The document discusses recognizing sparse perfect elimination bipartite graphs. It begins with an example of Gaussian elimination on a matrix that introduces new non-zero values. The key points are that perfect elimination bipartite graphs correspond to matrices that can be eliminated without creating new non-zeros, and this can be achieved by finding a sequence of bisimplicial edges in the corresponding bipartite graph. The document proposes using bisimplicial edges as pivots during elimination to avoid introducing new non-zeros.
Calculo y geometria analitica (larson hostetler-edwards) 8th ed - solutions m...ELMIR IVAN OZUNA LOPEZ
This document contains the solutions to odd-numbered exercises from Chapter P of a calculus textbook. It provides answers and work for 43 problems involving graphing functions, finding intercepts, determining symmetries, and other skills related to functions and their graphs. The problems progress from simple linear functions to more complex expressions involving square roots, cubes, and other operations.
This document contains mathematical formulas and definitions related to quantum mechanics, spin, angular momentum, spherical harmonics, and other physics concepts. It includes expressions for spin operators, spherical harmonics, raising and lowering operators, eigenstates of the quantum harmonic oscillator, and Legendre polynomials. Operators, matrices, eigenstates, and other quantum mechanical notation are used throughout.
This document contains mathematical expressions and definitions related to quantum mechanics, angular momentum, spherical harmonics, and other physics concepts. It includes:
1) Definitions of the Pauli spin matrices and their properties.
2) Expressions for spin operators and raising/lowering operators.
3) Spherical harmonic functions and their relationships to angular momentum quantum numbers.
4) Operators for the quantum harmonic oscillator and their eigenstates.
5) Additional equations for spin, magnetic fields, and other quantum mechanical systems.
The document describes methods for image registration and transformation. It discusses several transformation models (Tθ) and metrics (M) to measure misalignment. It defines compositions of multiple transformations and how to calculate the dice similarity between images after applying sequential transformations. Finally, it shows graphs comparing dice similarity for different numbers of atlas images and target images.
Bayesian Inference on a Stochastic Volatility model Using PMCMC methodspaperbags
This document summarizes a Bayesian inference method called particle Markov chain Monte Carlo (PMCMC) for estimating parameters in a stochastic volatility model of financial time series. PMCMC combines sequential Monte Carlo (SMC) methods and Markov chain Monte Carlo (MCMC) to sample the parameter posterior distribution. SMC is used to estimate likelihoods and simulate state trajectories, while MCMC proposals are accepted or rejected based on a Metropolis-Hastings ratio involving the estimated likelihoods. The document outlines the stochastic volatility model, parameter estimation using Gibbs sampling, SMC methods for simulation and filtering, and the particle MCMC algorithm for joint simulation of parameters and states.
The document describes a teaching presentation on displaying fractions in diagrams. It provides instructions on inserting or changing numerators and denominators to show equivalent, simplified, improper, and mixed numbers. Users must follow limitations such as entering only whole numbers between 1-10 as denominators to avoid errors in the diagrams. The goal is to demonstrate different fraction types visually through an interactive fraction diagram tool.
Diane Watson | Research to improve public confidence and views on quality in ...Sax Institute
Dr Diane Watson, (then International Visiting Health Services Research Fellow at the Sax Institute from the University of British Columbia, Canada) spoke with the HARC network in April 2009 about ways to strengthen public confidence in the hospital system through research and analysis.
HARC stands for the Hospital Alliance for Research Collaboration. HARC is a collaborative network of researchers, health managers, clinicians and policy makers based in NSW, Australia managed by the Sax Institute.
HARC Forums bring members of the HARC network together to discuss the latest research and analysis about important issues facing our hospitals.
For more information visit saxinstitute.org.au.
QUALCOMM had a record year in 2004 with increased revenue, earnings, and operating cash flows due to growing adoption of 3G CDMA technology and advanced devices. Key highlights include:
- CDMA2000 and WCDMA 3G networks expanded significantly worldwide, driving strong demand for QUALCOMM's chipsets. QUALCOMM shipped over 137 million chipsets in fiscal year 2004, more than doubling the prior year's shipments.
- Mobile data usage increased as high-speed 3G networks and BREW-enabled devices enabled new multimedia services. Over 200 million BREW applications have been downloaded.
- South Korea and Japan led the rollout of 1xEV-DO wireless broadband networks, achieving over 10
Calculo y geometria analitica (larson hostetler-edwards) 8th ed - solutions m...ELMIR IVAN OZUNA LOPEZ
This document contains the solutions to odd-numbered exercises from Chapter P of a calculus textbook. It provides answers and work for 43 problems involving graphing functions, finding intercepts, determining symmetries, and other skills related to functions and their graphs. The problems progress from simple linear functions to more complex expressions involving square roots, cubes, and other operations.
This document contains mathematical formulas and definitions related to quantum mechanics, spin, angular momentum, spherical harmonics, and other physics concepts. It includes expressions for spin operators, spherical harmonics, raising and lowering operators, eigenstates of the quantum harmonic oscillator, and Legendre polynomials. Operators, matrices, eigenstates, and other quantum mechanical notation are used throughout.
This document contains mathematical expressions and definitions related to quantum mechanics, angular momentum, spherical harmonics, and other physics concepts. It includes:
1) Definitions of the Pauli spin matrices and their properties.
2) Expressions for spin operators and raising/lowering operators.
3) Spherical harmonic functions and their relationships to angular momentum quantum numbers.
4) Operators for the quantum harmonic oscillator and their eigenstates.
5) Additional equations for spin, magnetic fields, and other quantum mechanical systems.
The document describes methods for image registration and transformation. It discusses several transformation models (Tθ) and metrics (M) to measure misalignment. It defines compositions of multiple transformations and how to calculate the dice similarity between images after applying sequential transformations. Finally, it shows graphs comparing dice similarity for different numbers of atlas images and target images.
Bayesian Inference on a Stochastic Volatility model Using PMCMC methodspaperbags
This document summarizes a Bayesian inference method called particle Markov chain Monte Carlo (PMCMC) for estimating parameters in a stochastic volatility model of financial time series. PMCMC combines sequential Monte Carlo (SMC) methods and Markov chain Monte Carlo (MCMC) to sample the parameter posterior distribution. SMC is used to estimate likelihoods and simulate state trajectories, while MCMC proposals are accepted or rejected based on a Metropolis-Hastings ratio involving the estimated likelihoods. The document outlines the stochastic volatility model, parameter estimation using Gibbs sampling, SMC methods for simulation and filtering, and the particle MCMC algorithm for joint simulation of parameters and states.
The document describes a teaching presentation on displaying fractions in diagrams. It provides instructions on inserting or changing numerators and denominators to show equivalent, simplified, improper, and mixed numbers. Users must follow limitations such as entering only whole numbers between 1-10 as denominators to avoid errors in the diagrams. The goal is to demonstrate different fraction types visually through an interactive fraction diagram tool.
Diane Watson | Research to improve public confidence and views on quality in ...Sax Institute
Dr Diane Watson, (then International Visiting Health Services Research Fellow at the Sax Institute from the University of British Columbia, Canada) spoke with the HARC network in April 2009 about ways to strengthen public confidence in the hospital system through research and analysis.
HARC stands for the Hospital Alliance for Research Collaboration. HARC is a collaborative network of researchers, health managers, clinicians and policy makers based in NSW, Australia managed by the Sax Institute.
HARC Forums bring members of the HARC network together to discuss the latest research and analysis about important issues facing our hospitals.
For more information visit saxinstitute.org.au.
QUALCOMM had a record year in 2004 with increased revenue, earnings, and operating cash flows due to growing adoption of 3G CDMA technology and advanced devices. Key highlights include:
- CDMA2000 and WCDMA 3G networks expanded significantly worldwide, driving strong demand for QUALCOMM's chipsets. QUALCOMM shipped over 137 million chipsets in fiscal year 2004, more than doubling the prior year's shipments.
- Mobile data usage increased as high-speed 3G networks and BREW-enabled devices enabled new multimedia services. Over 200 million BREW applications have been downloaded.
- South Korea and Japan led the rollout of 1xEV-DO wireless broadband networks, achieving over 10
The document discusses recognizing sparse perfect elimination bipartite graphs through matrix elimination. It provides an example of Gaussian elimination on a matrix that introduces new non-zero values. The key points are:
- Perfect elimination bipartite graphs correspond to matrices that allow elimination without creating new non-zeros.
- Existing algorithms have time complexity of O(n^5) or O(n^3/log n) but may produce dense matrices from sparse ones.
- A new algorithm is proposed that avoids this issue by working directly with the sparse matrix structure.
The document presents a polynomial-time algorithm for finding a minimal conflicting set of rows (MCSR) in a binary matrix that contains a given row. It defines MCSR as a set of rows that does not have the consecutive ones property but where any proper subset does have the property. The algorithm works by representing the binary matrix as a vertex-colored bipartite graph and detecting forbidden substructures called Tucker configurations that characterize when the consecutive ones property does not hold. It finds an MCSR containing the given row by pruning rows from the graph until a Tucker configuration exists using the current set but not with any proper subset.
In most of the algorithms analyzed until now, we have been looking and studying problems solvable in polynomial time. The polynomial time algorithm class P are algorithms that on inputs of size n have a worst case running time of O(n^k) for some constant k. Thus, informally, we can say that the Non-Polynomial (NP) time algorithms are the ones that cannot be solved in O(n^k) for any constant k
.
Demonstration of cancelling down a fraction to its lowest terms. Single step examples using the 3 and 7 times table. How to cancel fractions with larger numbers in several steps. How to use the prime factors of the numerator and denominator to cancel large fractions.
This document discusses algorithmic game theory and its applications to internet markets. It provides historical context on market equilibrium theory starting from Adam Smith and outlines some classic models like those of Fisher, Arrow, and Debreu. It then discusses how these economic theories provide mathematical foundations for modeling new internet-based markets and computational problems in auction algorithms. Specific examples discussed include the AdWords auction market and algorithms for maximizing revenue in these markets.
The document discusses computational models for algebraic decision trees and algebraic computation trees over a ground field F. It describes how algebraic decision trees use polynomials of degree ≤ d to branch at each node, while algebraic computation trees allow testing polynomials to be calculated from previous polynomials along the path. The document then covers existing lower bounds on the complexity C(S) of the membership problem for a set S in terms of topological invariants of S, such as the number of connected components, Euler characteristic, and sum of Betti numbers.
1. While PowerPoint may seem simple, there is actually a lot to learn about how to create truly great presentations.
2. Many presentations are ineffective because firms do not utilize PowerPoint's full potential or properly train their staff.
3. External training is often too general or expensive for companies.
1. While PowerPoint may seem simple, there is actually a lot to learn about how to create truly great presentations.
2. Many presentations are ineffective because firms do not utilize PowerPoint's full potential or properly train their staff.
3. External training is often too general or expensive for companies.
This document contains a grading table for a Mechanics course at the Universidad Nacional Experimental "Francisco de Miranda". It lists the names and identification numbers of 71 students, along with their grades on tests, exams, and other evaluations throughout the course. The professor, Ing. Joan Gil, teaches the course in the Structures Department of the Civil Engineering program.
Amth250 octave matlab some solutions (3)asghar123456
The document contains solutions to 6 questions on interpolation and curve fitting.
Question 1 estimates life expectancies in 1977, 1983 and 1988 for two countries using polynomial interpolation and cubic spline interpolation.
Question 2 finds an interpolating function that fits given data points by solving a linear system.
Question 3 compares errors between polynomial, cubic spline and pchip cubic interpolation on a dataset and analyzes properties of cubic spline and pchip interpolants.
Question 4 plots a cubic spline interpolant and its derivatives, showing it satisfies properties of being cubic on subintervals with continuous derivatives.
Question 5 uses linear least squares to fit a linear model to some data.
Question 6 fits quadratic, exponential and power
The Origin of Diversity - Thinking with Chaotic WalkTakashi Iba
We will show that diverse complex patterns can emerge even in the universe governed by deterministic laws. See the details of this study on our paper: Iba, T. & Shimonishi, K. (2011), "The Origin of Diversity: Thinking with Chaotic Walk," in Unifying Themes in Complex Systems Volume VIII: Proceedings of the Eighth International Conference on Complex Systems, New England Complex Systems Institute Series on Complexity (Sayama, H., Minai, A. A., Braha, D. and Bar-Yam, Y. eds., NECSI Knowledge Press, 2011), pp.447-461.
Conversor nº binarios a decimales y viceversa 2Jaime914
This document contains a table with numerical values in both decimal and binary formats. The table has four columns listing the decimal value, 18-bit binary representation, and indicators for whether the number is positive or negative. It provides conversions between decimal and binary numbers up to the value 262,143.
This document contains a class roster from the Mechanical Rationality program at the Francisco de Miranda National Experimental University. It lists the names of 27 students along with their identification numbers and grades on tests, assignments, and final exams throughout the semester. The grades are presented as percentages and determine the students' final letter grades for the course.
The document discusses the method of multiplicities, which is a technique for combinatorics using algebra. It involves finding a polynomial that vanishes on a set with high multiplicity. This is applied to problems in list decoding of Reed-Solomon codes, bounding the size of Kakeya sets, and constructing randomness extractors. Specifically, the method is used to improve bounds on list decoding, show that certain Kakeya sets must be large, and allow extraction of more randomness from weak sources. Propagating multiplicities of derivatives allows tighter analysis of these problems.
The document summarizes research on multiple-conclusion calculi for first-order Gödel logic. It introduces Gödel logic and describes its semantics using both many-valued semantics based on truth values in the interval [0,1] and Kripke-style semantics. It then outlines proof theory for Gödel logic, including early sequent calculi and more recent hypersequent calculi. The hypersequent calculus introduced in 1991 uses standard rules and has been extended to the first-order case. The document provides details on the structural and logical rules of this single-conclusion hypersequent system.
The document summarizes a talk on polynomial identity testing (PIT). PIT is the problem of determining if a polynomial computed by an arithmetic circuit is identical to the zero polynomial. The talk outlines the definition of PIT, its connection to circuit lower bounds, and surveys positive results for restricted circuit classes. It also provides examples of proof techniques for PIT on depth-3 and depth-4 circuits and discusses the relationship between PIT and polynomial factorization.
This document summarizes an algorithm for maximizing throughput in online scheduling of equal length jobs. The algorithm aims to schedule incoming jobs with the goal of maximizing total value of completed jobs by their deadlines. It uses a charging scheme and potential function to prove it is (2+√5)-competitive, an improvement over prior algorithms. The algorithm handles jobs arriving online with weights, processing times, deadlines, and considers models where preemption allows restarting or resuming previously completed work. Open questions remain around settling the exact competitive ratio and developing new algorithmic methods.
The document discusses efficient algorithms for performing approximate matching queries on strings that have been grammar-compressed. It introduces the concept of implicit unit-Monge matrices which can represent permutation matrices in a space-efficient way using a range tree data structure. This representation allows dominance counting queries, needed for string comparison, to be performed in O(log2 n) time after an O(n log n) preprocessing step. More advanced data structures can improve these asymptotic time and space bounds further.
This document presents an overview of the consensus problem from an informal and formal perspective. It discusses how consensus requires representativity, where the decision reflects a sufficient number of individual opinions, and stability, where the decision is robust to individual opinion variations. It also presents some key formalizations, including defining consensus as a function from the set of sensor inputs and memory states to decisions. It introduces the concept of a geodesic to measure stability as the maximum number of state transitions needed to return to the starting configuration along a trajectory where each sensor changes at most once.
This document summarizes research on the combinatorial properties of Burrows-Wheeler Transforms (BWT). It discusses prior work that characterized words with simple BWT image forms. It also introduces two general decision problems about BWT images and claims to provide efficient solutions to these problems. Specifically, it presents a theorem providing a criterion to check whether a given word is a valid BWT image based on analyzing the number of orbits in the word's stable sorting.
The document discusses locally decodable codes, which allow recovery of individual data symbols from a coded data set even after erasures. Reed-Muller codes and multiplicity codes were early constructions that provided locality but only up to a rate of 0.5. Matching vector codes were later introduced and can achieve locality r for codes of positive rate and length n=O(r^2). However, the optimal tradeoff between rate, length, and locality remains an open problem.
The document discusses recognizing sparse perfect elimination bipartite graphs through matrix elimination. It provides an example of Gaussian elimination on a matrix that introduces new non-zero values. The key points are:
- Perfect elimination bipartite graphs correspond to matrices that allow elimination without creating new non-zeros.
- Existing algorithms have time complexity of O(n^5) or O(n^3/log n) but may produce dense matrices from sparse ones.
- A new algorithm is proposed that avoids this issue by working directly with the sparse matrix structure.
The document presents a polynomial-time algorithm for finding a minimal conflicting set of rows (MCSR) in a binary matrix that contains a given row. It defines MCSR as a set of rows that does not have the consecutive ones property but where any proper subset does have the property. The algorithm works by representing the binary matrix as a vertex-colored bipartite graph and detecting forbidden substructures called Tucker configurations that characterize when the consecutive ones property does not hold. It finds an MCSR containing the given row by pruning rows from the graph until a Tucker configuration exists using the current set but not with any proper subset.
In most of the algorithms analyzed until now, we have been looking and studying problems solvable in polynomial time. The polynomial time algorithm class P are algorithms that on inputs of size n have a worst case running time of O(n^k) for some constant k. Thus, informally, we can say that the Non-Polynomial (NP) time algorithms are the ones that cannot be solved in O(n^k) for any constant k
.
Demonstration of cancelling down a fraction to its lowest terms. Single step examples using the 3 and 7 times table. How to cancel fractions with larger numbers in several steps. How to use the prime factors of the numerator and denominator to cancel large fractions.
This document discusses algorithmic game theory and its applications to internet markets. It provides historical context on market equilibrium theory starting from Adam Smith and outlines some classic models like those of Fisher, Arrow, and Debreu. It then discusses how these economic theories provide mathematical foundations for modeling new internet-based markets and computational problems in auction algorithms. Specific examples discussed include the AdWords auction market and algorithms for maximizing revenue in these markets.
The document discusses computational models for algebraic decision trees and algebraic computation trees over a ground field F. It describes how algebraic decision trees use polynomials of degree ≤ d to branch at each node, while algebraic computation trees allow testing polynomials to be calculated from previous polynomials along the path. The document then covers existing lower bounds on the complexity C(S) of the membership problem for a set S in terms of topological invariants of S, such as the number of connected components, Euler characteristic, and sum of Betti numbers.
1. While PowerPoint may seem simple, there is actually a lot to learn about how to create truly great presentations.
2. Many presentations are ineffective because firms do not utilize PowerPoint's full potential or properly train their staff.
3. External training is often too general or expensive for companies.
1. While PowerPoint may seem simple, there is actually a lot to learn about how to create truly great presentations.
2. Many presentations are ineffective because firms do not utilize PowerPoint's full potential or properly train their staff.
3. External training is often too general or expensive for companies.
This document contains a grading table for a Mechanics course at the Universidad Nacional Experimental "Francisco de Miranda". It lists the names and identification numbers of 71 students, along with their grades on tests, exams, and other evaluations throughout the course. The professor, Ing. Joan Gil, teaches the course in the Structures Department of the Civil Engineering program.
Amth250 octave matlab some solutions (3)asghar123456
The document contains solutions to 6 questions on interpolation and curve fitting.
Question 1 estimates life expectancies in 1977, 1983 and 1988 for two countries using polynomial interpolation and cubic spline interpolation.
Question 2 finds an interpolating function that fits given data points by solving a linear system.
Question 3 compares errors between polynomial, cubic spline and pchip cubic interpolation on a dataset and analyzes properties of cubic spline and pchip interpolants.
Question 4 plots a cubic spline interpolant and its derivatives, showing it satisfies properties of being cubic on subintervals with continuous derivatives.
Question 5 uses linear least squares to fit a linear model to some data.
Question 6 fits quadratic, exponential and power
The Origin of Diversity - Thinking with Chaotic WalkTakashi Iba
We will show that diverse complex patterns can emerge even in the universe governed by deterministic laws. See the details of this study on our paper: Iba, T. & Shimonishi, K. (2011), "The Origin of Diversity: Thinking with Chaotic Walk," in Unifying Themes in Complex Systems Volume VIII: Proceedings of the Eighth International Conference on Complex Systems, New England Complex Systems Institute Series on Complexity (Sayama, H., Minai, A. A., Braha, D. and Bar-Yam, Y. eds., NECSI Knowledge Press, 2011), pp.447-461.
Conversor nº binarios a decimales y viceversa 2Jaime914
This document contains a table with numerical values in both decimal and binary formats. The table has four columns listing the decimal value, 18-bit binary representation, and indicators for whether the number is positive or negative. It provides conversions between decimal and binary numbers up to the value 262,143.
This document contains a class roster from the Mechanical Rationality program at the Francisco de Miranda National Experimental University. It lists the names of 27 students along with their identification numbers and grades on tests, assignments, and final exams throughout the semester. The grades are presented as percentages and determine the students' final letter grades for the course.
The document discusses the method of multiplicities, which is a technique for combinatorics using algebra. It involves finding a polynomial that vanishes on a set with high multiplicity. This is applied to problems in list decoding of Reed-Solomon codes, bounding the size of Kakeya sets, and constructing randomness extractors. Specifically, the method is used to improve bounds on list decoding, show that certain Kakeya sets must be large, and allow extraction of more randomness from weak sources. Propagating multiplicities of derivatives allows tighter analysis of these problems.
The document summarizes research on multiple-conclusion calculi for first-order Gödel logic. It introduces Gödel logic and describes its semantics using both many-valued semantics based on truth values in the interval [0,1] and Kripke-style semantics. It then outlines proof theory for Gödel logic, including early sequent calculi and more recent hypersequent calculi. The hypersequent calculus introduced in 1991 uses standard rules and has been extended to the first-order case. The document provides details on the structural and logical rules of this single-conclusion hypersequent system.
The document summarizes a talk on polynomial identity testing (PIT). PIT is the problem of determining if a polynomial computed by an arithmetic circuit is identical to the zero polynomial. The talk outlines the definition of PIT, its connection to circuit lower bounds, and surveys positive results for restricted circuit classes. It also provides examples of proof techniques for PIT on depth-3 and depth-4 circuits and discusses the relationship between PIT and polynomial factorization.
This document summarizes an algorithm for maximizing throughput in online scheduling of equal length jobs. The algorithm aims to schedule incoming jobs with the goal of maximizing total value of completed jobs by their deadlines. It uses a charging scheme and potential function to prove it is (2+√5)-competitive, an improvement over prior algorithms. The algorithm handles jobs arriving online with weights, processing times, deadlines, and considers models where preemption allows restarting or resuming previously completed work. Open questions remain around settling the exact competitive ratio and developing new algorithmic methods.
The document discusses efficient algorithms for performing approximate matching queries on strings that have been grammar-compressed. It introduces the concept of implicit unit-Monge matrices which can represent permutation matrices in a space-efficient way using a range tree data structure. This representation allows dominance counting queries, needed for string comparison, to be performed in O(log2 n) time after an O(n log n) preprocessing step. More advanced data structures can improve these asymptotic time and space bounds further.
This document presents an overview of the consensus problem from an informal and formal perspective. It discusses how consensus requires representativity, where the decision reflects a sufficient number of individual opinions, and stability, where the decision is robust to individual opinion variations. It also presents some key formalizations, including defining consensus as a function from the set of sensor inputs and memory states to decisions. It introduces the concept of a geodesic to measure stability as the maximum number of state transitions needed to return to the starting configuration along a trajectory where each sensor changes at most once.
This document summarizes research on the combinatorial properties of Burrows-Wheeler Transforms (BWT). It discusses prior work that characterized words with simple BWT image forms. It also introduces two general decision problems about BWT images and claims to provide efficient solutions to these problems. Specifically, it presents a theorem providing a criterion to check whether a given word is a valid BWT image based on analyzing the number of orbits in the word's stable sorting.
The document discusses locally decodable codes, which allow recovery of individual data symbols from a coded data set even after erasures. Reed-Muller codes and multiplicity codes were early constructions that provided locality but only up to a rate of 0.5. Matching vector codes were later introduced and can achieve locality r for codes of positive rate and length n=O(r^2). However, the optimal tradeoff between rate, length, and locality remains an open problem.
The document discusses locally decodable codes, which allow recovery of individual data symbols from a coded data set even after erasures. Reed-Muller codes and multiplicity codes were early constructions that provided locality but only up to a rate of 0.5. Matching vector codes were later introduced and can achieve locality r for codes of positive rate and length n=O(r^2). However, the optimal tradeoff between rate, length, and locality remains an open problem.
This document discusses the relationships between orbits of linear maps and regular languages. It shows that the chamber hitting problem (CHP) and permutation filter realizability problem are Turing equivalent. It also shows that the injective filter and surjective filter realizability problems are decidable by reducing them to problems about orbits. However, the regular realizability problem for the track product of the periodic and permutation filters is undecidable, as it can reduce the undecidable zero in the upper right corner problem.
The document summarizes precedence automata and languages. It provides historical background on operator precedence grammars and Floyd languages. It then discusses how precedence parsing works using an example arithmetic expression. Key points include using a precedence table to determine parentheses insertion and defining three types of moves for an automata model based on symbol precedence: push, mark, and flush. The example demonstrates the automata processing a Dyck language expression.
The document discusses the constraint satisfaction problem (CSP) and the dichotomy conjecture regarding the complexity of CSP instances. It provides definitions and examples of CSPs. It explains the role of polymorphisms in determining the complexity, identifying semilattice, majority and affine polymorphisms as "good". It outlines the dichotomy conjecture that CSPs are either solvable in polynomial time or NP-complete depending on the presence of certain types of local structure defined by polymorphisms. The document also discusses algorithms and results for various constraint languages.
This document describes a Synchronized Alternating Pushdown Automaton (SAPDA) that accepts the language of reduplication with a center marker (RCM). The SAPDA utilizes recursive conjunctive transitions to check that the nth letter before the center marker '$' is the same as the nth letter from the end of the string, for all letters n. This allows the SAPDA to accept strings of the form w$w, where w is any string over the alphabet {a,b}. The construction of the SAPDA involves states that check specific letters at specific positions relative to the center marker.
The document discusses the constraint satisfaction problem (CSP) and the dichotomy conjecture in computational complexity theory. It defines CSP and provides examples. It discusses the role of polymorphisms - operations that preserve constraints. The presence or absence of certain polymorphisms like semilattice, majority, and affine operations determines the complexity of CSP for a given constraint language. The document outlines a proposed dichotomy - CSP is either solvable in polynomial time or NP-complete, depending on the polymorphisms. It surveys partial results proving this conjecture and algorithms for certain constraint languages.
The document discusses shared-memory systems and charts. It provides definitions and concepts related to modeling shared-memory concurrency using partial orders of events called pomsets. Specifically, it defines:
- Shared-memory systems as consisting of registers, data, processes, actions, and rules for updating configurations.
- Pomsets as labeled partial orders used to model executions.
- The may-occur-concurrently relation for rules in a shared-memory system.
- Partial-order semantics for runs of pomsets in a shared-memory system.
- Shared-memory charts (SMCs) as pomsets with gates used to model specifications.
This document discusses the relationships between orbits of linear maps and regular languages. It shows that the chamber hitting problem (CHP) and permutation filter-realizability problem are Turing equivalent. It also shows that the injective filter-realizability problem and surjective filter-realizability problem are decidable, while the track product of the periodic and permutation filter-realizability problem is undecidable. The zero in the upper right corner problem, which is undecidable, can be reduced to the latter regular realizability problem.
The document discusses precedence automata and languages. It provides historical background on operator precedence grammars and related families of languages. As an example, it explains how parsing an arithmetic expression like 4+5×6 works according to an implicit context-free grammar and by respecting the precedence of operators. It introduces the concept of a precedence table to determine the admissible parentheses generators between pairs of symbols in a grammar.
Locally decodable codes allow recovery of individual data symbols even after data loss by accessing only a small number of codeword symbols. Reed-Muller codes provide locality but only up to a rate of 0.5, while multiplicity codes achieve higher rates but have weaker locality guarantees. Matching vector codes can match the best known locality bounds, constructing codes of length n with locality r for constant r, but the optimal tradeoff between rate, length and locality remains an open problem.
This document summarizes research on the combinatorial properties of Burrows-Wheeler Transforms (BWT). It discusses prior work that characterized words with simple BWT image forms. It also defines two general decision problems regarding whether a word is a valid BWT image or can form a specific BWT image pattern. The authors then present efficient solutions to these two problems, including a theorem providing a criterion for determining if a word is a BWT image based on the number of orbits in its stable sorting.
8. Recognizing Sparse Perfect Elimination Bipartite Graphs
Introduction and Motivation
Simplification
‘Regularity’ assumption: If we add some multiple of row i to row j,
at most one non-zero value is turned into a zero.
9. Recognizing Sparse Perfect Elimination Bipartite Graphs
Introduction and Motivation
Simplification
‘Regularity’ assumption: If we add some multiple of row i to row j,
at most one non-zero value is turned into a zero.
Exact values are not important
A problem instance is a n × n (0, 1)-matrix M (with m
non-zeroes, n ≤ m ≤ n2 ):
1 1 1 1
1 1 0 0
1 0 1 0
1 0 0 1
10. Recognizing Sparse Perfect Elimination Bipartite Graphs
Introduction and Motivation
Simplification
‘Regularity’ assumption: If we add some multiple of row i to row j,
at most one non-zero value is turned into a zero.
Exact values are not important
A problem instance is a n × n (0, 1)-matrix M (with m
non-zeroes, n ≤ m ≤ n2 ):
1 1 1 1
1 1 0 0
1 0 1 0
1 0 0 1
. . . or an equivalent bipartite graph GM (with m edges):
r1 r2 r3 r4
c1 c2 c3 c4
11. Recognizing Sparse Perfect Elimination Bipartite Graphs
Introduction and Motivation
Suitable Pivots
Remark
A pivot (i, j) (with Mi,j = 1) does not create additional
non-zeroes, if for every i , j we have that if Mi,j = 1 and
Mi ,j = 1, then Mi ,j = 1.
1 1 1 0
0 1 1 0
1 1 0 1
1 1 1 0
If we can find a sequence of n such pivots in distinct rows and
columns, we can perform elimination without creating new
non-zeroes.
12. Recognizing Sparse Perfect Elimination Bipartite Graphs
Introduction and Motivation
Bisimplicial Edges
Definition
An edge e of a bipartite graph G is called bisimplicial if the
neighbors of the vertices incident to it induce a complete bipartite
subgraph.
r1 r2 r3 r4 r1 r2 r3 r4
c1 c2 c3 c4 c1 c2 c3 c4
Bisimplicial edges in GM correspond to pivots that avoid new
non-zeroes in M.
13. Recognizing Sparse Perfect Elimination Bipartite Graphs
Perfect Elimination Bipartite Graphs
Perfect Elimination Bipartite Graphs
Definition
(Golumbic and Goss, (1978,1980)) A graph G is called perfect
elimination bipartite if there exists a sequence of edges
[e1 , e2 , . . . , en ] such that:
1 e1 is a bisimplicial edge in G and ei is bisimplicial in
G − [e1 , . . . , ei−1 ] for 2 ≤ i ≤ n;
2 G − [e1 , e2 , . . . , en ] is empty.
Perfect elimination bipartite graphs correspond to matrices
that allow elimination without creating new non-zeroes.
Naive algorithm for recognition: O n5
14. Recognizing Sparse Perfect Elimination Bipartite Graphs
Perfect Elimination Bipartite Graphs
A faster algorithm
Remark
(Goh and Rotem (1982)) Consider the matrix Q = MM T : Qi,j
contains the inner product of rows Mi,∗ and Mj,∗ . Let li equal the
number of elements in row Qi,∗ with value equal to Qi,i . Denote
by sj the column sums in M. (i, j) is bisimplicial in GM iff Mi,j = 1
and li = sj .
This leads to a O n3 algorithm
Spinrad (2004) subsequently improves this to O n3 / log n
15. Recognizing Sparse Perfect Elimination Bipartite Graphs
Perfect Elimination Bipartite Graphs
A faster algorithm
Remark
(Goh and Rotem (1982)) Consider the matrix Q = MM T : Qi,j
contains the inner product of rows Mi,∗ and Mj,∗ . Let li equal the
number of elements in row Qi,∗ with value equal to Qi,i . Denote
by sj the column sums in M. (i, j) is bisimplicial in GM iff Mi,j = 1
and li = sj .
This leads to a O n3 algorithm
Spinrad (2004) subsequently improves this to O n3 / log n
Unfortunately, a sparse M may lead to a dense Q:
1 1 1 1 1 1 1 1 4 1 1 1
1 0 0 0 1 0 0 0 1 1 1 1
1 0 0 0 × 1 0 0 0 = 1 1 1 1
1 0 0 0 1 0 0 0 1 1 1 1
16. Recognizing Sparse Perfect Elimination Bipartite Graphs
Perfect Elimination Bipartite Graphs
Summary so far
Matrices that allow elimination without new non-zeroes
correspond to perfect elimination bipartite graphs
Recognition algorithms (time complexity):
naive: O n5
based on matrix-multiplication: O n3 / log n
17. Recognizing Sparse Perfect Elimination Bipartite Graphs
Perfect Elimination Bipartite Graphs
Summary so far
Matrices that allow elimination without new non-zeroes
correspond to perfect elimination bipartite graphs
Recognition algorithms (time complexity):
naive: O n5
based on matrix-multiplication: O n3 / log n
However, the result of matrix-multiplication may be a dense
matrix, while avoiding new non-zeroes is mainly useful for
sparse matrices. . .
25. Recognizing Sparse Perfect Elimination Bipartite Graphs
A New Recognition Algorithm
Algorithm Outline
1 1 1 0
0 1 1 0
1 1 1
1 1 0 1 0 1 1
1 1 1 0 1 1 1
Up to n iterations (one for each pivot)
Each iteration, for every edge:
Continue checking other edges until blocked
If we finish checking other edges, we found a pivot
If all edges block, there is no pivot
26. Recognizing Sparse Perfect Elimination Bipartite Graphs
A New Recognition Algorithm
Implementation Details
Rows/columns of M stored as lists of column/row numbers
Consider a single edge over the entire algorithm:
1 1
1 1 step: O (m)
0 1 2
1 0
3 3 block: O (n)
27. Recognizing Sparse Perfect Elimination Bipartite Graphs
A New Recognition Algorithm
Implementation Details
Rows/columns of M stored as lists of column/row numbers
Consider a single edge over the entire algorithm:
1 1
1 1 step: O (m)
0 1 2
1 0
3 3 block: O (n)
Total work for a single edge: O (m) steps and O (n) blocks
For each pivot, we update all list items: O (m)
28. Recognizing Sparse Perfect Elimination Bipartite Graphs
A New Recognition Algorithm
Time and Space Complexities
Time complexity: O m2
initialization: O n2
steps: O m2
blocks: O (nm)
updates: O (nm)
Space complexity: O (m)
lists: O (m)
edge states: O (m)
row/column data: O (n)
pivots: O (n)
29. Recognizing Sparse Perfect Elimination Bipartite Graphs
Conclusion
Conclusion
Existing literature: focus on time complexity
However: space complexity is important in practice
Our new algorithm:
O m2 time
O (m) space
Both time and space improvement for sparse M
m < n n/ log n
30. Recognizing Sparse Perfect Elimination Bipartite Graphs
Conclusion
Conclusion
Existing literature: focus on time complexity
However: space complexity is important in practice
Our new algorithm:
O m2 time
O (m) space
Both time and space improvement for sparse M
m < n n/ log n
Work in progress:
Thinking about possible further time complexity improvements
Work on alternative elimination procedures