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1st Assesment
MTH 327
Normed spaces are vectors space having some additional structure properties due to the norm defined
on them. In fact the norm give the topological and analysis structure to vector space , so in this normed
spaces have algebraic , topological and analysis structure while vector spaces have only algebraic
structure. in some deep sense the norm defined on them tries to connect the topological properties and
algebraic properties i.e they are defined in term of each others .
MTH 375
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Numerical methods can give us an approximate solution to a problem but tell us next to nothing about the
structure of the solution space. Numerical analysis may not be able to give us anything but trivial solutions,
but in many cases it can tell us what the overall structure of the solutions has to look like.
A numerical method attempts to approximately solve questions about a system, such as the solutions to the
equations of motion of a system. An analytic solution obtained through mathematical analysis would be an
exact solution, but mathematical analysis can yield other answers about a system, whether it is correlation
functions, inequalities, or what have you
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MTH327
Normed spaces are vectors space having some additional structure properties due
to the norm defined on them. In fact the norm give the topological and analysis
structure to vector space , so in this normed spaces have algebraic , topological and
analysis structure while vector spaces have only algebraic structure. in some deep
sense the norm defined on them tries to connect the topological properties and
algebraic properties i.e they are defined in term of each others .
MTH467
OR became an established discipline during World War II, when the British government recruited
scientists to solve problems in critical military operations. Mathematical methods were developed to
determine the most effective use of radar and other new defence technologies at the time. OR groups were
later formed in the U.S. to meet needs of wartime operations, such as the optimal movement of troops,
supplies, and equipment.
Examples of where OR has been successful in recent years are the following:
Airline Industry (routing and flight plans, crew scheduling, revenue management)
Telecommunications (network routing, queue control);
Manufacturing Industry (system throughput and bottleneck analysis, inventory control, production
scheduling, capacity planning);
Healthcare (hospital management, facility design); and
Transportation (traffic control, logistics, network flow, airport terminal layout, location planning).
MTH365
The variance is a numerical value used to indicate how widely individuals in a group vary. If individual observations vary greatly
from the group mean, the variance is big; and vice versa.
It is important to distinguish between the variance of a population and the variance of a sample. They have different notation,
and they are computed differently. The variance of a population is denoted by σ2; and the variance of a sample, by s2.
The variance of a population is defined by the following formula:
σ2 = Σ ( Xi - X )2 / N
where σ2 is the population variance, X is the population mean, Xi is the ith element from the population, and N is the number of
elements in the population.
The variance of a sample is defined by slightly different formula:
s2 = Σ ( xi - x )2 / ( n - 1 )
where s2 is the sample variance, x is the sample mean, xi is the ith element from the sample, and n is the number of elements
in the sample. Using this formula, the variance of the sample is an unbiased estimate of the variance of the population.
And finally, the variance is equal to the square of the standard deviation.
MTH426
A set having no element is called empty set. Its cardinality is zero. The empty set may also be called the void set.
Common notations for the empty set include "{}", "∅".
Role Of Empty set in Set Theory:
Every power set of a set has a empty set for example A={x, y} then P (A) = {∅, {x}, {y}, {x, y}}
And empty set is the sub set of every set. In set theory some time when we take intersection of two or more sets
answer comes an empty set. Moreover empty set has a vital role in topology. For example in
Topology: Considered as a subset of the real number line or topological space the empty set is both closed and
open it is an example of a clopen set. All its boundary points (of which there are none) are in the empty set, and
the set is therefore closed; while for every one of its points (of which there are again none), there is an open
neighbourhood in the empty set, and the set is therefore open. Moreover, the empty set is a compact set by the
fact that every finite set is compact.
STA365
In probability theory and statistics, variance measures how far a set of numbers are spread out. A variance of
zero indicates that all the values are identical. The variance measures how far each number in the set is from the
mean. Variance is calculated by taking the differences between each number in the set and the mean, squaring
the differences (to make them positive) and dividing the sum of the squares by the number of values in the set.
USES OF VARIANCE: Variance is used in statistics for probability distribution. Since variance measures the
variability (volatility) from an average or mean, and volatility is a measure of risk, the variance statistic can help
determine the risk an investor might take on when purchasing a specific security.
Statisticians use variance to see how individual numbers relate to each other within a data set, rather than using
broader mathematical techniques such as arranging numbers into quartiles.
DRAWBACK:
A drawback to variance is that it gives added weight to numbers far from the mean (outliers), since squaring
these numbers can skew interpretations of the data.