This document provides information about a Discrete Mathematics course. It outlines the course objectives, which are to provide students with basic discrete mathematics knowledge and skills in mathematical reasoning, problem solving, and understanding abstract structures. The course learning outcomes include understanding logic, proofs, sets, counting principles, relations, functions, graphs, and trees. The course code, title, credit hours, books, and assessment breakdown are also noted. The document concludes with an introduction to the topics that will be covered, including logic and proofs, counting, graph theory, and number theory.
The solution to the single-source shortest-path tree problem in graph theory. This slide was prepared for Design and Analysis of Algorithm Lab for B.Tech CSE 2nd Year 4th Semester.
The solution to the single-source shortest-path tree problem in graph theory. This slide was prepared for Design and Analysis of Algorithm Lab for B.Tech CSE 2nd Year 4th Semester.
One of the main reasons for the popularity of Dijkstra's Algorithm is that it is one of the most important and useful algorithms available for generating (exact) optimal solutions to a large class of shortest path problems. The point being that this class of problems is extremely important theoretically, practically, as well as educationally.
This is a short presentation on Vertex Cover Problem for beginners in the field of Graph Theory...
Download the presentation for a better experience...
In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects.Graph theory is also important in real life.
One of the main reasons for the popularity of Dijkstra's Algorithm is that it is one of the most important and useful algorithms available for generating (exact) optimal solutions to a large class of shortest path problems. The point being that this class of problems is extremely important theoretically, practically, as well as educationally.
This is a short presentation on Vertex Cover Problem for beginners in the field of Graph Theory...
Download the presentation for a better experience...
In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects.Graph theory is also important in real life.
Object oriented analysis_and_design_v2.0Ganapathi M
This is the presentation I have been using to discuss OOAD concepts with the new joiners of the my company. Quick refresher, but will give the paradigm shift for the participants on how OOAD is different in theory & practice.
Entity relationship model, Components of ER model, Mapping E-R model to Relational schema, Network and Object-Oriented Data models, Storage Strategies: Detailed Storage Architecture, Storing Data, Magnetic Disk, RAID, Other Disks, Magnetic Tape, Storage Access, File & Record Organization, File Organizations & Indexes, Order Indices, B+ Tree Index Files, Hashing Data Dictionary
A brief information about the SCOP protein database used in bioinformatics.
The Structural Classification of Proteins (SCOP) database is a comprehensive and authoritative resource for the structural and evolutionary relationships of proteins. It provides a detailed and curated classification of protein structures, grouping them into families, superfamilies, and folds based on their structural and sequence similarities.
This pdf is about the Schizophrenia.
For more details visit on YouTube; @SELF-EXPLANATORY;
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THE IMPORTANCE OF MARTIAN ATMOSPHERE SAMPLE RETURN.Sérgio Sacani
The return of a sample of near-surface atmosphere from Mars would facilitate answers to several first-order science questions surrounding the formation and evolution of the planet. One of the important aspects of terrestrial planet formation in general is the role that primary atmospheres played in influencing the chemistry and structure of the planets and their antecedents. Studies of the martian atmosphere can be used to investigate the role of a primary atmosphere in its history. Atmosphere samples would also inform our understanding of the near-surface chemistry of the planet, and ultimately the prospects for life. High-precision isotopic analyses of constituent gases are needed to address these questions, requiring that the analyses are made on returned samples rather than in situ.
The increased availability of biomedical data, particularly in the public domain, offers the opportunity to better understand human health and to develop effective therapeutics for a wide range of unmet medical needs. However, data scientists remain stymied by the fact that data remain hard to find and to productively reuse because data and their metadata i) are wholly inaccessible, ii) are in non-standard or incompatible representations, iii) do not conform to community standards, and iv) have unclear or highly restricted terms and conditions that preclude legitimate reuse. These limitations require a rethink on data can be made machine and AI-ready - the key motivation behind the FAIR Guiding Principles. Concurrently, while recent efforts have explored the use of deep learning to fuse disparate data into predictive models for a wide range of biomedical applications, these models often fail even when the correct answer is already known, and fail to explain individual predictions in terms that data scientists can appreciate. These limitations suggest that new methods to produce practical artificial intelligence are still needed.
In this talk, I will discuss our work in (1) building an integrative knowledge infrastructure to prepare FAIR and "AI-ready" data and services along with (2) neurosymbolic AI methods to improve the quality of predictions and to generate plausible explanations. Attention is given to standards, platforms, and methods to wrangle knowledge into simple, but effective semantic and latent representations, and to make these available into standards-compliant and discoverable interfaces that can be used in model building, validation, and explanation. Our work, and those of others in the field, creates a baseline for building trustworthy and easy to deploy AI models in biomedicine.
Bio
Dr. Michel Dumontier is the Distinguished Professor of Data Science at Maastricht University, founder and executive director of the Institute of Data Science, and co-founder of the FAIR (Findable, Accessible, Interoperable and Reusable) data principles. His research explores socio-technological approaches for responsible discovery science, which includes collaborative multi-modal knowledge graphs, privacy-preserving distributed data mining, and AI methods for drug discovery and personalized medicine. His work is supported through the Dutch National Research Agenda, the Netherlands Organisation for Scientific Research, Horizon Europe, the European Open Science Cloud, the US National Institutes of Health, and a Marie-Curie Innovative Training Network. He is the editor-in-chief for the journal Data Science and is internationally recognized for his contributions in bioinformatics, biomedical informatics, and semantic technologies including ontologies and linked data.
Professional air quality monitoring systems provide immediate, on-site data for analysis, compliance, and decision-making.
Monitor common gases, weather parameters, particulates.
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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.
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.
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.
3. Course Objectives
The main objective of this course is to provide
students with the basic knowledge of discrete
mathematics.
Other objectives are as follows:
• understand mathematical reasoning, logically
and mathematically
• improve problem-solving skills of enumerating
objects using combinatorial analysis
• know the abstract mathematical structures
used to represent discrete objects and
relationships between these objects
3
4. Course Learning Outcomes
Upon completion of the course, Students will be able to:
• Write an argument using logical notation and determine if the
argument is or is not valid.
• Demonstrate the ability to write and evaluate a proof or outline
the basic structure of and give examples of each proof
technique described.
• Understand the basic principles of sets and operations in sets.
• Prove basic set equalities.
• Apply counting principles to determine probabilities.
• Demonstrate an understanding of relations and functions and
be able to determine their properties.
• Demonstrate different traversal methods for trees and graphs.
• Model problems in Computer Science using graphs and trees.
4
5. Course Code: GSC-221
Course Title: Discrete Mathematics
Credit Hours: 3
Abbreviation: DM
Prerequisite: -
Type of Course: Core
Course Description:
Propositional statements, predicate logic and its truth values,
quantifiers, methods of proofs, composition, Sequences, types of
sequences, Elementary number theory, mathematical Induction,
Recursive definition, recursively defined sets and structures, Basic
counting rules, pigeon hole principle, permutation, combination,
Relations, reflexive, symmetry, transitive, equivalence relations,
Graphs, terminologies, graph models, types of graphs, representation
About Theory Course
5
7. Books
• “DISCRETE MATHEMATICS AND ITS
APPLICATIONS” BY Kenneth H
Rosen. 7TH ED
• “DISCRETE MATHEMATICS WITH
APPLICATION” by Susanna S Epp.
4th ED
• “DISCRETE MATHEMATICS” by
Richard Johnson Baugh. 7th ED
7
8. Reference Books
• “DISCRETE MATHEMATICAL STRUCTURES” by
Kolman, busby & Ross. 4th ED
• “DISCRETE AND COMBINATORIAL
MATHEMATICS: AN APPLIED
INTRODUCTION” by Ralph P. Grimaldi.
• “LOGIC AND DISCRETE MATHEMATICS: A
COMPUTER SCIENCE PERSPECTIVE ” by
Winifred Grassman
8
9. Copying someone else’s work (partial or
complete) and submitting it as if it were one’s
own
Zero tolerance forplagiarism
Plagiarism
10. What we learn Next !!
Why DM?
What is DM?
History
Uses DM
Applications
10
14. • Discrete mathematics deals with objects
that come in discrete bundles, such as
integers, graphs and statements in logics
• e.g.,1 or 2 books
• Topics include probability, set theory
etc.
• Continuous mathematics deals with objects that vary
continuously,such as real numbers-vary smoothly
• e.g.,3.42 inches from a wall.
• Topic include calculus
• Think of digital watches versus analog watches
(ones where the second hand loops around
continuously withoutstopping)
Discrete Mathematics
15. Founder
Montes Archimedes is known
as the Father of Mathematics.
Mathematics is one of the
ancient sciences developed in
time immemorial
15
Paul Erdos is known as the father
of discrete mathematics.
In 1980s Discrete Mathematics
was introduce as a computer
science support course.
18. Cryptography The field of cryptography, which is the study of
how to create security structures and passwords for computers
and other electronic systems, is based entirely on discrete
mathematics.
This is partly because computers send information in discrete --
or separate and distinct -- bits. Number theory, one important
part of discrete math, allows cryptographers to create and break
numerical passwords. Because of the quantity of money and the
amount of confidential information involved, cryptographers must
first have a solid background in number theory to show they can
provide secure passwords and encryption methods.
Why Study Discrete
Mathematics/Structures
19. Relational Databases Relational databases play a part in almost
every organization that must keep track of employees, clients or
resources.
A relational database connects the traits of a certain piece of
information.
For example, in a database containing client information, the
relational aspect of this database allows the computer system to
know how to link the client’s name, address, phone number and
other pertinent information. This is all done through the discrete
math concept of sets.
Sets allow information to be grouped and put in order. Since each
piece of information and each trait belonging to that piece of
information is discrete, the organization of such information in a
database requires discrete mathematical methods.
Why Study Discrete
Mathematics/Structures
20. Computer Algorithms: Algorithms are the rules by which a
computer operates.
• These rules are created through the laws of discrete
mathematics.
• A computer programmer uses discrete math to design efficient
algorithms.
• This design includes applying discrete math to determine the
number of steps an algorithm needs to complete, which implies
the speed of the algorithm. Because of discrete mathematical
applications in algorithms, today’s computers run faster than
ever before.
Why Study Discrete
Mathematics/Structures
21. Image Processing Image processing is a method to convert an
image into digital form and perform some operations on it
• In order to get an enhanced image or to extract some useful
information from it. It convert image as two dimensional
signals
Graph Theory Google Maps uses discrete mathematics to
determine fastest driving routes and times.
• There is a simpler version that works with small maps and
technicalities involved in adapting to large maps.
• Used in Data Mining and Networking as well.
Why Study Discrete
Mathematics/Structures
22. Why Discrete Mathematics?
• How many ways are there to choose a valid password on a computer
system?
• What is the probability of winning a lottery?
• Is there a link between two computers in a network?
• How can I identify spam e-mail messages?
• How can I encrypt a message so that no unintended recipient can read it?
• What is the shortest path between two cities using a transportation
system?
• How can a list of integers be sorted so that the integers are in increasing
order?
• How many steps are required to do such a sorting?
• How can it be proved that a sorting algorithm correctly sorts a list?
• How can a circuit that adds two integers be designed?
• How many valid Internet addresses are there? 22
23. Applications
Design efficient computer systems.
•How did Google manage to build a fast search engine?
•What is the foundation of internet security?
algorithms, data structures, database,
parallel computing, distributed systems,
cryptography, computer networks…
Logic, sets/functions, counting, graph theory…
23
24. Topic 1: Logic and Proofs
Logic: propositional logic, first order logic
Proof: induction, contradiction
How do computers think?
Artificial intelligence, database, circuit, algorithms
24
