(c) H/(H \N) _= (HN)/N.
We end this section with one fact about groups which doesnt have an obvious
analogue for rings.
78 CHAPTER 3. GROUPS
Proposition 3.27 Let H and K be subgroups of the group G.
(a) HK is a subgroup of G if and only if HK = KH.
natural homomorphism from G to G/H. So normal subgroups are the same thing
as kernels of homomorphisms.
Example Let G = S3, and let H be the subgroup cfw_1, (1,2,3), (1,3,2). We have
observed that H is a normal subgroup. It has two cosets, namely H1 = H a
Notes on Algebraic Structures
Peter J. Cameron
These are the notes of the second-year course Algebraic Structures I at Queen
Mary, University of London, as I taught it in the second semester 20052006.
After a short introductory chapter consisti
/* remove an element from a list */
/* Make sure you keep a pointer to some other element! */
/* does not free the removed element */
/* splice e out */
e->next[RIGHT]->next[LEFT] = e->next[LEFT];
The region in use is specified by a base index pointing
* to the first element, and a length count giving the number
* of elements. A size field specifies the number of slots
* in the block.
-* |7|8|9| | | | | | | | | | | | | | | | |1|2|3|4|5
/* do a transplant from d2 to d */
/* but save old contents so we can free them */
oldContents = d->contents;
*d = *d2; /* this is equivalent to copying the components one by one */
/* these are the pieces we don't need any more */
1. Steam electric drive
In steam electric locomotives, the steam turbine is employed for driving a
generator used to feed the electric motors. Such types of locomotives are
not generally used for traction because of some mechanical diffic
circulating current to produce the torque in the reverse direction. This torque is
known as braking torque. This braking torque helps to bring the motor to rest.
Rheostatic or dynamic braking
In this method of braking, the electric motor is disconnected f
Dynamic and regenerative braking
Traction motors should be able to provide easy simple rehostatic and regenerative
braking subjected to higher voltages so that system must have the capability of
withstanding voltage fluctuations.
4. A stop of 7 min.
Then, determine the distance between two stations, the average, and the schedule
Acceleration () = 6.5 kmphps.
Acceleration period t1 = 20 s.
Maximum speed Vm = t1
= 6.5 20 = 130 kmph.
Free-running time (t2) = 12 60
applied at low speeds for providing better saturation where electric braking is
ineffective, during the normal service. The erection and maintenance costs of
tramways are high since the cost of overhead distribution structure is costlier and
Circular-type heating element
Initially when the heating element is connected to the supply, the temperature goes on
and finally reaches high temperature.
Let V be the supply voltage of the system and R be the resistance of the element, then el
3 Adhesive weight
The total weight to be carried out on the drive in wheels of a locomotive is known
as adhesive weight.
4 Coefficient of adhesion
It is defined as the ratio of the tractive effort required to propel the wheel of a
locomotive to its adhesi
is measured in newton.
The net effective force or the total tractive effort (Ft) on the wheel of a
locomotive or a train to run on the track is equals to the sum of tractive effort:
1. Required for linear and angular acceleration (Fa).
2. To overcome the
6. Wire brush to clean the weld.
7. Earth clamp and protective clothing.
COMPARISON BETWEEN AC AND DC WELDING
AC welding DC welding
1 Motor generator set or rectifier is required in case of
the availability of AC supply.
Only transformer is required.
seduced frequencies. In addition to the above advantages, the induction motors
suffer from some drawbacks; they are low-starting torque, high-starting current,
and the absence of speed control. The main disadvantage of such track
electrification system is
Maximum speed Vm = 45 kmph.
Losses per motor = 3,000 W.
Copper losses = I2Rm = I2 0.5
Motor input = motor output + constant loss + copper losses
450 I = 20,000 + 3,000 + 0.5I2
0.5 I2 450I + 23,000 = 0.
After solving, we get I = 54.39 A.
Total current draw
o The submerged arc welding can be done manually or automatically.
o The submerged arc welding is widely used in the heavy steel plant fabrication work.
o It can be employed for welding high strength steel, corrosion resistance steel, and low
The laser beam emitting from the flash tube, passing through the focusing lens, where it is
pinpointed on the workpiece. The heat so developed by the laser beam melts the work-piece
the weld is completed. The welding characteristics of the laser are s
of providing additional generators.
o The maintenance and running costs are comparatively low.
o The speed control of the electric motor is easy.
o Regenerative braking is possible so that the energy can be fed back to the supply
system during the braking
showing Equation (10.17)will be required to oppose the force due to gravitational
force, but while going down the gradient, the same force will be added to the total
The total tractive effort required for the propulsion of train Ft = Fa
precisely, the expected 2 number of trials is bounded by 2. So, in (expected) two trials,
we will find a good splitter that reduces the size of the problem to at most 3
argument can be repeated for the recursive calls, namely, the expected number
Example 10.22: Calculate the energy consumption if a maximum speed of 12
m/sec and for a given run of 1,500 m, an acceleration of 0.36 m/s2 desired. The
tractive resistance during acceleration is 0.052 N/kg and during the coasting is 6.12
N/1,000 kg. A
is desirable that for any algorithm, you get an estimate of the maximum size of the
numbers to ensure that operands do not exceed (log n) so that it is safe to use the
1.4 Other models
There is clear trade-off between the simplicity and the fid
The energy returned to the line = 0.811.08
= 88.707 kW-hr.
Example 10.19: An electric train has an average speed of 50 kmph on a level
track betweenstops 1,500 m a part. It is accelerated at 2 kmphs and is braked at 3
kmphs. Estimate the energy consumptio
transformer through iron cored reactors RE1 andRE2 and commutating pole winding
An auxiliary transformer is provided to excite the field winding of the traction
motor. Let us assume V be the voltage of tap-changing transformer and If is the
The above is easily achieved by applying integer sorting in the range [1.m] starting
from the least significant digits - note that the maximum number of digits in radix
m representation is k. If we apply the same algorithm for sorting words, then the