This document contains a student assignment submission for a course on Perspective in Informatics. It includes the student's responses to 3 questions:
1) Analyzing different functions to determine if they satisfy the properties of a distance measure, including max(x,y), diff(x,y), and sum(x,y).
2) Computing sketches of vectors using different random vectors and analyzing the estimated vs. true angles between the vectors.
3) Calculating the expected Jaccard similarity of two randomly selected subsets R and S of a universe U with n elements and size m.
Application of Derivative Class 12th Best Project by Shubham prasadShubham Prasad
Application of Derivative Class 12th Best Project by Shubham prasad, Student of Nalanda English Medium School Kurud Bhilai Durg Chhattisgarh.
Art Integrated Learning on Mathematics branch Application of Derivatives Class 12th Ncert
Introduction Stochastic Processes.
Markov Chains.
Chapman-Kolmogorov Equations
Classification of States
Recurrence and Transience
Limiting Probabilities
Principal Component Analysis, or PCA, is a factual method that permits you to sum up the data contained in enormous information tables by methods for a littler arrangement of "synopsis files" that can be all the more handily envisioned and broke down.
Accelerating Dynamic Time Warping Subsequence Search with GPUDavide Nardone
Many time series data mining problems require
subsequence similarity search as a subroutine. While this can
be performed with any distance measure, and dozens of
distance measures have been proposed in the last decade, there
is increasing evidence that Dynamic Time Warping (DTW) is
the best measure across a wide range of domains. Given
DTW’s usefulness and ubiquity, there has been a large
community-wide effort to mitigate its relative lethargy.
Proposed speedup techniques include early abandoning
strategies, lower-bound based pruning, indexing and
embedding. In this work we argue that we are now close to
exhausting all possible speedup from software, and that we
must turn to hardware-based solutions if we are to tackle the
many problems that are currently untenable even with stateof-
the-art algorithms running on high-end desktops. With this
motivation, we investigate both GPU (Graphics Processing
Unit) and FPGA (Field Programmable Gate Array) based
acceleration of subsequence similarity search under the DTW
measure. As we shall show, our novel algorithms allow GPUs,
which are typically bundled with standard desktops, to achieve
two orders of magnitude speedup. For problem domains which
require even greater scale up, we show that FPGAs costing just
a few thousand dollars can be used to produce four orders of
magnitude speedup. We conduct detailed case studies on the
classification of astronomical observations and similarity
search in commercial agriculture, and demonstrate that our
ideas allow us to tackle problems that would be simply
untenable otherwise.
Dar a conocer la importancia de los espacios y sub espacios vectoriales en la rama de la electrónica y automatización, también plantearemos ejercicios aplicando el teorema de wronksiano
Application of Derivative Class 12th Best Project by Shubham prasadShubham Prasad
Application of Derivative Class 12th Best Project by Shubham prasad, Student of Nalanda English Medium School Kurud Bhilai Durg Chhattisgarh.
Art Integrated Learning on Mathematics branch Application of Derivatives Class 12th Ncert
Introduction Stochastic Processes.
Markov Chains.
Chapman-Kolmogorov Equations
Classification of States
Recurrence and Transience
Limiting Probabilities
Principal Component Analysis, or PCA, is a factual method that permits you to sum up the data contained in enormous information tables by methods for a littler arrangement of "synopsis files" that can be all the more handily envisioned and broke down.
Accelerating Dynamic Time Warping Subsequence Search with GPUDavide Nardone
Many time series data mining problems require
subsequence similarity search as a subroutine. While this can
be performed with any distance measure, and dozens of
distance measures have been proposed in the last decade, there
is increasing evidence that Dynamic Time Warping (DTW) is
the best measure across a wide range of domains. Given
DTW’s usefulness and ubiquity, there has been a large
community-wide effort to mitigate its relative lethargy.
Proposed speedup techniques include early abandoning
strategies, lower-bound based pruning, indexing and
embedding. In this work we argue that we are now close to
exhausting all possible speedup from software, and that we
must turn to hardware-based solutions if we are to tackle the
many problems that are currently untenable even with stateof-
the-art algorithms running on high-end desktops. With this
motivation, we investigate both GPU (Graphics Processing
Unit) and FPGA (Field Programmable Gate Array) based
acceleration of subsequence similarity search under the DTW
measure. As we shall show, our novel algorithms allow GPUs,
which are typically bundled with standard desktops, to achieve
two orders of magnitude speedup. For problem domains which
require even greater scale up, we show that FPGAs costing just
a few thousand dollars can be used to produce four orders of
magnitude speedup. We conduct detailed case studies on the
classification of astronomical observations and similarity
search in commercial agriculture, and demonstrate that our
ideas allow us to tackle problems that would be simply
untenable otherwise.
Dar a conocer la importancia de los espacios y sub espacios vectoriales en la rama de la electrónica y automatización, también plantearemos ejercicios aplicando el teorema de wronksiano
This upload is actually experimental, so sorry for the lost animations. This is my first post on SlideShare. Future presentations will take into account the loss of animation.
Also, I saw that the titles of all my slides got covered by something, so I'll never use this theme again. The titles of the slides are:
Slide 1: Vectors and Scalars
Slide 2: In this lecture, you will learn
Slide 3: What are vectors?
Slide 4: What are scalars?
Slide 5: A joke
Slide 6: A joke
Slide 7: What was that for?
Slide 8: What was that for?
Slide 9: Vectors
Slide 10: Geometric Representation
Slide 11: Vector Addition
Slide 12: Scalar Multiplication
Slide 13: The Zero Vector
Slide 14: The Negative of a Vector
Slide 15: Vector Subtraction
Slide 16: More Properties of Vector Algebra
Slide 17: Magnitude of a Vector
Slide 18: Vectors in a Coordinate System
Slide 19: Unit Vectors
Slide 20: Algebraic Representation of Vectors
Slide 21: Algebraic Addition of Vectors
Slide 22: Algebraic Multiplication of a Vector by a Scalar
Slide 23: Example 1
Slide 24: Example 2
Slide 25: A few words of caution
Slide 26: Problems
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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.
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.
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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.
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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.
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Perspective in Informatics 3 - Assignment 1 - Answer Sheet
1. Subject: Perspective in Informatics 3 – Fall Semester 2014
Professor: Davood Rafiei
Assignment No.1 HOANG Nguyen Phong
Submitted on November 3rd ID number: 6930-26-1264
Question 1 [30 marks]
• 3.5.1: on the space of nonnegative integers, which of the following functions are
distance measures? If so, prove it; if not, prove that it fails to satisfy one or more of the
axioms.
a) max(x, y) = the larger of x and y.
This function is distance measure function because of the following reasons:
• In the space of nonnegative integers as given from the beginning, the function would
never return a negative value.
• If x and y are at the same position in the space, then no larger value is defined, which
would return a null value (which is 0). That satisfies the reflexive property of distance
measure function.
• Measuring both distances from x to y and x < y, and from y to x would only return one
larger value. It satisfies the symmetric property of distance measure function.
• Let x and y are 2 separate nodes, and a is a random node (different from x and y).
Then, the triangle-inequality can be proved as shown in the below table:
3 Possible cases of a max(x,a) + max(y,a) > max(x,y) Check
a ∈ [x,y] a + y ≥ y true
a < (x,y) x + y ≥ y true
a > (x,y) a + a ≥ y true(since a≥y => 2a≥y)
• Actually, this function is the L∞-norm Euclidean distance measuring function, which is
used when x and y have many dimensions (where the dimension ~> ∞). Then, the
distance between x and y is approximately equal to the max(x,y).
b) diff(x, y) = |x − y| (the absolute magnitude of the difference between x and y).
• By proving in the same manner of the above case, this function is also a distance
measure function, because of the following reasons:
• Since the absolute-value function, it would always return a nonnegative value.
• If x and y is a same point, the function will return 0. That satisfies the reflexive
property.
• Let x and y are 2 separate nodes, and a is a random node (different from x and y).
Then, the triangle-inequality can be proved as shown in the below table:
3 Possible cases of a diff(x,a) + diff(y,a) > diff(x,y) Check
a ∈ [x,y]
(a – x) + (y – a) ≥ y – x
y – x ≥ y – x
true
a < (x,y)
(x – a) + (y – a) > y – x
x + y – 2a > y – x
x > a
true (since a<x as given in
the initial condition of a )
a > (x,y) a + a > y
true(since a>y as given in
the initial condition of a
=> 2a>y)
• Actually, we can imagine that this function is a L1-norm Euclidean Distance function
for measuring x and y in 1 dimension.
c) sum(x, y) = x + y.
It is easily proved that this function is not a distance measure function, since it does not
satisfies the reflexive property. For instance, if x and y are a same point (≠0), the function
would return a positive value in lieu of 0 because they are both in nonnegative space.
1
2. Subject: Perspective in Informatics 3 – Fall Semester 2014
Professor: Davood Rafiei
• 3.7.2: Let us compute sketches using the following four “random” vectors:
V1= [+1,+1,+1,-1] V2=[+1,+1,-1,+1]
V3=[+1,-1,+1,+1] V4=[-1,+1,+1,+1]
Compute the sketches of the following vectors.
• [2,3,4,5]
Random vector Dot product Sketch value
V1= [+1,+1,+1,-1] 4 +1
V2=[+1,+1,-1,+1] 6 +1
V3=[+1,-1,+1,+1] 8 +1
V4=[-1,+1,+1,+1] 10 +1
(b)[-2,3,-4,5]
Random vector Dot product Sketch value
V1= [+1,+1,+1,-1] -8 -1
V2=[+1,+1,-1,+1] 10 +1
V3=[+1,-1,+1,+1] -4 -1
V4=[-1,+1,+1,+1] 6 +1
(c)[2,-3,4,-5]
Random vector Dot product Sketch value
V1= [+1,+1,+1,-1] 8 +1
V2=[+1,+1,-1,+1] -10 -1
V3=[+1,-1,+1,+1] 4 +1
V4=[-1,+1,+1,+1] -6 -1
For each pair, what is the estimated angle between them, according to the sketches? What are
the true angles?
The following 2 formulas are employed to calculate the Estimated angle and true angles:
• Estimated angle = 180O(1 – sim( Sketches of 2 vectors))
• True Angle =
𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷(𝑡𝑡ℎ𝑒𝑒 2 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣)
𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑜𝑜𝑜𝑜 𝑡𝑡ℎ𝑒𝑒 2 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣
Pair Estimated angle True angles
∠ (a)(b) 90o 90o-15o=75o
∠ (b)(c) 180o 180o
∠ (a)(c) 90o 90o+15o=105o
• 3.7.3: suppose we form sketches by using all sixteen of the vectors of length 4, whose
components are each +1 or -1. Compute the sketches of the three vectors in Exercise
3.7.2.
*at dot product = 0, sketch value is randomly chosen to be 1 or +1 as highlighted in gray.
2
4. Subject: Perspective in Informatics 3 – Fall Semester 2014
Professor: Davood Rafiei
Vector c 2 -3 4 -5
Random vector dot Product Sketch value
v1 -1 -1 -1 -1 2 1
v2 -1 -1 -1 1 -8 -1
v3 -1 -1 1 -1 10 1
v4 -1 -1 1 1 0 1
v5 -1 1 -1 -1 -4 -1
v6 -1 1 -1 1 -14 -1
v7 -1 1 1 -1 4 1
v8 -1 1 1 1 -6 -1
v9 1 -1 -1 -1 6 1
v10 1 -1 -1 1 -4 -1
v11 1 -1 1 -1 14 1
v12 1 -1 1 1 4 1
v13 1 1 -1 -1 0 -1
v14 1 1 -1 1 -10 -1
v15 1 1 1 -1 8 1
v16 1 1 1 1 -2 -1
How do the estimates of the angles between each pair compare with the true angles?
Pair Estimated angle True angles
∠ (a)(b) ½ => 90o 90o-15o=75o
∠ (b)(c) 11/12 => approximate 180o 180o
∠ (a)(c) ½ => 90o 90o+15o=105o
Then it can be deduced that even all of 16 random vectors are chosen, the estimates of the
angles between each pair compare with the true angles do not change compared with the result
in problem 3.7.2.
4
5. Subject: Perspective in Informatics 3 – Fall Semester 2014
Professor: Davood Rafiei
Question 2 [10 marks] 3.7.4(A): Suppose we form sketches using the four vectors from
Exercise 3.7.2. What are the constrains on a, b, c, and d that will cause the sketch of the vector
[a, b, c, d] to be [+1,+1,+1,+1]? (write your constrains in as simple form as possible)
The dot products of four random vectors and [a, b, c, d] can be represented in form of matrix as
following equation:
�
1 1 1 −1
1 1 −1 1
1 −1 1 1
−1 1 1 1
�。 �
𝑎𝑎
𝑏𝑏
𝑐𝑐
𝑑𝑑
� = �
𝑥𝑥1
𝑥𝑥2
𝑥𝑥3
𝑥𝑥4
�
the sketch of [a, b, c, d] is [+1 , +1, +1, +1] where all of x1,x2,x3,x4 ≥ 0
�
1 1 1 −1
1 1 −1 1
1 −1 1 1
−1 1 1 1
�
−1
�
1 1 1 −1
1 1 −1 1
1 −1 1 1
−1 1 1 1
�。 �
𝑎𝑎
𝑏𝑏
𝑐𝑐
𝑑𝑑
� = �
1 1 1 −1
1 1 −1 1
1 −1 1 1
−1 1 1 1
�
−1
�
𝑥𝑥1
𝑥𝑥2
𝑥𝑥3
𝑥𝑥4
�
�
𝑎𝑎
𝑏𝑏
𝑐𝑐
𝑑𝑑
� = �
1 1 1 −1
1 1 −1 1
1 −1 1 1
−1 1 1 1
�
−1
�
𝑥𝑥1
𝑥𝑥2
𝑥𝑥3
𝑥𝑥4
�
We have�
1 1 1 −1
1 1 −1 1
1 −1 1 1
−1 1 1 1
�
−1
=
1
4
�
1 1 1 −1
1 1 −1 1
1 −1 1 1
−1 1 1 1
�
So a, b, a and d can be constrained by the following equation:
�
𝑎𝑎
𝑏𝑏
𝑐𝑐
𝑑𝑑
� =
1
4
�
1 1 1 −1
1 1 −1 1
1 −1 1 1
−1 1 1 1
� �
𝑥𝑥1
𝑥𝑥2
𝑥𝑥3
𝑥𝑥4
� where x1,x2,x3,x4 ≥ 0�
𝑎𝑎 + 𝑏𝑏 + 𝑐𝑐 − 𝑑𝑑 ≥ 0
𝑎𝑎 + 𝑏𝑏 − 𝑐𝑐 + 𝑑𝑑 ≥ 0
𝑎𝑎 − 𝑏𝑏 + 𝑐𝑐 + 𝑑𝑑 ≥ 0
−𝑎𝑎 + 𝑏𝑏 + 𝑐𝑐 + 𝑑𝑑 ≥ 0
5
6. Subject: Perspective in Informatics 3 – Fall Semester 2014
Professor: Davood Rafiei
Question 3 [10 marks]
a) Consider a universe U with n elements, and let R and S be subsets of U both of size m,
chosen uniformly at random.
What is the expected value of the Jaccard similarity of R and S?
The Expectation of an event x is calculated as Ε(x) = ∑x. P(x)
In this case, Jaccard Similarity of R and S is calculated as:
Sim(R,S)=
|𝑅𝑅⋂𝑆𝑆|
|𝑅𝑅⋃𝑆𝑆|
=
𝑘𝑘
2𝑚𝑚−𝑘𝑘
(𝑤𝑤ℎ𝑒𝑒𝑒𝑒𝑒𝑒 0 ≤ 𝑘𝑘 ≤ 𝑚𝑚 𝑖𝑖𝑖𝑖 𝑡𝑡ℎ𝑒𝑒 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 𝑜𝑜𝑜𝑜 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑒𝑒𝑒𝑒𝑒𝑒 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 𝑅𝑅 𝑎𝑎𝑎𝑎𝑎𝑎 𝑆𝑆)
Next, the probability of Sim(R,S) is calculated as following:
P(sim(R,S)=(
𝑘𝑘
2𝑚𝑚−𝑘𝑘
))=
𝐶𝐶 𝑚𝑚
𝑘𝑘 𝐶𝐶𝑛𝑛−𝑚𝑚
𝑚𝑚−𝑘𝑘
𝐶𝐶𝑛𝑛
𝑚𝑚
Since:
• To create set R, we combine m element(s) from n elements of the universal set U. It is
calculated as: 𝐶𝐶𝑛𝑛
𝑚𝑚
• Next, to create set S, we need to take k common element(s) from set R first, which is
calculated as 𝐶𝐶𝑚𝑚
𝑘𝑘
. Then the left (m-k) element(s) are chosen from (n-m) elements, since
m element(s) have been chosen to create set R at the beginning. The formula is:
𝐶𝐶𝑛𝑛−𝑚𝑚
𝑚𝑚−𝑘𝑘
As a result, Expectation of Jaccard Similarity sim(S,T) is estimated as:
E(sim(S,T))=∑
𝑘𝑘
2𝑚𝑚−𝑘𝑘
𝐶𝐶 𝑚𝑚
𝑘𝑘
𝐶𝐶𝑛𝑛−𝑚𝑚
𝑚𝑚−𝑘𝑘
𝐶𝐶𝑛𝑛
𝑚𝑚 =𝑚𝑚
𝑘𝑘=0 ∑
𝑘𝑘
2𝑚𝑚−𝑘𝑘
�
𝑚𝑚
𝑘𝑘��
𝑛𝑛−𝑚𝑚
𝑚𝑚−𝑘𝑘�
� 𝑛𝑛
𝑚𝑚�
𝑚𝑚
𝑘𝑘=0
b) How does your answer to part (a) change if R and S must include a certain element (say z)
of U?
It means k ~> z, then the answer is changed to be:
E(sim(S,T))=∑
𝑧𝑧
2𝑚𝑚−𝑧𝑧
𝐶𝐶 𝑚𝑚
𝑧𝑧
𝐶𝐶𝑛𝑛−𝑚𝑚
𝑚𝑚−𝑧𝑧
𝐶𝐶𝑛𝑛
𝑚𝑚 =𝑧𝑧
𝑘𝑘=0 ∑
𝑧𝑧
2𝑚𝑚−𝑧𝑧
� 𝑚𝑚
𝑧𝑧 �� 𝑛𝑛−𝑚𝑚
𝑚𝑚−𝑧𝑧�
� 𝑛𝑛
𝑚𝑚�
𝑧𝑧
𝑘𝑘=0
c) How does your answer to part (a) change if R and S must be disjoint?
It means k=0, then the answer is changed to be:
E(sim(S,T))=∑
𝑘𝑘
2𝑚𝑚−𝑘𝑘
𝐶𝐶 𝑚𝑚
𝑘𝑘
𝐶𝐶𝑛𝑛−𝑚𝑚
𝑚𝑚−𝑘𝑘
𝐶𝐶𝑛𝑛
𝑚𝑚 =0
𝑘𝑘=0 0
6