Contents
SECOND SHIFTING THROEM and its example.
DERIVATIVE PROPERTY (multiplication by t property) with examples.
INTEGRAL PROPERTY (division by t property) with examples.
Merge sort is a sorting technique based on divide and conquer technique. With worst-case time complexity being Ο(n log n), it is one of the most respected algorithms.
Merge sort first divides the array into equal halves and then combines them in a sorted manner.
Brian Covello: Research in Mathematical Group Representation Theory and SymmetryBrian Covello
Brian Covello's research review on group representation theory and symmetry. In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication. Representations of groups are important because they allow many group-theoretic problems to be reduced to problems in linear algebra, which is well understood. They are also important in physics because, for example, they describe how the symmetry group of a physical system affects the solutions of equations describing that system.
The term representation of a group is also used in a more general sense to mean any "description" of a group as a group of transformations of some mathematical object. More formally, a "representation" means a homomorphism from the group to the automorphism group of an object. If the object is a vector space we have a linear representation. Some people use realization for the general notion and reserve the term representation for the special case of linear representations.
In mathematics, representation theory is a technique for analyzing abstract groups in terms of groups of linear transformations.
For the symmetric groups, a graphical method exists to determine their finite representations that associates with each representation a Young tableau (also known as a Young diagram). The direct product of two representations may easily be decomposed into a direct sum of irreducible representation by a set of rules for the "direct product" of two Young diagrams. Each diagram also contains information about the dimension of the representation to which it corresponds. Young tableaux provide a far cleaner way of working with representations than the algebraic methods that underlie their use.
Merge sort is a sorting technique based on divide and conquer technique. With worst-case time complexity being Ο(n log n), it is one of the most respected algorithms.
Merge sort first divides the array into equal halves and then combines them in a sorted manner.
Brian Covello: Research in Mathematical Group Representation Theory and SymmetryBrian Covello
Brian Covello's research review on group representation theory and symmetry. In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication. Representations of groups are important because they allow many group-theoretic problems to be reduced to problems in linear algebra, which is well understood. They are also important in physics because, for example, they describe how the symmetry group of a physical system affects the solutions of equations describing that system.
The term representation of a group is also used in a more general sense to mean any "description" of a group as a group of transformations of some mathematical object. More formally, a "representation" means a homomorphism from the group to the automorphism group of an object. If the object is a vector space we have a linear representation. Some people use realization for the general notion and reserve the term representation for the special case of linear representations.
In mathematics, representation theory is a technique for analyzing abstract groups in terms of groups of linear transformations.
For the symmetric groups, a graphical method exists to determine their finite representations that associates with each representation a Young tableau (also known as a Young diagram). The direct product of two representations may easily be decomposed into a direct sum of irreducible representation by a set of rules for the "direct product" of two Young diagrams. Each diagram also contains information about the dimension of the representation to which it corresponds. Young tableaux provide a far cleaner way of working with representations than the algebraic methods that underlie their use.
On Spaces of Entire Functions Having Slow Growth Represented By Dirichlet SeriesIOSR Journals
In this paper spaces of entire function represented by Dirichlet Series have been considered. A
norm has been introduced and a metric has been defined. Properties of this space and a characterization of
continuous linear functionals have been established.
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New Method for Finding an Optimal Solution of Generalized Fuzzy Transportatio...BRNSS Publication Hub
In this paper, a proposed method, namely, zero average method is used for solving fuzzy transportation problems by assuming that a decision-maker is uncertain about the precise values of the transportation costs, demand, and supply of the product. In the proposed method, transportation costs, demand, and supply are represented by generalized trapezoidal fuzzy numbers. To illustrate the proposed method, a numerical example is solved. The proposed method is easy to understand and apply to real-life transportation problems for the decision-makers.
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Traction is the force that allows a vehicle to move forward or backward on a surface. It is the result of friction between the tires and the ground. Traction is important for vehicle safety and performance, as it affects acceleration, braking, and cornering.
The theory predicts that failure occurs when the maximum tensile stress reaches a critical value. This critical value is determined by the same factors as in shear, namely the friction angle and the cohesion of the material.
The Mohr-Coulomb failure envelope in traction is a plot of the tensile stress versus the normal stress acting on the material. The slope of the envelope still represents the friction angle, while the intercept on the tensile stress axis represents the tensile strength of the material.
factors affecting
Tire type
Surface conditions
Vehicle weight
Driving style
Road grade and slope
Temperature
tire pressure
Soil stabilization is the permanent physical and chemical alteration of soils to enhance their physical properties.
Stabilization can increase the shear strength of a soil and control the shrink-swell properties of a soil, thus improving the load-bearing capacity of a sub-grade to support pavements and foundations.
Stabilization can be used to treat a wide range of sub-grade materials from expansive clays to granular materials.
Stabilization can be achieved with a variety of chemical additives including lime, fly ash, and Portland cement, as well as by-products such as lime-kiln dust and cement-kiln dust.
1) Mechanical Soil Stabilization Technique:
Dense and well graded material can be achieved by mixing and compacting two or more soils of different grades.
Addition of a small amount of fine materials such as silts or clays enables binding of the non-cohesive soils which increases strength of the material.
Factors affecting the mechanical stability of mixed soil may include:
The mechanical strength and purity of the constituent materials
The percentage of materials and its gradation in the mix
The degree of soil binding taking place
The mixing, rolling, and compaction procedures adopted in the field
The environmental and climatic conditions
2) Compaction Soil Stabilisation Technique:
Uses mechanical means for expulsion of air voids within the soil mass resulting in soil that can bear load subsequently without further immediate compression.
Dynamic compaction is one of the major types of soil stabilization; in this procedure, a heavyweight is dropped repeatedly onto the ground at regular intervals to quite literally pound out deformities and ensure a uniformly packed surface.
1) Moisture Content. 2) Specific gravity of soil. 3) Atterberg’s limit. 4) Liquid limit. 5) Particle size distribution. 6) Preparation of reinforced soil sample. 7) Determination of shear strength.
1) Moisture Content
Soil tests natural moisture content of the soil is to be determined. The natural water content also called the natural moisture content is the ratio of the weight of water to the weight of the solids in a given mass of soil.
2) Specific gravity of soil.
The specific gravity of soil is defined as the unit weight of the soil mass divided by the unit weight of distilled water at 4°C.
3) Atterberg’s limit
Atterberg's limits are a set of tests used in soil mechanics to determine the plasticity and compressibility characteristics of soil
1. It improves the strength of the soil, thus, increasing the soil bearing capacity.
2. It is a lot of economical each in terms of price and energy to extend.
3. Bearing capacity of the soil instead of going for deep foundation or raft foundation.
4. It offers more stability to the soil in slopes or other such places.
5. Sometimes soil stabilization is also stop soil erosion or formation of mud, which is extremely helpful particularly in dry and arid weather.
Generally, there are five types of plant hormones, namely, auxin, gibberellins (GAs), cytokinins, abscisic acid (ABA) and ethylene. In addition to these, there are more derivative compounds, both natural and synthetic, which also act as plant growth regulators.
Cottage cheese is a curdled milk product with a mild flavor and a creamy, heterogenous, soupy texture. It is made from skimmed milk by draining curds, but retaining some of the whey and keeping the curds loose
AERODYNAMIC PROPERTIES OF FOOD MATERIALS.pptxRanit Sarkar
Aero and /or hydrodynamic properties are very important characters in hydraulic transport and handling as well as hydraulic sorting of agricultural products. To provide basic data for the development of equipment for sorting and sizing of agro commodities, several properties such as: physical characteristics and terminal velocity are needed. The two important aerodynamic characteristics of a body are its terminal velocity and aerodynamic drag. By defining the terminal velocity of different threshed materials, it is possible to determine and set the maximum possible air velocity in which material out of grain can be removed without loss of grain or the principle can be applied to classify grain into different size groups. In addition, agricultural materials and food products are routinely conveyed using air. For such operations, the interaction between the solid particles and the moving fluids determine the forces applied to the particles. The interaction is affected by the density, shape, and size of the particle along with the density, viscosity, and velocity of the fluid. This chapter discusses briefly with the different aerodynamic properties and their methods of measurement.
Choose The Right Rotavator
It is important to select the correct size rotavator for your field or garden. There is little point arranged a large rotavator for a small garden. Also, consider if you will have sufficient space to access the area. if necessary consult with an expert to ensure you choose the correct rotavator for your needs.
When To Rotavate
It is generally advised to rotavate in spring or autumn. These seasons offer softer soil and will result in more aeration than in the summer months.
Check Soil Moisture
Your soil moisture can play a large part in how successful your rotavating is. Sandy soil will rotavate in a very similar way whether dry or wet and so the moisture level is not as important.
In comparison, clay soil must be done when the moisture is favourable. if the soil is too dry it will be very hard and difficult to break apart. In contrast, when the soil is too moist the clay can stick to your rotavator cause unnecessary mess and potential damage to your requirement.
Weed Control
Weed removal is very important when rotavating. If left you will find the weed will quickly grow out of control and the seeds have been mixed throughout the soil of your entire field or garden.
Control The Rotavator Properly
When you are using your rotavator you must ensure you maintain full control of the equipment. A rotavator is a powerful piece of equipment and it can easily course damage or harm if not used properly.
Rotavate The Land In Strips
When Rotavating your land plan head, it is advised to rotavate in strips to ensure the best result. Make a few passes over each strip, and repeat the process at right angles to the original rotavated strips. Don’t dig much deeper than two or three inches deep on the first pass. You can then set the rotavator to dig deeper on each pass after that. You should rotavate offer the course of several hours.
Almost all automobiles employ liquid cooling systems for their engines. A typical automotive cooling system comprises (1) a series of channels cast into the engine block and cylinder head, surrounding the combustion chambers with circulating water or other coolant to carry away excessive heat, (2) a radiator, consisting of many small tubes equipped with a honeycomb of fins to radiate heat rapidly, which receives and cools hot liquid from the engine, (3) a centrifugal-type water pump with which to circulate coolant, (4) a thermostat, which maintains constant temperature by automatically varying the amount of coolant passing into the radiator, and (5) a fan, which draws fresh air through the radiator. For operation at temperatures below 0 °C (32 °F), it is necessary to prevent the coolant from freezing. This is usually done by adding some compound, such as ethylene glycol, to depress the freezing point of the coolant. By varying the amount of additive, it is possible to protect against freezing of the coolant down to any minimum temperature normally encountered. Coolants contain corrosion inhibitors designed to make it necessary to drain and refill the cooling system only every few years.
Thermal processing of fruits and vegetables.pptxRanit Sarkar
There are two main temperature categories employed in thermal processing: Pasteurization and Sterilization. The basic purpose for the thermal processing of foods is to reduce or destroy microbial activity, reduce or destroy enzyme activity and to produce physical or chemical changes to make the food meet a certain quality standard. e.g. gelatinization of starch & denaturation of proteins to produce edible food.
CAGE SYSTEM for POULTRY FARMING.pptx.pptxRanit Sarkar
Types and Specification of Poultry Cage system. This system involves rearing of poultry on raised wire netting floor in smaller compartments, called cages. Description and knowledge about present and past systems used in poultry farming in cage system. Advantages and disadvantages of cage system. Difference between different cage system based on description.
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...Dr. Vinod Kumar Kanvaria
Exploiting Artificial Intelligence for Empowering Researchers and Faculty,
International FDP on Fundamentals of Research in Social Sciences
at Integral University, Lucknow, 06.06.2024
By Dr. Vinod Kumar Kanvaria
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
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This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
Normal Labour/ Stages of Labour/ Mechanism of LabourWasim Ak
Normal labor is also termed spontaneous labor, defined as the natural physiological process through which the fetus, placenta, and membranes are expelled from the uterus through the birth canal at term (37 to 42 weeks
3. CONTENTS
SECOND SHIFTING THROEM and its example.
DERIVATIVE PROPERTY (multiplication by t property) with examples.
INTEGRAL PROPERTY (division by t property) with examples.
4. If 𝐿{𝑓(𝑡)}=𝐹(𝑠) and 𝑢(𝑡−𝑎)=
then we define a new function
𝐿{𝑓(𝑡−𝑎)𝑢(𝑡−𝑎)} = e−sa F(s)
And its Inverse Laplace is
𝐿−1{e−sa F(s)} = 𝑓(𝑡−𝑎)𝑢(𝑡−𝑎)
0
1
{ if t a
if t a
5. Second shifting throem
Example :-
𝐿{𝑓(𝑡−𝑎) 𝑢(𝑡−𝑎)}
Comparing with second shift throem
𝑢(𝑡−𝑎) = 𝑢(𝑡−2) 𝑖.𝑒 𝑎 = 2
𝑓(𝑡−𝑎) = 𝑓(𝑡−2) = (𝑡−1)2
Put 𝑡=𝑡+2 then 𝑓(𝑡) = (𝑡+1)2 = 𝑡2+2𝑡+1
𝐹(𝑠)=𝐿{𝑡2+2𝑡+1 }= Hence by the second shifting theorem, we get
𝐿 {𝑓(𝑡−𝑎)𝑢(𝑡−𝑎)}=e−sa F(s)
𝐿 {(𝑡−1)2𝑢(𝑡−2)}=e−2s
3 2
2 2 1
s s s
3 2
2 2 1
s s s
6. 𝐿−1{e−sa F(s)} = 𝑓(𝑡−𝑎)𝑢(𝑡−𝑎)
here e-as= e-3s i.e. a=3
i.e u(t-3)
And f(s) =
f(t)= L-1 = sint
f(t-3) = sin (t-3)
i.e ans = sin(t-3)u(t-3)
Example of L-1 using
second shifting throem
3
1
2
[ ]
1
s
e
L
s
2
1
1
s
2
1
1
s
7. DERIVATIVE PROPERTY (multiplication by t property)
{ ( )} ( )
L f t F s
if
then
{ ( )} ( 1) { ( )}
n
n n
n
d
L t f t F s
ds
{ ( )} ( 1) { ( )}
d
L t f t F s
ds
2
2 2
2
{ ( )} ( 1) { ( )}
d
L t f t F s
ds
8. Example of derivitive property (multiplication by t property)
Find the laplace of
2 3
( )
t
L t e
{ ( )} ( 1) { ( )}
n
n n
n
d
L t f t F s
ds
Comparing with derivative property
2
1 ( 3)
( 3)
d s
s ds
1
( ) ( )
3
d d
f s
ds ds s
3
2 ( 3)
( )
( 3)
d s
s ds
2
1
( 1)
s
3
2
( 3)
s
2 2
3
2
{ ( )} ( 1)
( 3)
L t f t
s
2
2 2
1
{ ( )}
( 3)
d d
f s
ds ds s
3
{ ( )} { } ( )
t
L f t L e F s
1
( )
3
F s
s
2
3
2
{ ( )}
( 3)
L t f t
s
9. INTEGRAL PROPERTY (DIVISION by t property)
{ ( )} ( )
L f t F s
( )
{ } ( )
s
f t
L F s ds
t
2
( )
{ } ( )
s
s
f t
L F s ds ds
t
if
then
10. Example of integral property (division by t property)
Find the laplace of
comparing it with equation we get
1 1
{ ( )} ( )
4 6
L f t F s
s s
4 6
( ) ( ) ( )
t t
f t L e L e
4 6
( ) ( )
t t
f t e e
4 6
{ ( )} ( )
t t
L f t L e e
4 4
limln( ) ln( )
6 6
s
s s
s s
4
[ln( )]
6
s
s
s
[ln( 4) ln( 6)]s
s s
1 1
( ) ( )
4 6
s s
F s ds
s s
4
0 ln
6
s
s
4
1
4
limln( ) ln( )
6 6
1
s
s
s
s
s
1
4
ln( )
6
s
s
6
ln( )
4
s
s
4 6
t t
e e
t
4 6
{ } ( )
t t
s
e e
L F s ds
t
11. The knowledge of Laplace transform has in recent years become an
essential part of mathematical background required of engineers and
scientists. This is because the transform method an easy and effective
means for the solution of many problems arising in engineering.
The properties presented are very useful to solve specfic find of
laplace transform problems in a easy and more enhanced way.
conclusion