A Characterization of Twin Prime PairsJeffrey Gold
Proceedings - NCUR V. (1991), Vol. I, pp. 362-366. Jeffrey F. Gold Department of Mathematics, Department of Physics University of Utah Don H. Tucker Department of Mathematics University of Utah The basic idea of these remarks is to give a tight characterization of twin primes greater than three. It is hoped that this might lead to a decision on the conjecture that infinitely many twin prime pairs exist; that is, number pairs (p; p+ 2) in which both p and p + 2 are prime integers.
Implicit differentiation allows us to find slopes of lines tangent to curves that are not graphs of functions. Almost all of the time (yes, that is a mathematical term!) we can assume the curve comprises the graph of a function and differentiate using the chain rule.
A Characterization of Twin Prime PairsJeffrey Gold
Proceedings - NCUR V. (1991), Vol. I, pp. 362-366. Jeffrey F. Gold Department of Mathematics, Department of Physics University of Utah Don H. Tucker Department of Mathematics University of Utah The basic idea of these remarks is to give a tight characterization of twin primes greater than three. It is hoped that this might lead to a decision on the conjecture that infinitely many twin prime pairs exist; that is, number pairs (p; p+ 2) in which both p and p + 2 are prime integers.
Implicit differentiation allows us to find slopes of lines tangent to curves that are not graphs of functions. Almost all of the time (yes, that is a mathematical term!) we can assume the curve comprises the graph of a function and differentiate using the chain rule.
SET
A set is a well defined collection of objects, called the “elements” or “members” of the set.
A specific set can be defined in two ways-
If there are only a few elements, they can be listed individually, by writing them between curly braces ‘{ }’ and placing commas in between. E.g.- {1, 2, 3, 4, 5}
The second way of writing set is to use a property that defines elements of the set.
e.g.- {x | x is odd and 0 < x < 100}
If x is an element o set A, it can be written as ‘x A’
If x is not an element of A, it can be written as ‘x A’
Special types of sets-
Standard notations used to define some sets:
N- set of all natural numbers
Z- set of all integers
Q- set of all rational numbers
R- set of all real numbers
C- set of all complex numbers
TYPES OF SETS
-subset
-singleton set
-universal set
-empty set
-finite set
-infinte set
-eual set
-disjoint set
-cardinal set
-power set
OPERATIONS ON SET
The four basic operations are:
1. Union of Sets
2. Intersection of sets
3. Complement of the Set
4. Cartesian Product of sets
Union of two given sets is the smallest set which contains all the elements of both the sets.
A B = {x | x A or x B}
Let a and b are sets, the intersection of two sets A and B, denoted by A B is the set consisting of elements which are in A as well as in B
A B = {X | x A and x B}
If A B= , the sets are said to be disjoint.
If U is a universal set containing set A, then U-A is called complement of a set.
This is meant for age group 11 to 14 years.
For Class VIII CBSE.
Some viewers have requested me to send the file through mail.
So I allowed everybody to download.My request is whenever you are using plz acknowledge me.
Pratima Nayak ,Teacher,Kendriya Vidyalaya,Fort William,Kolkata
pnpratima@gmail.com
Based on Text book
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Embracing GenAI - A Strategic ImperativePeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
Honest Reviews of Tim Han LMA Course Program.pptxtimhan337
Personal development courses are widely available today, with each one promising life-changing outcomes. Tim Han’s Life Mastery Achievers (LMA) Course has drawn a lot of interest. In addition to offering my frank assessment of Success Insider’s LMA Course, this piece examines the course’s effects via a variety of Tim Han LMA course reviews and Success Insider comments.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
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.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
Palestine last event orientationfvgnh .pptxRaedMohamed3
An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
3. Essential Questions
How do you write polynomials in standard form?
How do you add and subtract polynomials?
Where you’ll see this:
Part-time jobs, travel, geography, modeling
5. Vocabulary
1. Monomial: An expression that has one term (a
number, variable, or a combination of both a
number and variables without any addition or
subtraction)
2. Coefficient:
3. Constant:
4. Polynomial:
5. Term:
6. Vocabulary
1. Monomial: An expression that has one term (a
number, variable, or a combination of both a
number and variables without any addition or
subtraction)
2. Coefficient: The number that is with the variable
3. Constant:
4. Polynomial:
5. Term:
7. Vocabulary
1. Monomial: An expression that has one term (a
number, variable, or a combination of both a
number and variables without any addition or
subtraction)
2. Coefficient: The number that is with the variable
3. Constant: A number without a variable
4. Polynomial:
5. Term:
8. Vocabulary
1. Monomial: An expression that has one term (a
number, variable, or a combination of both a
number and variables without any addition or
subtraction)
2. Coefficient: The number that is with the variable
3. Constant: A number without a variable
4. Polynomial: A collection of terms that are
combined by addition or subtraction
5. Term:
9. Vocabulary
1. Monomial: An expression that has one term (a
number, variable, or a combination of both a
number and variables without any addition or
subtraction)
2. Coefficient: The number that is with the variable
3. Constant: A number without a variable
4. Polynomial: A collection of terms that are
combined by addition or subtraction
5. Term: Each monomial within a polynomial
12. Vocabulary
6. Binomial: A polynomial with two terms
7. Trinomial: A polynomial with three terms
8. Standard Form:
9. Like Terms:
13. Vocabulary
6. Binomial: A polynomial with two terms
7. Trinomial: A polynomial with three terms
8. Standard Form: When a polynomial is written from
highest to lowest degree (highest to lowest
exponent)
9. Like Terms:
14. Vocabulary
6. Binomial: A polynomial with two terms
7. Trinomial: A polynomial with three terms
8. Standard Form: When a polynomial is written from
highest to lowest degree (highest to lowest
exponent)
9. Like Terms: Terms that have the same variable
parts (variables and exponents)
15. Example 1
Tell the variable for which the polynomial is
arranged in standard form.
3 2
a. 2a + 3ab − 4b
3 2
b. 2(a + b) + 3(a + b) − 4(a + b) + 7
16. Example 1
Tell the variable for which the polynomial is
arranged in standard form.
3 2
a. 2a + 3ab − 4b
a
3 2
b. 2(a + b) + 3(a + b) − 4(a + b) + 7
17. Example 1
Tell the variable for which the polynomial is
arranged in standard form.
3 2
a. 2a + 3ab − 4b
a
3 2
b. 2(a + b) + 3(a + b) − 4(a + b) + 7
(a + b)
18. Example 2
Add the polynomials.
2 2
a. (2x − 3x + 7) + (−2x − 8) + ( x − 7x)
2 2 2 2
b. (3x − 4 xy ) + (− x + 4 y ) + (2xy − y )
19. Example 2
Add the polynomials.
2 2
a. (2x − 3x + 7) + (−2x − 8) + ( x − 7x)
2 2
2x − 3x + 7 − 2x − 8 + x − 7x
2 2 2 2
b. (3x − 4 xy ) + (− x + 4 y ) + (2xy − y )
20. Example 2
Add the polynomials.
2 2
a. (2x − 3x + 7) + (−2x − 8) + ( x − 7x)
2 2
2x − 3x + 7 − 2x − 8 + x − 7x
2
3x
2 2 2 2
b. (3x − 4 xy ) + (− x + 4 y ) + (2xy − y )
21. Example 2
Add the polynomials.
2 2
a. (2x − 3x + 7) + (−2x − 8) + ( x − 7x)
2 2
2x − 3x + 7 − 2x − 8 + x − 7x
2
3x −12x
2 2 2 2
b. (3x − 4 xy ) + (− x + 4 y ) + (2xy − y )
22. Example 2
Add the polynomials.
2 2
a. (2x − 3x + 7) + (−2x − 8) + ( x − 7x)
2 2
2x − 3x + 7 − 2x − 8 + x − 7x
2
3x −12x −1
2 2 2 2
b. (3x − 4 xy ) + (− x + 4 y ) + (2xy − y )
23. Example 2
Add the polynomials.
2 2
a. (2x − 3x + 7) + (−2x − 8) + ( x − 7x)
2 2
2x − 3x + 7 − 2x − 8 + x − 7x
2
3x −12x −1
2 2 2 2
b. (3x − 4 xy ) + (− x + 4 y ) + (2xy − y )
2 2 2 2
3x − 4 xy − x + 4 y + 2xy − y
24. Example 2
Add the polynomials.
2 2
a. (2x − 3x + 7) + (−2x − 8) + ( x − 7x)
2 2
2x − 3x + 7 − 2x − 8 + x − 7x
2
3x −12x −1
2 2 2 2
b. (3x − 4 xy ) + (− x + 4 y ) + (2xy − y )
2 2 2 2
3x − 4 xy − x + 4 y + 2xy − y
2 2
2x − 2xy + 3y
26. Example 3
Subtract 4x + y from the sum of x + 3y and 8x - 2y.
( x + 3y ) + (8 x − 2y ) − (4 x + y )
27. Example 3
Subtract 4x + y from the sum of x + 3y and 8x - 2y.
( x + 3y ) + (8 x − 2y ) − (4 x + y )
x + 3y + 8 x − 2y − 4 x − y
28. Example 3
Subtract 4x + y from the sum of x + 3y and 8x - 2y.
( x + 3y ) + (8 x − 2y ) − (4 x + y )
x + 3y + 8 x − 2y − 4 x − y
5x
29. Example 4
Simplify.
3 2 3 2
a. (6 x + 3x − 11x) + (2x − 9 x − 5 x)
2 2 2 2
b. ( x y − 2xy + 8) − (−7x y + 2xy − 4)
30. Example 4
Simplify.
3 2 3 2
a. (6 x + 3x − 11x) + (2x − 9 x − 5 x)
3 2 3 2
6 x + 3x − 11x + 2x − 9 x − 5 x
2 2 2 2
b. ( x y − 2xy + 8) − (−7x y + 2xy − 4)
31. Example 4
Simplify.
3 2 3 2
a. (6 x + 3x − 11x) + (2x − 9 x − 5 x)
3 2 3 2
6 x + 3x − 11x + 2x − 9 x − 5 x
3 2
8 x − 6 x − 16 x
2 2 2 2
b. ( x y − 2xy + 8) − (−7x y + 2xy − 4)
32. Example 4
Simplify.
3 2 3 2
a. (6 x + 3x − 11x) + (2x − 9 x − 5 x)
3 2 3 2
6 x + 3x − 11x + 2x − 9 x − 5 x
3 2
8 x − 6 x − 16 x
2 2 2 2
b. ( x y − 2xy + 8) − (−7x y + 2xy − 4)
2 2 2 2
x y − 2xy + 8 + 7x y − 2xy + 4
33. Example 4
Simplify.
3 2 3 2
a. (6 x + 3x − 11x) + (2x − 9 x − 5 x)
3 2 3 2
6 x + 3x − 11x + 2x − 9 x − 5 x
3 2
8 x − 6 x − 16 x
2 2 2 2
b. ( x y − 2xy + 8) − (−7x y + 2xy − 4)
2 2 2 2
x y − 2xy + 8 + 7x y − 2xy + 4
2 2
8 x y − 4 xy + 12
34. Example 4
Simplify.
2 2 2 2 2
c. ( x y + x − xy ) − (− y + y + xy + 4 x y )
35. Example 4
Simplify.
2 2 2 2 2
c. ( x y + x − xy ) − (− y + y + xy + 4 x y )
2 2 2 2 2
x y + x − xy + y − y − xy − 4 x y
36. Example 4
Simplify.
2 2 2 2 2
c. ( x y + x − xy ) − (− y + y + xy + 4 x y )
2 2 2 2 2
x y + x − xy + y − y − xy − 4 x y
2 2 2
−3x y + x − 2xy + y − y
