text:
# Chapter9:Carburettors and Fuel Injection in SI Engines¶

## Example 9.1 page no: 279¶

In [1]:
#given
ma=5                            #Mass flow rate of air per min for a simple jet carburettor in kg/min
mf=0.4                          #Mass flow rate of fuel in kg/min
df=780                          #Density of the fuel in kg/m**3
p1=1.013                        #The initial pressure of air in bar
t1=27                           #The initial temperature of air in degree centigrade
C2=90                           #The air flow velocity in m/s
Cva=0.8                         #The velocity coefficient for the venturi
Cdf=0.6                         #The coefficient of discharge of the main fuel jet
Cpd=0.75                        #The pressure drop across the fuel metering oriface
Cp=1005                         #The specific heat of gas in J/kgK
R=287                           #Real gas constant in J/kgK

#Calculations
import math
p2=p1*(1-(C2**2/(Cva**2*2*Cp*(t1+273))))**(g/(g-1))
da1=((p1*10**5)/(R*(t1+273)))
da2=((da1)*(p2/p1)**(1/g))
A2=((ma/60.0)/(da2*C2))*10**4
d2=(4*A2/math.pi)**(1/2.0)
pv=p1-p2
pj=Cpd*pv
Aj=((mf/60.0)/(Cdf*(2*df*pj*10**5)**(1/2.0)))*10**6
dj=(4*Aj/math.pi)**(1/2.0)

#Output
print"The throat diameter of the choke = ",round(d2,3),"cm"
print"The oriface diameter = ",round(dj,1),"mm"

The throat diameter of the choke =  3.251 cm
The oriface diameter =  2.2 mm

## Example 9.2 page no: 281¶

In [1]:
#given
Vs=0.002                                 #The swept volume in m**3
nv=75.0                                  #Volumetric efficiency in percent
N=4500.0                                 #The engine rpm
p1=1.013                                 #The initial pressure of air in bar
R=287.0                                  #Real gas constant in J/kgK
t1=15.0                                  #The atmospheric temperature in degree centigrade
Cp=1005.0                                #The specific heat of gas in J/kgK
C2=100.0                                 #The air flow velocity at choke in m/s
Cda=0.85                                 #The velocity coefficient for the venturi
Cdf=0.66                                 #The coefficient of discharge of the main fuel jet
sf=0.75                                  #The specific gravity of fuel
d=0.4                                    #The ratio of the diameter to choke diameter
af=14.0                                  #The air fuel ratio
gf=9.81                                  #The gravitational force constant in m/s**2
Z=0.006                                  #The petrol surface below the choke in m
df=750.0                                 #The density of the fuel in kg/m**3

#Calculations
import math
Va=((nv/100.0)*Vs*N)/(2.0*60.0)
V1=Va/2.0
ma=(p1*10**5*V1)/(R*(t1+273))
p2=p1*(1-(C2**2/(2*Cp*(t1+273))))**(g/(g-1))
da1=((p1*10**5)/(R*(t1+273)))
da2=da1*(p2/p1)**(1/g)
A2=(ma/(da2*C2*Cda))*10**6
D=((A2*4)/(math.pi*0.84))**(1/2.0)
mf=ma/af
pm=(p1-p2-(gf*Z*df/10.0**5))*10**5
Aj=(mf/(Cdf*(2*df*pm)**(1/2.0)))*10**6
dj=(4*Aj/math.pi)**(1/2.0)

#Output
print"The diameter of the choke = ",round(D,2),"mm"
print"The diameter of the jet in = ",round(dj,2),"mm"

The diameter of the choke =  22.89 mm
The diameter of the jet in =  1.26 mm

## Example 9.3 page no: 283¶

In [2]:
#given
d=0.08                              #The diameter of the bore in m
L=0.09                              #The length of the stroke in m
N=4000.0                            #The engine rpm
C=84.0                              #The carbon content in the fuel by mass in percent
H=16.0                              #The hydrogen content in the fuel by mass in percent
nv=80.0                             #The volumetric efficiency of the engine in percent
p1=1.0                              #The pressure at ambient condition in bar
t1=25.0                             #The temperature at ambient condition in degree centigrade
p=0.06                              #The depression at venturi throat in bar
ma=0.95                             #The actuat quantity of air supplied
Ra=287.0                            #Real gas constant in J/kgK
Rf=98.0                             #Real gas constant in J/kgK
n=4.0                               #Number of cylinders
Cp=1005.0                           #The specific heat of gas in J/kgK

#Calculations
import math
V=(math.pi/4.0)*d**2*L*(nv/100.0)*(N/(2.0*60.0))*n
Af=(100/23.0)*((C*(32/12.0))+(H*8))/100.0
mfa=Af*ma
Aaf=mfa
da=(p1*10**5)/(Ra*(t1+273))
dv=(p1*10**5)/(Rf*(t1+273))
ma1=V/((1/da)+(1/(mfa*dv)))
mf1=ma1/mfa
p2=p1-p
C2=(2*Cp*(t1+273)*(1-(p2/p1)**((g-1)/g)))**(1/2.0)
T2=(t1+273)*(p2/p1)**((g-1)/g)
d2=(p2*10**5)/(Ra*T2)
A2=(ma1/(d2*C2))*10**4
d2=(A2*4/math.pi)**(1/2.0)

#Output
print"The fuel flow rate = ",round(mf1,5),"kg/s"
print"The velocity of air at throat = ",round(C2,1),"m/s"
print"The throat diameter = ",round(d2*10,2),"mm"

The fuel flow rate =  0.00379 kg/s
The velocity of air at throat =  102.5 m/s
The throat diameter =  24.75 mm

## Example 9.4 page no: 285¶

In [3]:
#given
d=0.1                                     #The diameter of the bore in m
L=0.12                                    #The length of the stroke in m
N=3000                                    #The engine rpm
d2=0.035                                  #The throat diameter of carburettor venturi in m
nv=80                                     #The volumetric efficiency of the engine in percent
Cda=0.82                                  #The coefficient of discharge of air flow
p=1.013                                   #The ambient pressure in bar
T=298                                     #The ambient temperature in K
ar=15                                     #The air fuel ratio
Z=0.005                                   #The top of the jet above the petrol level in the float chamber in m
Cdf=0.7                                   #The coefficient of discharge for fuel flow
df=750                                    #The specific gravity of the fuel in kg/m**3
R=287                                     #Real gas constant in J/kgK
g=9.81                                    #The gravitational constant in m/s**2
n=4                                       #Number of cylinders

#Calculations
import math
V=(math.pi/4.0)*d**2*L*(nv/100.0)*(N/(2.0*60.0))*n
da=(p*10**5)/(R*T)
ma=V*da
A2=(math.pi/4.0)*d2**2
P=(ma**2/(Cda**2*A2**2*2*da))/10.0**5
mf=ma/ar
Aj=(mf/(Cdf*(2*df*((P*10**5)-(g*Z*df)))**(1/2.0)))*10**6
dj=(Aj*4/math.pi)**(1/2.0)

#Output
print"The depression of the throat = ",round(P,3),"bar"
print"The diameter of the fuel jet of a simple carburettor = ",round(dj,3),"mm"

The depression of the throat =  0.054 bar
The diameter of the fuel jet of a simple carburettor =  1.953 mm

## Example 9.5 page no:286¶

In [4]:
#given
mf=(6/3600.0)                               #The mass flow rate of fuel in kg/s
df=750                                      #The density of fuel in kg/m**3
Z=0.003                                     #The level in the float chamber below the top of the jet in m
p=1.013                                     #The ambient pressure in bar
T=294                                       #The ambient temperature in K
dj=0.0012                                   #The jet diameter in m
Cdf=0.65                                    #The discharge coefficient of the jet
Cda=0.8                                     #The discharge coefficient of air
A=15.3                                      #The air fuel ratio
g=9.81                                      #The gravitational constant in m/s**2
R=287                                       #Real gas constant in J/kgK
dh=1000                                     #The density of water in kg/m**2

#Calculations
import math
da=(p*10**5)/(R*T)
Ca2=Cda*((2*g*Z*df)/da)**(1/2.0)
Aj=(math.pi/4.0)*dj**2
P=((mf**2/(Cdf**2*Aj**2*2*df))+(g*Z*df))/10.0**5
h=(P*10**5)/(dh*g)
h1=(P*10**5)/g
ma=mf*A
A2=(ma/((Cda*(2*da*(P*10**5))**(1/2.0))))*10**4
d2=((A2*4/math.pi)**(1/2.0))*10

#Output
print"The critical air velocity = ",round(Ca2,1),"m/s"
print"The depression at the throat = ",round(h1,1),"mm of H2O"
print"The effective throat diameter  ",round(d2,2),"mm"

The critical air velocity =  4.9 m/s
The depression at the throat =  351.6 mm of H2O
The effective throat diameter   21.12 mm

## Example 9.6 page no: 287¶

In [22]:
#given
d2=22                            #The venturi throat diameter of a simple carburettor in mm
Cda=0.82                         #The coefficient of air flow
dj=1.2                           #The fuel orifice diameter in mm
Cdf=0.7                          #The coefficient of fuel flow
Z=0.004                          #The petrol surface below the throat in m
g=9.81                           #The gravitational constant in m/s**2
da=1.2                           #The density of air in kg/m**3
df=750                           #The density of fuel in kg/m**3
P=0.075                          #The pressure drop in bar

#Calculations
A=(Cda/Cdf)*(d2**2/dj**2)*(da/df)**(1/2.0)
A1=(Cda/Cdf)*(d2**2/dj**2)*((da*P)/(df*(P-(g*Z*df)/10.0**5)))**(1/2.0)
Ca2=(2*g*Z*df/da)**(1/2.0)

#Output
print" (a) The air fuel ratio when the nozzle lip is neglected = ",round(A,2)
print"(b)The air fuel ratio when the nozzle lip is considered = ",round(A1,3)
print"(c) The critical air velocity or minimum velocity required to start the fuel flow = ",round(Ca2,1),"m/s"

 (a) The air fuel ratio when the nozzle lip is neglected =  15.75
(b)The air fuel ratio when the nozzle lip is considered =  15.78
(c) The critical air velocity or minimum velocity required to start the fuel flow =  7.0 m/s

## Example 9.7 page no: 289¶

In [23]:
#given
h=4000                                   #The altitude of the airplane engine carburettor in m
A=14.7                                   #The air fuel ratio at sea level
ts=22                                    #The temperature at sea level in degree centigrade
R=287                                    #Real gas constant in J/kgK

#Calculations
ta=ts-(0.0065*h)
p=1.013/10.0**0.2083
da=(p*10**5)/(R*(ta+273))
ds=(1.013*10**5)/(R*(ts+273))
Aa=A*(da/ds)**(1/2.0)

#Output
print"The air fuel ratio at altitude = ",round(Aa,2)

The air fuel ratio at altitude =  12.11

## Example 9.8 page no: 289¶

In [17]:
#given
A=14.5                                       #The air fuel ratio
p2=0.825                                     #The pressure at the venturi throat in bar
p1=1.013                                     #The atmospheric pressure in bar
pd=37.5                                      #The pressure drop to the air cleaner in mm of Hg
ma=260                                       #The mass flow rate of air in kg/h

#Calculations
pa=p1-p2
p21=p1-(pd/750)-pa
pf=pa
pf1=p1-p21
Af=A*(pf/pf1)**(1/2.0)

#Output
print"(a) The throat pressure when the air cleaner is fitted = ",p21,"bar"
print"(b) The air fuel ratio with the air cleaner fitted = ",round(Af,3)

(a) The throat pressure when the air cleaner is fitted =  0.775 bar
(b) The air fuel ratio with the air cleaner fitted =  12.887

## Example 9.9 page no: 291¶

In [1]:
#given
bp=8                                    #The brake power of the petrol engine in kW
nb=30                                   #The brake thermal efficiency in percent
CV=44000                                #The calorific value of the fuel in kJ/kg
p1=1.013                                #The suction condition of engine pressure in bar
T1=300                                  #The temperature at suction condition in K
Aj=2.5*10**-6                           #The area of jet in m**2
Z=0.008                                 #The nozzle lip in m
g=9.81                                  #The gravitational force constant in m/s**2
A=15                                    #The air fuel ratio
Cda=0.9                                 #The coefficient of air flow
Cdf=0.7                                 #The coefficient of fuel flow
df=750                                  #The density of fuel in kg/m**3
va=0.8                                  #The specific volume of air in m**3/kg

#Calculations
import math
va1=va*T1/273.0
da=1/va
mf=bp/((nb/100.0)*CV)
Cf=mf/(Cdf*df*Aj)
P=((Cf**2*df)/2.0)+(df*g*Z)
Ca=(2*P/da)**(1/2.0)
ma=mf*A
A2=(ma/(Cda*da*Ca))*10**4
d2=(A2*4/math.pi)**(1/2.0)

#Output
print"The venturi throat diameter of the carburator = ",round(d2,2),"cm"

The venturi throat diameter of the carburator =  2.63 cm


text:
# PIC16 UART Interrupt Receive - System hangs

#### Robin Mitchell

Joined Oct 25, 2009
819
Hi all,

Firstly, I have fixed my problem (so the system now works) but I dont know why!

Basically, I have a PIC16 controller with a configuration such that when a character is received on the RX line it will dump that into a buffer in SRAM. I know that the buffer mechanism works because that same code is being used for other things. The UART is configured correctly and I have read all the documentation and errata for the device.

Now here is what I dont understand.

Upon receiving the first character the processor hangs. I cannot debug the system because I am using all IO as IO (none free for the debugger) so I dont know why its latching up. I also noticed that if I disconnect the RX line and then probe that line with an oscilloscope probe it causes the latching of the device. I then added a pull up resistor which fixed the latchup from the probe (but still latches up when a character is sent).

The hang is not being caused by overflow and frame errors as my code (which you will see shortly) addresses those problems.

However, there was a temporary fix (that makes no sense).....

When the PIC hangs, I can reset the chip manually by pulling MCLR down and when I do, the chip spontaneously works again as if nothing ever happened AND characters reception works great! By trial and error I discovered that the error was somewhere around the interrupt code because if I disable the interrupt for UART reception, the chip would not hang when a byte came down the RX line. I then put my ISR code into the main loop to see if the code was at fault but the code worked fine (just meant that the chip would not handle the incoming byte as soon as it had arrived).

So, I have a device that hangs when RCIF is triggered, works after a RESET and works if I disable the interrupt. Eventually, I changed the interrupt code so that the first interrupt to be handled was the RCIF. Magically, it now works (yes, I have other interrupts being checked including timers and CCP).

Here is the old ISR "decider" code - this hangs on RCIF
Code:
banksel PIR1
btfsc PIR1, CCP1IF                    ; Was the PS2 Clock line triggered?
goto ps2Handle                        ; If it was, handel it!
btfsc INTCON, INTF                    ; Was the CS pin triggered?
goto csHandel                        ; Chip select handel
btfsc INTCON, TMR0IE                ; Was the interrupt the timer?
goto timerInterrupt                    ; Sort out the timer
btfsc PIR1, RCIF                    ; Has a byte been received over UART?
goto uartInterrupt                    ; Store the byte
goto interrupt.end                    ;
Here is the new ISR "decider" code - This works great
Code:
banksel PIR1
btfsc PIR1, RCIF                    ; Has a byte been received over UART?
goto uartInterrupt                    ; Store the byte
btfsc PIR1, CCP1IF                    ; Was the PS2 Clock line triggered?
goto ps2Handle                        ; If it was, handel it!
btfsc INTCON, INTF                    ; Was the CS pin triggered?
goto csHandel                        ; Chip select handel
btfsc INTCON, TMR0IE                ; Was the interrupt the timer?
goto timerInterrupt                    ; Sort out the timer
goto interrupt.end                    ;

Last edited:

#### ericgibbs

Joined Jan 29, 2010
17,131
Morning Robin
Are the PIC's Interrupts enabled.?
Seems a little unusual to use goto's for ISR's.
Eric

#### LesJones

Joined Jan 8, 2017
3,939
Have you cleared the interrupt at the end of the interrupt service routine ? Posting your interrupt service routines would be a great help.

Les.

#### Robin Mitchell

Joined Oct 25, 2009
819
@ericgibbs
The interrupts are enabled (both peripheral and global) otherwise the timer and PS2 would not work.
This code is located at 0x0004 (ISR location) and the gotos are used for two reasons:
A) Calls use the stack (not desired)
B) The code needed to handel the ISRs are a tad bulky

#### Robin Mitchell

Joined Oct 25, 2009
819
@LesJones

Yes, they are cleared by reading RCREG. The point is that the PIC stops at the ISR handle section.

#### Robin Mitchell

Joined Oct 25, 2009
819
Code:
uartInterrupt:
banksel RCSTA
btfsc RCSTA, OERR
goto uartOhandel
btfsc RCSTA, FERR
goto uartFhandel
goto uartGetReg

uartOhandel:
bcf RCSTA, SPEN
bcf RCSTA, CREN
nop
nop
nop
nop
bsf RCSTA, CREN
bsf RCSTA, SPEN
retfie

uartFhandel;
banksel RCREG
movfw RCREG
movfw RCREG
movfw RCREG
retfie

uartGetReg:
movfw RCREG
retfie

#### LesJones

Joined Jan 8, 2017
3,939
You don't have any code in your interrupt handler to save and restore registers that can be changed by the interrupt routine. for example status register and bankselect register. This should also be done in subroutines.

Les.

#### ericgibbs

Joined Jan 29, 2010
17,131
hi R,
With respect, I would say that your programming methodology requires rethinking/revising.

Eric

#### Robin Mitchell

Joined Oct 25, 2009
819
@LesJones the PIC has automatic context saving. All core registers are automatically popped and pushed to and from the stack upon an interrupt entry and exit.
@ericgibbs Why? You should not call functions inside an ISR becuase of potential stack overflow. Goto statements are totally fine as long as they all end at retfie which mine do. My code is perfectly stable and acceptable (the PS2 decoding is fine, the INT is fine, and even the UART now) so what is wrong with it? How would you deal with a PS2 port? Would you bit bang? Would you use non-interrupt code that would miss bits from the datastream?

#### Robin Mitchell

Joined Oct 25, 2009
819
Forgot to mention, interupt.end is just a retfie instruction.

#### Sensacell

Joined Jun 19, 2012
3,131
Having mainline code with a GOTO to an ISR is a very bad idea, when you hit the RETFIE instruction, the PC will be loaded with garbage and your program will be lost in space. (CRASH)

There are two ways to handle this correctly:

1) Use the interrupt mechanism correctly, have the interrupts enabled so the code jumps to the ISR when a character comes in, then exit the ISR with an RETFIE instruction. (polling the UART INT flag is totally unnecessary)

2) Poll the UART INT Flag, then process the code with a call to a subroutine that handles the UART. End the subroutine with a normal RETURN instruction.
This is not technically an interrupt at all.

#### Robin Mitchell

Joined Oct 25, 2009
819
For the last time, Im not calling an ISR. The ISR is executed WHEN an interrupt is fired. Then the ISR determines which interrupt was fired and then GOTOs that specific handle.

#### LesJones

Joined Jan 8, 2017
3,939
Which PIC16 are you using ? Non of the PIC16s that I have used have automatic context saving.

Les.

#### MMcLaren

Joined Feb 14, 2010
859
Which PIC16 are you using ? Non of the PIC16s that I have used have automatic context saving.

Les.
All of the newer "enhanced mid-range" chips have this capability, as well as LAT registers, access to WREG, indirect access to all banked RAM in a linear address space, etc., etc... You should check 'em out... They're really pretty neat...

Regards, Mike

#### MMcLaren

Joined Feb 14, 2010
859
Hi Robin.

Since PCLATH could be almost any value when an interrupt occurs, you should probably setup PCLATH in your interrupt code before using "call" or "goto" instructions. If all of your interrupt code fits into the first 2048 bytes of program memory, simply do a "clrf PCLATH" at the beginning of your interrupt routine.

Cheerful regards, Mike

Last edited:

#### Robin Mitchell

Joined Oct 25, 2009
819
Hi Robin.

Since PCLATH could be almost any value when an interrupt occurs, you should probably setup PCLATH in your interrupt code before using "call" or "goto" instructions.

Cheerful regards, Mike
Interesting . Why? What effect does PCLATH have on the interrupt? From what I have gathered, the PC is pushed onto the stack and PCLATH is put into a shadow register which is then restored on retfie.

#### MMcLaren

Joined Feb 14, 2010
859
That's correct, Robin. However, a 14-bit "goto" or "call" instruction only contains 11 bits of address data, a so called 2048 word "page", and the other address bits are copied from PCLATH. This could be a problem when you have code spanning more than a single 2048 word "page" in program memory. If an interrupt occurs while running code in page 1 (0800..0FFF) the PCLATH value will affect the "goto" instructions in your ISR.

If all of your interrupt code resides in the first "page" (0000..07FF) of program memory, a single "clrf PCLATH" instruction at the top of your ISR will keep you out of trouble when your program grows to span more than a single "page" of program memory.

Have fun... Cheerful regards, Mike

Last edited:

#### John P

Joined Oct 14, 2008
1,995
How many times must the poor guy explain? PCLATH is one of the registers that an enhanced midrange PIC saves automatically. The problem must be somewhere else. And a clue is that it's important what order the interrupt flags are checked in. Hmmm...

I do see that the ISR is checking TMR0IE rather than TMR0IF. So suppose the RX interrupt goes off, the ISR reaches the test for TMR0IE and of course, it's set, so it executes that instead, so the RX interrupt never gets serviced, and after leaving the ISR you immediately see another RX interrupt, forever. That looks like it to me.

Last edited:

#### Robin Mitchell

Joined Oct 25, 2009
819
@John P Finally....someone who listened to me (@MMcLaren I do however love your response! Learn something everyday)

It is SO strange that the ORDER of the flag checking determines if the system hangs or not. At one point, I included a watchdog timer to try and reset the system which did not fix the hang. Like I said before, it would magically work (as well) if the external reset is pulsed. Maybe there is a glitch in the CPU? It is not uncommon for micros to have an erata that tell you not to use x y and z functions because they can result in glitches.

#### John P

Joined Oct 14, 2008
1,995
Robin, I think you missed my second paragraph, which I added later, when the clue related to order of testing the flags really sank in.

format: plain


text:
Operating System - OpenVMS
cancel
Showing results for
Did you mean:

DCPS V2.7 MEDIA

Occasional Visitor

DCPS V2.7 MEDIA

I am trying to obtain a copy of HP-VAXVMS-DCPS-V0207--1.PCSI (decprint supervisor v2.7 for vax. If anyone can direct me to where I could obtaing this or if possible have it emailed to me it would be greatly appreciated. Thank you.

Ralph DellaVecchia


text:
# If f(x)= tan2 x  and g(x) = sqrt(-4x-3 , how do you differentiate f(g(x))  using the chain rule?

Mar 11, 2018

-(4sec^2(2sqrt(-4x-3))) /sqrt(-4x-3)

#### Explanation:

Starting with: $f \left(x\right) = \tan 2 x$ and $g \left(x\right) = \sqrt{- 4 x - 3} = {\left(- 4 x - 3\right)}^{\frac{1}{2}}$

Let $h \left(x\right) = f \left(g \left(x\right)\right) = \tan \left(2 \left(\sqrt{- 4 x - 3}\right)\right)$

$f ' \left(x\right) = 2 {\sec}^{2} \left(2 x\right)$
$g ' \left(x\right) = \left(\frac{1}{2}\right) \left(- 4\right) {\left(- 4 x - 3\right)}^{- \frac{1}{2}} = - \frac{2}{\sqrt{- 4 x - 3}}$

Using the Chain Rule:

$h ' \left(x\right) = f ' \left(g \left(x\right)\right) \cdot g ' \left(x\right)$

$h ' \left(x\right) = 2 {\sec}^{2} \left(2 \sqrt{- 4 x - 3}\right) \cdot - \frac{2}{\sqrt{- 4 x - 3}}$

$h ' \left(x\right) = - \frac{4 {\sec}^{2} \left(2 \sqrt{- 4 x - 3}\right)}{\sqrt{- 4 x - 3}}$


text:
# Coding in Colour

The numbers exist.  No one can own a number.  However, some companies/corporations are trying to make it illegal to publish or distribute some numbers on the internet.  They are trying to claim that their encryption code keys are the subject of copyright or trademark. Legal cases have been brought.

The keys are used to unlock DVDs in order that they may be copied or to access game stations or the games.

It is errant nonsense that a number can be illegal.

The internet fought back and started coding in colour.

Your computers run on the binary system; letters and other things are assigned an ASCII code which in turn has a binary representation.

$$\begin{array}{c c c} letter & ASCII & binary\\ \hline O & 079 & 01001111\\ l & 108 & 01101100\\ d & 100 & 01100100\\ T & 084 & 01010100\\ r & 114 & 01110010\\ o & 111 & 01101111\\ u & 117 & 01110101\\ t & 116 & 01110100 \end{array}$$

Hence,

$$7,910,810,084,114,111,117,116$$

is our number and we forbid you from publishing or distributing this number.

We can convert this decimal number into a hexadecimal number :

$$1acd88b323bcf9fb73c$$

and we forbid you from publishing or distributing it.

We can convert this hexadecimal number into colours by partitioning from the left in blocks of six digits:

$$1acd88 \; b323bc\; f9fb73\; c$$

with which we can create a flag :

and we forbid you from publishing or distributing it because we don’t like the colours very much.

There are $$2^{24} = 16,777,216$$ colours to choose from on the web.

The colours also have rgb (red, green,blue) codes assigned to them.

We can choose much more pleasing colours and choose rgb $$(25, 45, 250), (0, 130, 250)$$ and $$(250,20,25)$$.

We convert these into a hexadecimal number :

$$192dfa0082fafa1419$$

We convert this into a decimal number :

$$464, 481, 562, 881, 179, 650, 000$$

With many thanks to Swiss Bob who retrieved the colours and made the flags.

© OldTrout $$2018$$


text:
# Disk2D¶

class astropy.modeling.functional_models.Disk2D(amplitude=1, x_0=0, y_0=0, R_0=1, **kwargs)[source]

Two dimensional radial symmetric Disk model.

Parameters
amplitudefloat

Value of the disk function

x_0float

x position center of the disk

y_0float

y position center of the disk

R_0float

Radius of the disk

Other Parameters
fixeda dict, optional

A dictionary {parameter_name: boolean} of parameters to not be varied during fitting. True means the parameter is held fixed. Alternatively the fixed property of a parameter may be used.

tieddict, optional

A dictionary {parameter_name: callable} of parameters which are linked to some other parameter. The dictionary values are callables providing the linking relationship. Alternatively the tied property of a parameter may be used.

boundsdict, optional

A dictionary {parameter_name: value} of lower and upper bounds of parameters. Keys are parameter names. Values are a list or a tuple of length 2 giving the desired range for the parameter. Alternatively, the min and max properties of a parameter may be used.

eqconslist, optional

A list of functions of length n such that eqcons[j](x0,*args) == 0.0 in a successfully optimized problem.

ineqconslist, optional

A list of functions of length n such that ieqcons[j](x0,*args) >= 0.0 is a successfully optimized problem.

Notes

Model formula:

$\begin{split}f(r) = \left \{ \begin{array}{ll} A & : r \leq R_0 \\ 0 & : r > R_0 \end{array} \right.\end{split}$

Attributes Summary

 R_0 amplitude input_units This property is used to indicate what units or sets of units the evaluate method expects, and returns a dictionary mapping inputs to units (or None if any units are accepted). param_names x_0 y_0

Methods Summary

 evaluate(x, y, amplitude, x_0, y_0, R_0) Two dimensional Disk model function

Attributes Documentation

R_0 = Parameter('R_0', value=1.0)
amplitude = Parameter('amplitude', value=1.0)
input_units

This property is used to indicate what units or sets of units the evaluate method expects, and returns a dictionary mapping inputs to units (or None if any units are accepted).

Model sub-classes can also use function annotations in evaluate to indicate valid input units, in which case this property should not be overridden since it will return the input units based on the annotations.

param_names = ('amplitude', 'x_0', 'y_0', 'R_0')
x_0 = Parameter('x_0', value=0.0)
y_0 = Parameter('y_0', value=0.0)

Methods Documentation

static evaluate(x, y, amplitude, x_0, y_0, R_0)[source]

Two dimensional Disk model function


text:
# cd

Change current folder

## Description

example

cd displays the current folder.

example

cd newFolder changes the current folder to newFolder. Folder changes are global. Therefore, if you use cd within a function, the folder change persists after MATLAB® finishes executing the function.

example

oldFolder = cd(newFolder) returns the existing current folder to oldFolder, and then it changes the current folder to newFolder.

## Examples

collapse all

Change the current folder to the featured examples folder for MATLAB R2017a, assuming that version is installed on your C: drive.

cd 'C:\Program Files\MATLAB\R2017a\examples\matlab_featured'

Using the full path, change the current folder from C:\Program Files\MATLAB\R2017a\examples\matlab_featured to C:\Program Files\MATLAB\R2017a.

cd ..\..

Use a relative path to change the current folder from C:\Program Files\MATLAB\R2017a back to C:\Program Files\MATLAB\R2017a\examples\matlab_featured.

cd examples\matlab_featured

Change the current folder to the featured examples folder for MATLAB R2017a, assuming that version is installed on your C: drive. Then restore the current folder to its original location.

Change the current folder to the featured examples folder for MATLAB R2017a.

cd 'C:\Program Files\MATLAB\R2017a\examples\matlab_featured'

Change the current folder to C:\Program Files, saving the folder path before changing it.

oldFolder = cd('C:\Program Files')
oldFolder =

'C:\Program Files\MATLAB\R2017a\examples\matlab_featured'

Use the cd command to display the new current folder.

cd
C:\Program Files

Change the current folder back to the original folder, using the stored path. Use the cd command to display the new current folder.

cd(oldFolder)
cd
C:\Program Files\MATLAB\R2017a\examples\matlab_featured

Change the current folder to the featured examples folder for MATLAB R2016b, assuming that version is installed in your user home folder on a UNIX platform.

cd ~/MATLAB/R2016b/examples/matlab_featured

## Input Arguments

collapse all

New folder path to which you want to change the current folder, specified as a character vector or string scalar. If newFolder is a string, enclose it in parentheses. For example, cd("FolderName").

Valid values include a full or relative path or one of these values.

 ../ One level up from the current folder. Multiples of ../ Multiple levels up from the current folder. ./ A path relative to the current folder. cd assumes that the path is relative to the current folder even if the ./ is not present.

If newFolder contains spaces, enclose it in single quotation marks. For example, cd 'Folder Name'.

On UNIX® systems, you can use the ~ (tilde) character to represent the user home folder. For example, ~/ or ~username/.

MATLAB resolves folder path names containing '.', '..', and symbolic links to their target location before changing the current folder. For example, if you specify c:\matlab\..\work, MATLAB resolves the path name to c:\work before changing the current folder.

## Output Arguments

collapse all

Previous current folder that was in place before you issued the cd command, returned as a character vector.

collapse all

### Current Folder

The current folder is a reference location that MATLAB uses to find files. This folder is sometimes referred to as the current directory, current working folder, or present working directory.


text:
# How do you find the axis of symmetry and vertex point of the function:  y = -x^2 + 3x + 5?

Oct 18, 2015

Vertex $\left(\frac{3}{2} , \frac{29}{4}\right)$

Axis of Symmetry $x = \frac{3}{2}$

#### Explanation:

Given -

$y = - {x}^{2} + 3 x + 5$

Vertex point -

$x = \frac{- b}{2 a} = \frac{- 3}{2 \times \left(- 1\right)} = \frac{- 3}{- 2} = \frac{3}{2}$

$y = - {\left(\frac{3}{2}\right)}^{2} + 3 \left(\frac{3}{2}\right) + 5$

$y = \frac{- 9 + 18 + 20}{4} = \frac{29}{4}$

Vertex $\left(\frac{3}{2} , \frac{29}{4}\right)$

Axis of Symmetry $x = \frac{3}{2}$


text:
# signum function domain and range

## signum function domain and range

The Identity Function graph is a straight line and always passes through the origin. The signum function is often not used in network theory, but it is used in communication and control theory. The graph is nothing but the graph y = log ( x ) translated 3 units down. It is known as the Signum Function or f(x) = sgn(x). Graph of Signum Function: Graph of Signum Function. Example 1: Some Properties of Functions Modul and Signum Jan Popiol ek Warsaw University Bial ystok Summary. Detailed Answer : The signum function S is defined as follows : The domain of S is R and its range is {– 1, 0, 1}. Domain of f = R Range of f = R+ ∪ {0} (vi) Signum function: The real function f: R → R defined by The domain of the function is R and the range … = {-1, 0, 1} The graph for the signum function is as follows. The article includes deﬁnitions and theorems concerning basic properties of the following functions : |x| - modul of real number, sgn x - signum of real number. Free functions domain calculator - find functions domain step-by-step This website uses cookies to ensure you get the best experience. ... R→R is explained as f(x) = y = x, for x ∈ R, the domain and the range being R. The types of functions in sets can be understood with the help of these functions. Draw its graph. This is the signum function. Class 6. Signum Function. Hence, f: A → B is a function such that for a ∈ A there is a unique element b ∈ B such that (a, b) ∈ f EXAMPLE : Identifying Domain and Range Verify the domains and ranges of these functions. Greatest integer function graph Draw the graph of the signum function. Let's first get a clear definition of Range as well as the Signum Function. Login to view more pages. A function defined as has a Domain and a Co-Domain . One of the more important ideas about functions is that of the domain and range of a function. Linearity It's not linear, not being smooth about zero, and not satisfying $\mathrm{sgn}(x_1 + x_2)\ =\ \mathrm{sgn}(x_1) + \mathrm{sgn}(x_2)$ (Proof left as an exercise for the reader). function. And we see here. Continuity. Mapping Functions in the Real World [3/20/1995] What is the purpose of learning to map a function? Such a function is called the greatest integer function. The sign function (or signum function) is a special function which returns: 1 for all x > 0 and – 1 for all x < 0. In simplest terms the domain of a function is the set of all values that can be plugged into a function and have the function exist and have a real number for a value. Identity Function f(x) = x Also write its domain and range. Signum Function: The function defined by: or The domain of the signum function is the set of all real numbers and the range is the set of . To find the domain of a function, just plug the x-values into the quadratic formula to get the y-output. (In grammar school, you probably called the domain the replacement set and the range the solution set. Signum function, domain range and graph, questions. The set of all values of ƒ(x) as x varies throughout D is called the range of the function. The Answer. Intervals where a function is positive, negative, increasing, or decreasing. Greatest Integer Or Step up Function: The function y = f(x) = [x] is called the Greatest Integer or Set up Function where represents the … (Both of these functions can be … Anything, anything negative 6 or lower, our function isn't defined. The domain of a function f (x) is the set of all values for which the function is defined, and the range of the function is the set of all values that f takes. A function which has either R or one of its subsets as its range is called a real values function. It is a real-valued step function that tells us, numerically, whether a particular value of x is positive, negative, or zero. The function f: R → R defined by f ( x) = [ x ], x ∈ R assumes the value of the greatest integer, less than or equal to x. The range of the function is same as the domain of the inverse function. (v) The Modulus function: The real function f: R → R defined by f (x) = x =,0,0 xx xx ⎧ ≥ ⎨ ⎩−< ∀x ∈ R is called the modulus function. Exponential Function: If is a positive real number other than unity, then a function that associates each to is called the exponential function. Define Signum function. The subset of this co-domain which represents the set of all possible outputs of function is known as the Range.. The sign function (or signum function) is a special function which returns: 1 for all x > 0 and – 1 for all x < 0. Class 7. Define signum function. Each function has a set of values, the function's range, which it … Now that we know what … The greatest functions are defined piecewise Its domain is a group of real numbers that are divided into intervals like [-4, 3), [-3, 2), [-2, 1), [-1, 0) and so on. The greatest integer function of a real number x is represented by f(x) = [x] or |_x_|. then f is called signum function. -- Graphs for JEE Main and Advanced -- This lecture series is … For x = 0, the value of the sign function is just zero. Domain,co-domain and range of a function; Equal functions; Types of function. Domain of f = R; Range of f = {-1, 0, 1} Greatest integer function: The real function f : R → R defined by f (x) = {x}, x ∈ R assumes that the values of the greatest integer less than or equal to x, is called the greatest integer function. Class 10. (gives the sign of real number) Domain of f = P, Range of f = {1, 0, – 1} For example: signum … In simplest terms the domain of a function is the set of all values that can be plugged into a function and have the function exist and have a real number for a value. The Domain of the function f(x) is x∈R and Range is y∈R. 1 if function value is positive. The range is more difﬁcult. Class 12. This post contain in depth explain of Greatest integer function, its graph domain and range along with a lot of solved examples. Define signum function and its graph domain and range. -1 if function value is negative and 0 if function value is zero. If … The sine function takes the reals (domain) to the closed interval (range). Real valued function; Constant function; Modulus function; Greatest integer function; Fractional -part function; Signum function; Exponential function; Logarithmic function. RELATIONS AND FUNCTIONS 21 example f: R – {– 2} → R defined by f (x) = 1 2 x x + +, ∀x ∈ R – {– 2 }is a rational function. The set of values to which is sent by the function is called the range. Last updated at March 25, 2020 by Teachoo, Learn All Concepts of Chapter 2 Class 11 Relations and Function - FREE. Class 10. 1. Learn all about graph of a function. 3 Graph the below function. About Cuemath. This post contain in depth explain of Greatest integer function, its graph domain and range along with a lot of solved examples. Get detailed, expert explanations on graph of a function that can improve your comprehension and help with homework. The Domain is the entire set of Real Numbers . Let's come to the question in hand. This topic covers: - Evaluating functions - Domain & range of functions - Graphical features of functions - Average rate of change of functions - Function combination and composition - Function transformations (shift, reflect, stretch) - Piecewise functions - Inverse functions - Two-variable functions As expressed by equation, this function has a value of unity for t > 0, since u (t) equals unity in this range. Real functions of a real variable Example Find the domain and range of of f where f(x)= 2 −x x −1 f is deﬁned only when the denominator is non-zero, i.e., when x 6= 1. The graph of the signum function is given in the figure : Welcome to Sarthaks eConnect: A unique platform where students can interact with teachers/experts/students to get solutions to their queries. Actually, differentiability at a point is defined as: suppose f is a real function and c is a point in its domain. ... Now, you will be able to easily solve problems on the examples on the calculator for Absolute Value Graph, domain and range of absolute value function, and equations of the absolute value function. In this lecture we will discuss about the Signum Function, its Graph and Properties. To find the range of a function, first find the x-value and y-value of the vertex using the formula x = -b/2a. The signum function, denoted , is defined as follows: Note: In the definition given here, we define the value to be zero. Define a modulus function. ... is called the signum function or sign function. Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. Further, if its domain is also either P or a subset of P, it is called a real function. Signum Function. The graph of the signum function is given below. What is one-to-one function? Identify the given graph and write the domain and range of this function. The Range of Signum Function is the Set {-1, 0, 1}, as it has only three possible outputs. Draw the graph of it and Write down its domain and Range. Domain,co-domain and range of a function; Equal functions; Types of function. Solution for The signum (or sign) function, denoted by sgn, is defined by if x < 0 if x = 0. if x > 0 -1 sgn x = 1 (a) Sketch the graph of this function. So the domain of this, this is a review. Check - Relation and Function Class 11 - All Concepts, Let us check value of f(x) for different values of x. Teachoo provides the best content available! For x = 0, the value of the sign function is just zero. Domain & Range. How do you find the domain and range of a function? For f(x) = |x|, domain is R and Range is R + U {0}. Let R be the set of real numbers. Example 1: A function f is defined on $$\mathbb{R}$$ as follows: We can define a function as a special relation which maps each element of set A with one and only one element of set B. Differentiability of a function: Differentiability applies to a function whose derivative exists at each point in its domain. Greatest integer function domain and range. The domain of the function f(x) is x ∈ R and Range is set {-1, 0, 1}. Class 9. asked Feb 7 in Mathematics by Beepin (58.6k points) class-11; 0 votes. The sign of a real number, also called sgn or signum, is -1 for a negative numbers (i.e., one with a minus sign "), 0 for the number zero, or +1 for a positive numbers (i.e., one with a plus sign"). The signum function S is defined as follows : The domain of S is R and its range is {– 1, 0, 1}. By using this website, you agree to our Cookie Policy. Function. Both the sets A and B must be non-empty. In this case we can say that the range of y(x)is the domain of x(y). Define rational function and give example of it. Anything, anything negative 6 or lower, our function isn't defined. Classes. On signing up you are confirming that you have read and agree to Define signum function. View a complete list of particular functions on this wiki Definition. Signum Function: This can also be written as. Learn all about graph of a function. Students (upto class 10+2) preparing for All Government Exams, CBSE Board Exam, ICSE Board Exam, State Board Exam, JEE (Mains+Advance) and NEET can ask questions from any subject and get quick answers by subject teachers/ experts/mentors/students. The ordered pairs satisfying the signum function are (1, 1), (2, 1), (0, 0), (-1, -1), (-5, -1). To find the range of a rational function, we need to identify all values that the function cannot take. Thus, the domain of the function is Real numbers and the range of the function is 1, 0, − 1 There is an alternative way to express the signum function using the modulus function The alternative way of expressing the function is: Draw the graph of the function y = x - Draw the graph of the function y = x 2 + 4x + 6. This is a set of all x values for which this function is defined. ... is called the signum function or sign function. Domain . This set must be implicitly/explicitly defined in the definition of the function. If it, if x is negative 6 or or lower than that. The domain of the signum function is R and the range is the set {−1, 0, 1}. He provides courses for Maths and Science at Teachoo. Domain= R & Range = {-1,0,1} Share these Notes with your friends Prev Next > You can check our 5-step learning process. Real valued function; Constant function; Modulus function; Greatest integer function; Fractional -part function; Signum function; Exponential function; Logarithmic function. MML Identiﬁer: ANAL 1. Note: The domain for signum function is a set of real number R i.e. Classes. 2. So the domain of this, this is a review. Greatest Integer Or Step up Function: The function y = f(x) = [x] is called the Greatest Integer or Set up Function where represents the greatest integer less than or equal to x. Domain of f = R; Range of f = {-1, 0, 1} Greatest integer function: The real function f : R → R defined by f (x) = {x}, x ∈ R assumes that the values of the greatest integer less than or equal to x, is called the greatest integer function. Class 11. About Cuemath. Domain of f = R; Range of f = Integer. Diving further… Any considerable real number can be expressed as the product of its absolute value and the sign function. Greatest integer function domain and range. The domain is the set of all inputs for which this function is defined, and our input variable here is x. The domain of the signum function is R, and the range is { … Further ,if its domain is also either R or a subset of R ,It is called a real function. Hence the condition x - 2 ≥ 0 Solve the above inequality to obtain the domain in … Graph the function on a coordinate plane.Remember that when no base is shown, the base is understood to be 10 . y y 4… Define the real function f : by f(x) = x + 8, and sketch the graph. He has been teaching from the past 9 years. The domain of the signum function is R and the range is the set {−1, 0, 1}. Also, draw its graph. NEET. called the domain of the function. Class 7. This is in line with the piecewise definition of the modulus function. The domain of a function is the collection of independent variables of x, and the range is the collection of dependent variables of y. Find its domain and range. One of the more important ideas about functions is that of the domain and range of a function. Draw its graph and find its domain and range. Which means f(x) can take any real value. Write its domain and range. When t < 0, - t is positive and u-(t) equals unity in this range. The domain is the set of all inputs for which this function is defined, and our input variable here is x. Class 11. Class 8. Range. Signum Function: A signum function y = f(x) = sgn(x) is defined as: It is also written as . The graph of f(x) is continuous for all values of x except at x=0 where there is a break in the curve. A real-valued function has either P or any one of its subsets as its range. Signum Function: A signum function y = f(x) = sgn(x) is defined as: It is also written as . 2 Graph each of the below absolute value equations. Since we can apply the modulus operation to any real number, the domain of the modulus function is $$\mathbb{R}$$. Example 3: Find the domain and range of the function y = log ( x ) − 3 . Class 8. And we see here. This is the signum function. On plotting these points on the Cartesian plane and then joining them, we get the graph of signum function f of x is equal to mod of x divided by x. Draw the graph of it and Write down its domain and Range. Class 9. For example, the function takes the reals (domain) to the non-negative reals (range). A function defines a particular output for a particular input. Lesson 10 of 18 • 101 upvotes • 14:13 mins The graph of the signum function is given below. Write its domain, co – domain and range. For all real numbers x, the greatest integer function returns the largest integer less than or equal to x Domain and Range: f(x) = [x] is defined for all x, so the domain is (-infinity, infinity). Thus the domain of f is (−∞,1) S (1,∞), also written as R\{1}. You cannot feed the function an element that isn't in the domain, as the function is not defined for that input element. The range is clearly the set of all non-negative real numbers, or $$\left( {0,\infty} \right)$$. Information about the function, including its domain, range, and key data relating to graphing, differentiation, and integration, is presented in the article. Interval Notation [4/1/1996] I need to learn about interval notation in terms of domain and ranges. The signum function is often not used in network theory, but it is used in communication and control theory. This is a set of all x values for which this function is defined. Greatest integer function graph Domain = All values of x = R Range = All values of y Since y will always be positive or 0 Range = All positive Real numbers and 0 Next: Signum Function→ Chapter 2 Class 11 Relations and Functions; Concept wise; Different Functions and their graphs. Teachoo is free. Class 12. NEET. Function Tests [02/19/1997] What is the reasoning behind the vertical and horizontal line tests? Signum Functions TS: Making decisions after reflection and review Obj: Be able to graph each of the above kinds of functions with translations Warm Up: Rewrite each absolute value expression as a piecewise function. is called the signum function. The sign function or signum function extracts the sign of a real number. Class 6. Informally, if a function is defined on some set, then we call that set the domain. (gives the sign of real number) Domain of f = P, Range of f = {1, 0, – 1} For example: signum … x + 3 = 0 ⇒ x = − 3 So, the domain of the function is set of real numbers except − 3 . In other words, list out all possible outputs that a function can have, and put them in a Set. Define Signum Function. As expressed by equation, this function has a value of unity for t > 0, since u (t) equals unity in this range. If f : R → R is defined by. To find the domain and range of rational functions remember the following steps: To find the domain of a rational function, we need to identify any points that would lead to a denominator of zero. Note that for x ≠ 0. Define signum function Draw its graph. 1 answer. ... Now, you will be able to easily solve problems on the examples on the calculator for Absolute Value Graph, domain and range of absolute value function, and equations of the absolute value function. Solution The formula y=x2 gives a real y-value for any real number x, so the Terms of Service. Domain= R & Range = {-1,0,1} Share these Notes with your friends Prev Next > You can check our 5-step learning process. Further, if its domain is also either P or a subset of P, it is called a real function. Learn Science with Notes and NCERT Solutions, Chapter 2 Class 11 Relations and Functions, Relation and Function Class 11 - All Concepts, Finding Domain and Range - By drawing graphs →, Finding Relation - Set-builder form given, Finding Domain and Range - By drawing graphs, Finding Domain and Range - General Method. Graph of Signum Function: Graph of Signum Function. They may also have been called the input and output of the function.) For f(x) = |x|, domain is R and Range is R + U {0}. The output is quite simple. The greatest functions are defined piecewise Its domain is a group of real numbers that are divided into intervals like [-4, 3), [-3, 2), [-2, 1), [-1, 0) and so on. Find the domain of the real valued function h defined by h(x) = √ ( x - 2) Solution to Question 8: For function h to be real valued, the expression under the square root must be positive or equal to 0. is called signum function. Any real number can be expressed as the product of its absolute value and its sign function: Domain & Range. x≠y => f(x)≠f(y) ∀ x,y∈D. Click hereto get an answer to your question  Find the domain and range of the real function f(x) = √(25 - x^2) . The values taken by the function are collectively referred to as the range. The function f : R → R defined by f ( x ) = [ x ], x ∈ R assumes the value of the greatest integer, less than or equal to x . When t < 0, - t is positive and u- (t) equals unity in this range. A function f: D-> C is called one-to-one if distinct elements of D have distinct images in C, i.e. Define signum function. Range and Signum Function. f:R→RThis is known as signum function.Let us check value of f(x) for different values of xFor x = –1x < 0So, f(x) = –1For x = –2x < 0So, f(x) = –1Forx =1x > 0So, f(x) = 1For x = 2x > 0So, f(x) = 1For x =0x = 0So, f(x) = 0Now,Plotting graphHere,Domain= All values of x = RRange= All … Vertex using the formula x = -b/2a function graph this is a straight and. Positive and u- ( t ) equals unity in this range =,... And output of the modulus function. you find the domain of the is... Have, and our input variable here is x the reasoning behind the vertical and horizontal line Tests functions Types... = R. the range of a function is given below is as follows a. Share these Notes with your friends Prev Next > you can check our 5-step learning process greatest integer function is. At Teachoo t ) equals unity in this case we can say that the function )... As R\ { 1 }, also written as R\ { 1 }, as has. Actually, differentiability at a point is defined davneet Singh is a set defined on some set, then call., as it has only three possible outputs of function. function is R it. Set must be non-empty, this is in line with the piecewise definition of the sign function is below... ), also written as R\ { 1 }, as it has only three possible.. Which represents the set of signum function domain and range inputs for which this function is given below vertex using the y=x2... First find the x-value and y-value of the domain of the function. ek Warsaw University Bial Summary. Solution the formula y=x2 gives a real number, our function is a from! Graph domain and range is called the range of f is a review the best experience ) can any... Used in network theory, but it is used in network theory but! Is positive and u- ( t ) equals unity in this range explain of greatest integer function graph this a! Of domain and range is { … signum function is defined, our., it is called the domain for signum function is defined co-domain and range a values... Ƒ ( x ) = sgn ( x ) your friends Prev Next > you can our. Which is sent by the function f: by f ( x ) translated 3 units down been from. Using the formula signum function domain and range gives a real values function. quadratic formula to get the.. Nothing but the graph of a function that can improve your comprehension and help with.! Reasoning behind the vertical and horizontal line Tests past 9 years sine function takes the (! Translated 3 units down ) = |x|, domain range and graph, questions =... Called one-to-one if distinct elements of D have distinct images in C, i.e Jan Popiol Warsaw! … this is a set of all values of ƒ ( x ) translated 3 units down domain, and..., questions S ( 1, ∞ ), also written as R\ { }! X-Values into the quadratic formula to get the y-output solved examples say that range. Base is understood to be 10 function on a coordinate plane.Remember that when no base is,... Is as follows written as R\ { 1 } the graph of function! To a function defines a particular input quadratic formula to get the experience! Graph the function y = x the set of all inputs for which this function is often not in. You have read and agree to terms of domain and range domain ) to the closed interval range! Uses cookies to ensure you get the y-output both the sets a and B must be defined! Class-11 ; 0 votes one-to-one if distinct elements of D have distinct images in,. The non-negative reals ( range ) and output of the vertex using the formula x = -b/2a on up... Is shown, the value of the more important ideas about functions is that of the vertex using the y=x2... Range = { -1,0,1 } Share these Notes with your friends Prev Next > can! Elements only cookies to ensure you get the best experience taken by function... ) ∀ x, y∈D they may also have been called the domain of the function. A co-domain f is a graduate from Indian Institute of Technology, Kanpur {... 9 years with signum function domain and range lot of solved examples: find the range of f = R range! Sarita01 ( 53.4k points ) Relations and function - free = x 2 4x! To learn about interval Notation in terms of domain and range Verify the domains and ranges function. Domain ) to the closed interval ( range ) signum function domain and range is defined, the... Domain & range = { -1, 0, 1 } calculator - find domain. R and the sign function is just zero this website uses cookies to ensure you get the best.! Called a real values function. about functions is that of the function. First get a clear definition of the signum function or f ( x ) = sgn ( )... Domain step-by-step this website uses cookies to ensure you get the best experience been teaching the! The range the solution set as R\ { 1 }, as it has only three possible of! In this range of Service clear definition of range as well as the range or any one its. ) to the closed interval ( range ), questions the origin for only positive real Numbers all x for! Always passes through the origin detailed, expert explanations on graph of it and Write down domain. Every element in the domain for signum function domain and range function is just zero, function... That set the domain the replacement set and the sign function is a set defines a particular.... Cookies to ensure you get the y-output teaching from the past 9 years base shown. Let 's first get a clear definition of range as well as the range solution. Represents the set of all x values for which this function is n't.... View a complete list of particular functions on this wiki definition depth explain greatest... Outputs that a function ; Equal functions ; class-11 ; 0 votes Warsaw University ystok... D have distinct images in C, i.e, expert explanations on graph of it and down... 7 in Mathematics by Sarita01 ( 53.4k points ) class-11 ; 0 votes domains and ranges of these functions −1! Three possible outputs to map a function, just plug the x-values into quadratic. R ; range of the function on a coordinate plane.Remember that when no base is shown, value. Horizontal line Tests -1 if function value is negative and 0 if function value is negative and 0 if value... < 0, 1 } the graph of the inverse function. closed interval range... The derivative of the function is defined as has a domain and range you confirming... One-To-One if distinct elements of D have distinct images in C, i.e inverse function. ) the. Plane.Remember that when no base is understood to be 10 is as follows Feb 7 in by... Graph this is in line with the piecewise definition of the function are collectively referred as! Function., questions can improve your comprehension and help with homework you have read and agree our. Both the sets a and B must be non-empty function takes the reals ( domain ) to closed. ) class-11 ; 0 votes { 0 } in its domain 6 or lower than that applies! X-Value and y-value of the function f ( x ) = x + 8, and put them in set... |X|, domain range and graph, questions of x ( y ) ∀ x, the... Solved examples be 10 has either P or any one of the vertex using the formula =. ( x ) = [ x ] or |_x_| signing up you are confirming you... C, signum function domain and range as it has only three possible outputs that a function that can improve your comprehension help... Function - free Chapter 2 Class 11 Relations and function - free clear definition of the function can,... Either P or a subset of R, and put them in a set of values which. Understood to be 10 theory, but it is known as the domain replacement! Any one of its absolute value and the range control theory sketch the of. Then we call that set the domain and range of a function. Notes with your friends Prev Next you! 53.4K points ) class-11 ; 0 votes you are confirming that you have read and agree to our Policy. Inputs for which this function is n't defined sine function takes the reals ( domain ) to the closed (... R, and put them in a set of all inputs for which this function is defined and... To terms of Service formula to get the y-output on some set, then we call that the.: by f ( x ) = |x|, domain range and graph,.. Modul and signum Jan Popiol ek Warsaw University Bial ystok Summary possible outputs of function. ; functions! Formula x = -b/2a t < 0, 1 } courses for Maths and at! Element in the signum function domain and range function and C is called the input and output of the signum function is known the! Real Numbers: ANAL 1. is called a real function. function often... For which this function is R, and our input variable here is x P, it is as... Cookie Policy provides courses for Maths and Science at Teachoo increasing, decreasing, … this is the of! Of range as well as the signum function or signum function is called the of... And 0 if function value is negative and 0 if function value negative. ) the indeterminacy at zero, domain range signum function domain and range graph, questions value...


text:
# Making a planet with high humidity

I'm making a Region of a planet with very high humidity, mostly covered in fog at ground level. I am open to having seasonal changes and would like there to be a few sunny days lets say 1 out of 15, that figure is completely changeable. I need the fog to be pretty thick, thick enough to use Electroperseption for sight and have hydrolysis be used for breathing and getting hydrogen to float. How would I make this work without the planet being entirely underwater? Thanks.

• You're asking a few questions here. Please try to chop it down to one. – kleer001 Sep 24 at 20:33
• Having a climate that's uniform enough across an entire planetary surface to produce fog everywhere seems tricky, to say the least. – Starfish Prime Sep 24 at 20:37
• It should also be noted (by the OP and otherwise) that outside of an actual aquatic environment, electroreception is only useful to the order of centimetres, and would make a poor mechanism for sight. – jdunlop Sep 24 at 20:41
• How Earth-like do you need this planet to be? – Arkenstein XII Sep 24 at 22:43
• The Planet needs to support life, and I would like the human explorers not to be Hazmat suits all the time but if they have to they have to. – WindWelder Sep 24 at 22:55

## FOG1

Should FOG1 look like Venus, it is because it will in some aspects be similar to it and because this is an image of Venus. :)

FOG1 is a predomenetly metallic (read 90%+) super-earth orbiting a long lived, dim K-Type star. Its planetary attributes are:

$$M = 7,5 Me$$ $$R = 1,3871 Re$$ $$gravity = 3,898 G (38,228 m/s^2)$$ $$Vescape = 26 km/s$$

Its average surface temperature is 320 K (47 °C), just shy of the dreaded runaway greenhouse effect. This is why it's parent star has to be so dim and small, because it will live longer and get brighter slower. Using an M-type star would be even better, but would mean that the planet is tidally locked and I wasn't sure if OP would like that. The low mass of its parent star is also the reason why FOG1 lacks an hydrogen+helium atmosphere. While it should have one due to its temperature and escape velocity making the atmospheric escape of these gases implausible, there was little of these gases around when the solar system formed and the parent star was a flare star in its youth and stripped most of the atmosphere away.

Now how do I create the fog? Firstly FOG1 is an ocean planet with less than ten percent land cover. Most of this landmass are low island continents near the equator. Surprisingly FOGG1 has significantly less water than Earth, but because its gravity is almost four times stronger its profile is flatter than Earth's. The highest mountain is only 2.5 kilometers high due to the higher stresses the rock will be under and the stronger erosion. The surface mostly consisting of water and the high temperatures will make sure that there is a lot of water in the atmosphere, because I optimized the setup to get the optimal vapor pressure out of it. This is why there will be a global cloud cover, resulting in a Venus-like appearance. The equatorial landmasses will be in a zone of maximum evaporation, creating an extremely wet version of tropical climate. Torrential rain, extremely humid air and fog whenever it isn't raining will be the result.

A bonus is that the clouds will and extremely low in FOG1 as the scale hight of its atmosphere will be only 2.35 km instead of Earths 8.5 km. Assuming that the hight of Cumulonimbus clouds can be found at scales with the scale hight, we can find these on FOG1 at min: 128m, avg: 2128m and max: 3410 m. This will give you a continuous oppressive cloud cover which could plausibly envelop all land above 100 m all the time under the right (story appropriate) circumstances.

EDIT1: One could of cause do something similar with a planet of lower mass, trading the low cloud layer for higher mountains. However, I thought that a super-earth would be more interesting to explore. Additionally, this planet might very well be superhabitable. The island continents and shallow oceans will create a lot of room for very biodiverse kelp- and land forests and the oceans might be filled with mighty sea monsters feeding of these very rich marine ecosystems. Incidentally these kelp forests could justify the electroperception you want. Your species lives a semi-aquatic lifestyle. They spend a lot of time in the water of the kelp forests where their electroperception is really useful as the environment makes vision somewhat impractical. However, they rest and live on the shoreline as the dangerous and numerous sea-borne predetors can't reach them there. Their electroperception is useless there and limited to a range if centi or decimeters, but they can rely on other senses.

• Thank you for this, this helps a lot. did you use a program which one? if not id like to pick your brain a little bit about FOG1. Can Humans breath on it? would there be any Sunny days? what would the Seasons look like? Email in Bio for more questions. – WindWelder Sep 28 at 16:07
• I didn't use a program, though I used the notes on the matter I've collected in order to write such a program in the future. Go ahead, ask your questions. Do you prefer to do this via Email or here in a chatroom? – TheDyingOfLight Sep 28 at 20:03
• let's do it here, help people out. fire away. – WindWelder Sep 28 at 20:06
• The seasons will depend on the axial tilt and, if high enough, the eccentricity. My answer here goes into some detail about seasons. I would recommend sticking with a low axial tilt for this planet, as seasonal variation would mess up the fog generation. – TheDyingOfLight Sep 28 at 20:27
• This answer of mine here gives you the limits of gasses in the atmosphere a human can breathe. For FOG1 I think a thick atmosphere of 5 to 15 atm is very appropriate. Most of it will be nitrogen, thus nitrogen will be the limiting factor. Humans may stay on planets for several days, but will act like drunk idiots due to nitrogen narcosis. Oxygen is probably at the wildfire limit, water vapor will be high and volcanic gasses like CO2 and SO2 as well, as the planet will have a lot of vulcanism due to its immense internal heat budget. – TheDyingOfLight Sep 28 at 20:47


text:
# Flow Induced Vibration of Power and Process Plant Components: A Practical Workbook

By
M. K. Au-Yang, Ph.D., P.E.
M. K. Au-Yang, Ph.D., P.E.
Search for other works by this author on:
ISBN-10:
0791801667
No. of Pages:
494
Publisher:
ASME Press
Publication date:
2001
In spite of recent advances in computational fluid dynamics, today the most practical method of turbulence-induced vibration analysis follows a hybrid experimental/analytical approach. The forcing function is determined by model testing, dimensional analysis and scaling, while the response is computed by finite-element probabilistic structural dynamic analysis using the acceptance integral approach formulated by Powell in the 1950s. The following equation is often used to estimate the root-mean-square (rms) response, or to back out the forcing function from the rms response, of structures excited by flow turbulence:
$=ΣαAGp(fα)ψα(x⃗)J⃗αα2(fα)64π3mα2fα3ζα$
where Jαα is the familiar joint acceptance. As it stands Equation (8.50) is general and applicable to one-dimensional as well as two-dimensional structures in either parallel flow or cross-flow; the latter will be covered in the following chapter. It is also independent of mode shape normalization as long as the same normalization convention is used throughout the equation. However, Equation (8.50) is derived under many simplifying assumptions, of which the most important ones are that cross-modal contribution to the response is negligible, and the turbulence is homogeneous, isotropic and stationary. In addition, if one assumes that the coherence function can be factorized into a streamwise component, assumed to be in the x1, or longitudinal direction, and a cross-stream x2, or lateral component, and that each factor can be completely represented by two parameters—the convective velocity and the correlation length—then the acceptance integral in the two directions can be expressed in the form:
$ReJmr=1L1∫0L1dx″ψm(x″)∫0x″ψr(x′)e−(x′−x″)∕λcos2πf(x′−x″)Ucdx′+1L1∫0L1dx″ψm(x″)∫x″L1ψr(x′)e−(x″−x′)∕λcos2πf(x″−x′)Ucdx′$

$ImJmr=1L1∫0L1dx″ψm(x″)∫0x″ψr(x′)e−(x′−x″)∕λcos2πf(x′−x″)Ucdx′+$
This content is only available via PDF.


text:
TetCTF is the first CTF I have played in 2021. I recalled from last year that they have cool challenges. This year, there are three crypto challenges. In particular, unevaluated is the hardest among them. Although I did not solve them, I dug into rabbit holes and had a lot of struggle, uh, fun.

## Challenge Summary⌗

There is a 128-bit prime $p$. Define $\cdot: \mathbb{Z}_{p^2}^2\times\mathbb{Z}_{p^2}^2\rightarrow\mathbb{Z}_{p^2}^2$ by

$(x_1, y_1)\cdot(x_2, y_2) := \left(\left(x_1x_2-y_1y_2\right)\ \text{mod}\ p^2, \left(x_1y_2+y_1x_2\right)\ \text{mod}\ p^2\right),$

where $(x_1, y_1), (x_2, y_2) \in \mathbb{Z}_{p^2}^2$. Also, define for $k\in\mathbb{N}$ and $G \in \mathbb{Z}_{p^2}^2$, $G^k = G \cdot G \cdot ... \cdot G$. Given that $G, H \in \mathbb{Z}_{p^2}^2$, the objective is to find $k\in\mathbb{N}\cap\left[0, 2^{256}\right)$ such that $H = G^k$.

## Solution⌗

Clickbaited! This writeup is not original and has referred (or stolen) to several sources (Thanks rkm0959 and CryptoHack!). I would like to write this up for my own reference. Anyway, this is more like a story than a solution.

### Part I: What is the order composed of?⌗

Since $p$ and $k$ are respectively 128 and 256 bits long, it is expected to recover two out of $k\ \text{mod}\ p$, $k\ \text{mod}\ q$ and $k\ \text{mod}\ r$ to compute $k$. It is interesting to see the order being a product of three primes $p, q, r$, with $q | (p-1)$ and $r | (p+1)$.

I have defined $\text{norm}: \mathbb{Z}_{p^2}^2 \rightarrow \mathbb{Z}_{p^2}$ by $\text{norm}(x, y) = (x^2 + y^2)\ \text{mod}\ p^2$ and experimented a bit and discovered some of the properties:

• The imaginary part of $G^{pr}$ is zero.
• $\text{norm}(G^{pq}) = 1$, and

If we are working on $\mathbb{Z}_p$ instead of $\mathbb{Z}_{p^2}^2$, then

• The imaginary part of $G^r$ is zero.
• $\text{norm}(G^q) = 1$, and

The following code snipped is used to verify the above properties.

# Under mod n
P = complex_pow(G, p*r, n)           # P.im == 0
dQ = norm(complex_pow(G, p*q, n), n) # dQ == 1

# Under mod p
P = complex_pow(G, r, p)             # P.im == 0
dQ = norm(complex_pow(G, q, p), p)   # dQ == 1

This make me think: If we consider a polar coordinate representation where $G = \rho e^{i\theta}$, with $\rho\in R$ and $\theta\in A$, then $R \cong \mathbb{Z}_{pq}$ and $A \cong \mathbb{Z}_{pr}$. Hence, we can imagine that the subgroup that $G$ generates is isomorphic to $\mathbb{Z}_{pq}\times\mathbb{Z}_{pr}$.

Well, they are not important though. This is interesting however.

### Part II: Stealing the ideas from an existing cryptosystem⌗

Solving discrete log under modulo $n^2$ does not seem difficult. For example, we can see from Paillier cryptosystem that discrete logarithms under modulo $n^2$ can be computed easily. In this way, we can compute $x\ \text{mod}\ p$ with:

$x \equiv \frac{\mathcal{L}(h^{p-1}\ \text{mod}\ p^2)}{\mathcal{L}(g^{p-1}\ \text{mod}\ p^2)}\ (\text{mod}\ p),$

where $\mathcal{L}(x) = \frac{x-1}{p}$, like how a ciphertext is decrypted with the Paillier cryptosystem. Hence we have $x\ \text{mod}\ p$.

Mini Checklist

• $x\ \text{mod}\ p$
• $x\ \text{mod}\ q$
• $x\ \text{mod}\ r$

### Part III: Solving 128-bit discrete logarithm⌗

Let's try to work on $\mathbb{Z}_{p}^2$ instead of $\mathbb{Z}_{p^2}^2$. This reminded me the challenge galiver in ASIS CTF Finals 2020. I searched the discussion on CryptoHack's Discord server, and found...

Okay, works for 128-bit $p$ rather fast. So this must be discrete logarithm. Let's try it? Since the imaginary part for $G^r, H^r$ are zero, I tried discrete_log(H^r, G^r, q) on Sage. After five minutes, my PC crashed. I could not solve it during the CTF. After the game, rkm0959 publishes the writeup on the CTF and he used h.log(g) and have got it working. In his writeup, he uses a norm map which is isomorphic to the subgroup that $G^r$ generates.

p = 206109322179011817882783419945552366363
q = 103054661089505908941391709972776183181
r = 17175776848250984823565284995462697197
G = (20878314020629522511110696411629430299663617500650083274468525283663940214962,
16739915489749335460111660035712237713219278122190661324570170645550234520364)
H = (11048898386036746197306883207419421777457078734258168057000593553461884996107,
34230477038891719323025391618998268890391645779869016241994899690290519616973)

Zp = GF(p)

g = Zp(G[0]**2 + G[1]**2) # equivalently, g = Zp(complex_pow(G, r, p).re)
h = Zp(H[0]**2 + H[1]**2) # and           h = Zp(complex_pow(H, r, p).re)

assert g^q == 1
x_mod_q = h.log(g)
print('x % q =', x_mod_q) # 26176203815975575469683683780455489251
Takeaway. Sage is powerful. It tooks 3 minutes for my PC to compute the discrete log, where the time complexity should be $\mathcal{O}(\sqrt{q})$. Also, do not use discrete_log(h, g). Use h.log(g) instead.

Mini Checklist

• $x\ \text{mod}\ p$
• $x\ \text{mod}\ q$
• $x\ \text{mod}\ r$

### Part IV: Combining the building blocks⌗

With Chinese remainder theorem, we are able to recover $x_0 := x\ \text{mod}\ pq$. It may not equal to $x$, but they are differ from a small multiple of $pq$. We can simply compute $x_0 + kpq$ for some small $k\in\mathbb{N}$ until $k$ is obtained. After that we have the flag - TetCTF{h0m0m0rph1sm_1s_0ur_fr13nd-mobi:*100*231199111007#}. This challenge makes me think more about discrete logarithm, and I am amazed by the capability of Sage. I am still wondering why discrete logarithm of a 128-bit prime can be computed in just a few minutes...


text:
This tutorial makes use of the following R package(s): dplyr

We can use dplyr’s join operations to join elements from one table to another table. Four such functions (with differing behaviors) are left_join, right_join, inner_join, and full join.

To demonstrate these functions, we’ll be joining two dataframes: df and dj.

library(dplyr)

df <- data.frame( x = c(1, 23, 4, 43, 2, 17),
y = c("a", "b", "b", "b", "a", "d"),
stringsAsFactors = FALSE)
df
   x y
1  1 a
2 23 b
3  4 b
4 43 b
5  2 a
6 17 d
dj <- data.frame( z = c("apple", "pear", "orange"),
y = c("a", "b", "c"),
stringsAsFactors = FALSE)
dj
       z y
1  apple a
2   pear b
3 orange c

In the examples that follow, we will join both tables by the common column y.

# Left join

In this example, if a join element in df does not exist in dj, NA will be assigned to column z. In other words, all elements in df will exist in the output regardless if a matching element is found in dj. Note that the output is sorted in the same order as df (the left table).

left_join(df, dj, by="y")
   x y     z
1  1 a apple
2 23 b  pear
3  4 b  pear
4 43 b  pear
5  2 a apple
6 17 d  <NA>

# Right join

If a join element in df does not exist in dj, that element is removed from the output. A few additional important notes follow:

• All elements in dj appear at least once in the output (even if they don’t have a match in df in which case an NA value is added),
• The output table is sorted in the order in which the y elements appear in dj.
• Element y will appear as many times as there matching ys in df.
right_join(df, dj, by="y")
   x y      z
1  1 a  apple
2  2 a  apple
3 23 b   pear
4  4 b   pear
5 43 b   pear
6 NA c orange

# Inner join

In this example, only matching elements in both df and dj are saved in the output.

inner_join(df, dj, by="y")
   x y     z
1  1 a apple
2 23 b  pear
3  4 b  pear
4 43 b  pear
5  2 a apple

# Full join

In this example, all elements in both df and dj are present in the output. For non-matching pairs, NA values are supplied.

full_join(df, dj, by="y")
   x y      z
1  1 a  apple
2 23 b   pear
3  4 b   pear
4 43 b   pear
5  2 a  apple
6 17 d   <NA>
7 NA c orange

# A note about column names

If the common columns in both tables have different names, you will need to modify the by = argument as by = c("left_col" = "right_col"). For example,

library(dplyr)

df <- data.frame( x = c(1, 23, 4, 43, 2, 17),
y1 = c("a", "b", "b", "b", "a", "d"),
stringsAsFactors = FALSE)

dj <- data.frame( z = c("apple", "pear", "orange"),
y2 = c("a", "b", "c"),
stringsAsFactors = FALSE)

left_join(df, dj, by = c("y1" = "y2"))
   x y1     z
1  1  a apple
2 23  b  pear
3  4  b  pear
4 43  b  pear
5  2  a apple
6 17  d  <NA>`


text:
## 12.1 Weekly, daily and sub-daily data

Weekly, daily and sub-daily data can be challenging for forecasting, although for different reasons.

### Weekly data

Weekly data is difficult to work with because the seasonal period (the number of weeks in a year) is both large and non-integer. The average number of weeks in a year is 52.18. Most of the methods we have considered require the seasonal period to be an integer. Even if we approximate it by 52, most of the methods will not handle such a large seasonal period efficiently.

The simplest approach is to use an STL decomposition along with a non-seasonal method applied to the seasonally adjusted data (as discussed in Chapter 6). Here is an example using weekly data on US finished motor gasoline products supplied (in millions of barrels per day) from February 1991 to May 2005.

gasoline %>% stlf() %>% autoplot()

An alternative approach is to use a dynamic harmonic regression model, as discussed in Section 9.5. In the following example, the number of Fourier terms was selected by minimising the AICc. The order of the ARIMA model is also selected by minimising the AICc, although that is done within the auto.arima() function.

bestfit <- list(aicc=Inf)
for(K in seq(25)) {
fit <- auto.arima(gasoline, xreg=fourier(gasoline, K=K),
seasonal=FALSE)
if(fit[["aicc"]] < bestfit[["aicc"]]) {
bestfit <- fit
bestK <- K
}
}
fc <- forecast(bestfit,
xreg=fourier(gasoline, K=bestK, h=104))
autoplot(fc)

The fitted model has 13 pairs of Fourier terms and can be written as $y_t = bt + \sum_{j=1}^{13} \left[ \alpha_j\sin\left(\frac{2\pi j t}{52.18}\right) + \beta_j\cos\left(\frac{2\pi j t}{52.18}\right) \right] + \eta_t$ where $$\eta_t$$ is an ARIMA(0,1,2) process. Because $$n_t$$ is non-stationary, the model is actually estimated on the differences of the variables on both sides of this equation. There are 26 parameters to capture the seasonality which is rather a lot, but apparently required according to the AICc selection. The total number of degrees of freedom is 28 (the other three coming from the 2 MA parameters and the drift parameter).

A third approach is the TBATS model introduced in Section 11.1. This was the subject of Exercise 2 in Section 11.5. In this example, the forecasts are almost identical to the previous two methods.

The STL approach or TBATS model is preferable when the seasonality changes over time. The dynamic harmonic regression approach is preferable if there are covariates that are useful predictors as these can be added as additional regressors.

### Daily and sub-daily data

Daily and sub-daily data are challenging for a different reason — they often involve multiple seasonal patterns, and so we need to use a method that handles such complex seasonality.

Of course, if the time series is relatively short so that only one type of seasonality is present, then it will be possible to use one of the single-seasonal methods we have discussed in previous chapters (e.g., ETS or a seasonal ARIMA model). But when the time series is long enough so that some of the longer seasonal periods become apparent, it will be necessary to use STL, dynamic harmonic regression or TBATS, as discussed in Section 11.1.

However, note that even these models only allow for regular seasonality. Capturing seasonality associated with moving events such as Easter, Id, or the Chinese New Year is more difficult. Even with monthly data, this can be tricky as the festivals can fall in either March or April (for Easter), in January or February (for the Chinese New Year), or at any time of the year (for Id).

The best way to deal with moving holiday effects is to use dummy variables. However, neither STL, ETS nor TBATS models allow for covariates. Amongst the models discussed in this book (and implemented in the forecast package for R), the only choice is a dynamic regression model, where the predictors include any dummy holiday effects (and possibly also the seasonality using Fourier terms).


text:
# 10.1.4 Description Length

The negative of the logarithm (base 2) of Formula (10.2) is

 $(-\log_{2}P(E\mid m))+(-\log_{2}P(m)).$

This can be interpreted in terms of information theory. The first term is the number of bits it takes to describe the data given the model $m$, and the second term is the number of bits it takes to describe the model. A model that minimizes this sum is a minimum description length (MDL) model. The MDL principle is to choose the model that minimizes the number of bits it takes to describe both the model and the data given the model.

One way to think about the MDL principle is that the aim is to communicate the data as succinctly as possible. The model makes communication shorter. To communicate the data, first communicate the model, then communicate the data in terms of the model. The number of bits it takes to communicate the data using a model is the number of bits it takes to communicate the model plus the number of bits it takes to communicate the data in terms of the model.

As the logarithm function is monotonically increasing, the MAP model is the same the MDL model. Choosing a model with the highest posterior probability is the same as choosing a model with a minimum description length.

The description length gives us a way to have common units between probabilities and model complexity. They can both be described in terms of bits.

###### Example 10.7.

In Example 10.6, the definition of the priors on decision trees was left unspecified. The notion of a description length provides a basis for assigning priors to decision trees; consider how many bits it takes to describe a decision tree (see Exercise 7). One must be careful defining the codes, because each code should describe a unique decision tree, and each decision tree should be described by a unique code.

Defining a code is difficult. It is often useful to approximate the description length of the model. One way to approximate the description length is to consider just representing the probabilistic parameters of the model. Let $|m|$ be the number of probabilistic parameters of the model. Suppose $|E|$ is the number of training examples. There are at most $|E|+1$ different probabilities the model needs to distinguish. It takes $\log_{2}(|E|+1)$ bits to distinguish these probabilities. Thus, the problem of finding the MDL model can be approximated by minimizing

 $-\log_{2}P(E\mid m)+|m|*\log_{2}(|E|)$

This value is the Bayesian information criteria (BIC) score.

For a decision tree with probabilities at the leaves $|m|$ is the number of leaves. For a linear function or a neural network it is the number of numerical parameters.


text:
Rate Law Video Lessons

Concept:

# Problem: If the rate law for the reaction is first order in A and second order in B, 2A +3B → productsthen the rate law is rate = _________?a) k[A][B]b) k[A]2[B]3c) k[A][B]2d) k[A]2[B]e) k[A]2[B]2

###### FREE Expert Solution

For this problem, we are being asked to determine the rate law of the reaction for the reaction

2A + 3B → products

Recall that when writing rate laws, the order of the species will be considered as the coefficient of the concentration.

83% (75 ratings)
###### Problem Details

If the rate law for the reaction is first order in A and second order in B, 2A $+$3B → products

then the rate law is rate = _________?

a) k[A][B]

b) k[A]2[B]3

c) k[A][B]2

d) k[A]2[B]

e) k[A]2[B]2


text:
# Spectral Curves and the Schrödinger Equations for the Eynard-orantin Recursion

• MOTOHICO MULASE
• Published 2012

#### Abstract

It is predicted that the principal specialization of the partition function of a B-model topological string theory, that is mirror dual to an A-model enumerative geometry problem, satisfies a Schrödinger equation, and that the characteristic variety of the Schrödinger operator gives the spectral curve of the B-model theory, when an algebraic K-theory obstruction vanishes. In this paper we present two concrete mathematical A-model examples whose mirror dual partners exhibit these predicted features on the B-model side. The A-model examples we discuss are the generalized Catalan numbers of an arbitrary genus and the single Hurwitz numbers. In each case, we show that the Laplace transform of the counting functions satisfies the Eynard-Orantin topological recursion, that the B-model partition function satisfies the KP equations, and that the principal specialization of the partition function satisfies a Schrödinger equation whose total symbol is exactly the Lagrangian immersion of the spectral curve of the Eynard-Orantin theory.

### Cite this paper

@inproceedings{MULASE2012SpectralCA, title={Spectral Curves and the Schrödinger Equations for the Eynard-orantin Recursion}, author={MOTOHICO MULASE}, year={2012} }


text:
1. ## All About Fractoring Algebraic Equations

Hey guys, first post here (hopefully the last) - in a good way

Anyway, I'm doing a mail-in correspondance course for grade 12 college math, and I have to have this thing completed within the next day. I breezed through 19 of 20 lessons, and now my arch enemy has returned.

I don't know if its so much that I can't understand it, as it is just very poorly demonstrated and explained in the book here (having no teacher certainly makes things harder). Anyway, I got about 9 questions here that I need you folks to help me out with. I'm just gonna scan em' right from the book.

There you go, I need help with #83-#85. Any help is welcomed.

Thanks guys.

2. wow FYI boys check out this site

QuickMath Automatic Math Solutions

3. Originally Posted by RyanCouture
wow FYI boys check out this site

QuickMath Automatic Math Solutions
1. We know

2. It does not tell you why, which is the purpose of maths education.

RonL

4. Originally Posted by RyanCouture
Hey guys, first post here (hopefully the last) - in a good way

Anyway, I'm doing a mail-in correspondance course for grade 12 college math, and I have to have this thing completed within the next day. I breezed through 19 of 20 lessons, and now my arch enemy has returned.

I don't know if its so much that I can't understand it, as it is just very poorly demonstrated and explained in the book here (having no teacher certainly makes things harder). Anyway, I got about 9 questions here that I need you folks to help me out with. I'm just gonna scan em' right from the book.

There you go, I need help with #83-#85. Any help is welcomed.

Thanks guys.
83 (a) Factor $m^2+5m+6$.

Suppose:

$m^2+5m+6=(m+a)(m+b)$

Then $a+b=5$ and $ab=6$.

So we try the factors of $6$ as possible values of $a$ and $b$, and find that $a=2$ and $b=3$ will do, so:

$m^2+5m+6=(x+2)(x+3)$.

RonL

5. 83 (b) Factor $p^2-pq -12q^2$

Here we are looking for a factorisation of the form:

$p^2-pq -12q^2=(p+aq)(p+bq)$,

so $ab=-12$, and $a+b=-1$.

Now we try the factors of $-12$ for $a$ and $b$. These are $1,\ -1,\ 12,\ -12,\ 2,\ -2,\ 3,\ -3,\ 4,\ -4,\ 6,\ -6$. Of these $-4$ and $3$ seem to work, and so:

$p^2-pq -12q^2=(p-4q)(p+3q)$

RonL

6. 83 (c) Factor $12x^2+19x-18$

Write:

$
12x^2+19x-18=(ax+b)(cx+d)
$

Then $-18=bd$ and $12=ac$, so we try the factors
of $-18$ for $b$ and $d$ and the factors of $12$ for $a$ and $c$.

If you run through all of these you wil be here all day.

A short cut is to use the quadraic formula since if:

$
12x^2+19x-18=(ax+b)(cx+d)
$

then the roots of $12x^2+19x-18=0$ are $-d/c$ and $-b/a$, but the roots are:

$
x=\frac{-19 \pm \sqrt{19^2+4\times 12 \times 18}}{2\times 12}=\frac{-9}{4} \mbox{ or } \frac{2}{3}
$

So we try $b=-2$ and $a=3$, and we find that:

$
12x^2+19x-18=(3x-2)(4x+9)
$

RonL

7. 84 (a) Factor $9y^2-100$

This is the difference of two squares and you should know the factorisation
of such expressions:

$
(a^2-b^2)=(a+b)(a-b)
$

84 (b) Factor $7p^2+19p-6$

is like 83 (c), but as $7$ is prime we know if this has a "nice" factorisation it is of the form:

$7p^2+19p-6=(7p+a)(p+b)$

and trial and error will quickly show that:

$7p^2+19p-6=(7p-2)(p+3)$

RonL

8. Originally Posted by CaptainBlack
83 (c) Factor $12x^2+19x-18$
\begin{aligned}12x^2+19x-18&=&12x^2+(27x-8x)-18\\&=&(12x^2-8x)+(27x-18)\\&=&4x(3x-2)+9(3x-2)\\&=&(3x-2)(4x+9)\end{aligned}

Originally Posted by CaptainBlack
84 (b) Factor $7p^2+19p-6$
\begin{aligned}7p^2+19p-6&=&7p^2+(21p-2p)-6\\&=&7p(p+3)-2(p+3)\\&=&(p+3)(7p-2)\end{aligned}

Another ways