This document provides an overview of mathematical functions in MATLAB, including:
1) Common math functions such as absolute value, rounding, floor/ceiling, exponents, logs, and trigonometric functions.
2) How to write custom functions and use programming constructs like if/else statements and for loops.
3) Data analysis functions including statistics and histograms.
4) Complex number representation and basic complex functions in MATLAB.
Olivier Hudry (INFRES-MIC2 Télécom ParisTech)
A Branch and Bound Algorithm to Compute a Median Permutation
Algorithms & Permutations 2012, Paris.
http://igm.univ-mlv.fr/AlgoB/algoperm2012/
Olivier Hudry (INFRES-MIC2 Télécom ParisTech)
A Branch and Bound Algorithm to Compute a Median Permutation
Algorithms & Permutations 2012, Paris.
http://igm.univ-mlv.fr/AlgoB/algoperm2012/
1. Introduction
• Preliminaries
• Some Useful Definitions
• Types of fuzzy sets
• Degree of Fuzzy Sets
• Operators of Fuzzy Sets
• Conditions & Limitations
• Multiplication
• Summation
• Operators of Theory Sets
• Characteristics of S & T
• Some definitions for T & S
• Unity and Community Defs.
• Mean Operators
• Fuzzy AND & OR
• Combinations of Fuzzy AND & OR
2. Fuzzy Measurement & Measurement of Fuzzy Sets
• Fuzzy Measurement
• Dr. ASGARI Zadeh Possibility Definition
• SUGENO Definition
• Possibility Definition
• Graph of S Function
• Measurement of Fuzzy Sets
• Entropy of Fuzzy Sets (De Luca & Termini)
• YAGER Definition for Ã
3. Propagation principle
• Propagation principle & Applications
• Propagation principle and Second Types of Fuzzy Sets
• Fuzzy Numbers & Algebraic Operations
• Fuzzy Numbers Intervals
• L-R Interval Function (Asymmetric)
• L-R Interval Function
• L-R Interval Function Operations
4. Functions & Fuzzy Analyzing
• Functions & Fuzzy Analyzing
• Functions & Fuzzy Analyzing
• Fuzzy functions Extremes
• Integral of Fuzzy Functions
• Integral of Type 2 fuzzy function with definite interval
• Differentiation of Definite functions With Fuzzy Domains & Ranges
• Integral of fuzzy function with definite interval
• Properties of fuzzy Integral
• Integral of Definite functions with fuzzy interval
5. Relations & Fuzzy Graphs
• Fuzzy Relations
• Fuzzy Graphs in Fuzzy Sets.
• Fuzzy Images in 2-D Graphs
• Fuzzy Images in n-D Graphs
• Operations in Fuzzy Graphs
• Fuzzy Forests
This is the entrance exam paper for ISI MSQE Entrance Exam for the year 2008. Much more information on the ISI MSQE Entrance Exam and ISI MSQE Entrance preparation help available on http://crackdse.com
1. Introduction
• Preliminaries
• Some Useful Definitions
• Types of fuzzy sets
• Degree of Fuzzy Sets
• Operators of Fuzzy Sets
• Conditions & Limitations
• Multiplication
• Summation
• Operators of Theory Sets
• Characteristics of S & T
• Some definitions for T & S
• Unity and Community Defs.
• Mean Operators
• Fuzzy AND & OR
• Combinations of Fuzzy AND & OR
2. Fuzzy Measurement & Measurement of Fuzzy Sets
• Fuzzy Measurement
• Dr. ASGARI Zadeh Possibility Definition
• SUGENO Definition
• Possibility Definition
• Graph of S Function
• Measurement of Fuzzy Sets
• Entropy of Fuzzy Sets (De Luca & Termini)
• YAGER Definition for Ã
3. Propagation principle
• Propagation principle & Applications
• Propagation principle and Second Types of Fuzzy Sets
• Fuzzy Numbers & Algebraic Operations
• Fuzzy Numbers Intervals
• L-R Interval Function (Asymmetric)
• L-R Interval Function
• L-R Interval Function Operations
4. Functions & Fuzzy Analyzing
• Functions & Fuzzy Analyzing
• Functions & Fuzzy Analyzing
• Fuzzy functions Extremes
• Integral of Fuzzy Functions
• Integral of Type 2 fuzzy function with definite interval
• Differentiation of Definite functions With Fuzzy Domains & Ranges
• Integral of fuzzy function with definite interval
• Properties of fuzzy Integral
• Integral of Definite functions with fuzzy interval
5. Relations & Fuzzy Graphs
• Fuzzy Relations
• Fuzzy Graphs in Fuzzy Sets.
• Fuzzy Images in 2-D Graphs
• Fuzzy Images in n-D Graphs
• Operations in Fuzzy Graphs
• Fuzzy Forests
This is the entrance exam paper for ISI MSQE Entrance Exam for the year 2008. Much more information on the ISI MSQE Entrance Exam and ISI MSQE Entrance preparation help available on http://crackdse.com
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
I am Stacy W. I am a Statistical Physics Assignment Expert at statisticsassignmenthelp.com. I hold a Masters in Statistics from, University of McGill, Canada
I have been helping students with their homework for the past 7years. I solve assignments related to Statistical.
Visit statisticsassignmenthelp.com or email info@statisticsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Statistical Physics Assignments.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
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.
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
Safalta Digital marketing institute in Noida, provide complete applications that encompass a huge range of virtual advertising and marketing additives, which includes search engine optimization, virtual communication advertising, pay-per-click on marketing, content material advertising, internet analytics, and greater. These university courses are designed for students who possess a comprehensive understanding of virtual marketing strategies and attributes.Safalta Digital Marketing Institute in Noida is a first choice for young individuals or students who are looking to start their careers in the field of digital advertising. The institute gives specialized courses designed and certification.
for beginners, providing thorough training in areas such as SEO, digital communication marketing, and PPC training in Noida. After finishing the program, students receive the certifications recognised by top different universitie, setting a strong foundation for a successful career in digital marketing.
Acetabularia Information For Class 9 .docxvaibhavrinwa19
Acetabularia acetabulum is a single-celled green alga that in its vegetative state is morphologically differentiated into a basal rhizoid and an axially elongated stalk, which bears whorls of branching hairs. The single diploid nucleus resides in the rhizoid.
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
Digital Tools and AI for Teaching Learning and Research
1010n3a
1. ECE 1010 ECE Problem Solving I
MATLAB 3
Functions
Overview
In this chapter we start studying the many, many, mathematical
functions that are included in MATLAB. Some time will be spent
introducing complex variables and how MATLAB handles them.
Later we will learn how to write custom (user written) functions
which are a special form of m-file. Two dimensional plotting
functions will be introduced. We will also explore programming
constructs for flow control (if-else-elseif code blocks)
and looping (for loop etc.). Next data analysis functions will be
investigated, which includes sample statistics and histogram plot-
ting functions.
Chapter 3: Overview 3–1
2. ECE 1010 ECE Problem Solving I
Mathematical Functions
Common Math Functions
Table 3.1: Common math functions
Function Description Function Description
abs(x) x sqrt(x) x
round(x) nearest integer fix(x) nearest integer
floor(x) nearest integer ceil(x) nearest integer
toward – ∞ toward ∞
sign(x) rem(x,y) the remainder
– 1, x < 0
of x ⁄ y
0, x = 0
1, x > 0
exp(x) x log(x) natural log ln x
e
log10(x) log base 10
log 10 x
Examples:
» x = [-5.5 5.5];
» round(x)
ans = -6 6
» fix(x)
ans = -5 5
» floor(x)
ans = -6 5
» ceil(x)
Chapter 3: Mathematical Functions 3–2
3. ECE 1010 ECE Problem Solving I
ans = -5 6
» sign(x)
ans = -1 1
» rem(23,6)
ans = 5
Trigonometric and Hyperbolic Functions
• Unlike pocket calculators, the trigonometric functions always
assume the input argument is in radians
• The inverse trigonometric functions produce outputs that are
in radians
Table 3.2: Trigonometric functions
Function Description Function Description
sin(x) sin ( x ) cos(x) cos ( x )
tan(x) tan ( x ) asin(x) –1
sin ( x )
acos(x) –1 atan(x) –1
cos ( x ) tan ( x )
atan2(y, the inverse tan-
x) gent of y ⁄ x
including the
correct quad-
rant
Examples:
2 2
• A simple verification that sin ( x ) + cos ( x ) = 1
Chapter 3: Mathematical Functions 3–3
4. ECE 1010 ECE Problem Solving I
» x = 0:pi/10:pi;
» [x' sin(x)' cos(x)' (sin(x).^2+cos(x).^2)']
ans =
0 0 1.0000 1.0000
0.3142 0.3090 0.9511 1.0000
0.6283 0.5878 0.8090 1.0000
0.9425 0.8090 0.5878 1.0000
1.2566 0.9511 0.3090 1.0000
1.5708 1.0000 0.0000 1.0000
1.8850 0.9511 -0.3090 1.0000
2.1991 0.8090 -0.5878 1.0000
2.5133 0.5878 -0.8090 1.0000
2.8274 0.3090 -0.9511 1.0000
3.1416 0.0000 -1.0000 1.0000
• Parametric plotting:
– Verify that by plotting sin θ versus cos θ for 0 ≤ θ ≤ 2π we
obtain a circle
» theta = 0:2*pi/100:2*pi; % Span 0 to 2pi with 100 pts
» plot(cos(theta),sin(theta)); axis('square')
» title('Parametric Plotting','fontsize',18)
» ylabel('y - axis','fontsize',14)
» xlabel('x - axis','fontsize',14)
» figure(2)
» hold
Current plot held
» plot(cos(5*theta).*cos(theta), ...
cos(5*theta).*sin(theta)); % A five-leaved rose
Chapter 3: Mathematical Functions 3–4
5. ECE 1010 ECE Problem Solving I
Parametric Plotting
1
Circle
0.8
0.6
0.4 Rose
0.2
y - axis
0
-0.2
-0.4
-0.6
-0.8
-1
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
x - axis
• The trig functions are what you would expect, except the fea-
tures of atan2(y,x) may be unfamiliar
y y
4
2 x 2 x
φ 1 = 0.4636
4
φ 2 = – 2.0344
Chapter 3: Mathematical Functions 3–5
6. ECE 1010 ECE Problem Solving I
» [atan(2/4) atan2(2,4)]
ans = 0.4636 0.4636 % the same
» [atan(-4/-2) atan2(-4,-2)]
ans = 1.1071 -2.0344 % different; why?
x
• The hyperbolic functions are defined in terms of e
Table 3.3: Hyperbolic functions
Function Description Function Description
sinh(x) x –x cosh(x) x –x
e –e e –e
-----------------
- -----------------
-
2 2
tanh(x) x –x asinh(x) 2
e –e
------------------ ln ( x + x + 1 )
x –x
e +e
acosh(x) 2 atanh(x) 1+x
ln ( x + x – 1 ) ln -----------, x ≤ 1
-
1–x
• There are no special concerns in the use of these functions
except that atanh requires an argument that must not
exceed an absolute value of one
Complex Number Functions
• Before discussing these functions we first review a few facts
about complex variable theory
• In electrical engineering complex numbers appear frequently
• A complex number is an ordered pair of real numbers1
1. Tom M. Apostle, Mathematical Analysis, second edition, Addison Wesley,
p. 15, 1974.
Chapter 3: Mathematical Functions 3–6
7. ECE 1010 ECE Problem Solving I
denoted ( a, b )
– The first number, a, is called the real part, while the second
number, b, is called the imaginary part
– For algebraic manipulation purposes we write ( a, b )
= a + ib = a + jb where i = j = – 1 ; electrical
engineers typically use j since i is often used to denote cur-
rent
Note: –1 × –1 = –1 ⇒ j × j = –1
• For complex numbers z 1 = a 1 + jb 1 and z 2 = a 2 + jb 2
we define/calculate
z1 + z2 = ( a1 + a2 ) + j ( b1 + b2 ) (sum)
z 1 – z 2 = ( a 1 – a 2 ) + j ( b 1 – b 2 ) (difference)
z 1 z 2 = ( a 1 a 2 – b 1 b 2 ) + j ( a 1 b 2 + b 1 a 2 ) (product)
z1 ( a1 a2 + b1 b2 ) – j ( a1 b2 – b1 a2 )
---- = --------------------------------------------------------------------------- (quotient)
-
z2 2 2
a +b
2 2
2 2
z1 = a 1 + b 1 (magnitude)
–1
∠z 1 = tan ( b 1 ⁄ a 1 ) (angle)
z 1∗ = a 1 – jb 1 (complex conjugate)
• MATLAB is consistent with all of the above, starting with the
fact that i and j are predefined to be – 1
Chapter 3: Mathematical Functions 3–7
8. ECE 1010 ECE Problem Solving I
• The rectangular form of a complex number is as defined
above,
z = ( a, b ) = a + jb
• The corresponding polar form is
z = r ∠θ
2 2 –1
where r = a + b and θ = tan ( b ⁄ a )
Imaginary Axis
Complex Plane b
r
θ Real
a Axis
• MATLAB has five basic complex functions, but in reality
most all of MATLAB’s functions accept complex arguments
and deal with them correctly
Table 3.4: Basic complex functions
Function Description
conj(x) Computes the conjugate of z = a + jb
which is z∗ = a – jb
real(x) Extracts the real part of z = a + jb
which is real ( z ) = a
imag(x) Extracts the imaginary part of
z = a + jb which is imag ( z ) = b
Chapter 3: Mathematical Functions 3–8
9. ECE 1010 ECE Problem Solving I
Table 3.4: Basic complex functions
Function Description
angle(x) computes the angle of z = a + jb using
atan2 which is
atan2(imag(z),real(z))
Euler’s Formula: A special mathematical result of, special
importance to electrical engineers, is the fact that
jb
e = cos b + j sin b (3.1)
• Turning (3.1) around yields
jθ – jθ
e –e
sin θ = ---------------------
- (3.2)
2j
jθ – jθ
e +e
cos θ = ---------------------
- (3.3)
2
• It also follows that
jθ
z = a + jb = re (3.4)
where
2 2 –1 b
r= a + b , θ = tan --, a = r cos θ, b = r sin θ
-
a
Chapter 3: Mathematical Functions 3–9
10. ECE 1010 ECE Problem Solving I
• Some examples:
» z1 = 2+j*4; z2 = -5+j*7;
» [z1 z2]
ans =
2.0000 + 4.0000i -5.0000 + 7.0000i
» [real(z1) imag(z1) abs(z1) angle(z1)]
ans =
2.0000 4.0000 4.4721 1.1071
» [conj(z1) conj(z2)]
ans =
2.0000 - 4.0000i -5.0000 - 7.0000i
» [z1+z2 z1-z2 z1*z2 z1/z2]
ans =
-3.0000 +11.0000i 7.0000 - 3.0000i
-38.0000 - 6.0000i 0.2432 - 0.4595i
Polar Plots: When dealing with complex numbers we often deal
with the polar form. The plot command which directly plots vec-
tors of magnitude and angle is polar(theta,r)
• This function is also useful for general plotting
• As an example the equation of a cardioid in polar form, with
parameter a, is
r = a ( 1 + cos θ ), 0 ≤ θ ≤ 2π
» theta = 0:2*pi/100:2*pi; % Span 0 to 2pi with 100 pts
» r = 1+cos(theta);
» polar(theta,r)
Chapter 3: Mathematical Functions 3–10