hsu and J. X. Zhou, Chaotic vibrations of the one-dimensional wave equation due to a self-excitation boundary condition. Ⅲ: Natural hysteresis memory effects, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 8 (1998), 447-470.  doi: 10.1142/S0218127498000292.  Google Scholar [4] G. Chen, S.-B. Hsu and J. X. Zhou, Nonisotropic spatiotemporal chaotic vibration of the wave equation due to mixing energy transport and a van der Pol boundary condition, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 12 (2002), 535-559.  doi: 10.1142/S0218127402004504.  Google Scholar [5] G. Chen, T. W. Huang and Y. Huang, Chaotic behavior of interval maps and total variations of iterates, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 14 (2004), 2161-2186.  doi: 10.1142/S0218127404010540.  Google Scholar [6] G. Chen, B. Sun and T. W. Huang, Chaotic oscillations of solutions of the Klein-Gordon equation due to inbalance of distributed and boundary energy flows, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 24 (2014), 1430021, 19 pp. doi: 10.1142/S0218127414300213.  Google Scholar [7] X. P. Dai, Chaotic dynamics of continuous-time topological semiflow on Polish spaces, J. Differential Equations, 258 (2015), 2794-2805.  doi: 10.1016/j.jde.2014.12.027.  Google Scholar [8] M. W. Hirsch, S. Smale and R. L. Devaney, Differential Equations, Dynamical Systems, and an Introduction to Chaos, Second edition, Pure and Applied Mathematics (Amsterdam), 60. Elsevier/Academic Press, Amsterdam, 2004. doi: 10.1016/C2009-0-61160-0.  Google Scholar [9] C.-C. Hu, Chaotic vibrations of the one-dimensional mixed wave system, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 19 (2009), 579-590.  doi: 10.1142/S0218127409023202.  Google Scholar [10] Y. Huang, Growth rates of total variations of snapshots of the 1D linear wave equation with composite nonlinear boundary reflection relations, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 1183-1195.  doi: 10.1142/S0218127403007138.  Google Scholar [11] Y. Huang, J. Luo and Z. L. Zhou, Rapid fluctuations of snapshots of one-dimensional linear wave equations with a van der Pol nonlinear boundary condition, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 567-580.  doi: 10.1142/S0218127405012223.  Google Scholar [12] L. L. Li, Y. L. Chen and Y. Huang, Nonisotropic spatiotemporal chaotic vibrations of the one-dimensional wave equation with a mixing transport term and general nonlinear boundary condition, J. Math. Phys., 51 (2010), 102703, 22 pp. doi: 10.1063/1.3486070.  Google Scholar [13] L. L. Li, Y. Huang, G. Chen and T. W. Huang, Chaotic oscillations of second order linear hyperbolic equations with nonlinear boundary conditions: A factorizable but noncommutative case, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 25 (2015), 1530032, 20 pp. doi: 10.1142/S0218127415300323.  Google Scholar [14] L. L. Li, T. W. Huang and X. Y. Huang, Chaotic oscillations of the 1D wave equation due to extreme imbalance of self-regulations, J. Math. Anal. Appl., 450 (2017), 1388-1400.  doi: 10.1016/j.jmaa.2017.01.095.  Google Scholar [15] Y. C. Li, Chaos in Partial Differential Equations, Graduate Series in Analysis. International Press, omerville, MA, 2004.  doi: 10.1002/cnm.650.  Google Scholar [16] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2$^nd$ edition, Texts in Applied Mathematics, 2. Springer-Verlag, New York, 2003. doi: 10.1063/1.4822950.  Google Scholar [17] Q. M. Xiang and Q. G. Yang, Nonisotropic chaotic oscillations of the wave equation due to the interaction of mixing transport term and superlinear boundary condition, J. Math. Anal. Appl., 462 (2018), 730-746.  doi: 10.1016/j.jmaa.2018.02.031.  Google Scholar [18] Q. M. Xiang and Q. G. Yang, Chaotic oscillations of a linear hyperbolic PDE with a general nonlinear boundary condition, J. Math. Anal. Appl., 472 (2019), 94-111.  doi: 10.1016/j.jmaa.2018.10.083.  Google Scholar [19] Q. G. Yang, G. R. Jiang and T. S. Zhou, Chaotification of linear impulsive differential systems with applications, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 22 (2012), 1250297, 12 pp. doi: 10.1142/S0218127412502975.  Google Scholar [20] Q. G. Yang and Q. M. Xiang, Existence of chaotic oscillations in second-order linear hyperbolic PDEs with implicit boundary conditions, J. Math. Anal. Appl., 457 (2018), 751-775.  doi: 10.1016/j.jmaa.2017.08.018.  Google Scholar [21] Z. B. Yin and Q. G. Yang, Distributionally scrambled set for an annihilation operator, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 25 (2015), 1550178, 13 pp. doi: 10.1142/S0218127415501783.  Google Scholar [22] Z. B. Yin and Q. G. Yang, Generic distributional chaos and principal measure in linear dynamics, Ann. Pol. Math., 118 (2016), 71-94.  doi: 10.4064/ap3908-9-2016.  Google Scholar [23] Z. B. Yin and Q. G. Yang, Distributionally n-chaotic dynamics for linear operators, Rev. Mat. Complut., 31 (2018), 111-129.  doi: 10.1007/s13163-017-0226-5.  Google Scholar [24] Z. B. Yin and Q. G. Yang, Distributionally n-scrambled set for weighted shift operators, J. Dyn. Control Syst., 23 (2017), 693-708.  doi: 10.1007/s10883-017-9359-6.  Google Scholar

show all references

##### References:
 [1] G. Chen, S.-B. Hsu and J. X. Zhou, Chaotic vibrations of the one-dimensional wave equation due to a self-excitation boundary condition. Ⅰ: Controlled hysteresis, Trans. Amer. Math. Soc., 350 (1998), 4265-4311.  doi: 10.1090/S0002-9947-98-02022-4.  Google Scholar [2] G. Chen, S.-B. Hsu and J. X. Zhou, Chaotic vibrations of the one-dimensional wave equation due to a self-excitation boundary condition. Ⅱ: Energy injection, period doubling and homoclinic orbits, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 8 (1998), 423-445.  doi: 10.1142/S0218127498000280.  Google Scholar [3] G. Chen, S.-B. Hsu and J. X. Zhou, Chaotic vibrations of the one-dimensional wave equation due to a self-excitation boundary condition. Ⅲ: Natural hysteresis memory effects, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 8 (1998), 447-470.  doi: 10.1142/S0218127498000292.  Google Scholar [4] G. Chen, S.-B. Hsu and J. X. Zhou, Nonisotropic spatiotemporal chaotic vibration of the wave equation due to mixing energy transport and a van der Pol boundary condition, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 12 (2002), 535-559.  doi: 10.1142/S0218127402004504.  Google Scholar [5] G. Chen, T. W. Huang and Y. Huang, Chaotic behavior of interval maps and total variations of iterates, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 14 (2004), 2161-2186.  doi: 10.1142/S0218127404010540.  Google Scholar [6] G. Chen, B. Sun and T. W. Huang, Chaotic oscillations of solutions of the Klein-Gordon equation due to inbalance of distributed and boundary energy flows, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 24 (2014), 1430021, 19 pp. doi: 10.1142/S0218127414300213.  Google Scholar [7] X. P. Dai, Chaotic dynamics of continuous-time topological semiflow on Polish spaces, J. Differential Equations, 258 (2015), 2794-2805.  doi: 10.1016/j.jde.2014.12.027.  Google Scholar [8] M. W. Hirsch, S. Smale and R. L. Devaney, Differential Equations, Dynamical Systems, and an Introduction to Chaos, Second edition, Pure and Applied Mathematics (Amsterdam), 60. Elsevier/Academic Press, Amsterdam, 2004. doi: 10.1016/C2009-0-61160-0.  Google Scholar [9] C.-C. Hu, Chaotic vibrations of the one-dimensional mixed wave system, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 19 (2009), 579-590.  doi: 10.1142/S0218127409023202.  Google Scholar [10] Y. Huang, Growth rates of total variations of snapshots of the 1D linear wave equation with composite nonlinear boundary reflection relations, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 1183-1195.  doi: 10.1142/S0218127403007138.  Google Scholar [11] Y. Huang, J. Luo and Z. L. Zhou, Rapid fluctuations of snapshots of one-dimensional linear wave equations with a van der Pol nonlinear boundary condition, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 567-580.  doi: 10.1142/S0218127405012223.  Google Scholar [12] L. L. Li, Y. L. Chen and Y. Huang, Nonisotropic spatiotemporal chaotic vibrations of the one-dimensional wave equation with a mixing transport term and general nonlinear boundary condition, J. Math. Phys., 51 (2010), 102703, 22 pp. doi: 10.1063/1.3486070.  Google Scholar [13] L. L. Li, Y. Huang, G. Chen and T. W. Huang, Chaotic oscillations of second order linear hyperbolic equations with nonlinear boundary conditions: A factorizable but noncommutative case, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 25 (2015), 1530032, 20 pp. doi: 10.1142/S0218127415300323.  Google Scholar [14] L. L. Li, T. W. Huang and X. Y. Huang, Chaotic oscillations of the 1D wave equation due to extreme imbalance of self-regulations, J. Math. Anal. Appl., 450 (2017), 1388-1400.  doi: 10.1016/j.jmaa.2017.01.095.  Google Scholar [15] Y. C. Li, Chaos in Partial Differential Equations, Graduate Series in Analysis. International Press, omerville, MA, 2004.  doi: 10.1002/cnm.650.  Google Scholar [16] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2$^nd$ edition, Texts in Applied Mathematics, 2. Springer-Verlag, New York, 2003. doi: 10.1063/1.4822950.  Google Scholar [17] Q. M. Xiang and Q. G. Yang, Nonisotropic chaotic oscillations of the wave equation due to the interaction of mixing transport term and superlinear boundary condition, J. Math. Anal. Appl., 462 (2018), 730-746.  doi: 10.1016/j.jmaa.2018.02.031.  Google Scholar [18] Q. M. Xiang and Q. G. Yang, Chaotic oscillations of a linear hyperbolic PDE with a general nonlinear boundary condition, J. Math. Anal. Appl., 472 (2019), 94-111.  doi: 10.1016/j.jmaa.2018.10.083.  Google Scholar [19] Q. G. Yang, G. R. Jiang and T. S. Zhou, Chaotification of linear impulsive differential systems with applications, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 22 (2012), 1250297, 12 pp. doi: 10.1142/S0218127412502975.  Google Scholar [20] Q. G. Yang and Q. M. Xiang, Existence of chaotic oscillations in second-order linear hyperbolic PDEs with implicit boundary conditions, J. Math. Anal. Appl., 457 (2018), 751-775.  doi: 10.1016/j.jmaa.2017.08.018.  Google Scholar [21] Z. B. Yin and Q. G. Yang, Distributionally scrambled set for an annihilation operator, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 25 (2015), 1550178, 13 pp. doi: 10.1142/S0218127415501783.  Google Scholar [22] Z. B. Yin and Q. G. Yang, Generic distributional chaos and principal measure in linear dynamics, Ann. Pol. Math., 118 (2016), 71-94.  doi: 10.4064/ap3908-9-2016.  Google Scholar [23] Z. B. Yin and Q. G. Yang, Distributionally n-chaotic dynamics for linear operators, Rev. Mat. Complut., 31 (2018), 111-129.  doi: 10.1007/s13163-017-0226-5.  Google Scholar [24] Z. B. Yin and Q. G. Yang, Distributionally n-scrambled set for weighted shift operators, J. Dyn. Control Syst., 23 (2017), 693-708.  doi: 10.1007/s10883-017-9359-6.  Google Scholar
The spatiotemporal profiles of system (23) with $(\alpha_1,\beta_1) = (0.1,1)$, $(\alpha_2,\beta_2) = (0.5,1)$, $x\in [0,1]$ and $t\in [60,64]$: (a) $w_{x}(x,t)$; (b) $w_{t}(x,t)$.
The spatiotemporal profiles of system (23) with $\gamma_1 = 1.1\pi$ and $\gamma_2 = 0.4\pi$, $x\in [0,1]$ and $t\in [60,64]$: (a) $w_{x}(x,t)$; (b) $w_{t}(x,t)$.
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# A company has $6.10 per unit in variable costs and$4.90 per unit in fixed cost at a volume of...

## Question:

A company has $6.10 per unit in variable costs and$4.90 per unit in fixed cost at a volume of 50,000 units. if the company marks up the total cost by 0.52, what price should be charged if 57,000 units are expected to be sold?

## Markup and Selling Price:

The selling price per unit for a given product is usually determined by taking into account the variable cost per unit, fixed cost per unit and the markup per unit. The variable cost per unit and the fixed cost per unit further consist of two components, namely, manufacturing cost per unit and selling & administrative expense per unit.

The calculated selling price per unit to be charged for 57,000 units is $10.92 per unit. The total fixed cost is given by: • = Fixed cost per unit * corresponding number of units sold • =$4.90 * 50,000
• = $245,000 The revised fixed cost per unit if 57,000 units are sold is given by: • = Total fixed cost / number of units sold • {eq}= \dfrac{\$245,000}{57,000} {/eq}
• = $4.30 per unit The selling price per unit for 57,000 units is given by: • = Revised fixed cost per unit + variable cost per unit + markup per unit • =$4.30 + $6.10 +$0.52
• = \$10.92 per unit


text:
# Given a circle and a total length 'L', calculate point where 2 tangent lines curve

(Robinmouwen) #1

This gif shows what I want to achieve, except the fact that 1 input and output are switched.

Instead of giving the point where the tangent lines converge/end (Y value slider) and receiving the total curvelength from Grasshopper.

I’d like to be able to input the total length and receive the point.
Does anyone have any ideas?

I can achieve by making use of Galapagos and it’s accurate enough for my application but I’d rather have a real time solution instead of letting Galapagos solve for a minute.

(David Rutten) #2

Middelbare school wiskunde. Weet je nog?

The distance d from the circle centre to the intersection point p is the long edge of a right triangle. The circle radius r and the tangent line t connecting the circle with the intersection point are the right sides of that same triangle.

Thus (given Pythagoras) p^2 = r^2 + t^2 and p = \sqrt{r^2 + t^2}.

pythagoras.gh (9.4 KB)

(Robinmouwen) #3

Thank you for the quick answer, this is what I previously had.
But it’s only part of of what I search. Maybe I didn’t explain it clearly.

I want to be able to have the total length as input.
i.e. the length of (2) tangents, like you provided + the length of arc around the circle between 2 tangents.

(David Rutten) #4

Yeah that’s a lot harder. Haven’t gotten to an analytic solution yet, maybe my approach is flawed.

I assumed a unit circle (so we just scale up our answer by the actual radius) and a total length L for the part of the circle s and the two tangent line lengths t, giving us L = s + 2t.

The angle \alpha is defined as the angle between the horizontal axis and the line connecting the centre of the circle to the tangent point on the circle. Since we’re working with a unit circle, this angle is tan(\alpha) = t, \alpha = tan^{-1}(t).

Furthermore s is merely the part of the circle not covered by 2\alpha. This leads us to the compound expression L = (\pi - \alpha) + 2tan(\alpha), but I haven’t been able to solve for \alpha.

I have the feeling I’m missing some easy solution.

(David Rutten) #5

Here’s a dirty approximate solution. Instead of rewriting the equation L = 2 \left(\sqrt{p^2-1} - cos^{-1}\left(\frac{1}{p} \right) + \pi \right) so that p is expressed in terms of L, I just graphed a portion of the graph and intersected it with a horizontal line at the L elevation. This could be solved much more accurately by a small loop in a C# component though.

approximatesolution.gh (20.5 KB)

(Robinmouwen) #6

This is amazing and definitely accurate enough for me.
Could you explain how you got to this formula from the one in the previous comment?

I tried adapting it to accomodate for a variable radius, but the lengths keep diverging if I change the radius with this formule.


text:
# How can I express this answer with 3 oscillations in terms of 2 oscillations?

I want to express the equation $\frac{2}{9}\sin(9t) + \frac{9}{56}\cos(5t) + \frac{271}{56}\cos(9t)$ in terms of two oscillations of cosine so it'll have the form:

_cos(_t - _) + _cos(_t)

Is this possible?

Thank you!

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## 1 Answer

Using $$\cos\left(y - x\right) = \sin x \sin y + \cos x \cos y$$ with $y=9t$, $$\frac{2}{9} \sin\left(9t\right)+\frac{271}{56}\cos\left(9t\right) = \cos\left(9t - x\right)$$ where $$x = \tan^{-1} \left(\tan x\right) = \tan^{-1}\left(\frac{\sin x}{\cos x}\right) = \tan^{-1}\left(\frac{2 / 9}{ 271 / 56}\right) = \tan^{-1} \left( \frac{112}{2439}\right).$$

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