This ppt covers the topic of Abstract Algebra in M.Sc. Mathematics 1 semester topic of Algebraic closed field is Introduction , field & sub field , finite extension field , algebraic element.
A presentation on famous set Cantor Set. it describes the properties of cantor set. which the most important set of early era. it is explined with proof and theorems. references are given. ppt is somewhat plane. it would not cover the area of applications of cantor set
This ppt covers the topic of Abstract Algebra in M.Sc. Mathematics 1 semester topic of Algebraic closed field is Introduction , field & sub field , finite extension field , algebraic element.
A presentation on famous set Cantor Set. it describes the properties of cantor set. which the most important set of early era. it is explined with proof and theorems. references are given. ppt is somewhat plane. it would not cover the area of applications of cantor set
2. Linear Algebra for Machine Learning: Basis and DimensionCeni Babaoglu, PhD
The seminar series will focus on the mathematical background needed for machine learning. The first set of the seminars will be on "Linear Algebra for Machine Learning". Here are the slides of the second part which is discussing basis and dimension.
Here is the link of the first part which was discussing linear systems: https://www.slideshare.net/CeniBabaogluPhDinMat/linear-algebra-for-machine-learning-linear-systems/1
2. Linear Algebra for Machine Learning: Basis and DimensionCeni Babaoglu, PhD
The seminar series will focus on the mathematical background needed for machine learning. The first set of the seminars will be on "Linear Algebra for Machine Learning". Here are the slides of the second part which is discussing basis and dimension.
Here is the link of the first part which was discussing linear systems: https://www.slideshare.net/CeniBabaogluPhDinMat/linear-algebra-for-machine-learning-linear-systems/1
Module 1 (Part 1)-Sets and Number Systems.pdfGaleJean
1. Fossil records of whale evolution: Search for "whale evolution fossils" or "transitional fossils of whales."2. Comparative anatomy of homologous structures: Look for images of "homologous structures in different species" or specific examples like "homologous forelimbs in vertebrates."3. Molecular biology and genetic similarities: Search for "DNA sequences in different species," "genetic similarities between primates," or "genetic code comparison."4. Biogeography and species distribution: Look for images of "marsupials in Australia and placental mammals," or "species distribution maps showing evolution and migration."5. Artificial selection examples: Search for images of "domesticated plants vs. wild ancestors" or "different dog breeds through selective breeding."By using these keywords, you should be able to find suitable images that can visually enhance your presentation and help your classmates better grasp the concepts of descent with modification and evolutionary processes.
A Probabilistic Algorithm for Computation of Polynomial Greatest Common with ...mathsjournal
In the earlier work, Knuth present an algorithm to decrease the coefficient growth in the Euclidean algorithm of polynomials called subresultant algorithm. However, the output polynomials may have a small factor which can be removed. Then later, Brown of Bell Telephone Laboratories showed the subresultant in another way by adding a variant called 𝜏 and gave a way to compute the variant. Nevertheless, the way failed to determine every 𝜏 correctly.
In this paper, we will give a probabilistic algorithm to determine the variant 𝜏 correctly in most cases by adding a few steps instead of computing 𝑡(𝑥) when given 𝑓(𝑥) and𝑔(𝑥) ∈ ℤ[𝑥], where 𝑡(𝑥) satisfies that 𝑠(𝑥)𝑓(𝑥) + 𝑡(𝑥)𝑔(𝑥) = 𝑟(𝑥), here 𝑡(𝑥), 𝑠(𝑥) ∈ ℤ[𝑥]
this is a presentation on a a number theory topic concerning primes, it discusses three topics, the sieve of Eratosthenes, the euclids proof that primes is infinite, and solving for tau (n) primes.
Conjuntos y valor absoluto, valor absoluto con desifgualdadesYerelisLiscano
definición de conjuntos, conjuntos reales propiedades de los conjuntos, desigualdades, propiedades de las desigualdades, valor absoluto, propiedades del valor absoluto y valor absolito con desigualdades.
In this paper, the concepts of sequences and series of complement normalized fuzzy numbers are introduced in terms of 𝛾-level, so that some properties and characterizations are presented, and some convergence theorems are proved
Richard's aventures in two entangled wonderlandsRichard Gill
Since the loophole-free Bell experiments of 2020 and the Nobel prizes in physics of 2022, critics of Bell's work have retreated to the fortress of super-determinism. Now, super-determinism is a derogatory word - it just means "determinism". Palmer, Hance and Hossenfelder argue that quantum mechanics and determinism are not incompatible, using a sophisticated mathematical construction based on a subtle thinning of allowed states and measurements in quantum mechanics, such that what is left appears to make Bell's argument fail, without altering the empirical predictions of quantum mechanics. I think however that it is a smoke screen, and the slogan "lost in math" comes to my mind. I will discuss some other recent disproofs of Bell's theorem using the language of causality based on causal graphs. Causal thinking is also central to law and justice. I will mention surprising connections to my work on serial killer nurse cases, in particular the Dutch case of Lucia de Berk and the current UK case of Lucy Letby.
Nutraceutical market, scope and growth: Herbal drug technologyLokesh Patil
As consumer awareness of health and wellness rises, the nutraceutical market—which includes goods like functional meals, drinks, and dietary supplements that provide health advantages beyond basic nutrition—is growing significantly. As healthcare expenses rise, the population ages, and people want natural and preventative health solutions more and more, this industry is increasing quickly. Further driving market expansion are product formulation innovations and the use of cutting-edge technology for customized nutrition. With its worldwide reach, the nutraceutical industry is expected to keep growing and provide significant chances for research and investment in a number of categories, including vitamins, minerals, probiotics, and herbal supplements.
Comparing Evolved Extractive Text Summary Scores of Bidirectional Encoder Rep...University of Maribor
Slides from:
11th International Conference on Electrical, Electronics and Computer Engineering (IcETRAN), Niš, 3-6 June 2024
Track: Artificial Intelligence
https://www.etran.rs/2024/en/home-english/
This presentation explores a brief idea about the structural and functional attributes of nucleotides, the structure and function of genetic materials along with the impact of UV rays and pH upon them.
Seminar of U.V. Spectroscopy by SAMIR PANDASAMIR PANDA
Spectroscopy is a branch of science dealing the study of interaction of electromagnetic radiation with matter.
Ultraviolet-visible spectroscopy refers to absorption spectroscopy or reflect spectroscopy in the UV-VIS spectral region.
Ultraviolet-visible spectroscopy is an analytical method that can measure the amount of light received by the analyte.
Earliest Galaxies in the JADES Origins Field: Luminosity Function and Cosmic ...Sérgio Sacani
We characterize the earliest galaxy population in the JADES Origins Field (JOF), the deepest
imaging field observed with JWST. We make use of the ancillary Hubble optical images (5 filters
spanning 0.4−0.9µm) and novel JWST images with 14 filters spanning 0.8−5µm, including 7 mediumband filters, and reaching total exposure times of up to 46 hours per filter. We combine all our data
at > 2.3µm to construct an ultradeep image, reaching as deep as ≈ 31.4 AB mag in the stack and
30.3-31.0 AB mag (5σ, r = 0.1” circular aperture) in individual filters. We measure photometric
redshifts and use robust selection criteria to identify a sample of eight galaxy candidates at redshifts
z = 11.5 − 15. These objects show compact half-light radii of R1/2 ∼ 50 − 200pc, stellar masses of
M⋆ ∼ 107−108M⊙, and star-formation rates of SFR ∼ 0.1−1 M⊙ yr−1
. Our search finds no candidates
at 15 < z < 20, placing upper limits at these redshifts. We develop a forward modeling approach to
infer the properties of the evolving luminosity function without binning in redshift or luminosity that
marginalizes over the photometric redshift uncertainty of our candidate galaxies and incorporates the
impact of non-detections. We find a z = 12 luminosity function in good agreement with prior results,
and that the luminosity function normalization and UV luminosity density decline by a factor of ∼ 2.5
from z = 12 to z = 14. We discuss the possible implications of our results in the context of theoretical
models for evolution of the dark matter halo mass function.
Earliest Galaxies in the JADES Origins Field: Luminosity Function and Cosmic ...
Cantor Infinity theorems
1. To Infinity and Beyond!
Cantor’s Infinity theorems
Oren Ish-Am
2. Georg Cantor (1845-1918)
His theories were so counter-intuitive that met with much resistance from
contemporary mathematicians (Poincaré, Kronecker)
They were referred to as “utter nonsense” “laughable” and “a challenge to the
uniqueness of God”.
Invented Set Theory – fundamental in mathematics.
Used in many areas of mathematics including topology,
algebra and the foundations of mathematics.
Before Cantor – only finite sets in math - infinity was a
philosophical issue
Cantor, a devout Lutheran, believed the theory had been
communicated to him by God
3. Set Theory – basic terminology
Set – collection of unique elements – order is not important
A = 3, , B = 8,
Member: 𝒂 ∈ 𝑨 means 𝑎 is a member of the set 𝐴
∈ A 8 ∈ B
Subset: 𝑺 ⊂ 𝑨 means 𝑆 contains some of the elements of the set 𝐴
S = 3, S ⊂ A
Union: 𝑪 = 𝑨 ∪ 𝑩 means 𝐶 contains all the elements 𝐴 and all elements of 𝐵
C = 3, , 8,,
4. Set Theory – basic terminology
Set Size – |𝑨| is the number of elements in 𝐴
Empty Set: {} or 𝝓 is the empty set – set with no items. 𝜙 = 0
|A| = 3 |B| = 2
Natural Numbers: ℕ = {1,2,3…}
Rational Numbers: ℚ =
1
3
,
101
23
,
6
6
,
10
5
, …
Real Numbers : ℝ = ℚ ∪ all numbers that are not rationals like π,e, 2 …
6. Counting (it’s as easy as 1,2,3)
How can we tell the size of a set of items?
Just count them:
21 3 4
A B
For finite sets, if A is a proper subset of B (A ⊂ B)
Then the size of A is smaller then the size of B ( A < B )
7. Counting (it’s not that easy…)
Surely there are less squares than natural numbers (1,2,3,…) – right?
What can be say about an infinite subset of an infinite set?
Let’s try another counting method - comparison
1 2 3 4 5 6 7 8 9 …
…
Can we count infinite sets?
For example - what is the “size” of the set of all squares?
8. Assume we have a classroom full of chairs and students
We know all chairs are full and - there are students standing.
Without having to count, we know there are more students then chairs!
Counting (it’s not that easy…)
Similarly, if there is exactly one student on each chair – the
set of students is the size of the set of chairs.
This type of mapping is called bijection ( חדחדועל ערכי )
9. Cardinality )עוצמה(
Definition: Two sets have the same cardinality iff there exists a
bijection between them
We can now measure the size of infinite sets! Let’s try an example:
The set of natural numbers and the set of even numbers
2 4 6 8 10 12 14 16 …f(x)
1 2 3 4 5 6 7 8 …x
A simple function exists: 𝑓 𝑥 = 2𝑥, a bijection.
Therefore the two sets have the same cardinality.
10. Cardinality )עוצמה(
How about natural number and integers?
1 -1 2 -2 3 -3 4 …f(x) 0
1 2 3 4 5 6 7 …x 0
𝑓 𝑥 =
𝑥/2 𝑥 ∈ odd
− 𝑥 2 𝑥 ∈ even
Cantor was sick of Greek and Latin symbols
Decided to note this cardinality as ℵ 𝟎
This is the cardinality of the natural numbers, called countably infinite
11. Cardinality of ℚ
The rationales are dense;
there is a fraction between every two fractions.
There are infinite fractions between 0 and 1 alone!
In fact, there are infinite fractions between any two fractions!
Could we find a way to match a natural number to each fraction?
Ok, we get it, so ℵ0 + 𝑐𝑜𝑛𝑠𝑡 = ℵ0 and ℵ0 ⋅ 𝑐𝑜𝑛𝑠𝑡 = ℵ0
What about the rationals - ℚ?
12. Cardinality of ℚ
Naturals 1 2 3 4 5 6 7 8 9 10
numerator 1 1 2 3 2 1 1 2 3 4
denominator 1 2 1 1 2 3 4 3 2 1
We map a natural number to a numerator and denominator (fraction)
Eventually, we will reach all possible fractions – Bijection! Hurray!
13. Countably Infinite - ℵ 𝟎
Challenge: find the bijection from natural numbers to fractions.
This is the cardinality of
All finite strings
All equations
All functions you can write down a description of
All computer programs.
Seems like one can find a clever bijection from the natural numbers
to any infinite set….right? Are we done here?
Well… not so fast…
14. Cardinality of Reals - ℝ
List might start like this:
1. .1415926535...
2. .5820974944...
3. .3333333333...
4. .7182818284...
5. .4142135623...
6. .5000000000...
7. .8214808651...
Cantor had a hunch that the Reals were not countable.
Proof by contradiction – let’s assume they are and make a
list:
This method is called
Proof By Diagonalization
15. Cardinality of Reals - ℝ
New Number = 0.0721097… isn’t in the list! Contradiction!
The reals are not countable – more than one type of infinity!!
“Complete List”
1. .1415926535...
2. .5820974944...
3. .3333333333...
4. .7182818284...
5. .4142135623...
6. .5000000000...
7. .8214808651...
etc.
Make a new number:
1. .0415926535...
2. .5720974944...
3. .3323333333...
4. .7181818284...
5. .4142035623...
6. .5000090000...
7. .8214807651...
etc.
16. Cardinality of Reals
Easy! Just de-interleave the digits. For example
0.1809137360 … → 0.1017664 … , 0.893301 …
Therefore - ℵ is the cardinality of any continuum in ℝ 𝑛
Cantor: “I see this but I do not believe!”
0 1
Cantor called this cardinality ℵ (or ℵ1 as we will soon see).
ℵ is the cardinality of the continuum –all the points on the line:
0,1 , (−∞, ∞) both have cardinality ℵ. Can you find the bijection?
What about the Unit Square? Can we map 0,1 → 0,1 × [0,1]?
17. Cardinality of Reals
This leads to some interesting “paradoxes” like the famous Banach-Tarsky:
Given a 3D solid ball, you can break it into a finite number of disjoint subsets,
which can be put back together in a different way to yield two identical copies
of the original ball.
The reconstruction can work with as few as five “pieces”.
Not pieces in the normal sense – more like sparse infinite sets of points
18. Cardinality of Reals
ספור אין או סוף אין?
Implications of ℵ0 < ℵ:
Most real numbers do not have a name.
They will not be the result of any formula or computer program.
Can we say they actually “exist” if they can never be witnessed??
This has actual implications in mathematics!
We need to add the Axiom of Choice
19. More Power!
So are we done now? Countable infinity and continuum?
Power Set: For a set 𝑆 the power set 𝑃(𝑆) is the set of all subsets of 𝑆
including the empty set (𝜙 or {}) and 𝑆 itself.
Example:
𝑆 = 1,2,3
𝑃 𝑆 = 𝜙, 1 , 2 , 3 , 1,2 , 1,3 , 2,3 , 1,2,3
Challenge: prove that for any finite set 𝑆, 𝑃 𝑆 = 2 𝑆 (hint – think binary)
What about infinite sets? Here Cantor came up with a simple and ingenious
proof that shows for any infinite set 𝐴: 𝑃 𝐴 > |𝐴|
20. Cantor’s Theorem
Theorem: Let 𝑓 be a map from set 𝐴 to 𝑃(𝐴) then 𝑓: 𝐴 → 𝑃(𝐴) is not
surjective ()על
על חח"ע חח"עועל
21. Cantor’s Theorem (proof from “the book”)
Proof:
a. Consider the set 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑓 𝑥 }.
Set of all items from 𝐴 which are not an element of the subset they
are mapped to by 𝑓
a. Note that 𝐵 is a subset of 𝐴 the therefore B ∈ 𝑃(𝐴)
b. Therefore there exists an element 𝑦 ∈ 𝐴 such that 𝑓 𝑦 = 𝐵
c. Two options:
𝑦 ∈ 𝐵 in this case by definition of 𝐵, 𝑦 ∉ 𝑓 𝑦 = 𝐵:
contradiction!
𝑦 ∉ 𝐵 in this case 𝑦 ∉ 𝑓(𝑦) which means 𝑦 ∈ 𝐵: contradiction!
So such a function 𝑓 cannot exist and there can not be a bijection:
𝑃 𝐴 > 𝐴
1
2
3
4
5
.
.
.
{1,2}
{3,17,5}
{8}
{17,4}
{1,2,5}
.
.
.
y
.
B={2,3,…}
.
A P(A)
22. Cardinality of Reals – connection to subsets of ℕ
Challenge: Prove that 2 ℕ = |ℝ| (hint – think binary. Again)
Let’s look at [0,1]: The binary representation of all numbers in [0,1] are all
the subsets of ℕ:
0.11000101… is the set {1,2,6,8,…}
23. Other cardinalities
Why not go further? Define ℶ = 2ℵ
the cardinality of the power set of ℝ. This
is the cardinality of all functions 𝑓: ℝ → ℝ
We can continue forever increasing the cardinality:
ℵ0, ℵ1 = 2ℵ0, ℵ2 = 2ℵ1, … , ℵ 𝑛+1 = 2ℵ 𝑛, …
There is no largest cardinality!
Wait – what about the increments? Is there a cardinality between the
integers and the continuum?
This was Cantor’s famous Continuum Hypothesis
24. Continuum Hypothesis
CH is independent of the axioms of set theory. That is, either CH or its
negation can be added as an axiom to ZFC set theory, with the resulting
theory being consistent.
Continuum Hypothesis (CH): There is no set whose cardinality is strictly
between that of the integers and the real numbers.
Stated by Cantor in 1878 is became one of the most famous unprovable
hypotheses and drove Cantor crazy.
Only in 1963 it was proved by Paul Cohen that….
25. Cardinality of Cardinalities
Just one last question to ask – exactly how many different cardinalities are
there?
By iterating the power set we received the sets
ℕ , 𝑃 ℕ , 𝑃 𝑃 ℕ , P P 𝑃 ℕ , …
With the cardinalities
ℵ0, ℵ1, ℵ2, ℵ3, …
There are at least ℵ0 different cardinalities.
Are we good Vincent?
26. 𝑩 = 1,17, 4,9 , 102,4,22 , 2 , 3,6 , 96,3,21 , …
Cardinality of Cardinalities
Ok, hold on to your hats…
Let’s look at the set 𝐴 that contains all the power sets we just saw:
𝐴 = ℕ , 𝑃 ℕ , 𝑃 𝑃 ℕ , P P 𝑃 ℕ , …
And now let’s look at the set 𝑩 = ⋃𝑨 which is the union of all the elements
of all the sets in 𝐴. Here is what 𝐵 may look like
Elements
from ℕ
Elements from
𝐏 ℕ
Element from
𝐏 𝐏 ℕ
𝐵’s cardinality is as large as the largest set in 𝐴
27. Cardinality of Cardinalities
The set 𝐵 𝑏𝑖𝑔𝑔𝑒𝑟 = 𝑃 𝐵 = 𝑃(⋃𝐴) - that cardinality is not in the ℵ 𝑛 list
In fact – for any set 𝑆 of sets, the set 𝑆 𝑏𝑖𝑔𝑔𝑒𝑟 = 𝑃 ⋃𝑆 will have a cardinality
not in 𝑆!
No set, no matter how large cannot hold all cardinalities.
So the collection of possible cardinalities is… is not even a set because it does
not have a cardinality.
Mathematicians call this a proper class, something like the set of all sets
(which is not permitted in set theory to avoid things like Russel’s paradox).