The presentation on simple mathematics of random walk. This mathematical concept have applications in calculations of Brownian motion and signal processing.
Random walks are stochastic processes that can model many natural phenomena. A random walk is generated by successive random steps on a mathematical structure like integers or graphs. Random walks can simulate processes like molecular motion or animal foraging. They have applications in fields like recommender systems, investment theory, and generating fractal images. A random walk on a graph corresponds to a Markov chain, with transition probabilities defined by the graph structure. Random walks approach a unique stationary distribution if the graph is connected and aperiodic. The mixing time measures how fast this convergence occurs. Random walk algorithms are used for tasks like ranking genes by likelihood of having a property or learning vertex embeddings in networks.
1) The document describes a canonical transformation from the original (x, p) coordinates to new canonical coordinates (X, P) for the harmonic oscillator Hamiltonian.
2) An explicit transformation is found using a generating function of type 1. The new coordinates (X, P) correspond to an action-angle pair where P is the action (energy) and X is the cyclic coordinate proportional to time.
3) In the new coordinates, the Hamiltonian and equations of motion simplify greatly, with P being a constant of motion and X varying linearly with time.
1. The document presents an overview of Markov chains and processes, with a focus on their applications in marketing.
2. It provides an example of brand switching over time as a Markov process and outlines some key aspects of Markov processes including finite states, stationarity, and uniform time periods.
3. The document also gives an example problem using a transition matrix to determine the market share of two breakfast brands in a steady state based on customer purchase patterns.
1. Hartree-Fock theory describes molecules using a linear combination of atomic orbitals to approximate molecular orbitals. It treats electrons as independent particles moving in the average field of other electrons.
2. The Hartree-Fock method involves iteratively solving the Fock equations until self-consistency is reached between the input and output orbitals. This approximates electron correlation by including an average electron-electron repulsion term.
3. The Hartree-Fock method satisfies the Pauli exclusion principle through the use of Slater determinants, which are antisymmetric wavefunctions that go to zero when the spatial or spin coordinates of any two electrons are identical.
The document discusses entropy from statistical and thermodynamic perspectives. It defines entropy statistically as the natural logarithm of the number of microscopic configurations of a system. The document outlines how entropy is a state function that always increases for irreversible processes according to the second law of thermodynamics. It also discusses how entropy is additive for combined systems and how the conditions of thermal, mechanical, and chemical equilibrium can be defined in terms of entropy being maximized. The Gibbs paradox regarding mixing of ideal gases is also summarized.
Introduction Stochastic Processes.
Markov Chains.
Chapman-Kolmogorov Equations
Classification of States
Recurrence and Transience
Limiting Probabilities
Random walks are stochastic processes that can model many natural phenomena. A random walk is generated by successive random steps on a mathematical structure like integers or graphs. Random walks can simulate processes like molecular motion or animal foraging. They have applications in fields like recommender systems, investment theory, and generating fractal images. A random walk on a graph corresponds to a Markov chain, with transition probabilities defined by the graph structure. Random walks approach a unique stationary distribution if the graph is connected and aperiodic. The mixing time measures how fast this convergence occurs. Random walk algorithms are used for tasks like ranking genes by likelihood of having a property or learning vertex embeddings in networks.
1) The document describes a canonical transformation from the original (x, p) coordinates to new canonical coordinates (X, P) for the harmonic oscillator Hamiltonian.
2) An explicit transformation is found using a generating function of type 1. The new coordinates (X, P) correspond to an action-angle pair where P is the action (energy) and X is the cyclic coordinate proportional to time.
3) In the new coordinates, the Hamiltonian and equations of motion simplify greatly, with P being a constant of motion and X varying linearly with time.
1. The document presents an overview of Markov chains and processes, with a focus on their applications in marketing.
2. It provides an example of brand switching over time as a Markov process and outlines some key aspects of Markov processes including finite states, stationarity, and uniform time periods.
3. The document also gives an example problem using a transition matrix to determine the market share of two breakfast brands in a steady state based on customer purchase patterns.
1. Hartree-Fock theory describes molecules using a linear combination of atomic orbitals to approximate molecular orbitals. It treats electrons as independent particles moving in the average field of other electrons.
2. The Hartree-Fock method involves iteratively solving the Fock equations until self-consistency is reached between the input and output orbitals. This approximates electron correlation by including an average electron-electron repulsion term.
3. The Hartree-Fock method satisfies the Pauli exclusion principle through the use of Slater determinants, which are antisymmetric wavefunctions that go to zero when the spatial or spin coordinates of any two electrons are identical.
The document discusses entropy from statistical and thermodynamic perspectives. It defines entropy statistically as the natural logarithm of the number of microscopic configurations of a system. The document outlines how entropy is a state function that always increases for irreversible processes according to the second law of thermodynamics. It also discusses how entropy is additive for combined systems and how the conditions of thermal, mechanical, and chemical equilibrium can be defined in terms of entropy being maximized. The Gibbs paradox regarding mixing of ideal gases is also summarized.
Introduction Stochastic Processes.
Markov Chains.
Chapman-Kolmogorov Equations
Classification of States
Recurrence and Transience
Limiting Probabilities
This document provides an overview of statistical mechanics. It defines microstates and macrostates, and explains that statistical mechanics studies systems with many microstates corresponding to a given macrostate. The Boltzmann distribution is derived, which gives the probability of finding a system in a particular microstate as being proportional to the exponential of the negative of the energy of that microstate divided by the temperature. Maxwell-Boltzmann statistics are described as applying to classical distinguishable particles, yielding the Maxwell-Boltzmann distribution. References for further reading are also included.
Quantum cryptography uses properties of quantum mechanics to securely distribute encryption keys. It allows two users to generate a shared secret key with information-theoretic security. This is accomplished through quantum key distribution, which exploits the quantum mechanical principle that measuring a quantum system can disturb the system. Even if an eavesdropper has unlimited computing power, the laws of physics guarantee the security of the key exchange. The paper introduces cryptography, traditional techniques, and the differences between traditional and quantum cryptography.
An introduction to logistic regression for physicians, public health students and other health workers. Logistic regression is a way to look at effect of a numeric independent variable on a binary (yes-no) dependent variable. For example, you can analyze or model the effect of birth weight on survival.
In computational physics and Quantum chemistry, the Hartree–Fock (HF) method also known as self consistent method is a method of approximation for the determination of the wave function and the energy of a quantum many-body system or many electron system in a stationary state
Stochastic Processes describe the system derived by noise.
Level of graduate students in mathematics and engineering.
Probability Theory is a prerequisite.
For comments please contact me at solo.hermelin@gmail.com.
For more presentations on different subjects visit my website at http://www.solohermelin.com.
This document defines key concepts in probability and provides examples. It discusses probability vocabulary like sample space, outcome, trial, and event. It defines probability as the number of times a desired outcome occurs over total trials. Events are independent if the outcome of one does not impact others, and mutually exclusive if they cannot occur together. The addition and multiplication rules for probability are explained. Conditional probability describes the probability of a second event depending on the first occurring. Counting techniques are discussed for finding total possible outcomes of combined experiments. Review questions are provided to test understanding of the material.
1. The Ising model is a statistical mechanics model of ferromagnetism. It represents magnetic dipole moments as "spins" on a lattice that can point in one of two directions and interact with neighboring spins.
2. The Ising model can explain phase transitions like ferromagnetism, anti-ferromagnetism, gas-liquid transitions, and liquid-solid transitions.
3. The statistical mechanics of the Ising model are studied using the Hamiltonian, which includes terms for spin-spin interaction energy and the energy of an external magnetic field interacting with the magnetic moments. Partition functions are then used to calculate thermodynamic properties.
The Born-Oppenheimer approximation, proposed in 1927 by physicists Max Born and J. Robert Oppenheimer, treats the motions of nuclei and electrons in molecules separately. It approximates that the nuclei in a molecule are stationary relative to the rapidly moving electrons. This allows molecular structure and properties to be determined by first solving the electronic Schrodinger equation at fixed nuclear positions, and then adding the internuclear repulsion energy to obtain the total internal energy of the molecule. As a result of this approximation, molecules have well-defined shapes determined by the equilibrium positions of their nuclei.
This document summarizes key concepts related to Markov chains and linear algebra. It provides an example of using a transition matrix to model the probabilities of television viewers switching between two stations over time. The transition matrix allows calculating the probability vectors for future weeks through matrix multiplication. A steady-state vector can also be determined by solving the equation A*p=p, representing the long-term probabilities once the system reaches equilibrium.
The document discusses the construction of Brillouin zones. It begins by defining a crystal structure as a periodic array of atoms consisting of a lattice and basis. A Brillouin zone is then defined as the Wigner-Seitz primitive cell in the reciprocal lattice. The document goes on to explain that Brillouin zones are constructed from planes that are perpendicular bisectors of all reciprocal lattice vectors. The first Brillouin zone is the smallest volume around the origin enclosed by these planes, and higher order zones are volumes between the first zone and subsequent planes. The document provides examples of determining k-values that satisfy Bragg's equation for the first and second Brillouin zones of a simple rectangular lattice.
This document provides an overview of magnetostatics and key concepts related to magnetism. It begins with a top ten list of magnetism principles. It then discusses the properties of magnetic poles, fields, and materials. Key points made include that every magnet has both a north and south pole, magnetic fields are generated by moving charges, and materials can be classified based on their magnetic permeability. The document also introduces critical magnetism concepts such as the Biot-Savart law, Ampere's law, magnetic dipoles, and the forces and energy associated with magnetic fields.
The normal distribution is a continuous probability distribution defined by its probability density function. A random variable has a normal distribution if its density function is defined by a mean (μ) and standard deviation (σ). The normal distribution is symmetrical and bell-shaped. It is commonly used to approximate other distributions when the sample size is large.
This document discusses Markov chains, which are a type of stochastic process used to model randomly changing systems. It defines Markov chains and their key properties, like the Markov property and transition probabilities. It provides examples like modeling customer purchases over time and inventory management. It also covers concepts like steady state probabilities, transition matrices, and mean first passage times.
This document discusses evidence for covalent bonding in metal complexes from three perspectives:
1. The nephelauxetic effect shows that electron-electron repulsion is less in complexes than free metal ions due to delocalization of electrons over ligand orbitals.
2. Nephelauxetic parameters (β) quantify this effect, with softer ligands having smaller β values.
3. Electron paramagnetic resonance (EPR) spectroscopy reveals hyperfine splitting in complex spectra, showing interaction between ligand nuclear spins and metal electron spins, further indicating covalent bonding.
Classification of magnetic materials on the basis of magnetic momentVikshit Ganjoo
I made this presentation for my own college assignment and i had referred contents from websites and other presentations and made it presentable and reasonable hope you will like it!!!
The document provides an overview of probability concepts including:
- Probability is a measure of how likely an event is, defined as the number of favorable outcomes divided by the total number of possible outcomes.
- Theoretical probability predicts outcomes without performing experiments, dealing with events as combinations of elementary outcomes.
- Random experiments may have different results each time while deterministic experiments always produce the same outcome.
- Elementary events are individual outcomes, and compound events combine multiple elementary outcomes.
- Theoretical probability of an event is the number of favorable elementary events divided by the total number of possible events.
- The probabilities of an event and its negation must sum to 1.
This document presents on the topics of quantum mechanics, including the Schrodinger equation, potential barriers and tunneling, and the infinite potential well. It discusses how classically a particle cannot pass through a potential barrier with energy lower than the barrier potential, but quantum mechanically there is a probability of tunneling. It derives the wave functions and transmission coefficients for particles incident on potential barriers and wells. The transmission probability for tunneling decreases exponentially with increasing barrier thickness and potential. The infinite potential well confines particles within fixed boundaries, with allowed quantum states and energies determined by the boundary conditions.
Eigenvalues and eigenfunctions are key concepts in linear algebra. An eigenfunction is a function that when operated on by a linear operator produces a constant multiplied version of itself. The constant is the corresponding eigenvalue. Eigenvalues are the solutions to the characteristic polynomial of the linear operator. Eigenfunctions are not unique as any constant multiple of an eigenfunction is also an eigenfunction with the same eigenvalue. The spectrum of an operator is the set of all its eigenvalues.
This document discusses counting techniques, probability, statistics, and graphical representations of data. It defines key terms like sample space, permutations, combinations, theoretical and experimental probability. It also describes common sampling methods, measures of central tendency including mean, median and mode, measures of variability like range and standard deviation, and graphical representations such as histograms, bar charts, frequency polygons and pie charts.
This document provides an overview of statistical mechanics. It defines microstates and macrostates, and explains that statistical mechanics studies systems with many microstates corresponding to a given macrostate. The Boltzmann distribution is derived, which gives the probability of finding a system in a particular microstate as being proportional to the exponential of the negative of the energy of that microstate divided by the temperature. Maxwell-Boltzmann statistics are described as applying to classical distinguishable particles, yielding the Maxwell-Boltzmann distribution. References for further reading are also included.
Quantum cryptography uses properties of quantum mechanics to securely distribute encryption keys. It allows two users to generate a shared secret key with information-theoretic security. This is accomplished through quantum key distribution, which exploits the quantum mechanical principle that measuring a quantum system can disturb the system. Even if an eavesdropper has unlimited computing power, the laws of physics guarantee the security of the key exchange. The paper introduces cryptography, traditional techniques, and the differences between traditional and quantum cryptography.
An introduction to logistic regression for physicians, public health students and other health workers. Logistic regression is a way to look at effect of a numeric independent variable on a binary (yes-no) dependent variable. For example, you can analyze or model the effect of birth weight on survival.
In computational physics and Quantum chemistry, the Hartree–Fock (HF) method also known as self consistent method is a method of approximation for the determination of the wave function and the energy of a quantum many-body system or many electron system in a stationary state
Stochastic Processes describe the system derived by noise.
Level of graduate students in mathematics and engineering.
Probability Theory is a prerequisite.
For comments please contact me at solo.hermelin@gmail.com.
For more presentations on different subjects visit my website at http://www.solohermelin.com.
This document defines key concepts in probability and provides examples. It discusses probability vocabulary like sample space, outcome, trial, and event. It defines probability as the number of times a desired outcome occurs over total trials. Events are independent if the outcome of one does not impact others, and mutually exclusive if they cannot occur together. The addition and multiplication rules for probability are explained. Conditional probability describes the probability of a second event depending on the first occurring. Counting techniques are discussed for finding total possible outcomes of combined experiments. Review questions are provided to test understanding of the material.
1. The Ising model is a statistical mechanics model of ferromagnetism. It represents magnetic dipole moments as "spins" on a lattice that can point in one of two directions and interact with neighboring spins.
2. The Ising model can explain phase transitions like ferromagnetism, anti-ferromagnetism, gas-liquid transitions, and liquid-solid transitions.
3. The statistical mechanics of the Ising model are studied using the Hamiltonian, which includes terms for spin-spin interaction energy and the energy of an external magnetic field interacting with the magnetic moments. Partition functions are then used to calculate thermodynamic properties.
The Born-Oppenheimer approximation, proposed in 1927 by physicists Max Born and J. Robert Oppenheimer, treats the motions of nuclei and electrons in molecules separately. It approximates that the nuclei in a molecule are stationary relative to the rapidly moving electrons. This allows molecular structure and properties to be determined by first solving the electronic Schrodinger equation at fixed nuclear positions, and then adding the internuclear repulsion energy to obtain the total internal energy of the molecule. As a result of this approximation, molecules have well-defined shapes determined by the equilibrium positions of their nuclei.
This document summarizes key concepts related to Markov chains and linear algebra. It provides an example of using a transition matrix to model the probabilities of television viewers switching between two stations over time. The transition matrix allows calculating the probability vectors for future weeks through matrix multiplication. A steady-state vector can also be determined by solving the equation A*p=p, representing the long-term probabilities once the system reaches equilibrium.
The document discusses the construction of Brillouin zones. It begins by defining a crystal structure as a periodic array of atoms consisting of a lattice and basis. A Brillouin zone is then defined as the Wigner-Seitz primitive cell in the reciprocal lattice. The document goes on to explain that Brillouin zones are constructed from planes that are perpendicular bisectors of all reciprocal lattice vectors. The first Brillouin zone is the smallest volume around the origin enclosed by these planes, and higher order zones are volumes between the first zone and subsequent planes. The document provides examples of determining k-values that satisfy Bragg's equation for the first and second Brillouin zones of a simple rectangular lattice.
This document provides an overview of magnetostatics and key concepts related to magnetism. It begins with a top ten list of magnetism principles. It then discusses the properties of magnetic poles, fields, and materials. Key points made include that every magnet has both a north and south pole, magnetic fields are generated by moving charges, and materials can be classified based on their magnetic permeability. The document also introduces critical magnetism concepts such as the Biot-Savart law, Ampere's law, magnetic dipoles, and the forces and energy associated with magnetic fields.
The normal distribution is a continuous probability distribution defined by its probability density function. A random variable has a normal distribution if its density function is defined by a mean (μ) and standard deviation (σ). The normal distribution is symmetrical and bell-shaped. It is commonly used to approximate other distributions when the sample size is large.
This document discusses Markov chains, which are a type of stochastic process used to model randomly changing systems. It defines Markov chains and their key properties, like the Markov property and transition probabilities. It provides examples like modeling customer purchases over time and inventory management. It also covers concepts like steady state probabilities, transition matrices, and mean first passage times.
This document discusses evidence for covalent bonding in metal complexes from three perspectives:
1. The nephelauxetic effect shows that electron-electron repulsion is less in complexes than free metal ions due to delocalization of electrons over ligand orbitals.
2. Nephelauxetic parameters (β) quantify this effect, with softer ligands having smaller β values.
3. Electron paramagnetic resonance (EPR) spectroscopy reveals hyperfine splitting in complex spectra, showing interaction between ligand nuclear spins and metal electron spins, further indicating covalent bonding.
Classification of magnetic materials on the basis of magnetic momentVikshit Ganjoo
I made this presentation for my own college assignment and i had referred contents from websites and other presentations and made it presentable and reasonable hope you will like it!!!
The document provides an overview of probability concepts including:
- Probability is a measure of how likely an event is, defined as the number of favorable outcomes divided by the total number of possible outcomes.
- Theoretical probability predicts outcomes without performing experiments, dealing with events as combinations of elementary outcomes.
- Random experiments may have different results each time while deterministic experiments always produce the same outcome.
- Elementary events are individual outcomes, and compound events combine multiple elementary outcomes.
- Theoretical probability of an event is the number of favorable elementary events divided by the total number of possible events.
- The probabilities of an event and its negation must sum to 1.
This document presents on the topics of quantum mechanics, including the Schrodinger equation, potential barriers and tunneling, and the infinite potential well. It discusses how classically a particle cannot pass through a potential barrier with energy lower than the barrier potential, but quantum mechanically there is a probability of tunneling. It derives the wave functions and transmission coefficients for particles incident on potential barriers and wells. The transmission probability for tunneling decreases exponentially with increasing barrier thickness and potential. The infinite potential well confines particles within fixed boundaries, with allowed quantum states and energies determined by the boundary conditions.
Eigenvalues and eigenfunctions are key concepts in linear algebra. An eigenfunction is a function that when operated on by a linear operator produces a constant multiplied version of itself. The constant is the corresponding eigenvalue. Eigenvalues are the solutions to the characteristic polynomial of the linear operator. Eigenfunctions are not unique as any constant multiple of an eigenfunction is also an eigenfunction with the same eigenvalue. The spectrum of an operator is the set of all its eigenvalues.
This document discusses counting techniques, probability, statistics, and graphical representations of data. It defines key terms like sample space, permutations, combinations, theoretical and experimental probability. It also describes common sampling methods, measures of central tendency including mean, median and mode, measures of variability like range and standard deviation, and graphical representations such as histograms, bar charts, frequency polygons and pie charts.
In three of the exercises below there is given a code of a method na.pdffeelinggift
In three of the exercises below there is given a code of a method named find, and a fourth one
named printMany. Analyze the codes through the following points.
Explain your choice of the input size n and in terms of O(n) scale determine the running time
(number of steps) T(n) of the algorithms represented by the methods.
Use the simplest and possibly the smallest valid Big-Oh expression.
T(n) can also be considered as the number elementary operations the algorithm must make.
If it applies, point out your estimates for the worst and best cases, and also for the average case
performance if available.
Document each method describing what you consider the method’s precondition and post-
condition.
It is not necessary to run these methods in actual programs, but if the task it performs is dubious,
testing the method with various input in actual applications of the code may help to find its
purpose and the big-Oh estimate.
1) int find( int[] list, int element ){
int answer = 0;
for(int k = 0; k < list.length; k++ )
if (element==list[k])
answer++;
return answer;
}//end method
Comments
What does the method do:
Input size n =
Worst case T(n) = O(________) Best case T(n) = O(________)
2) staticintfind(int[] arr){
zeroCounter = 0;
(intk = 0; k< arr.length; k++){
(arr[k]==0)
zeroCounter++;
}
(zeroCounter==arr.length)
0;
(zeroCounter < arr.length - 2){
//see maxIndex() definition below
max = maxIndex(arr);
arr[max] = 0;
//see display() definition below
display(arr);
zeroCounter ++;
}
maxIndex(arr);
//end method
//helper methods
3) staticint maxIndex(int[]arr){
int maxindex = 0;
for(int k = 0 ; k< arr.length; k++){
// note the use of absolute value
if(Math.abs(arr[maxindex]) < Math.abs(arr[k]))
maxindex = k;
}
return maxindex;
}
staticvoid display(int[]arr){
System.out.println();
for(int k = 0 ; k< arr.length; k++)
System.out.print(arr[k]+” “);
System.out.println();
}
Comments
What does the method do:
Input size n =
Worst case T(n) = O(________) Best case T(n) = O(________)
3) int find(int[] num){
int answer = 0;
for(int k = 0; k < num.length; k++ )
for(int j = k; j< num.length; j++){
int current = 0;
for(int i = k; i<=j; i++)
current += num[i];
if (current > answer)
answer = current;
}
return answer;
}
Note: Given two indices i<=j of an array of integers num, the sum
num[i]+ num[i+1] + …+ num[j] is called a sub-sum
Comments
What does the method do:
Input size n =
Worst case T(n) = O(________) Best case T(n) = O(________)
4) void printMany(int[]arr){
int N = arr.length;
for(int k = 0 ; k< N; k++){
int p = k;
while(p>0){
System.out.println(arr[p]+\" \");
p = p/2;
}
}
Comments
What does the method do:
Input size n =
Worst case T(n) = O(________) Best case T(n) = O(________)
Solution
1) int find( int[] list, int element ){
int answer = 0;
for(int k = 0; k < list.length; k++ ) //traversing array start to end
if (element==list[k]) //checking for element
answer++; // if element found, increase by 1
return answer;
}//end method
In the function given above, we can.
This document provides an overview of stochastic processes and Markov chains. It defines stochastic processes as families of random variables indexed by time. Markov chains are a type of stochastic process where the future state depends only on the present state, not on the past. The document discusses examples of Markov chains, transition matrices, classification of states as transient or persistent, and properties like irreducibility. It aims to introduce key concepts in stochastic processes and Markov chains.
Estimating the Evolution Direction of Populations to Improve Genetic AlgorithmsAnnibale Panichella
Meta-heuristics have been successfully used to solve a wide variety of problems. However, one issue many techniques have is their risk of being trapped into local optima, or to create a limited variety of solutions (problem known as ``population drift''). During recent and past years, different kinds of techniques have been proposed to deal with population drift, for example hybridizing genetic algorithms with local search techniques or using niche techniques.
This paper proposes a technique, based on Singular Value Decomposition (SVD), to enhance Genetic Algorithms (GAs) population diversity. SVD helps to estimate the evolution direction and drive next generations towards orthogonal dimensions.
The proposed SVD-based GA has been evaluated on 11 benchmark problems and compared with a simple GA and a GA with a distance-crowding schema. Results indicate that SVD-based GA achieves significantly better solutions and exhibits a quicker convergence than the alternative techniques.
The document provides an overview of key concepts in probability theory and stochastic processes. It defines fundamental terms like sample space, events, probability, conditional probability, independence, random variables, and common probability distributions including binomial, Poisson, exponential, uniform, and Gaussian distributions. Examples are given for each concept to illustrate how it applies to modeling random experiments and computing probabilities. The three main axioms of probability are stated. Key properties and formulas for expectation, variance, and conditional expectation are also summarized.
- Permutation refers to arrangements that consider order, while combination refers to selections where order does not matter.
- The number of permutations of n distinct objects taken r at a time is nPr = n!/(n-r)!, while the number of combinations is nCr = n!/r!(n-r)!.
- Examples are given to illustrate permutations involving restricted arrangements and circular permutations. Restricted permutations consider cases where certain objects are always or never included.
Deep learning and neural networks (using simple mathematics)Amine Bendahmane
The document provides an overview of machine learning and deep learning concepts through a series of diagrams and explanations. It begins by introducing concepts like regression, classification, and clustering. It then discusses supervised vs unsupervised learning before explaining neural networks and components like the perceptron, multi-layer perceptrons, and convolutional neural networks. It notes how neural networks learn representations and separate data through hidden layers.
Best-first search is a heuristic search algorithm that expands the most promising node first. It uses an evaluation function f(n) that estimates the cost to reach the goal from each node n. Nodes are ordered in the fringe by increasing f(n). A* search is a special case of best-first search that uses an admissible heuristic function h(n) and is guaranteed to find the optimal solution.
Quicksort is a divide and conquer sorting algorithm that works by partitioning an array around a pivot value and recursively sorting the subarrays. In the best case when the array is partitioned evenly, quicksort runs in O(n log n) time as the array is cut in half at each recursive call. However, in the worst case when the array is already sorted, each partition only cuts off one element, resulting in O(n^2) time as the recursion reaches a depth of n. Choosing a better pivot value can improve quicksort's performance on poorly sorted arrays.
The document provides an introduction to Brownian motion by starting with a one-dimensional discrete case modeled as a drunk walking randomly. It shows that Brownian motion has the properties of being memory-less, homogeneous in time and space. By taking the limit of discrete steps, the model arrives at continuous Brownian motion described by a partial differential equation. The document then briefly outlines the history of Brownian motion from its discovery to developments in modeling it as a stochastic process.
1. The document covers probability axioms and rules including the additive rule, conditional probability, independence, and Bayes' rule. It also defines discrete and continuous random variables and their probability distributions.
2. Important discrete distributions discussed include the Bernoulli distribution for a binary outcome experiment and the binomial distribution for repeated Bernoulli trials.
3. Techniques for counting permutations, combinations, and sequences of events are presented to handle probability problems involving counting.
Grade 10_Math-Lesson 2-3 Graphs of Polynomial Functions .pptxErlenaMirador1
The document discusses how to graph polynomial functions by determining:
1) The end behavior using the leading coefficient test
2) The maximum number of turning points from the degree of the polynomial
3) The x-intercepts by finding the zeros of the polynomial
4) The y-intercept by evaluating the polynomial at x=0
It provides examples of using these steps to graph various polynomial functions of degrees 1-5.
Grade 10_Math-Lesson 2-3 Graphs of Polynomial Functions .pptxErlenaMirador1
The document discusses how to graph polynomial functions by determining:
1) The end behavior using the leading coefficient test
2) The maximum number of turning points from the degree of the polynomial
3) The x-intercepts by finding the zeros of the polynomial
4) The y-intercept by evaluating the polynomial at x=0
It provides examples of using these steps to graph various polynomial functions of degrees 1-5.
2 Review of Statistics. 2 Review of Statistics.WeihanKhor2
This document provides an overview of discrete probability distributions, including the binomial and Poisson distributions.
1) It defines key concepts such as random variables, probability mass functions, and expected value as they relate to discrete random variables. 2) The binomial distribution describes independent Bernoulli trials with a constant probability of success, and is used to calculate probabilities of outcomes from events like coin flips. 3) The Poisson distribution approximates the binomial when the number of trials is large and the probability of success is small. It models rare, independent events with a constant average rate and can be used for problems involving traffic accidents or natural disasters.
The document describes and analyzes two algorithms for finding the convex hull of a set of points: a brute force algorithm and a divide and conquer algorithm.
The brute force algorithm iterates through all points three times, checking all possible line combinations, resulting in O(n3) time complexity.
The divide and conquer algorithm recursively divides the point set into halves at each step by finding the furthest point from the current left-right boundary line. It has O(n log n) time complexity.
An experiment comparing runtimes on sample point sets showed the divide and conquer approach was significantly faster than the brute force approach.
This document provides an overview of kernel methods for machine learning. It discusses the evolution of learning methods from perceptrons in the 1950s to kernel methods in the 1990s. Kernel methods embed data into a higher-dimensional Hilbert space to allow for linear classification of non-linear relationships. The kernel trick replaces the inner product in this space with a kernel function, avoiding the need to explicitly define the embedding. Common kernel functions include polynomial kernels and Gaussian RBF kernels. The document provides code examples of kernel ridge regression in Python and discusses applications of string kernels and normalization techniques.
The document discusses discrete probability concepts including sample spaces, events, axioms of probability, conditional probability, Bayes' theorem, random variables, probability distributions, expectation, and classical probability problems. It provides examples and explanations of key terms. The Monty Hall problem is used to demonstrate defining the sample space, event of interest, assigning probabilities, and computing the probability of winning by sticking or switching doors.
The document discusses various probability distributions including the binomial, Poisson, and normal distributions. It provides definitions and key properties of each distribution. It also discusses sampling with and without replacement as well as the Monte Carlo method for simulating physical systems using random sampling. The Monte Carlo method can be used to computationally estimate values like pi by simulating the throwing of darts at a circular target.
Different physical situation encountered in nature are described by three types of statistics-Maxwell-Boltzmann Statistics, Bose-Einstein Statistics and Fermi-Dirac Statistics
The interpretation of phase diagrams have application in petroleum industry, metallurgy, chemical industry, solvent separation and so on. This presentation guid you to understand phase diagrams.
Coordination complexes-bonding and magnetism.pdfAnjali Devi J S
This document discusses coordination complexes, their bonding properties, and magnetism. It covers several theories of bonding in coordination complexes including valence bond theory, crystal field theory, and ligand field theory. Valence bond theory describes coordinate covalent bonds formed between metal centers and ligands. Crystal field theory models ligand fields as point charges that split the metal's d orbitals into different energy levels, influencing complex properties. Magnetism arises from both spin and orbital contributions of unpaired electrons. Temperature and external fields can induce spin state changes between high and low spin configurations in some complexes.
This document provides information about the Poisson distribution, a discrete probability distribution that can be used to predict the probability of certain events occurring within a fixed time period or other interval. The key points are:
- The Poisson distribution gives the probability of a given number of discrete events happening in a fixed time interval, where the events are independent and cannot occur simultaneously.
- It is defined by one parameter, lambda (λ), which represents the average number of events occurring per interval. The mean and variance of the distribution are both equal to λ.
- Some examples of phenomena that can be modeled by the Poisson distribution include the number of phone calls received per minute or cars passing on a road per hour.
Probability is a numerical measure of how likely an event is to occur. It is defined as the number of favorable outcomes divided by the total number of possible outcomes. A random experiment is an action with some defined outcomes that may occur by chance. The sample space is the set of all possible outcomes. Conditional probability is the probability of one event occurring given that another event has occurred.
This document discusses the binomial distribution, an important discrete probability distribution that describes the number of successes in a fixed number of independent yes/no experiments, such as coin tosses, where the probability of success p is the same for each trial. The key points are:
- The binomial distribution formula is used to calculate the probability of getting x successes in n trials.
- Examples of binomial experiments include calculating the probability of getting a certain number of heads when tossing a coin multiple times.
- A binomial random variable X is notated as X~B(n, p) where n is the number of trials and p is the probability of success on each trial.
- The mean and variance of a binomial distribution
The gamma function is a generalization of the factorial function to complex and real number arguments. It is defined as Γ(n) = ∫_0^∞ e^{-x} x^{n-1} dx for n > 0. Some key properties of the gamma function are:
1) Γ(n+1) = nΓ(n) for n > 0
2) Γ(1/2) = √π
3) Values of the gamma function for positive integers n are equal to (n-1)!
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This document discusses Stirling's approximation, which provides an accurate way to calculate large factorials. It was developed by Scottish mathematician James Stirling. The derivation shows that the natural logarithm of N! can be approximated as NlnN - N. This allows factorials of huge numbers like Avogadro's number to be calculated. The document also provides a more accurate form of Stirling's approximation and uses it to calculate some large factorials with low error rates.
Combinatorics is a subfield of discrete mathematics that focuses on counting combinations and arrangements of discrete objects. It involves counting the number of ways to put things together into various combinations. Some key rules in combinatorics include the sum rule, which states that the number of ways to accomplish either of two independent tasks is the sum of the number of ways to accomplish each task individually. The product rule states that the number of ways to accomplish two independent tasks is the product of the number of ways to accomplish each task. Generating functions can be used to efficiently represent counting sequences by coding terms as coefficients of a variable in a formal power series. They allow problems involving counting and arrangements to be solved using operations with formal power series.
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(1)Chem. Soc. Rev., 2017,46,5771
(2)Chem. Soc. Rev., 2017,46,7706
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Or: Beyond linear.
Abstract: Equivariant neural networks are neural networks that incorporate symmetries. The nonlinear activation functions in these networks result in interesting nonlinear equivariant maps between simple representations, and motivate the key player of this talk: piecewise linear representation theory.
Disclaimer: No one is perfect, so please mind that there might be mistakes and typos.
dtubbenhauer@gmail.com
Corrected slides: dtubbenhauer.com/talks.html
Equivariant neural networks and representation theory
1. Random walk.pdf
1. Random walk
Dr. Anjali Devi J S
Assistant Professor (Contract Faculty) , Mahatma Gandhi University, Kerala.
2. Revision
• Random Experiment?
A process by which we observed something uncertain.
Example: Random Experiment- Toss a coin
Sample space{ Head, Tail}
• Random variable?
A numerical description or function of the outcome of a statistical
experiment.
• Random walk?
3. Random Walk- Applications
• Brownian motion
• Swimming of a E coli
• Polymer random coil
• Protein search for a binding site on DNA
Random motion of particles
4. What is Random walk?
• Radom walk is the process by which randomly- moving objects
wander away from where they started.
5. Brownian motion
Brownian motion of a big particle (dust particle) that collides with a large set of smaller particles (molecules of a
gas) which move with different velocities in different random directions.
7. Simplest random walk is 1- dimensional walk
Suppose that the black dot below is sitting on a number
line. The black dot starts in the center.
8. 1- dimensional Random walk…..
• Then, it takes a step, either forward or backward, with equal
probability. It keeps taking steps either forward or backward each
time.
• Let's call the 1st step a1, the second step a2, the third step a3 and so
on. Each "a" is either equal to +1 (if the step is forward) or -1 (if the
step is backward). The picture below shows a black dot that has taken
5 steps and ended up at -1 on the number line.
9. Random walk- Demonstration
x=0
x=-1 x=+1
Number of steps, N=0 Probability=1
Probability=1/2
N=1
N=2 x=-2 X=0 X=-2
Probability=1/3 2/3 1/3
N=3 x=-3 X=-1 X=+1 X=-3
Probability=1/4 2/4 2/4 1/4
10. Random walk
• At every step, object goes to right or left with probability ½
• There N steps, and 2 possible options at each step, so there are total
2N different possible paths.
• Note that every path has the same probability: (½)N
11. Random walk
• But we want to know the probability of object being at a certain
location after N steps.
• While every path has same probability, different ending locations has
different number of paths which lead to them.
12. Random walk- Demonstration
x=0
x=-1 x=+1
Number of steps, N=0 Probability=1
Probability=1/2
N=1
N=2 x=-2 X=0 X=-2
Probability=1/3 2/3 1/3
N=3 x=-3 X=-1 X=+1 X=-3
Probability=1/4 2/4 2/4 1/4
For N=2, there is only one path that leads to x=2
13. The position of object after N steps
• All paths are equally likely but we need to know the number of paths
that leads to object ending up at a particular value of x after N steps.
• The locations of the object after N steps is determined by ho many
steps it takes to the right , which we’ll call m, and how many it takes
to the left N-m
The position x of the object after these N steps is x =m-(N-m)=2m-N
14. Question
Find the position of foot ball which undergoes one dimensional
random walk after 10 steps. It is given that ball took 5 steps to right.
The position x of the object after these N steps is x =m-(N-m)=2m-N
15. Consider a drunkard whose motion is confined to the X-axis. For simplicity let us assume that
after every unit of time, he moves one step either to the right or to the left with probabilities p
and q respectively.
If he starts at the origin, how far his typical distance after N units of time have elapsed?
OR
What is the probability [PN(m)] that drunkard is at coordinate m?
Question
Answer
N=n1+n2
m =n1-n2
Let n1 be the number of steps to the right, n2 be the number of steps to left,
16. Answer (Contd)
Suppose we assume that drunkard has zero memory and that every step is completely
independent of the previous step and is only characterized by the probabilities p and q
The number of ways in which N steps can be composed of n1 right steps and n2 left
steps,
NCm
For each of these possibilities, probability is given by
pn1qn2
The overall probability of finding the drunkard at position m after N steps is,
PN(m)= NCmpn1qn2
PN(m)= NCmpn1qn2
22. Random walk in the plane and space
Theorem
In the symmetric random walks in one and two dimensions there is probability
one that the particle will sooner or later (and therefore infinitely often) return
to initial position. In three dimensions, however, this probability is <1
23. Random walk: Notions and
notations
Epoch: A point on the time axis. A physical experiments may take some
time, but our ideal trials are timeless and occur at epochs.
𝑺𝟏, 𝑺𝟐,……. , 𝑺𝒏 -points on vertical x axis; they will be called the
positions of a particle performing a random walk
The event at epoch n the particle is at the point r will be denoted by
{𝑺𝒏 = 𝐫}
( Simply, we call this event as a “visit” to r at epoch n.)
For its probability 𝒑𝒏,𝒓
24. Random walk: Notions and
notations
The number 𝑵𝒏,𝒓 . 𝒐𝒇 𝒑𝒂𝒕𝒉𝒔 𝒇𝒓𝒐𝒎 𝒕𝒉𝒆 𝒐𝒓𝒊𝒈𝒊𝒏 to the point (n,r)
𝒑𝒏,𝒓=P {𝑺𝒏 = 𝐫} =𝑛𝐶𝑛+𝑟
2
2−𝑛
25. Random walk: Notions and
notations
A return to the origin occurs at epoch k if 𝑺𝒌 = 𝟎
Hence k necessarily even, and for k=2ν
The probability of return to the origin equals 𝒑2ν ,𝟎
Because of frequent occurrence of this probability , 𝒑2ν ,𝟎is
denoted as 𝒖2ν,𝒓
27. First return to origin
A first return to origin occurs at epoch 2ν if
𝑆1 ≠ 0, … … . 𝑆2ν−1 ≠ 0, 𝑏𝑢𝑡 𝑆2ν = 0
The probability for this event will be denoted by 𝑓2ν
𝑓0=0
28. Basic Results
The Main Lemma
The probability that no return to the origin occurs up to and including epoch
2n is the same as the probability that a return occurs at epoch 2n.
In symbols,
P{𝑆1 ≠ 0, … … . , 𝑆2𝑛 ≠ 0}=𝑃{𝑆2𝑛 = 0}= 𝑢2𝑛
Lemma 1
29. Lemma 2
The Main Lemma
Basic Results
The probability that the first return to the origin (𝑓2𝑛) occurs at epoch 2n is
given by,
In symbols,
𝑓2𝑛 =
1
2𝑛 − 1
𝑢2𝑛
𝑢2𝑛 be the probability that the 2𝑛th trial
takes the particle to the initial position
31. The ballot theorem
Basic Results
Let n and x be positive integers. There are exactly
𝑥
𝑛
𝑁𝑛,𝑥 paths
(𝑠1, … … 𝑠𝑛 = 𝑥) from the origin to the point (n,x) such that
𝑠1>0,…. 𝑠1>0.
The theorem has received its name from the classical probability
problem: Ballot problem.
32. The ballot problem
In an election candidate A secures a votes and candidate B secures b votes.
What is the probability that A is ahead throughout the count?
P(A is ahead throughout the count)=
𝑁𝑎+𝑏
≠0
(0,𝑎−𝑏)
𝑁𝑎+𝑏(0,𝑎−𝑏)
=
𝑎−𝑏
𝑎+𝑏
33. Example:
If there are 5 voters, of whom 3 vote for candidate A
and 2 vote for candidate B, the probability that A will
always be in lead is,
1
5
34. Generating functions And Random walks
Recall, that if (𝑋𝑖) is independent and identically distributed, then
𝑆𝑛 = 𝑆0 +
𝑖=0
𝑛
𝑋𝑖
is a random walk
35. What is Generating functions?
• A generating function is an infinite series that has the ability to be
summed.
• These series act as filling cabinet where the information stored within
can be seen on the coefficient on 𝑧𝑛
𝐺 𝑎𝑛; 𝑧 =
𝑛=0
∞
𝑎𝑛𝑧𝑛
36. Probability Generating functions?
• A probability mass function is a list of probabilities for each possible
value the random variable can take on.
• These values can be extracted from special type of generating function
known as the probability generating function (p.g.f)
• The p.g.f is where the value of the coefficients on 𝑧𝑛
are the
probability the random variable takes the wanted value n.
38. Generating functions for random walk
If X has p.g.f G(z) then we have (when S0=0)
Gn(z)=E(𝑧𝑆𝑛)=(G(z))n
We can define the function H by
𝐻 𝑧, 𝑤 =
𝑛=0
∞
𝑤𝑛𝐺𝑛 𝑧 = (1 − 𝑤𝐺(𝑧))−1
This bivariate generating function tells everything about Sn
as P(Sn)=r is the coefficient of zrwn in 𝐻 𝑧, 𝑤
39. Further Reading
1. Elementary Probability, David Stirzaker,
Cambridge University Press.
2. Introduction to Probability Theory and Its
Applications, W. Feller (2nd Edn.), John
Wiley and Sons.