text:
# Return type of getchar(), fgetc() and getc() in C

Details about getchar(), fgetc() and getc() functions in C programming are given as follows −

## The getchar() function

The getchar() function obtains a character from stdin. It returns the character that was read in the form of an integer or EOF if an error occurs.

A program that demonstrates this is as follows −

## Example

Live Demo

#include <stdio.h>

int main (){
int i;

printf("Enter a character: ");
i = getchar();

printf("The character entered is: ");
putchar(i);

return(0);
}

## Output

The output of the above program is as follows −

Enter a character: G
The character entered is: G

Now let us understand the above program.

The value obtained using the getchar() function is stored in i which is an integer variable. Then the character value is displayed using putchar(). The code snippet that shows this is as follows −

int i;

printf("Enter a character: ");
i = getchar();

printf("The character entered is: ");
putchar(i);

## The fgetc() function

The fgetc() function obtains a character from a file stream which is a pointer to a FILE object. This function returns the character that was read in the form of an integer or EOF if an error occurs.

A program that demonstrates this is as follows −

## Example

Live Demo

#include <stdio.h>

int main (){
FILE *fp;
fp = fopen("file.txt", "w");
fprintf(fp, "Apple");
fclose(fp);

int i;

fp = fopen("file.txt","r");

if(fp == NULL){
perror("Error in opening file");
return(-1);
}

while((i=fgetc(fp))!=EOF){
printf("%c",i);
}

fclose(fp);
return(0);
}

## Output

The output of the above program is as follows −

Apple

Now let us understand the above program.

First, the file is created and the data "Apple" is stored inside it. Then the file is closed. The code snippet that shows this is as follows −

FILE *fp;
fp = fopen("file.txt", "w");
fprintf(fp, "Apple");
fclose(fp);

The file is opened again in reading mode. If fp is NULL then error message is displayed. Otherwise the contents of the file are displayed using the fgetc() function. The code snippet that shows this is as follows −

fp = fopen("file.txt","r");

if(fp == NULL){
perror("Error in opening file");
return(-1);
}

while((i=fgetc(fp))!=EOF){
printf("%c",i);
}

fclose(fp);

## The getc() function

The getc() function obtains a character from the stream that is specified. It returns the character that was read in the form of an integer or EOF if an error occurs.

A program that demonstrates this is as follows −

## Example

Live Demo

#include <stdio.h>

int main (){
int i;

printf("Enter a character: ");
i = getc(stdin);

printf("The character entered is: ");
putchar(i);

return(0);
}

## Output

The output of the above program is as follows −

Enter a character: K
The character entered is: K

Now let us understand the above program.

The getc() function obtains a character from the stream stdin as specified. This value is stored in int variable i. Then the character value is displayed using putchar(). The code snippet that shows this is as follows −

int i;

printf("Enter a character: ");
i = getc(stdin);

printf("The character entered is: ");
putchar(i);
karthikeya Boyini

I love programming (: That's all I know


text:
### Flight (5817 views - Transportation - Air Water Earth)

Flight is the process by which an object moves through an atmosphere (or beyond it, as in the case of spaceflight) without contact with the surface. This can be achieved by generating aerodynamic lift associated with propulsive thrust, aerostatically using buoyancy, or by ballistic movement. Many things can fly, from natural aviators such as birds, bats, and insects, to human inventions like aircraft, including airplanes, helicopters, balloons, and rockets which may carry spacecraft. The engineering aspects of flight are the purview of aerospace engineering which is subdivided into aeronautics, the study of vehicles that travel through the air, and astronautics, the study of vehicles that travel through space, and in ballistics, the study of the flight of projectiles.
Go to Article

## Flight

### Flight

Flight is the process by which an object moves through an atmosphere (or beyond it, as in the case of spaceflight) without contact with the surface. This can be achieved by generating aerodynamic lift associated with propulsive thrust, aerostatically using buoyancy, or by ballistic movement.

Many things can fly, from natural aviators such as birds, bats, and insects, to human inventions like aircraft, including airplanes, helicopters, balloons, and rockets which may carry spacecraft.

The engineering aspects of flight are the purview of aerospace engineering which is subdivided into aeronautics, the study of vehicles that travel through the air, and astronautics, the study of vehicles that travel through space, and in ballistics, the study of the flight of projectiles.

## Types of flight

### Buoyant flight

Humans have managed to construct lighter than air vehicles that raise off the ground and fly, due to their buoyancy in air.

An aerostat is a system that remains aloft primarily through the use of buoyancy to give an aircraft the same overall density as air. Aerostats include free balloons, airships, and moored balloons. An aerostat's main structural component is its envelope, a lightweight skin that encloses a volume of lifting gas[1][2] to provide buoyancy, to which other components are attached.

Aerostats are so named because they use "aerostatic" lift, a buoyant force that does not require lateral movement through the surrounding air mass to effect a lifting force. By contrast, aerodynes primarily use aerodynamic lift, which requires the lateral movement of at least some part of the aircraft through the surrounding air mass.

### Aerodynamic flight

#### Unpowered flight versus powered flight

Some things that fly do not generate propulsive thrust through the air, for example, the flying squirrel. This is termed gliding. Some other things can exploit rising air to climb such as raptors (when gliding) and man-made sailplane gliders. This is termed soaring. However most other birds and all powered aircraft need a source of propulsion to climb. This is termed powered flight.

#### Animal flight

The only groups of living things that use powered flight are birds, insects, and bats, while many groups have evolved gliding. The extinct Pterosaurs, an order of reptiles contemporaneous with the dinosaurs, were also very successful flying animals. Each of these groups' wings evolved independently. The wings of the flying vertebrate groups are all based on the forelimbs, but differ significantly in structure; those of insects are hypothesized to be highly modified versions of structures that form gills in most other groups of arthropods.[3]

Bats are the only mammals capable of sustaining level flight (see bat flight).[4] However, there are several gliding mammals which are able to glide from tree to tree using fleshy membranes between their limbs; some can travel hundreds of meters in this way with very little loss in height. Flying frogs use greatly enlarged webbed feet for a similar purpose, and there are flying lizards which fold out their mobile ribs into a pair of flat gliding surfaces. "Flying" snakes also use mobile ribs to flatten their body into an aerodynamic shape, with a back and forth motion much the same as they use on the ground.

Flying fish can glide using enlarged wing-like fins, and have been observed soaring for hundreds of meters. It is thought that this ability was chosen by natural selection because it was an effective means of escape from underwater predators. The longest recorded flight of a flying fish was 45 seconds.[5]

Most birds fly (see bird flight), with some exceptions. The largest birds, the ostrich and the emu, are earthbound, as were the now-extinct dodos and the Phorusrhacids, which were the dominant predators of South America in the Cenozoic era. The non-flying penguins have wings adapted for use under water and use the same wing movements for swimming that most other birds use for flight.[citation needed] Most small flightless birds are native to small islands, and lead a lifestyle where flight would offer little advantage.

Among living animals that fly, the wandering albatross has the greatest wingspan, up to 3.5 meters (11 feet); the great bustard has the greatest weight, topping at 21 kilograms (46 pounds).[6]

Most species of insects can fly as adults. Insect flight makes use of either of two basic aerodynamic models: creating a leading edge vortex, found in most insects, and using clap and fling, found in very small insects such as thrips.[7][8]

#### Mechanical

Mechanical flight is the use of a machine to fly. These machines include aircraft such as airplanes, gliders, helicopters, autogyros, airships, balloons, ornithopters as well as spacecraft. Gliders are capable of unpowered flight. Another form of mechanical flight is para-sailing where a parachute-like object is pulled by a boat. In an airplane, lift is created by the wings; the shape of the wings of the airplane are designed specially for the type of flight desired. There are different types of wings: tempered, semi-tempered, sweptback, rectangular and elliptical. An aircraft wing is sometimes called an airfoil, which is a device that creates lift when air flows across it.

##### Supersonic

Supersonic flight is flight faster than the speed of sound. Supersonic flight is associated with the formation of shock waves that form a sonic boom that can be heard from the ground,[9] and is frequently startling. This shockwave takes quite a lot of energy to create and this makes supersonic flight generally less efficient than subsonic flight at about 85% of the speed of sound.

##### Hypersonic

Hypersonic flight is very high speed flight where the heat generated by the compression of the air due to the motion through the air causes chemical changes to the air. Hypersonic flight is achieved by reentering spacecraft such as the Space Shuttle and Soyuz.

### Ballistic

#### Atmospheric

Some things generate little or no lift and move only or mostly under the action of momentum, gravity, air drag and in some cases thrust. This is termed ballistic flight. Examples include balls, arrows, bullets, fireworks etc.

#### Spaceflight

Essentially an extreme form of ballistic flight, spaceflight is the use of space technology to achieve the flight of spacecraft into and through outer space. Examples include ballistic missiles, orbital spaceflight etc.

Spaceflight is used in space exploration, and also in commercial activities like space tourism and satellite telecommunications. Additional non-commercial uses of spaceflight include space observatories, reconnaissance satellites and other earth observation satellites.

A spaceflight typically begins with a rocket launch, which provides the initial thrust to overcome the force of gravity and propels the spacecraft from the surface of the Earth.[10] Once in space, the motion of a spacecraft—both when unpropelled and when under propulsion—is covered by the area of study called astrodynamics. Some spacecraft remain in space indefinitely, some disintegrate during atmospheric reentry, and others reach a planetary or lunar surface for landing or impact.

## History

Many human cultures have built devices that fly, from the earliest projectiles such as stones and spears,[11][12] the boomerang in Australia, the hot air Kongming lantern, and kites.

### Aviation

George Cayley studied flight scientifically in the first half of the 19th century,[13][14][15] and in the second half of the 19th century Otto Lilienthal made over 200 gliding flights and was also one of the first to understand flight scientifically. His work was replicated and extended by the Wright brothers who made gliding flights and finally the first controlled and extended, manned powered flights.[16]

### Spaceflight

Spaceflight, particularly human spaceflight became a reality in the 20th century following theoretical and practical breakthroughs by Konstantin Tsiolkovsky and Robert H. Goddard. The first orbital spaceflight was in 1957[17], and Yuri Gagarin was carried aboard the first manned orbital spaceflight in 1961.[18]

## Physics

There are different approaches to flight. If an object has a lower density than air, then it is buoyant and is able to float in the air without expending energy. A heavier than air craft, known as an aerodyne, includes flighted animals and insects, fixed-wing aircraft and rotorcraft. Because the craft is heavier than air, it must generate lift to overcome its weight. The wind resistance caused by the craft moving through the air is called drag and is overcome by propulsive thrust except in the case of gliding.

Some vehicles also use thrust for flight, for example rockets and Harrier Jump Jets.

Finally, momentum dominates the flight of ballistic flying objects.

### Forces

Forces relevant to flight are[19]

These forces must be balanced for stable flight to occur.

#### Thrust

A fixed-wing aircraft generates forward thrust when air is pushed in the direction opposite to flight. This can be done in several ways including by the spinning blades of a propeller, or a rotating fan pushing air out from the back of a jet engine, or by ejecting hot gases from a rocket engine.[20] The forward thrust is proportional to the mass of the airstream multiplied by the difference in velocity of the airstream. Reverse thrust can be generated to aid braking after landing by reversing the pitch of variable-pitch propeller blades, or using a thrust reverser on a jet engine. Rotary wing aircraft and thrust vectoring V/STOL aircraft use engine thrust to support the weight of the aircraft, and vector sum of this thrust fore and aft to control forward speed.

#### Lift

In the context of an air flow relative to a flying body, the lift force is the component of the aerodynamic force that is perpendicular to the flow direction.[21] Aerodynamic lift results when the wing causes the surrounding air to be deflected - the air then causes a force on the wing in the opposite direction, in accordance with Newton's third law of motion.

Lift is commonly associated with the wing of an aircraft, although lift is also generated by rotors on rotorcraft (which are effectively rotating wings, performing the same function without requiring that the aircraft move forward through the air). While common meanings of the word "lift" suggest that lift opposes gravity, aerodynamic lift can be in any direction. When an aircraft is cruising for example, lift does oppose gravity, but lift occurs at an angle when climbing, descending or banking. On high-speed cars, the lift force is directed downwards (called "down-force") to keep the car stable on the road.

Lift can also occur in a different way if the air is not still, especially if there is an updraft due to heat ("thermals") or wind blowing along sloping terrain or other meteorological conditions. This form of lift permits soaring and is particularly important for gliding. It is used by birds and gliders to stay in the air for long periods with little effort.

#### Drag

For a solid object moving through a fluid, the drag is the component of the net aerodynamic or hydrodynamic force acting opposite to the direction of the movement.[22][23][24][25] Therefore, drag opposes the motion of the object, and in a powered vehicle it must be overcome by thrust. The process which creates lift also causes some drag.

#### Lift-to-drag ratio

Aerodynamic lift is created by the motion of an aerodynamic object (wing) through the air, which due to its shape and angle deflects the air. For sustained straight and level flight, lift must be equal and opposite to weight. In general, long narrow wings are able deflect a large amount of air at a slow speed, whereas smaller wings need a higher forward speed to deflect an equivalent amount of air and thus generate an equivalent amount of lift. Large cargo aircraft tend to use longer wings with higher angles of attack, whereas supersonic aircraft tend to have short wings and rely heavily on high forward speed to generate lift.

However, this lift (deflection) process inevitably causes a retarding force called drag. Because lift and drag are both aerodynamic forces, the ratio of lift to drag is an indication of the aerodynamic efficiency of the airplane. The lift to drag ratio is the L/D ratio, pronounced "L over D ratio." An airplane has a high L/D ratio if it produces a large amount of lift or a small amount of drag. The lift/drag ratio is determined by dividing the lift coefficient by the drag coefficient, CL/CD. [26]

The lift coefficient Cl is equal to the lift L divided by the (density r times half the velocity V squared times the wing area A). [Cl = L / (A * .5 * r * V^2)] The lift coefficient is also affected by the compressibility of the air, which is much greater at higher speeds, so velocity V is not a linear function. Compressibility is also affected by the shape of the aircraft surfaces. [27]

The drag coefficient Cd is equal to the drag D divided by the (density r times half the velocity V squared times the reference area A). [Cd = D / (A * .5 * r * V^2)] [28]

Lift-to-drag ratios for practical aircraft vary from about 4:1 for vehicles and birds with relatively short wings, up to 60:1 or more for vehicles with very long wings, such as gliders. A greater angle of attack relative to the forward movement also increases the extent of deflection, and thus generates extra lift. However a greater angle of attack also generates extra drag.

Lift/drag ratio also determines the glide ratio and gliding range. Since the glide ratio is based only on the relationship of the aerodynamics forces acting on the aircraft, aircraft weight will not affect it. The only effect weight has is to vary the time that the aircraft will glide for – a heavier aircraft gliding at a higher airspeed will arrive at the same touchdown point in a shorter time. [29]

#### Buoyancy

Air pressure acting up against an object in air is greater than the pressure above pushing down. The buoyancy, in both cases, is equal to the weight of fluid displaced - Archimedes' principle holds for air just as it does for water.

A cubic meter of air at ordinary atmospheric pressure and room temperature has a mass of about 1.2 kilograms, so its weight is about 12 newtons. Therefore, any 1-cubic-meter object in air is buoyed up with a force of 12 newtons. If the mass of the 1-cubic-meter object is greater than 1.2 kilograms (so that its weight is greater than 12 newtons), it falls to the ground when released. If an object of this size has a mass less than 1.2 kilograms, it rises in the air. Any object that has a mass that is less than the mass of an equal volume of air will rise in air - in other words, any object less dense than air will rise.

#### Thrust to weight ratio

Thrust-to-weight ratio is, as its name suggests, the ratio of instantaneous thrust to weight (where weight means weight at the Earth's standard acceleration ${\displaystyle g_{0}}$).[30] It is a dimensionless parameter characteristic of rockets and other jet engines and of vehicles propelled by such engines (typically space launch vehicles and jet aircraft).

If the thrust-to-weight ratio is greater than the local gravity strength (expressed in gs), then flight can occur without any forward motion or any aerodynamic lift being required.

If the thrust-to-weight ratio times the lift-to-drag ratio is greater than local gravity then takeoff using aerodynamic lift is possible.

### Flight dynamics

Flight dynamics is the science of air and space vehicle orientation and control in three dimensions. The three critical flight dynamics parameters are the angles of rotation in three dimensions about the vehicle's center of mass, known as pitch, roll and yaw (See Tait-Bryan rotations for an explanation).

The control of these dimensions can involve a horizontal stabilizer (i.e. "a tail"), ailerons and other movable aerodynamic devices which control angular stability i.e. flight attitude (which in turn affects altitude, heading). Wings are often angled slightly upwards- they have "positive dihedral angle" which gives inherent roll stabilization.

### Energy efficiency

To create thrust so as to be able to gain height, and to push through the air to overcome the drag associated with lift all takes energy. Different objects and creatures capable of flight vary in the efficiency of their muscles, motors and how well this translates into forward thrust.

Propulsive efficiency determines how much energy vehicles generate from a unit of fuel.[31][32]

### Range

The range that powered flight articles can achieve is ultimately limited by their drag, as well as how much energy they can store on board and how efficiently they can turn that energy into propulsion.[33]

For powered aircraft the useful energy is determined by their fuel fraction- what percentage of the takeoff weight is fuel, as well as the specific energy of the fuel used.

### Power-to-weight ratio

All animals and devices capable of sustained flight need relatively high power-to-weight ratios to be able to generate enough lift and/or thrust to achieve take off.

## Takeoff and landing

Vehicles that can fly can have different ways to takeoff and land. Conventional aircraft accelerate along the ground until sufficient lift is generated for takeoff, and reverse the process for landing. Some aircraft can take off at low speed; this is called a short takeoff. Some aircraft such as helicopters and Harrier jump jets can take off and land vertically. Rockets also usually take off and land vertically, but some designs can land horizontally.

Navigation is the systems necessary to calculate current position (e.g. compass, GPS, LORAN, star tracker, inertial measurement unit, and altimeter).

In aircraft, successful air navigation involves piloting an aircraft from place to place without getting lost, breaking the laws applying to aircraft, or endangering the safety of those on board or on the ground.

The techniques used for navigation in the air will depend on whether the aircraft is flying under the visual flight rules (VFR) or the instrument flight rules (IFR). In the latter case, the pilot will navigate exclusively using instruments and radio navigation aids such as beacons, or as directed under radar control by air traffic control. In the VFR case, a pilot will largely navigate using dead reckoning combined with visual observations (known as pilotage), with reference to appropriate maps. This may be supplemented using radio navigation aids.

### Guidance

A guidance system is a device or group of devices used in the navigation of a ship, aircraft, missile, rocket, satellite, or other moving object. Typically, guidance is responsible for the calculation of the vector (i.e., direction, velocity) toward an objective.

### Control

A conventional fixed-wing aircraft flight control system consists of flight control surfaces, the respective cockpit controls, connecting linkages, and the necessary operating mechanisms to control an aircraft's direction in flight. Aircraft engine controls are also considered as flight controls as they change speed.

#### Traffic

In the case of aircraft, air traffic is controlled by air traffic control systems.

Collision avoidance is the process of controlling spacecraft to try to prevent collisions.

## Flight safety

Air safety is a term encompassing the theory, investigation and categorization of flight failures, and the prevention of such failures through regulation, education and training. It can also be applied in the context of campaigns that inform the public as to the safety of air travel.

format: plain


text:
# What Happens As We Go Deeper?

Algebra Level 2

$\sqrt x , \qquad \sqrt{\sqrt x}, \qquad \sqrt{\sqrt{\sqrt x}} , \qquad \sqrt{\sqrt{\sqrt{\sqrt x}}}$

If $$0 < x < 1$$, which of the numbers above is the largest?

×


text:
Copied to
clipboard

## G = Dic14⋊10D4order 448 = 26·7

### 3rd semidirect product of Dic14 and D4 acting via D4/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — Dic14⋊10D4
 Chief series C1 — C7 — C14 — C2×C14 — C22×D7 — C2×C4×D7 — C2×Q8×D7 — Dic14⋊10D4
 Lower central C7 — C2×C14 — Dic14⋊10D4
 Upper central C1 — C22 — C4.4D4

Generators and relations for Dic1410D4
G = < a,b,c,d | a28=c4=d2=1, b2=a14, bab-1=a-1, ac=ca, dad=a13, cbc-1=dbd=a14b, dcd=c-1 >

Subgroups: 1356 in 290 conjugacy classes, 105 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C4.4D4, C4.4D4, C22×Q8, C2×C4○D4, Dic14, Dic14, C4×D7, D28, C2×Dic7, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×Q8, C22×D7, C22×C14, Q85D4, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, D14⋊C4, C23.D7, C4×C28, C7×C22⋊C4, C2×Dic14, C2×Dic14, C2×C4×D7, C2×C4×D7, C2×D28, D42D7, Q8×D7, C22×Dic7, C2×C7⋊D4, D4×C14, Q8×C14, C4×Dic14, C4×D28, C22⋊Dic14, Dic74D4, D14⋊D4, Dic7.D4, C282D4, D143Q8, C7×C4.4D4, C2×D42D7, C2×Q8×D7, Dic1410D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, C24, D14, C22×D4, C2×C4○D4, 2- 1+4, C22×D7, Q85D4, D4×D7, C23×D7, C2×D4×D7, D7×C4○D4, D4.10D14, Dic1410D4

Smallest permutation representation of Dic1410D4
On 224 points
Generators in S224
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28)(29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112)(113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140)(141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168)(169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196)(197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224)
(1 215 15 201)(2 214 16 200)(3 213 17 199)(4 212 18 198)(5 211 19 197)(6 210 20 224)(7 209 21 223)(8 208 22 222)(9 207 23 221)(10 206 24 220)(11 205 25 219)(12 204 26 218)(13 203 27 217)(14 202 28 216)(29 102 43 88)(30 101 44 87)(31 100 45 86)(32 99 46 85)(33 98 47 112)(34 97 48 111)(35 96 49 110)(36 95 50 109)(37 94 51 108)(38 93 52 107)(39 92 53 106)(40 91 54 105)(41 90 55 104)(42 89 56 103)(57 160 71 146)(58 159 72 145)(59 158 73 144)(60 157 74 143)(61 156 75 142)(62 155 76 141)(63 154 77 168)(64 153 78 167)(65 152 79 166)(66 151 80 165)(67 150 81 164)(68 149 82 163)(69 148 83 162)(70 147 84 161)(113 186 127 172)(114 185 128 171)(115 184 129 170)(116 183 130 169)(117 182 131 196)(118 181 132 195)(119 180 133 194)(120 179 134 193)(121 178 135 192)(122 177 136 191)(123 176 137 190)(124 175 138 189)(125 174 139 188)(126 173 140 187)
(1 84 97 123)(2 57 98 124)(3 58 99 125)(4 59 100 126)(5 60 101 127)(6 61 102 128)(7 62 103 129)(8 63 104 130)(9 64 105 131)(10 65 106 132)(11 66 107 133)(12 67 108 134)(13 68 109 135)(14 69 110 136)(15 70 111 137)(16 71 112 138)(17 72 85 139)(18 73 86 140)(19 74 87 113)(20 75 88 114)(21 76 89 115)(22 77 90 116)(23 78 91 117)(24 79 92 118)(25 80 93 119)(26 81 94 120)(27 82 95 121)(28 83 96 122)(29 171 224 156)(30 172 197 157)(31 173 198 158)(32 174 199 159)(33 175 200 160)(34 176 201 161)(35 177 202 162)(36 178 203 163)(37 179 204 164)(38 180 205 165)(39 181 206 166)(40 182 207 167)(41 183 208 168)(42 184 209 141)(43 185 210 142)(44 186 211 143)(45 187 212 144)(46 188 213 145)(47 189 214 146)(48 190 215 147)(49 191 216 148)(50 192 217 149)(51 193 218 150)(52 194 219 151)(53 195 220 152)(54 196 221 153)(55 169 222 154)(56 170 223 155)
(1 70)(2 83)(3 68)(4 81)(5 66)(6 79)(7 64)(8 77)(9 62)(10 75)(11 60)(12 73)(13 58)(14 71)(15 84)(16 69)(17 82)(18 67)(19 80)(20 65)(21 78)(22 63)(23 76)(24 61)(25 74)(26 59)(27 72)(28 57)(29 181)(30 194)(31 179)(32 192)(33 177)(34 190)(35 175)(36 188)(37 173)(38 186)(39 171)(40 184)(41 169)(42 182)(43 195)(44 180)(45 193)(46 178)(47 191)(48 176)(49 189)(50 174)(51 187)(52 172)(53 185)(54 170)(55 183)(56 196)(85 121)(86 134)(87 119)(88 132)(89 117)(90 130)(91 115)(92 128)(93 113)(94 126)(95 139)(96 124)(97 137)(98 122)(99 135)(100 120)(101 133)(102 118)(103 131)(104 116)(105 129)(106 114)(107 127)(108 140)(109 125)(110 138)(111 123)(112 136)(141 207)(142 220)(143 205)(144 218)(145 203)(146 216)(147 201)(148 214)(149 199)(150 212)(151 197)(152 210)(153 223)(154 208)(155 221)(156 206)(157 219)(158 204)(159 217)(160 202)(161 215)(162 200)(163 213)(164 198)(165 211)(166 224)(167 209)(168 222)

G:=sub<Sym(224)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196)(197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224), (1,215,15,201)(2,214,16,200)(3,213,17,199)(4,212,18,198)(5,211,19,197)(6,210,20,224)(7,209,21,223)(8,208,22,222)(9,207,23,221)(10,206,24,220)(11,205,25,219)(12,204,26,218)(13,203,27,217)(14,202,28,216)(29,102,43,88)(30,101,44,87)(31,100,45,86)(32,99,46,85)(33,98,47,112)(34,97,48,111)(35,96,49,110)(36,95,50,109)(37,94,51,108)(38,93,52,107)(39,92,53,106)(40,91,54,105)(41,90,55,104)(42,89,56,103)(57,160,71,146)(58,159,72,145)(59,158,73,144)(60,157,74,143)(61,156,75,142)(62,155,76,141)(63,154,77,168)(64,153,78,167)(65,152,79,166)(66,151,80,165)(67,150,81,164)(68,149,82,163)(69,148,83,162)(70,147,84,161)(113,186,127,172)(114,185,128,171)(115,184,129,170)(116,183,130,169)(117,182,131,196)(118,181,132,195)(119,180,133,194)(120,179,134,193)(121,178,135,192)(122,177,136,191)(123,176,137,190)(124,175,138,189)(125,174,139,188)(126,173,140,187), (1,84,97,123)(2,57,98,124)(3,58,99,125)(4,59,100,126)(5,60,101,127)(6,61,102,128)(7,62,103,129)(8,63,104,130)(9,64,105,131)(10,65,106,132)(11,66,107,133)(12,67,108,134)(13,68,109,135)(14,69,110,136)(15,70,111,137)(16,71,112,138)(17,72,85,139)(18,73,86,140)(19,74,87,113)(20,75,88,114)(21,76,89,115)(22,77,90,116)(23,78,91,117)(24,79,92,118)(25,80,93,119)(26,81,94,120)(27,82,95,121)(28,83,96,122)(29,171,224,156)(30,172,197,157)(31,173,198,158)(32,174,199,159)(33,175,200,160)(34,176,201,161)(35,177,202,162)(36,178,203,163)(37,179,204,164)(38,180,205,165)(39,181,206,166)(40,182,207,167)(41,183,208,168)(42,184,209,141)(43,185,210,142)(44,186,211,143)(45,187,212,144)(46,188,213,145)(47,189,214,146)(48,190,215,147)(49,191,216,148)(50,192,217,149)(51,193,218,150)(52,194,219,151)(53,195,220,152)(54,196,221,153)(55,169,222,154)(56,170,223,155), (1,70)(2,83)(3,68)(4,81)(5,66)(6,79)(7,64)(8,77)(9,62)(10,75)(11,60)(12,73)(13,58)(14,71)(15,84)(16,69)(17,82)(18,67)(19,80)(20,65)(21,78)(22,63)(23,76)(24,61)(25,74)(26,59)(27,72)(28,57)(29,181)(30,194)(31,179)(32,192)(33,177)(34,190)(35,175)(36,188)(37,173)(38,186)(39,171)(40,184)(41,169)(42,182)(43,195)(44,180)(45,193)(46,178)(47,191)(48,176)(49,189)(50,174)(51,187)(52,172)(53,185)(54,170)(55,183)(56,196)(85,121)(86,134)(87,119)(88,132)(89,117)(90,130)(91,115)(92,128)(93,113)(94,126)(95,139)(96,124)(97,137)(98,122)(99,135)(100,120)(101,133)(102,118)(103,131)(104,116)(105,129)(106,114)(107,127)(108,140)(109,125)(110,138)(111,123)(112,136)(141,207)(142,220)(143,205)(144,218)(145,203)(146,216)(147,201)(148,214)(149,199)(150,212)(151,197)(152,210)(153,223)(154,208)(155,221)(156,206)(157,219)(158,204)(159,217)(160,202)(161,215)(162,200)(163,213)(164,198)(165,211)(166,224)(167,209)(168,222)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196)(197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224), (1,215,15,201)(2,214,16,200)(3,213,17,199)(4,212,18,198)(5,211,19,197)(6,210,20,224)(7,209,21,223)(8,208,22,222)(9,207,23,221)(10,206,24,220)(11,205,25,219)(12,204,26,218)(13,203,27,217)(14,202,28,216)(29,102,43,88)(30,101,44,87)(31,100,45,86)(32,99,46,85)(33,98,47,112)(34,97,48,111)(35,96,49,110)(36,95,50,109)(37,94,51,108)(38,93,52,107)(39,92,53,106)(40,91,54,105)(41,90,55,104)(42,89,56,103)(57,160,71,146)(58,159,72,145)(59,158,73,144)(60,157,74,143)(61,156,75,142)(62,155,76,141)(63,154,77,168)(64,153,78,167)(65,152,79,166)(66,151,80,165)(67,150,81,164)(68,149,82,163)(69,148,83,162)(70,147,84,161)(113,186,127,172)(114,185,128,171)(115,184,129,170)(116,183,130,169)(117,182,131,196)(118,181,132,195)(119,180,133,194)(120,179,134,193)(121,178,135,192)(122,177,136,191)(123,176,137,190)(124,175,138,189)(125,174,139,188)(126,173,140,187), (1,84,97,123)(2,57,98,124)(3,58,99,125)(4,59,100,126)(5,60,101,127)(6,61,102,128)(7,62,103,129)(8,63,104,130)(9,64,105,131)(10,65,106,132)(11,66,107,133)(12,67,108,134)(13,68,109,135)(14,69,110,136)(15,70,111,137)(16,71,112,138)(17,72,85,139)(18,73,86,140)(19,74,87,113)(20,75,88,114)(21,76,89,115)(22,77,90,116)(23,78,91,117)(24,79,92,118)(25,80,93,119)(26,81,94,120)(27,82,95,121)(28,83,96,122)(29,171,224,156)(30,172,197,157)(31,173,198,158)(32,174,199,159)(33,175,200,160)(34,176,201,161)(35,177,202,162)(36,178,203,163)(37,179,204,164)(38,180,205,165)(39,181,206,166)(40,182,207,167)(41,183,208,168)(42,184,209,141)(43,185,210,142)(44,186,211,143)(45,187,212,144)(46,188,213,145)(47,189,214,146)(48,190,215,147)(49,191,216,148)(50,192,217,149)(51,193,218,150)(52,194,219,151)(53,195,220,152)(54,196,221,153)(55,169,222,154)(56,170,223,155), (1,70)(2,83)(3,68)(4,81)(5,66)(6,79)(7,64)(8,77)(9,62)(10,75)(11,60)(12,73)(13,58)(14,71)(15,84)(16,69)(17,82)(18,67)(19,80)(20,65)(21,78)(22,63)(23,76)(24,61)(25,74)(26,59)(27,72)(28,57)(29,181)(30,194)(31,179)(32,192)(33,177)(34,190)(35,175)(36,188)(37,173)(38,186)(39,171)(40,184)(41,169)(42,182)(43,195)(44,180)(45,193)(46,178)(47,191)(48,176)(49,189)(50,174)(51,187)(52,172)(53,185)(54,170)(55,183)(56,196)(85,121)(86,134)(87,119)(88,132)(89,117)(90,130)(91,115)(92,128)(93,113)(94,126)(95,139)(96,124)(97,137)(98,122)(99,135)(100,120)(101,133)(102,118)(103,131)(104,116)(105,129)(106,114)(107,127)(108,140)(109,125)(110,138)(111,123)(112,136)(141,207)(142,220)(143,205)(144,218)(145,203)(146,216)(147,201)(148,214)(149,199)(150,212)(151,197)(152,210)(153,223)(154,208)(155,221)(156,206)(157,219)(158,204)(159,217)(160,202)(161,215)(162,200)(163,213)(164,198)(165,211)(166,224)(167,209)(168,222) );

G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28),(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112),(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140),(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168),(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196),(197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224)], [(1,215,15,201),(2,214,16,200),(3,213,17,199),(4,212,18,198),(5,211,19,197),(6,210,20,224),(7,209,21,223),(8,208,22,222),(9,207,23,221),(10,206,24,220),(11,205,25,219),(12,204,26,218),(13,203,27,217),(14,202,28,216),(29,102,43,88),(30,101,44,87),(31,100,45,86),(32,99,46,85),(33,98,47,112),(34,97,48,111),(35,96,49,110),(36,95,50,109),(37,94,51,108),(38,93,52,107),(39,92,53,106),(40,91,54,105),(41,90,55,104),(42,89,56,103),(57,160,71,146),(58,159,72,145),(59,158,73,144),(60,157,74,143),(61,156,75,142),(62,155,76,141),(63,154,77,168),(64,153,78,167),(65,152,79,166),(66,151,80,165),(67,150,81,164),(68,149,82,163),(69,148,83,162),(70,147,84,161),(113,186,127,172),(114,185,128,171),(115,184,129,170),(116,183,130,169),(117,182,131,196),(118,181,132,195),(119,180,133,194),(120,179,134,193),(121,178,135,192),(122,177,136,191),(123,176,137,190),(124,175,138,189),(125,174,139,188),(126,173,140,187)], [(1,84,97,123),(2,57,98,124),(3,58,99,125),(4,59,100,126),(5,60,101,127),(6,61,102,128),(7,62,103,129),(8,63,104,130),(9,64,105,131),(10,65,106,132),(11,66,107,133),(12,67,108,134),(13,68,109,135),(14,69,110,136),(15,70,111,137),(16,71,112,138),(17,72,85,139),(18,73,86,140),(19,74,87,113),(20,75,88,114),(21,76,89,115),(22,77,90,116),(23,78,91,117),(24,79,92,118),(25,80,93,119),(26,81,94,120),(27,82,95,121),(28,83,96,122),(29,171,224,156),(30,172,197,157),(31,173,198,158),(32,174,199,159),(33,175,200,160),(34,176,201,161),(35,177,202,162),(36,178,203,163),(37,179,204,164),(38,180,205,165),(39,181,206,166),(40,182,207,167),(41,183,208,168),(42,184,209,141),(43,185,210,142),(44,186,211,143),(45,187,212,144),(46,188,213,145),(47,189,214,146),(48,190,215,147),(49,191,216,148),(50,192,217,149),(51,193,218,150),(52,194,219,151),(53,195,220,152),(54,196,221,153),(55,169,222,154),(56,170,223,155)], [(1,70),(2,83),(3,68),(4,81),(5,66),(6,79),(7,64),(8,77),(9,62),(10,75),(11,60),(12,73),(13,58),(14,71),(15,84),(16,69),(17,82),(18,67),(19,80),(20,65),(21,78),(22,63),(23,76),(24,61),(25,74),(26,59),(27,72),(28,57),(29,181),(30,194),(31,179),(32,192),(33,177),(34,190),(35,175),(36,188),(37,173),(38,186),(39,171),(40,184),(41,169),(42,182),(43,195),(44,180),(45,193),(46,178),(47,191),(48,176),(49,189),(50,174),(51,187),(52,172),(53,185),(54,170),(55,183),(56,196),(85,121),(86,134),(87,119),(88,132),(89,117),(90,130),(91,115),(92,128),(93,113),(94,126),(95,139),(96,124),(97,137),(98,122),(99,135),(100,120),(101,133),(102,118),(103,131),(104,116),(105,129),(106,114),(107,127),(108,140),(109,125),(110,138),(111,123),(112,136),(141,207),(142,220),(143,205),(144,218),(145,203),(146,216),(147,201),(148,214),(149,199),(150,212),(151,197),(152,210),(153,223),(154,208),(155,221),(156,206),(157,219),(158,204),(159,217),(160,202),(161,215),(162,200),(163,213),(164,198),(165,211),(166,224),(167,209),(168,222)]])

67 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 4A 4B 4C 4D 4E 4F 4G 4H ··· 4M 4N 4O 4P 7A 7B 7C 14A ··· 14I 14J ··· 14O 28A ··· 28R 28S ··· 28X order 1 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 ··· 4 4 4 4 7 7 7 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 size 1 1 1 1 4 4 14 14 28 2 2 2 2 4 4 4 14 ··· 14 28 28 28 2 2 2 2 ··· 2 8 ··· 8 4 ··· 4 8 ··· 8

67 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D7 C4○D4 D14 D14 D14 D14 2- 1+4 D4×D7 D7×C4○D4 D4.10D14 kernel Dic14⋊10D4 C4×Dic14 C4×D28 C22⋊Dic14 Dic7⋊4D4 D14⋊D4 Dic7.D4 C28⋊2D4 D14⋊3Q8 C7×C4.4D4 C2×D4⋊2D7 C2×Q8×D7 Dic14 C4.4D4 D14 C42 C22⋊C4 C2×D4 C2×Q8 C14 C4 C2 C2 # reps 1 1 1 2 2 2 2 1 1 1 1 1 4 3 4 3 12 3 3 1 6 6 6

Matrix representation of Dic1410D4 in GL6(𝔽29)

 17 0 0 0 0 0 2 12 0 0 0 0 0 0 11 1 0 0 0 0 27 21 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 26 22 0 0 0 0 18 3 0 0 0 0 0 0 25 3 0 0 0 0 24 4 0 0 0 0 0 0 28 0 0 0 0 0 0 28
,
 28 0 0 0 0 0 5 1 0 0 0 0 0 0 28 0 0 0 0 0 0 28 0 0 0 0 0 0 0 28 0 0 0 0 1 0
,
 1 0 0 0 0 0 24 28 0 0 0 0 0 0 25 3 0 0 0 0 24 4 0 0 0 0 0 0 0 1 0 0 0 0 1 0

G:=sub<GL(6,GF(29))| [17,2,0,0,0,0,0,12,0,0,0,0,0,0,11,27,0,0,0,0,1,21,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[26,18,0,0,0,0,22,3,0,0,0,0,0,0,25,24,0,0,0,0,3,4,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[28,5,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,0,1,0,0,0,0,28,0],[1,24,0,0,0,0,0,28,0,0,0,0,0,0,25,24,0,0,0,0,3,4,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

Dic1410D4 in GAP, Magma, Sage, TeX

{\rm Dic}_{14}\rtimes_{10}D_4
% in TeX

G:=Group("Dic14:10D4");
// GroupNames label

G:=SmallGroup(448,1130);
// by ID

G=gap.SmallGroup(448,1130);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,120,219,100,1571,297,192,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^28=c^4=d^2=1,b^2=a^14,b*a*b^-1=a^-1,a*c=c*a,d*a*d=a^13,c*b*c^-1=d*b*d=a^14*b,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽


text:
# More volume with more mass?

Regarding to the Schwarzschild solution, is there more volume with more mass?

Lets take for example a space shuttle (spheric for simplicity) in an stationary orbit at the position $$r_1$$. The black hole around which it is orbitting has a mass $$M$$. Now, the mass is increased (not interesting, how). $$M_2=2\cdot M_1$$for example. The position $$r_1$$ is kept the same. Doesn't the volume of the spaceshuttle increase due to the mass increase and curvature increase?

According to a comment, the question might be similar to the question whether the volume of the space shuttle increases while moving nearer to the center of the black hole. The curvature there is stronger then outside.

As the metric is $$ds^2 = -Bdt^2 + Adr^2 + \text{angular terms}$$, and A is increasing while decreasing r, the volume should increase either. However, this seems to be dependent on the sign convention: If (+---) is used instead of (-+++), then the volume is decreasing, isn't it? I read somewhere, that Einstein used (+---) and now we are using (-+++) mostly and it is pure convention - but isn't the volume change important to classify?

Addendum: To make the question clearer I want to change it to the following thought experiment: Mankind finally managed to develop a material which completely isolates from gravity (please, don't ask me how). Real hoverboards (see "Back to the future" 2) are used all-around the planet and, at last, the following experiment is done regularly in the physics undergrads course: They got an empty cube of this gravity-isolating material. In the middle of the cube, there is a (non-isolating) sphere which can be filled with a material of high gravitational mass. The volume of the cube is measured with and without the mass in the middle (probably by simply pressing water into it). Of course, the undergrads need to be very precise with the volume measurement, carefully keeping temperature the same and everything. Do they measure more volume inside the cube if the middle-sphere is filled with mass?

For clarification: volume is positive here, mass is also positive here. The material of the cube is magical. It just stays where it is and isolates gravity. (Hm, ok, I see: "It stays where it is" regarding inside the cube is different from "it stays where it is" regarding the room with the students. So, to define even this: It stays where it is regarding the room with the students)

• For me, it seems to be a real simple beginner question in general relativity. Very basic. I expected plenty of canonical answers. Why is the question downvoted? Is it to easy to answer? Not well defined? Not the actual question? I don't get it. I try to understand general relativity and this is one of the first questions I asked. Nov 26, 2021 at 5:13
• Please do not let posts look like revision histories. In particular, instead of saying "I want to change this question to the following question", just change it. On the other hand, please be mindful to not change the question in ways that invalidates existing answers. Nov 30, 2021 at 13:13
• If your magical water is weightless and can fill all the way to the horizon, then you can put an unlimited amount of water in the box. Dec 3, 2021 at 17:12

I'm not sure this fully answers your question, but I hope this can help to shed some light on the issue.

Instead of a spherical spaceship I consider a rod in free fall towards a Schwarzschild black hole.

In the Schwarzschild coordinates, the rod occupies the space between $$r=r_1$$ and $$r=r_2$$. The length $$L$$ of the rod can be found by considering the amount of time it would take a light signal to move from one end of the rod to the other. We have

$$L = c \int_{t_1}^{t_2} \text{ d}t,$$ where we assume that the light moves from $$(t_1,r_1)$$ to $$(t_2,r_2)$$.

If the light moves in a purely radial direction, we can describe its path by the coordinate functions $$t(\lambda)$$ and $$r(\lambda)$$. The equation of motion $$ds^2 =0$$ then takes the form $$g_{tt} \left(\frac{dt}{d\lambda}\right)^2 + g_{rr} \left(\frac{dr}{d\lambda}\right)^2 = 0,$$ which we can rewrite as $$\left(\frac{dt}{dr}\right)^2 = -\frac{g_{rr}}{g_{tt}}.$$

The length of the rod is then $$L = c \int_{r_1}^{r_2} \frac{dt}{dr} \text{ d}r = c \int_{r_1}^{r_2} \sqrt{-\frac{g_{rr}}{g_{tt}}} \text{ d}r,$$ where I have taken the positive square root because $$r_2 > r_1$$.

Notice that the length is independent of the signature of the metric, so whether you work with the (-+++) or (+---) metric is purely conventional and will not change the physics.

For the Schwarzschild metric, we obtain explicitly $$L = r_2 - r_1 + r_s \ln\left(\frac{r_2 - r_s}{r_1 - r_s}\right) > r_2 - r_1.$$

Now what happens if you magically, instantaneously increase the mass of the black hole? I think the length $$L$$ of the rod stays the same (I'm here assuming that the rod is infinitely stiff), but that it would now "appear shorter" to the distant observer - i.e. it would no longer occupy the entire space between $$r_1$$ and $$r_2$$.

Applying this reasoning to the case of a spaceship, the astronauts inside the ship should not feel its volume change when mass is added to the black hole. However, a distant observer might see that the spaceship now looks smaller than it did before.

In this analysis I have assumed that the rod/ship can be considered to be stationary for the duration of our experiment. I am not sure if all the same conclusions would hold if the rod/ship were in motion during the experiment.

• Thank you fir your answer! However, I'm still looking for the volume. Isn't the ricci tensor the volume gain? Can someone show rhis canonically? Nov 27, 2021 at 13:55
• Does this not depend on what we mean by length? The difference between coordinates does not in general provide a meaningful definition of length, consider for instance the distance between two points $(r,\phi_1)$ and $(r,\phi_2)$ in polar coordinates. On the other hand, the time taken by a photon to propagate from an observer A to a point B, and back again, provides a coordinate-independent measure of the distance between A and B (as measured by the observer A). Dec 3, 2021 at 11:45

@BarrierRemoval, you must be aware of what kind of curvature do you speak. Schwarzschild vacuum spacetime is indeed curved. Its Riemann curvature $$R^{\rho}_{\sigma \mu \nu}$$ is non-zero. On the other hand, its Ricci $$R_{\mu \nu}$$, and correspondingly $$R$$ curvatures are zero. Therefore, for a co-moving observer a spaceship does not change its volume. For reference see "Physical and Geometric Interpretations of the Riemann Tensor, Ricci Tensor, and Scalar Curvature", by Lee C. Loveridge https://arxiv.org/abs/gr-qc/0401099v1

• That's indeed important, thank you for pointing this out! I updated the question. Nov 29, 2021 at 18:55
• It's the Ricci-tensor with produces volume, isn't it? And the Weyl tensor is the difference between volumes, is that right? Therefore, the Ricci-tensor is zero outside the mass, but the Weyl-tensor is not. Nov 29, 2021 at 19:15
• And, together, they build up the Riemann tensor? Nov 29, 2021 at 19:16

For a Schwarzschild solution, the Ricci tensor vanishes identically. As the space shuttle moves towards the black hole (or if there is some change in the black hole's mass due to matter falling in) the volume of the shuttle won't change, since volume changes are governed by the Ricci tensor. The Weyl tensor (i.e. the components of the Riemann tensor which are not on the Ricci tensor) does not vanish, but it leads only to shape deformations. Hence, while the shuttle will undergo spaghettification, its volume remains constant while the shape changes. A similar statement will hold for any object in a region where the stress-energy tensor vanishes. However, there is no way to determine what would happen to the cube: since it is speculative, so is any affirmation we can make regarding it. Gravity as we understand can't be isolated. If you are assuming it can be isolated, then we are no longer working with GR and there is no meaning in discussing what would happen from the point of view of GR.

As for the sign conventions, they are just conventions. If you change your preferred metric signature, you must also change the signs on remaining formulae so that the predictions are still the same. Nevertheless, this is meaningless for a Schwarzschild solution, since the Ricci tensor vanishes and there is never any change in volume of infalling objects.

• Sorry for that, but, unfortunately, I suspect this to be wrong. The Ricci-tensor changes the volume and the Ricci-tensor is increased inside the mass. So, outside the mass, in my point of view, it's like lowering the deepest point of the Flamm's paraboloid. On the one hand (as you say), the steepness (spaghettification, shape change) is increased. But, on the other hand, a specific point of spacetime outside the mass is lowering it's position (which in my opinion belongs to the volume), too. Do you see this? Nov 30, 2021 at 8:51
• @BarrierRemoval I didn't really understand your comment. Could you rephrase it? Notice that everywhere outside of the mass the Ricci tensor vanishes identically, and hence the shuttle will not have volume changes. In the case of a black hole, the Ricci tensor vanishes everywhere (except at the singularity, which is not a point of spacetime), so there is no volume change. Nov 30, 2021 at 9:10
• I'm not absolutely sure (that's why I'm asking questions), but to me, the Ricci-tensor means volume gain at the position of the mass. OK, a black hole is a complicated thing - so, let's take something easier, just a sphere filled with mass (sun, neutron star). Then, inside the mass/sun, the ricci tensor is not zero. you add volume there. In infinity, the volume is still the same, the volume of the Minkowski space (probably, I get something wrong here). So in between it's basically as a potential difference. If you lift up potential at point 1 and are in between point 1 and 2, then... Nov 30, 2021 at 11:08
• you experience to things: 1. it is steeper a your point in between and 2. you are lift up a bit. That's all I tried to say. I'm not absolutely sure wether I'm right or not. It is named "general relativity" - so probably, there doesn't exist anything like "lifting up" and everything is only relative. Nov 30, 2021 at 11:10
• @BarrierRemoval for a localized mass, such as the sun, the Ricci tensor vanishes everywhere but at the star (recall Birkhoff's theorem: everywhere where $T_{ab} = 0$ and there is spherical symmetry, the solution is Schwarzschild). Hence, outside of the star, the change in volume is always zero. Once you are inside the star, things can get different and volume might change, but there is not change of volume at all outside of the star Nov 30, 2021 at 11:45

The volume of cube will change most probably. If there is matter or negative matter added into the center of the, special gravity isolated cube. The cube itself is made of exotic matter as it defies gravity in most probability, as normal matter can't defy gravity. In your question , it must be determined whether the spherical mass is

M , -M.

This type of a situation could also result in negative volume in the universe. The radius gets larger. That’s “how” it changes.

Oh, did you want to know “why” the radius of the Event Horizon changes when you add mass to a Black Hole? Well, then, why didn’t you say so?

The more mass a singularity (or any object) has, the greater its gravity. The greater its gravity, the farther away from the singularity is the point where light cannot escape the gravitational pull of the singularity. And, since the Event Horizon is defined as the maximum distance from a singularity where light cannot escape the gravitational pull of the singularity; that means the radius of the Event Horizon gets larger. The equation is, *R_schwarzschild=2GM/c^2 *
and if you understand equations you can see how R is directly proportional to M.

• Ok, probably that was not clear enough. I pointed out that the students add positive mass M to the cube. Nov 30, 2021 at 7:33
• Regarding the rest of your answer: It points in the right direction, but it is not really satisfying: You say that the event horizon of the black hole increases, ok. And, therefore, the volume inside the event horizon increases. However, that doesn't say anything about the volume of the space-time itself. You did not show that the event horizon isn't just moving outwards without changing the volume of the space-time itself. Nov 30, 2021 at 7:39
• I was actually trying to tell u that the cube maybe is made of exotic matter as it defies gravity according to you and treated the the spherical mass as normal matter Nov 30, 2021 at 7:39
• The volume of spacetime shall not change if I consider spacetime and matter differently Nov 30, 2021 at 7:41
• ok, thank you for the clarification. I did not get this because you wrote "In your question, it must be determined whether the spherical mass is M, -M. This type of a situation could also result in negative volume in the universe" Nov 30, 2021 at 7:45


text:
##### Page tree

An R-group query can involve several derivatives which differ in one or more substituents (fragments).

• R-group label

• R-group attachment point

• R-group definition

For R-group search, some conditions (so-called R-logic) can also be set to the R-group query.

You can also watch a video tutorial here, or you can meet the Smart R-group tool here.

## How to create R-group labels

##### by Smart R-group tool

Select Smart R-group tool. Click on a clear place on the canvas, or on the right place in the molecule (sprout drawing is also possible).

In this case, the default set is that the auto numbering is active, which means that R1, R2, an so on, groups will automatically appear in the place of the clicking in the right order.

##### by Keyboard shortcut

R+Space shortcut:

Opens the R-group dialog window, where you can set the number of the R-group (or you can leave the field empty to get an R-group without any number).

After closing the window, the chosen R-group will appear on the tip of the mouse pointer and this can be added to the root structure in the same way as atoms are inserted.

r keyboard shortcut:

Typing "r" results that "R" will appear on the tip of the cursor; it can be placed on the canvas with a left-click or pressing Enter as many times as you need.

How to edit R-group labels

1. There is an option to add or change R-groups to the required number by clicking on an R-group (when the Smart R-group tool is active). Clicking on an existing R-group opens the R-group dialog window, here you can turn off the auto numbering by unchecking the box and set the number of the R-group. When you check the auto numbering box again, numbering will carry on from the first number that is not yet represented on the canvas.
2. To change other symbols to R-group label (without number), hover over an atomic symbol or select it by the Selection tool, and type "r".
Existing atoms can be replaced by R-group labels using the Atom properties dialog as well, where the 'R-group' type should be chosen instead of 'Element'. In this case you can set the number, alias or map to the R-group.

How to add an attachment point by R-group attachment point button

Select the R-group attachment point button and then click on the atom. You can add more attachment points by clicking successively on the selected atom(s).

Right-click on the selected atom then choose + R-group attachment from the menu.

The attachment points on every separate fragment are automatically numbered beginning with "1".

## Edit the Attachment points

How to change the bond type of the attachment point
The bond type of the R-group attachment points can be changed using either the R-group attachment point tool or any of the bond tools.

## How to create R-group definitions

In order to create the definition for an R-group, draw the substituents from which the R-group should stand for, then you can choose from two options.

R-group definition import and export are supported only in ChemAxon Marvin Document (MRV) format in Marvin JS; besides, you can choose other formats if the relevant web service is available.
##### Smart R-group tool

Selecting with the Smart R-group tool results that the R-group label will be given to the fragment (or fragments), which appears on the canvas. When there is more than one R-group in the scaffold, the label of the fragment will get the smallest R-group number, which hasn't got a definition yet.

R+Space keyboard shortcut

Select the substituent(s) by one of the Selection tools, and then type R+Space shortcut. The R-group dialog opens, and you can set the relevant R-label.

## Adding/Removing elements to/from the definition

Smart R-group tool

Selecting the substituents together with the Smart R-group tool results that the same R-group label will be given to the fragments.

If you want to add one or more fragments to an R-group definition, select these fragments (with the Selection tool), right-click on the relevant R-group label, then choose the Add to this group option.

##### Remove

Smart R-group tool

You can remove fragments from the definition by selecting it with the Smart R-group tool. In this case, the selected element will get a new label.

In order to remove a fragment from a group, select this fragment, right-click on the R-label of the group and select Remove from this group option.

## Ungroup or Delete

If you do not wish to have an R-group definition any longer, you have some options to get rid of them.

How to ungroup an R-group definition

Ungrouping the definition results that the fragments will not be erased, but they will not belong to an R-group definition any longer.

Right-click on the R-group label, and use the Ungroup option in the pop-up menu.

Delete

In that case, the definition and all of its contents will be erased from the canvas. Deleting the definition can happen as with any other objects, by the Delete tool, or by the Delete keyboard button.


text:
# Does $\int_{1}^{\infty}\sin(x\log x)dx$ converge?

I'm trying to find out whether $\int_{1}^{\infty}\sin(x\log x)dx$ converges, I know that $\int_{1}^{\infty}\sin(x)dx$ diverges but $\int_{1}^{\infty}\sin(x^2)dx$ converges, more than that, $\int_{1}^{\infty}\sin(x^p)dx$ converges for every $p>0$, so it should be converges in infinity. I'd really love your help with this.

Thanks!

-

Since $x\log(x)$ is monotonic on $[1,\infty)$, let $f(x)$ be its inverse. That is, for $x\in[0,\infty)$ $$f(x)\log(f(x))=x\tag{1}$$ Differentiating implicitly, we get $$f'(x)=\frac{1}{\log(f(x))+1}\tag{2}$$ Then \begin{align} \int_1^\infty\sin(x\log(x))\;\mathrm{d}x &=\int_0^\infty\sin(x)\;\mathrm{d}f(x)\\ &=\int_0^\infty\frac{\sin(x)}{\log(f(x))+1}\mathrm{d}x\tag{3} \end{align} Since $\left|\int_0^M\sin(x)\;\mathrm{d}x\right|\le2$ and $\frac{1}{\log(f(x))+1}$ monotonically decreases to $0$, Dirichlet's test (Theorem 17.5) says that $(3)$ converges.

-

This is a version of the Van der Corput lemma, basically.

Note that it's enough to find some $n$ for which $\int_n^{\infty} \sin(x\log(x))\,dx$ converges. The key facts about $f(x) = x\log(x)$ that allow this are a) $\lim_{x \rightarrow \infty} f'(x) = \infty$ and b) $f''(x) > 0$ for $x$ large enough. Specifically, we write $$\int_n^{\infty} \sin(f(x))\,dx = \int_n^{\infty} f'(x) {\sin(f(x)) \over f'(x)}\,dx$$ $$= \lim_{N \rightarrow \infty} \int_n^{N} (f'(x) \sin(f(x)){1 \over f'(x)}\,dx$$ Integrating the integral on the right by parts you get $$\int_n^{N} (f'(x) \sin(f(x)){1 \over f'(x)}\,dx = -{\cos(f(N)) \over f'(N)} + {\cos(f(n)) \over f'(n)} + \int_n^N \cos(f(x)){d \over dx}{1 \over f'(x)}$$ $$= -{\cos(f(N)) \over f'(N)} + {\cos(f(n)) \over f'(n)} - \int_n^N \cos(f(x)) {f''(x) \over (f'(x))^2}$$ As $N$ goes to infinity the first term goes to zero since $f'(x)$ goes to $\infty$ as $x$ goes to $\infty$ and $|\cos(f(N))| \leq 1$. The third term is bounded in absolute value by $$\int_n^N\bigg|{f''(x) \over (f'(x))^2}\bigg|\,dx$$ Since $f''(x) > 0$ we can just take off the absolute values to get $$\int_n^N{f''(x) \over (f'(x))^2}\,dx$$ Integrating this becomes $${1 \over f'(N)} - {1 \over f'(n)}$$ Since $f'(N) \rightarrow \infty$ as $N$ goes to $\infty$ this converges as $N$ goes to infinity. Hence the integral $\int_n^{\infty} \cos(f(x)) {f''(x) \over (f'(x))^2}$ converges absolutely, and thus converges.

Hence we have shown the original integral converges.

-
+1. This kind of answer reminds all of us why we fell in love with calculus after mastering it either as honors undergraduates or after relearning it in graduate school. –  Mathemagician1234 Feb 7 '12 at 17:21
You might try $\int_0^\infty f(\sin(x \ln x))\ dx$ where $f(t) = t^3$ for $t\ge 0$ and $t$ for $t \le 0$. This also has humps of the same height and width, but it won't give you an alternating series. –  Robert Israel Feb 7 '12 at 17:41
@Robert: Furthermore, it diverges since the average of $|\sin^3(x)|$ is $2/3$ the average of $|\sin(x)|$. –  robjohn Feb 7 '12 at 18:44


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

 It is currently 22 Feb 2019, 10:53

### GMAT Club Daily Prep

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

Customized
for You

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

Track

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

Practice
Pays

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

## Events & Promotions

###### Events & Promotions in February
PrevNext
SuMoTuWeThFrSa
272829303112
3456789
10111213141516
17181920212223
242526272812
Open Detailed Calendar
• ### Free GMAT RC Webinar

February 23, 2019

February 23, 2019

07:00 AM PST

09:00 AM PST

Learn reading strategies that can help even non-voracious reader to master GMAT RC. Saturday, February 23rd at 7 AM PT
• ### FREE Quant Workshop by e-GMAT!

February 24, 2019

February 24, 2019

07:00 AM PST

09:00 AM PST

Get personalized insights on how to achieve your Target Quant Score.

# Bob is training for a fitness competition. In order to increase his ma

Author Message
TAGS:

### Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 53066
Bob is training for a fitness competition. In order to increase his ma  [#permalink]

### Show Tags

28 Jul 2017, 07:58
2
00:00

Difficulty:

5% (low)

Question Stats:

90% (01:55) correct 10% (01:53) wrong based on 83 sessions

### HideShow timer Statistics

Bob is training for a fitness competition. In order to increase his maximum number of pull-ups, he follows the following routine: he begins with 25 pull-ups, rests for thirty seconds, and then does 24 pull-ups and rests, dropping one pull-up each time (25, 24, 23, etc.) until his final set of 11 pull-ups. How many total pull-ups does Bob do?

(A) 55

(B) 150

(C) 270

(D) 275

(E) 325

_________________
Senior PS Moderator
Joined: 26 Feb 2016
Posts: 3343
Location: India
GPA: 3.12
Bob is training for a fitness competition. In order to increase his ma  [#permalink]

### Show Tags

28 Jul 2017, 09:58
The sum of n natural numbers is $$\frac{n(n+1)}{2}$$

In order to find the total number of pull ups that Bob does for the
fitness competition, we need to find the sum of the numbers from 11 to 25.
This can be done by subtracting the the sum of the first 10 natural numbers from the sum of the first 25 natural numbers.

Sum : $$\frac{25(26)}{2} - \frac{10(11)}{2} = 325 - 55 = 270$$(Option C)
_________________

You've got what it takes, but it will take everything you've got

Director
Joined: 21 Mar 2016
Posts: 521
Re: Bob is training for a fitness competition. In order to increase his ma  [#permalink]

### Show Tags

29 Jul 2017, 10:14
in case u r not aware of the formula,,,
adding will help,,,25 + 24 +....11

hope this helps
Manager
Joined: 24 Jun 2017
Posts: 122
Re: Bob is training for a fitness competition. In order to increase his ma  [#permalink]

### Show Tags

29 Jul 2017, 19:59
1
1
sum of consecutive integers = mean * number of terms
mean = (first term + last term) / 2 = (25 + 11)/2=18
number of terms = 25 - 11 + 1 (11 inclusively) = 15
18 * 15 = 270
Target Test Prep Representative
Status: Founder & CEO
Affiliations: Target Test Prep
Joined: 14 Oct 2015
Posts: 4955
Location: United States (CA)
Re: Bob is training for a fitness competition. In order to increase his ma  [#permalink]

### Show Tags

24 Jun 2018, 17:04
Bunuel wrote:
Bob is training for a fitness competition. In order to increase his maximum number of pull-ups, he follows the following routine: he begins with 25 pull-ups, rests for thirty seconds, and then does 24 pull-ups and rests, dropping one pull-up each time (25, 24, 23, etc.) until his final set of 11 pull-ups. How many total pull-ups does Bob do?

(A) 55

(B) 150

(C) 270

(D) 275

(E) 325

The number of pull-ups Bob does is the sum of integers from 11 to 25 inclusive. Thus, the total number of pull-ups he does is

15(11 + 25)/2 = 15 x 18 = 270

Alternate Solution:

We need to calculate the sum of the consecutive integers from 11 to 25, inclusive. Rather than calculate 11 + 12 + 13 + … + 24 + 25, we can calculate the average of the integers and then multiply the average by the number of integers to get the sum.

We know that there are (25 – 11) + 1 = 14 + 1 = 15 integers.

To calculate the average of the 15 integers, we use (smallest + largest)/2 = (11 + 25)/2 = 18.

The average is 18. Thus, the sum is 18 x 15 = 270.

_________________

Scott Woodbury-Stewart
Founder and CEO

GMAT Quant Self-Study Course
500+ lessons 3000+ practice problems 800+ HD solutions

EMPOWERgmat Instructor
Status: GMAT Assassin/Co-Founder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
Posts: 13577
Location: United States (CA)
GMAT 1: 800 Q51 V49
GRE 1: Q170 V170
Re: Bob is training for a fitness competition. In order to increase his ma  [#permalink]

### Show Tags

09 Oct 2018, 22:01
Hi All,

We're told that Bob is training for a fitness competition. In order to increase his maximum number of pull-ups, he follows the following routine: he begins with 25 pull-ups, rests for thirty seconds, and then does 24 pull-ups and rests, dropping one pull-up each time (25, 24, 23, etc.) until his final set of 11 pull-ups. We're asked for the total number of pull-ups that Bob completes. This question can be solved in several different ways, including by 'bunching.'

In simple terms, we are adding up all of the integers from 11 to 25, inclusive. Since there are 25 integers from 1 to 25 - and we are NOT including the first 10 integers, that means we are adding 25 - 10 = 15 individual numbers together. We can 'bunch' those numbers into seven groups of 2 and one left over number:

11 + 25 = 36
12 + 24 = 36
13 + 23 = 36
...
17 + 19 = 36
and there's one number left over: an 18

Thus, we have 7(36) + 18 = 252 + 18 = 270

GMAT assassins aren't born, they're made,
Rich
_________________

760+: Learn What GMAT Assassins Do to Score at the Highest Levels
Contact Rich at: Rich.C@empowergmat.com

# Rich Cohen

Co-Founder & GMAT Assassin

Special Offer: Save \$75 + GMAT Club Tests Free
Official GMAT Exam Packs + 70 Pt. Improvement Guarantee
www.empowergmat.com/

*****Select EMPOWERgmat Courses now include ALL 6 Official GMAC CATs!*****

Re: Bob is training for a fitness competition. In order to increase his ma   [#permalink] 09 Oct 2018, 22:01
Display posts from previous: Sort by


text:
# Evaluation functional

## In words...

For a space of real-valued functions over a given domain, the evaluation functional is the functional that, for any function of the space and any point in the domain, outputs the value of the function at that point.

...

## In maths...

The evaluation functional in a space $\H$ of functions from $\X$ to $\Y$ (i.e., $\H\subseteq \Y^\X$) is the functional defined as $$e : \X \times \H \rightarrow \Y$$ with $$\forall \g x \in \X,\ \forall f\in\H,\quad e_{\g x}(f) = f(\g x) .$$


text:
Courses
Courses for Kids
Free study material
Free LIVE classes
More

# Multiplying Mixed Numbers

## Multiplying Mixed Numbers with Like Denominators, Unlike Denominators, and Whole Numbers

Last updated date: 28th Mar 2023
Total views: 85.5k
Views today: 0.62k

## What Are Mixed Numbers?

Mixed numbers, also known as mixed fractions, are a combination of whole numbers and a proper fraction. The mixed number is partly a whole number and partly a fraction. The fractional part of a mixed number is always a proper fraction. For example, 534, 712, 723, etc are mixed numbers. Multiplying mixed numbers is similar to multiplying fractions. Here, we will learn the steps to multiply mixed numbers with the like denominators, unlike denominators, and whole numbers.

## Parts of Mixed Number

A mixed number is formed by combining three parts namely the whole number, a numerator, and a denominator.  Look at the image given to understand the parts of mixed numbers in a better way.

## How to Multiply Two Mixed Numbers with Like and Unlike Denominators?

Two fractions with the same denominator are known as the like denominators. Contrarily, two fractions with different denominators are known as unlike denominators. The steps to multiply the mixed fractions remains the same irrespective of the like or unlike denominators. The 4 steps given below describes how to multiply mixed numbers with like and unlike denominators:

Step 1: Convert the given mixed numbers into improper fractions.

Step 2: Multiply first the numerators and and then the denominators of the improper fraction.

Step 3: Simplify the fraction or result obtained in the previous step in lowest terms if possible.

Step 4: Convert the result obtained back to the mixed number.

## How to Convert a Mixed Number into an Improper Fraction?

To convert a mixed number into an improper fraction:

• Multiply the whole number with the denominator of the proper fraction.

• Add the numerator of the proper fraction to the product obtained in step 1.

• The denominator of the improper fraction will be the same as the denominator of the proper fraction.

• The resulting fraction will be an improper fraction.

Let us understand with an example:

Convert 512 into an improper fraction.

Solution:

Step 1: Multiplying the whole number with the denominator of the proper fraction.

5 2 = 10

Step 2: Adding the numerator of the proper fraction to the product obtained in step 1. Here, the numerator is 1. Accordingly, numerator of the  improper fraction will be  10 + 1 = 11

Step 3: The denominator of the improper fraction will stay the same. Accordingly, the denominator will be 2.

Hence, 512 in improper fraction will be  112.

As we have understood how to convert mixed number to an improper fraction, let us now understand multiplying mixed number with an example:

## Example on Multiplying Mixed Numbers with Like Denominators

What is 2 27 3 17 ?

Solution:

Step 1: Converting the mixed numbers into improper fractions, we have

### 3 17  = 3 7 + 17 = 227

Step 2: Multiplying the numerators and denominators of improper fractions obtained in step 1.

### 167227 = 35249

Step 3: Converting the final result back into the mixed fraction. Accordingly, 35249 = 7949

Therefore,  2 27 3 17 = 7949

## Example on Multiplying Mixed Numbers with Unlike Denominators

What is 1 23 2 34 ?

Solution:

Step 1: Converting the mixed numbers into improper fractions, we have

### 2 34  = 2 4 + 34 = 114

Step 2: Multiplying the numerators and denominators of improper fractions obtained in step 1.

### 53114 = 5512

Step 3: Converting the final result back into the mixed fraction. Accordingly, 5512 = 4712

Therefore, 1 23 2 33  = 4712

## How to Multiply Mixed Numbers with the Whole Number?

The steps on multiplying the mixed number with the whole number are discussed below:

Step 1: Convert the whole number into fraction assuming the denominator to be 1.

Step 2: Convert the given mixed numbers into improper fractions.

Step 3: Multiply the numerators and denominators of the fraction.

Step 4: Simplify the result or fraction obtained in the previous step in lowest terms if possible.

Step 5: Convert the result obtained back to the mixed number.

Let us understand with an example:

What is 216 3?

Solution:

Step 1: Converting the whole number into fraction assuming the denominator to be 1. Hence, 3 will be 31

Step 2:  Converting the given mixed number into improper fraction gives:   216 = 136

Step 3: Multiplying the numerators and denominators of the fraction gives:

136 31 = 396

Step 4: Simplifying  the fraction obtained in the previous step in lowest terms gives:

396 = 39 36 3  = 132

Step 5: Converting the result obtained back to the mixed number gives:

132 = 6 12

Therefore, 216 3 = 6 12

In short, multiplying mixed numbers with fractions is a four-step process. Here, first you have to convert a given mixed number into an improper fraction, multiply the numerator and denominator of the fraction. Simplify the result if possible and convert the result back to the mixed number.

## FAQs on Multiplying Mixed Numbers

1. How to convert improper fraction into mixed numbers?

To convert improper fraction into mixed number, divide the numerator by denominator. Now, consider the quotient as the whole number, and the remainder as the numerator of the proper fraction keeping the denominator same.  For example,  in 192, divide 19 by 2 to get quotient value as 9 and remainder value as 1. Accordingly, 192 = 9 12.

2. What is known as simplifying fraction in lowest possible terms in multiplying mixed numbers?

Simplifying fractions in lowest terms involves dividing both the numerator and denominator by a common factor to minimise the fraction to their equivalent fraction with the smallest possible numerator and denominator.


text:
## Chemistry: The Central Science (13th Edition)

Published by Prentice Hall

# Chapter 5 - Thermochemistry - Exercises - Page 206: 5.47a

#### Answer

$\Delta H = + 726.5 kJ$

#### Work Step by Step

Given the reaction : CH3OH + 3/2 O2 --> CO2 + 2 H2O $\Delta H = -726.5 kJ$ In order to find the enthalpy change for the reverse reaction, we must multiply the enthalpy by -1 Therefore, the enthalpy would be $\Delta H = -726.5 * -1 = 726.5 kJ$

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


text:
# Do not repeat yourselves or DRY way of working with AMC¶

Object oriented software design has a main principle, the do not repeat yourselves or DRY principle, that is,
Some people could say it is a way of being lazy, which is fine by me, as long as you compensate that with sufficient cleverness.

A way of use AMC in multiple courses is a follows:

Keep all questions in separate files, organised by course and subject.
Preferably also keep them in a version control system. At the moment that is subversion.

## Directory structure¶

Say you teach course LNX, MOD and SEN.
Then for instance my directory structure looks a lot like this.

├── lnx
│   └── exam
│       ├── builds
│       │   ├── 20110624
│       │   └── 20120627
│       └── questions
│           ├── chap1
│           ├── chap2
│           ├── chap3
│           ├── chap4
│           ├── chap5
│           ├── chap6
│           └── chap7
├── mod
│   └── exam
│       ├── builds
│       └── questions
└── sen
└── exam
├── builds
└── questions

Keep the question below the questions directory as in lnx/exam/questions/chap1/cmdline1.tex, which contain the normal
AMC question and questionmult environments. Prepare the questions with a normal editor.

### Role of files¶

For each exam that you build from this, create one exam.tex, one questions.tex and one students.csv file in a sub directory specific to the exam instance or exam event.
You could use the convention to use date in short iso8601 format, so 20120627 would be for the exam on 27th of June 2012.

The exam.tex file defines all the things that describe the exam and these details are put mostly on the frontpage of each exam.
Think of date, time of the exam, target group, subject title, teachers name and such.

The questions.tex file is a more or less simple file with one line specifying the question and the AMC-element group (the element group is what AMC uses to randomize the question order). A questions.tex file for a typical exam contains 30 or 40 such lines.
A line looks like this:

\element{general}{\inputQ{chap1/cmdline1.tex}}

The macro
\inputQ
is no more then an abbrivation of
\input{\QuestionBaseDir/#1}
, of which \QuestionBaseDir is explained below.

Keeping the questions separate from the exam file eases the way to create this file, for instance from a spread-sheet or a database query and also eases the comparison with earlier exams,
so one can easily see if there is not to much overlap in questions for instance for a resit of an exam with (a subset) of the same audience.

The students.csv file contains the id's and names (and in my case also the exam language) for the participating students.

## Using the AMC gui with the above setup¶

You can run (and do it for testing purposes of the exams) pdflatex on the exam.tex file, but for a real production, use the ~/MC-projects directory and the AMC gui to do all the
nice things that AMC provides.

To make this work you use a symbolic link source.tex pointing to exam.tex the tree explained above, for instance in the lnx exam that would be

hom@threehundred:~/MC-Projects/lnx_20120629
\$ ls -l
source.tex -> ../courses/lnx/exam/builds/20120629/exam.tex

but you could point anywhere.
To make sure AMC (or more precisely, pdflatex) can find the files to link and include, you should define a few macros that help latex.
These macros are
\def\CourseDir{/home/hom/fthv/lnx/exam}
\def\QuestionBaseDir{\CourseDir/questions}
\def\ExamInstanceDir{\CourseDir/builds/20120629}

It could even be dryer if you keep to conventions like questions and builds always under these names. Then you only need to set the coursedir once and the date for a specific exam instance.

\CourseDir specifies the course, the \QuestionsBaseDir should be obvious as is the \ExamInstanceDir.
Note that these specify absolute paths, so that pdflatex can find them, where ever your working directory and latex process directory is.

You can symbolically link the students.csv to the MC-projects subdirectory.

To make this exam known to AMC, start it via the gui and let it create a new empty exam with an appropriate name like lnx_20120629. Then press edit exam file, which brings up your editor with a new source.tex file. Source.tex is the default file that AMC uses. This is fine.

I save that file and close the editor, cd (in a terminal window) to the ~/MC-projects/lnx_20120629 subdirectory and create the symbolic link with

ln -s ../courses/lnx/exam/builds/20120629/exam.tex source.tex

This of course removes the auto generated source.tex file and replaces it with a link to the real exam file. You can however still use the edit button with AMC, which will then edit
the newly linked to file. Emacs even recognises the fact that it is a symbolic link to a versioned file and ask if you want to visit the original (pointed to) file. I always say yes.

Back in the AMC guy you can then complete all steps to create the pdf file, check the layouts, print the stuff and after the exam process all.

To make the automatic association and thus producing a personal exam per student, you can use the macros as proposed by Alexis Bienvenüe in ticket #32.

The toyexam.zip attached to this page is a copy of my current collection of test questions, which we can freely use and distribute (unless you teach European topology,
then you might want to use the questions in a real exam). Anyway, they are the same kind of questions that the maintainers of AMC use to test various features and question type.
Use them freely as a base to clarify questions and answers with respect to this DRY way of working.

### Use of the toyexamUnzip it somewhere conventient.Create a new exam in the AMC gui e.g., toyexam1 and let the source.tex file point to the .../toyexam/40_exam/builds/exam1/exam.tex file.The names of the directories may differ a bit from the description above, but it is the internal convention we use in our institute.¶

The subdir exam2 now contains a complete working example of using a students.csv and auto association.
Link the exam.tex file in the way as described above.


text:
# Software Tools in Haskell: echo

## write arguments to stdout

Posted on 2016-02-19 by nbloomf

This post is literate Haskell; you can load the source into GHCi and play along.

-- sth-echo: write arguments to stdout
module Main where

import System.Exit (exitSuccess)
import System.Environment (getArgs)

All the programs we’ve written so far are strictly filters: they read data from stdin and write data to stdout. The metaphor here is that small programs are chained together in a larger “pipeline”, and data flows from one end to the other; along the way, each filter changes the data in some way. By reading and writing from stdin and stdout, individual programs do not need to worry about where their data comes from and goes.

echo is the first program we’ve written that produces data without needing to take any from stdin; it is a source. (The converse, a program which consumes data without producing any, is a sink). echo simply takes a list of arguments at the command line and writes them to stdout. Although this is simple, like copy, we have to decide how to interpret the arguments to echo. By default, echo will treat its arguments as lines and write one per line. We use the --char argument to write raw characters.

main :: IO ()
main = do
args <- getArgs

case args of
("--char":xs) -> putStr (concat xs)
xs            -> sequence_ \$ map putStrLn xs

exitSuccess

We can now use echo to test our other programs. For instance, using the shell to run

sth-echo "hello" | sth-count --char

prints 6 to the terminal (including the trailing newline). Woo! But if we run

sth-echo --char "hello" | sth-count --char

we get 5.


text:
## vf321 Group Title With the initial conditions for a two mass system m1 and m2, where neither mass is significantly larger than the other, the resulting DE is unsolvable, right? Assume that they both start at rest: (see DE below) Do things get simpler if I give a starting velocity to both masses? What if I look at position relative to the com of the system? 2 years ago 2 years ago

1. vf321 Group Title

The DE for the two masses in 2D space in general:$m_1\frac{d^2\vec r_1}{dt^2}=G\frac{m_1 m_2 (\vec r_2-\vec r_1)}{|\vec r_2-\vec r_1|^3}$(switch necessary subscripts for m2) Making the assumption that they're not moving at first, we know movement will be linear. Make the linear axis "x": $m_1\frac{d^2x_1}{dt^2}=G\frac{m_1 m_2 (x_2-x_1)}{|x_2-x_1|^3}=\pm G\frac{m_1m_2}{(x_2-x_1)^2}$So we get a system of DEs: $m_1\frac{d^2x_1}{dt^2}=\pm G\frac{m_1m_2}{(x_2-x_1)^2}$$m_2\frac{d^2x_1}{dt^2}=\mp G\frac{m_1m_2}{(x_2-x_1)^2}$Still unsolvable to my knowledge...

2. vf321 Group Title

EDIT: last equation should be$m_2\frac{d^2x_2}{dt^2}=\mp G\frac{m_1m_2}{(x_2-x_1)^2}$

3. experimentX Group Title

consider one of the mass to be at rest and another is accelerating.

4. vf321 Group Title

You can't do that though! m_2 is accelerating too!

5. experimentX Group Title

well if you want a dynamic system then ... it's going to be a system of second order DE.

6. vf321 Group Title

Yeah... But that's the problem. The DE seems to be unsolvable. My question is asking if there's some clever symmetry (like the com or something) that we can use to get around the DE.

7. experimentX Group Title

not sure ... first order system is fairly complicated.

8. experimentX Group Title

plus ... this is non linear :(( looks like there must be some other approach.

9. vf321 Group Title

Yeah but its not a question from a textbook... I just made this up. There might not be an approach :(.

10. experimentX Group Title

let me try to ask mathematica. do you want general solution or ... do you have some boundary condition?

11. experimentX Group Title

didn't get any solution from mathematica.

12. vf321 Group Title

I think I have something that might help simplify a bit (N's 3rd law): $m_2\frac{d^2x_2}{dt^2}=-m_1\frac{d^2x_1}{dt^2}$$\int_0^tm_2\frac{d^2x_2}{dt^2}dt=\int_0^t-m_1\frac{d^2x_1}{dt^2}dt$$m_2(\frac{dx_2}{dt}-v_{i2})=-m_1(\frac{dx_1}{dt}-v_{i1})$$v_{i1}, v_{i2}=0 m/s$$\int_0^tm_2\frac{dx_2}{dt}dt=\int_0^t-m_1\frac{dx_1}{dt}dt$$m_2(x_2(t)-x_{i2})=-m_1(x_1(t)-x_{i1})$$x_2(t)=\frac{-m_1(x_1(t)-x_{i1})}{m_2}+x_{i2}$That seems right. If it is, then we don't have a system of DE's anymore...

13. vf321 Group Title

I think we get: $m_1\frac{d^2x_1}{dt^2}=\pm G\frac{m_1m_2}{(-(m_1x_1/m_2)-x_1+c_1)^2}$$c_1=\frac{m_1x_{i1}}{m_2}+x_{i2}$

14. experimentX Group Title

let dx/dt = v ... dv/dt = dv/dx *dx/dt = v dv/dx not sure if you integrate it wrt dt ...

15. vf321 Group Title

is $\int v(t)dt=x(t)$?

16. vf321 Group Title

Cause that's all i did.

17. experimentX Group Title

this doesn't solve x(t) at all ... we need x as explicit function of time. sorry ... i think integration method would work.

18. vf321 Group Title

It doesn't solve for x(t) as a function of time, but it makes x_2 represented as x_1. This way we have one DE to solve instead of a system, right?

19. experimentX Group Title

http://en.wikipedia.org/wiki/Two-body_problem i remember doing this kinda (central force) problem using reduced mass :((

20. experimentX Group Title

something like |dw:1345580217154:dw|

21. experimentX Group Title

still this system is interesting ... right now i'm working on first order systems. Maybe i'll start PDE right after it. If i encounter second order system ... i'll try to update.

22. experimentX Group Title

i think using symmetricity we can simplify this problem for two identical bodies. |dw:1345580536599:dw|

23. experimentX Group Title

for non identical bodies , the acceleration produced is equal to their ratio of masses |dw:1345580770508:dw|

24. experimentX Group Title

|dw:1345580931804:dw| choosing initial condition, and choosing a suitable coordinate system, i think .... this is pretty easy to solve x(t) as explicit function of time.

25. experimentX Group Title

I'm not sure for general case ... with non rest IC.

26. experimentX Group Title
27. vf321 Group Title

Okay thx i looked at the wiki link. I'm on to something on my own and I'll see if that works.

28. vf321 Group Title

OK no one else is helping at physics so I'll repost the problem in the math section...

29. vf321 Group Title

If any1's interested: http://math.stackexchange.com/questions/185593/solving-the-de-for-a-two-body-system

30. experimentX Group Title

try asking this instead. How to solve for general solution of system of this differential equation? |dw:1345676009886:dw|

31. experimentX Group Title

more over this is just 1D case ... for general 2d case the system of DE could me much worse.

32. vf321 Group Title

Yeah I know it's 1D I was trying just to tackle this, let alone the 2D.

33. experimentX Group Title

i guess CM frame of reference is a better choice to solve problem. later we can translate origin to any other choice after we find x and y. Try to simulate this graphically.

34. experimentX Group Title

since we know, the distance between the objects from CM with be at constant proportion of their masses.

35. vf321 Group Title

It's okay, I'll just do it out on my own using the CM. Don't worry about it.