Fullerenes are 3-valent plane graphs with faces of size 5 or 6. A space fullerene is a tiling of Euclidean space with fullerene tiles. The space fullerenes occur in metallurgy, bubble foams, and the solution of the Kelvin problem. Here we present enumeration techniques that allows to find many new space fullerenes.
CAPE PURE MATHEMATICS UNIT 2 MODULE 1 PRACTICE QUESTIONSCarlon Baird
dy/dx = (x - 3y)/(6x - 4)
The stationary points on the curve C occur when tan(x) = 2.
The equation of the tangent to C at the point where x=0 is y = 2ex.
This chapter introduces complex numbers. It defines a complex number as having the form x + iy, where x and y are real numbers. It describes how to represent complex numbers graphically on an Argand diagram and defines the modulus and argument of a complex number. It explains how to perform arithmetic operations like addition, subtraction, multiplication and division on complex numbers in both Cartesian (x + iy) and polar forms. It also introduces concepts like the conjugate of a complex number and using real and imaginary parts to solve equations. The chapter aims to explain the basic properties and manipulations of complex numbers.
CAPE PURE MATHEMATICS UNIT 2 MODULE 2 PRACTICE QUESTIONSCarlon Baird
This document contains practice questions on sequences, series, and approximations from a CAPE Pure Mathematics unit. Question 1 covers finding terms of sequences defined recursively and evaluating finite sums. Question 2 involves finding expressions for terms of sequences defined recursively and finding their sums. Later questions cover topics like proving identities using induction, evaluating infinite series, approximating functions using Taylor series, and finding roots of equations numerically. The questions provide worked examples of key concepts in sequences, series, and approximations.
This document contains information about an engineering mathematics examination, including five questions covering topics like numerical methods for solving differential equations, complex variables, orthogonal polynomials, and probability. It also provides materials data and stipulations for designing a M35 grade concrete mix according to Indian standards.
The first part of the document outlines five questions on the exam covering numerical methods like Euler's method, Picard's method, Runge-Kutta method, and Milne's predictor-corrector method for solving differential equations. It also includes questions on complex variables, orthogonal polynomials, and probability.
The second part provides test data for materials to be used in designing a concrete mix for M35 grade concrete according to Indian standards, including stipulations
The document contains questions from the Fourth Semester B.E. Degree Examination in Material Science and Metallurgy. It has two parts - Part A and Part B. Some of the key questions asked include defining atomic packing factor and calculating values for FCC structure, explaining different types of point defects, stating and explaining Fick's second law of diffusion,
This document appears to be part of an examination for a course in Building Materials and Construction Technology. It contains instructions to answer 5 full questions from the paper, selecting at least 2 questions from each part (Part A and Part B). Part A includes questions about foundations, masonry, lintels, stairs, and plasters/paints. Part B includes questions about doors, trusses, floors, and stresses/strains in materials. The document provides a list of potential exam questions within these topic areas.
This document contains 5 questions regarding a mathematics exam. It covers topics like algebra, geometry, calculus, differential equations, and matrices. Some key details:
- The exam has 5 questions worth a total of 80 marks.
- Question 1 has 8 short answer parts worth 16 marks total.
- Questions 2-4 have 4 medium length parts each worth 16 marks total.
- Question 5 has 2 long answer parts worth 16 marks total.
- The questions cover topics such as finding GCDs, eigenvalues, limits, differential equations, and geometry concepts.
This document appears to be an exam for the course Strength of Materials. It contains questions that ask students to:
- Define terms like "Bulk modulus"
- Derive expressions, like for the deformation of a member due to self weight
- Calculate things like the stress induced in a member due to an applied load
- Explain concepts such as principal stresses and maximum shear stress
- Solve problems involving things like eccentric loading on a beam and buckling of columns
The questions cover a wide range of topics in strength of materials including stress, strain, deformation, shear force and bending moment diagrams, principal stresses, and column buckling.
CAPE PURE MATHEMATICS UNIT 2 MODULE 1 PRACTICE QUESTIONSCarlon Baird
dy/dx = (x - 3y)/(6x - 4)
The stationary points on the curve C occur when tan(x) = 2.
The equation of the tangent to C at the point where x=0 is y = 2ex.
This chapter introduces complex numbers. It defines a complex number as having the form x + iy, where x and y are real numbers. It describes how to represent complex numbers graphically on an Argand diagram and defines the modulus and argument of a complex number. It explains how to perform arithmetic operations like addition, subtraction, multiplication and division on complex numbers in both Cartesian (x + iy) and polar forms. It also introduces concepts like the conjugate of a complex number and using real and imaginary parts to solve equations. The chapter aims to explain the basic properties and manipulations of complex numbers.
CAPE PURE MATHEMATICS UNIT 2 MODULE 2 PRACTICE QUESTIONSCarlon Baird
This document contains practice questions on sequences, series, and approximations from a CAPE Pure Mathematics unit. Question 1 covers finding terms of sequences defined recursively and evaluating finite sums. Question 2 involves finding expressions for terms of sequences defined recursively and finding their sums. Later questions cover topics like proving identities using induction, evaluating infinite series, approximating functions using Taylor series, and finding roots of equations numerically. The questions provide worked examples of key concepts in sequences, series, and approximations.
This document contains information about an engineering mathematics examination, including five questions covering topics like numerical methods for solving differential equations, complex variables, orthogonal polynomials, and probability. It also provides materials data and stipulations for designing a M35 grade concrete mix according to Indian standards.
The first part of the document outlines five questions on the exam covering numerical methods like Euler's method, Picard's method, Runge-Kutta method, and Milne's predictor-corrector method for solving differential equations. It also includes questions on complex variables, orthogonal polynomials, and probability.
The second part provides test data for materials to be used in designing a concrete mix for M35 grade concrete according to Indian standards, including stipulations
The document contains questions from the Fourth Semester B.E. Degree Examination in Material Science and Metallurgy. It has two parts - Part A and Part B. Some of the key questions asked include defining atomic packing factor and calculating values for FCC structure, explaining different types of point defects, stating and explaining Fick's second law of diffusion,
This document appears to be part of an examination for a course in Building Materials and Construction Technology. It contains instructions to answer 5 full questions from the paper, selecting at least 2 questions from each part (Part A and Part B). Part A includes questions about foundations, masonry, lintels, stairs, and plasters/paints. Part B includes questions about doors, trusses, floors, and stresses/strains in materials. The document provides a list of potential exam questions within these topic areas.
This document contains 5 questions regarding a mathematics exam. It covers topics like algebra, geometry, calculus, differential equations, and matrices. Some key details:
- The exam has 5 questions worth a total of 80 marks.
- Question 1 has 8 short answer parts worth 16 marks total.
- Questions 2-4 have 4 medium length parts each worth 16 marks total.
- Question 5 has 2 long answer parts worth 16 marks total.
- The questions cover topics such as finding GCDs, eigenvalues, limits, differential equations, and geometry concepts.
This document appears to be an exam for the course Strength of Materials. It contains questions that ask students to:
- Define terms like "Bulk modulus"
- Derive expressions, like for the deformation of a member due to self weight
- Calculate things like the stress induced in a member due to an applied load
- Explain concepts such as principal stresses and maximum shear stress
- Solve problems involving things like eccentric loading on a beam and buckling of columns
The questions cover a wide range of topics in strength of materials including stress, strain, deformation, shear force and bending moment diagrams, principal stresses, and column buckling.
This document contains the questions from an engineering mathematics exam with 8 questions divided into 2 parts (A and B). Part A contains 3 multi-part questions on topics related to differential equations, including using Taylor's series, Runge-Kutta method, and Milne's predictor-corrector method to solve initial value problems. Part B contains 5 multi-part questions covering additional topics such as Legendre polynomials, Bessel's differential equation, probability, hypothesis testing, and confidence intervals. The exam tests knowledge of numerical analysis techniques for solving differential equations as well as topics in advanced calculus, probability, and statistics.
The document contains the questions from the Fourth Semester B.E. Degree Examination in Engineering Mathematics - IV. It has two parts, Part A and Part B, with multiple choice questions in each part. Some of the questions in Part A ask students to use numerical methods like Picard's method, Euler's modified method, and Runge-Kutta method of fourth order to solve initial value problems and solve systems of simultaneous equations. Other questions in Part B involve topics like analytic functions, harmonic functions, and Legendre polynomials. Students are required to solve five full questions by selecting at least two from each part.
This document contains the questions and answers from an engineering physics exam. It covers topics like:
- Blackbody radiation and Planck's law
- De Broglie wavelength and particle-wave duality
- Quantum mechanics including the particle in a box model
- Normalization constants and probability distributions in quantum mechanics
The exam contains multiple choice and short answer questions testing understanding of fundamental concepts in modern physics including wave-particle duality, quantum mechanics, and blackbody radiation. It requires calculations of quantities like de Broglie wavelength and energies of the particle in a box model.
The document appears to be part of an engineering physics exam containing both multiple choice and written response questions.
Some key details:
- It contains 8 multiple choice questions testing concepts in physics including Wien's displacement law, de Broglie wavelength, Heisenberg's uncertainty principle, and properties of lasers.
- It also includes 4 written response questions requiring calculations and explanations relating to group velocity, Planck's law, Bragg reflection, and crystal structures.
In summary, the document presents an exam in engineering physics with both multiple choice and written response questions testing students on foundational concepts in areas such as quantum mechanics, thermodynamics, and solid state physics.
This document provides formulas and methods for solving ordinary differential equations and vector calculus problems that are covered in an Engineering Mathematics course. It includes:
1. Seven methods for finding the complementary function for ODEs with constant coefficients depending on the nature of the roots.
2. Methods for finding the particular integral for ODEs with constant coefficients, including four types of functions the right side could be.
3. An overview of key concepts in vector calculus including vector differential operators, gradient, divergence, curl, and theorems like Green's theorem, Stokes' theorem, and Gauss' divergence theorem.
This document appears to be an examination paper for Engineering Mathematics from a third semester B.E. degree program. It contains 10 questions across two parts - Part A and Part B. The questions cover a range of topics including Fourier series, differential equations, matrix eigenvalues, interpolation, and numerical methods. Students are instructed to answer any 5 full questions, selecting at least 2 from each part. The questions vary in marks from 4 to 10 marks each.
This document contains the solutions to an engineering mathematics exam. It asks the student to solve various problems related to differential equations using numerical methods like Picard's method, Euler's modified method, Adam Bashforth method, and 4th order Runge Kutta method. It also contains problems on complex numbers, analytic functions, and harmonic functions. Legendre polynomials and their properties are also discussed. Questions related to probability, random variables, and hypothesis testing are presented.
This document contains questions from engineering mathematics, strength of materials, and surveying exams. Some key questions include:
1) Finding Fourier transforms and series expansions of various functions.
2) Calculating stresses, strains, deflections, and loads in beams, columns, and other structural elements.
3) Explaining surveying concepts like bearings, triangulation, traversing, leveling, contours, and performing related calculations.
This document appears to be an examination paper containing 8 questions divided into two parts (Part A and Part B) related to the subject of Structural Analysis - I. The questions cover various topics like determinate and indeterminate structures, degree of redundancy, strain energy, deflections of beams using different methods, analysis of arches, cables and continuous beams. Students are instructed to answer 5 full questions by selecting at least 2 questions from each part. Standard notations and formulas can be used. Diagrams of beam and arch structures are provided with the questions.
1) The document provides information about a mathematics exam including 10 multiple choice questions from section 1. It provides the questions, solutions, and choices for 4 questions ranging in topics from complex numbers, probabilities, functions, and geometry.
2) Question 50 asks the reader to determine vectors for a parallelogram and parallelepiped to calculate the volume, which is 10.
The document appears to be a past examination paper for an advanced mathematics course. It contains 8 questions across two parts (Part A and Part B) related to topics in graph theory and combinatorics. The questions assess a range of skills, including proving theorems about graphs, analyzing graph properties, applying graph algorithms like Dijkstra's algorithm, and solving counting problems.
This document contains exam questions from multiple subjects including Engineering Mathematics, Material Science and Metallurgy, Applied Thermodynamics, and Production Technology and Tool Engineering. The questions cover a wide range of topics testing knowledge of calculus, differential equations, material properties, phase diagrams, thermodynamic cycles, refrigeration, and mechanisms. Students are instructed to answer 5 full questions by selecting at least 2 questions from each part of the exam.
1. The document contains a past exam paper for an Advanced Mathematics exam with 10 questions across two parts (A and B).
2. The questions cover a range of advanced mathematics topics including Taylor series, differential equations, probability, statistics, and linear algebra.
3. Students must answer 5 questions total, with at least 2 questions from each part. Questions involve calculating values, proving statements, finding probabilities, and more.
This document contains a mathematics exam with questions about complex numbers. It has four sections - straight objective type questions with single correct answers, multiple correct answer type questions, matrix matching questions, and a linked comprehension passage with follow up questions. The document tests knowledge of topics like complex number operations, properties of complex functions, geometry of complex numbers on the Argand plane, and relationships between complex number representations of geometric shapes.
The document is illegible and contains no discernible information. It appears to be random symbols and characters with no coherent words, sentences, or meaning.
This document contains questions from a Material Science and Metallurgy exam. It covers various topics:
- Crystal structures of BCC, FCC and HCP lattices and their properties. Diffusion of iron atoms in BCC lattice.
- Mechanical properties in the plastic region from stress-strain diagrams. True and conventional strain expressions. Twinning mechanism of plastic deformation.
- Fracture mechanisms based on Griffith's theory of brittle fracture. Factors affecting creep. Fatigue testing and S-N curves for materials.
- Solidification process and expression for critical nucleus radius. Cast metal structures. Solid solutions and Hume-Rothery rules. Phase diagrams and Gibbs phase rule.
The document discusses solving various differential equations using different numerical methods. It contains 6 questions related to numerical methods for solving differential equations. Specifically, it involves:
1) Using Taylor's series, Euler's method, and Adams-Bashforth method to solve differential equations.
2) Employing Picard's method and Runge-Kutta method to obtain approximate solutions of differential equations.
3) Using Milne's method to obtain an approximate solution of a differential equation.
4) Defining an analytic function and obtaining Cauchy-Riemann equations in polar form.
The questions cover a wide range of numerical methods for solving differential equations including Taylor series, Euler's method, Picard
This document appears to be an exam paper for an 8th semester software testing course. It contains 6 questions with subparts related to software testing topics. Question 1 asks about the definitions of error, fault, and failure and separation of actual vs observed behavior. Question 2 covers defect management, software vs hardware testing, and static testing. Question 3 is about cause-effect graphing and the BOR algorithm. Question 4 addresses infeasibility problems and structural testing criteria. Question 5 covers control and data dependence graphs, reaching definitions, and data flow analysis terms. Question 6 asks about test scaffolding, test oracles, and testing strategies like integration testing.
This document appears to be an exam for a Concrete Technology course, with questions covering various topics related to concrete materials and design. It includes two parts (A and B) with multiple choice questions. Part A questions cover topics like cement manufacturing processes, aggregate properties and testing, workability of concrete, and the role of chemical and mineral admixtures. Part B questions address factors influencing concrete strength, testing methods, elastic properties of concrete, durability, shrinkage and creep, and concrete mix design procedures. Students are instructed to answer any five full questions, selecting at least two from each part, and references are made to relevant Indian Standards for concrete.
Successful people replace words like "wish", "try", and "should" with "I will" in their speech and thinking. Ineffective people do not make this replacement and continue using weaker language. The document provides mathematical formulas and properties involving complex numbers, including formulas for roots of unity, sums of trigonometric series, representations of lines and circles using complex numbers, and other identities.
This document discusses fractal geometry and fractals. It provides a brief history of fractals, from their theoretical mathematical foundations to their modern discovery. Key properties of fractals are described, including self-similarity, iteration, and fractal dimension. Famous fractals like the Koch curve and Julia sets are examined. The Julia set is defined as the set of points that do not tend toward an attracting fixed point or infinity under iteration of a complex polynomial function. Overall, the document provides an introduction to fractal geometry and some of its most important concepts and examples.
1. Assume that an algorithm to solve a problem takes f(n) microse.docxSONU61709
1. Assume that an algorithm to solve a problem takes f(n) microseconds for some function f of the input size n. For each time t labeled across the top, determine the exact largest value of n which can be solved in time f(n) where f(n) ≤ t. Use a calculator! You will find it helpful to convert the t values to microseconds, and you may find it helpful to insert a row for n. Note that “lg n” is the log2 n. Note that the only row you can’t write out the values for fully is the “lg n” row—only there may you write 2x for the appropriate value of x. Use the Windows built-in scientific calculator (under Accessories menu) as necessary. A couple values are filled in to get you started. Important: “exact values” means precisely that. Check your answers with values above and below!
Time t =
f(n) =
1 second
1 hour
1 day
1 month
=30 days
n2
1,609,968
lg n
n3
2n
n lg n
2,755,147,513
2. Use loop counting to give a O( ) characterization of each of the following loops basing each upon the size of its input:
a. Algorithm Loop1(n):
s ← 0
for i ← 1 to n do
s ← s + i
b. Algorithm Loop2(p):
p ← 1
for i ← 1 to 2n do
p ← p * i
c. Algorithm Loop3(n):
p ← 1
for j ← 1 to n2 do
p ← p * i
d. Algorithm Loop4(n):
s ← 0
for j ← 1 to 2n do
for k ← 1 to j do
s ← s + j
e. Algorithm Loop5(n):
k ← 0
for r ← 1 to n2 do
for s ← 1 to r do
k ← k + r
3. Order the following functions from smallest to largest by their big-O notation—you can use the letters in your answer rather than copying each formula. Be clear which is smallest and which is largest, and which functions are asymptotically equivalent. For example, if g, h, and m are all O(n lg n), you would write g = h = m = O(n lg n).
a. 562 log3 108
b. n3
c. 2n lg n
d. lg nn
e. n3 lg n
f. (n3 lg n3)/2
g. nn
h. 56n
i. log5 (n!)
j. ncos n
k. n / lg n
l. lg* n
m.
4. a. Which of these equations is true, and why?
b. Which of these is smaller for very large n?
Trisecting the Circle: A Case for Euclidean Geometry
Author(s): Alfred S. Posamentier
Source: The Mathematics Teacher, Vol. 99, No. 6 (FEBRUARY 2006), pp. 414-418
Published by: National Council of Teachers of Mathematics
Stable URL: http://www.jstor.org/stable/27972006
Accessed: 09-02-2018 18:19 UTC
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This document contains the questions from an engineering mathematics exam with 8 questions divided into 2 parts (A and B). Part A contains 3 multi-part questions on topics related to differential equations, including using Taylor's series, Runge-Kutta method, and Milne's predictor-corrector method to solve initial value problems. Part B contains 5 multi-part questions covering additional topics such as Legendre polynomials, Bessel's differential equation, probability, hypothesis testing, and confidence intervals. The exam tests knowledge of numerical analysis techniques for solving differential equations as well as topics in advanced calculus, probability, and statistics.
The document contains the questions from the Fourth Semester B.E. Degree Examination in Engineering Mathematics - IV. It has two parts, Part A and Part B, with multiple choice questions in each part. Some of the questions in Part A ask students to use numerical methods like Picard's method, Euler's modified method, and Runge-Kutta method of fourth order to solve initial value problems and solve systems of simultaneous equations. Other questions in Part B involve topics like analytic functions, harmonic functions, and Legendre polynomials. Students are required to solve five full questions by selecting at least two from each part.
This document contains the questions and answers from an engineering physics exam. It covers topics like:
- Blackbody radiation and Planck's law
- De Broglie wavelength and particle-wave duality
- Quantum mechanics including the particle in a box model
- Normalization constants and probability distributions in quantum mechanics
The exam contains multiple choice and short answer questions testing understanding of fundamental concepts in modern physics including wave-particle duality, quantum mechanics, and blackbody radiation. It requires calculations of quantities like de Broglie wavelength and energies of the particle in a box model.
The document appears to be part of an engineering physics exam containing both multiple choice and written response questions.
Some key details:
- It contains 8 multiple choice questions testing concepts in physics including Wien's displacement law, de Broglie wavelength, Heisenberg's uncertainty principle, and properties of lasers.
- It also includes 4 written response questions requiring calculations and explanations relating to group velocity, Planck's law, Bragg reflection, and crystal structures.
In summary, the document presents an exam in engineering physics with both multiple choice and written response questions testing students on foundational concepts in areas such as quantum mechanics, thermodynamics, and solid state physics.
This document provides formulas and methods for solving ordinary differential equations and vector calculus problems that are covered in an Engineering Mathematics course. It includes:
1. Seven methods for finding the complementary function for ODEs with constant coefficients depending on the nature of the roots.
2. Methods for finding the particular integral for ODEs with constant coefficients, including four types of functions the right side could be.
3. An overview of key concepts in vector calculus including vector differential operators, gradient, divergence, curl, and theorems like Green's theorem, Stokes' theorem, and Gauss' divergence theorem.
This document appears to be an examination paper for Engineering Mathematics from a third semester B.E. degree program. It contains 10 questions across two parts - Part A and Part B. The questions cover a range of topics including Fourier series, differential equations, matrix eigenvalues, interpolation, and numerical methods. Students are instructed to answer any 5 full questions, selecting at least 2 from each part. The questions vary in marks from 4 to 10 marks each.
This document contains the solutions to an engineering mathematics exam. It asks the student to solve various problems related to differential equations using numerical methods like Picard's method, Euler's modified method, Adam Bashforth method, and 4th order Runge Kutta method. It also contains problems on complex numbers, analytic functions, and harmonic functions. Legendre polynomials and their properties are also discussed. Questions related to probability, random variables, and hypothesis testing are presented.
This document contains questions from engineering mathematics, strength of materials, and surveying exams. Some key questions include:
1) Finding Fourier transforms and series expansions of various functions.
2) Calculating stresses, strains, deflections, and loads in beams, columns, and other structural elements.
3) Explaining surveying concepts like bearings, triangulation, traversing, leveling, contours, and performing related calculations.
This document appears to be an examination paper containing 8 questions divided into two parts (Part A and Part B) related to the subject of Structural Analysis - I. The questions cover various topics like determinate and indeterminate structures, degree of redundancy, strain energy, deflections of beams using different methods, analysis of arches, cables and continuous beams. Students are instructed to answer 5 full questions by selecting at least 2 questions from each part. Standard notations and formulas can be used. Diagrams of beam and arch structures are provided with the questions.
1) The document provides information about a mathematics exam including 10 multiple choice questions from section 1. It provides the questions, solutions, and choices for 4 questions ranging in topics from complex numbers, probabilities, functions, and geometry.
2) Question 50 asks the reader to determine vectors for a parallelogram and parallelepiped to calculate the volume, which is 10.
The document appears to be a past examination paper for an advanced mathematics course. It contains 8 questions across two parts (Part A and Part B) related to topics in graph theory and combinatorics. The questions assess a range of skills, including proving theorems about graphs, analyzing graph properties, applying graph algorithms like Dijkstra's algorithm, and solving counting problems.
This document contains exam questions from multiple subjects including Engineering Mathematics, Material Science and Metallurgy, Applied Thermodynamics, and Production Technology and Tool Engineering. The questions cover a wide range of topics testing knowledge of calculus, differential equations, material properties, phase diagrams, thermodynamic cycles, refrigeration, and mechanisms. Students are instructed to answer 5 full questions by selecting at least 2 questions from each part of the exam.
1. The document contains a past exam paper for an Advanced Mathematics exam with 10 questions across two parts (A and B).
2. The questions cover a range of advanced mathematics topics including Taylor series, differential equations, probability, statistics, and linear algebra.
3. Students must answer 5 questions total, with at least 2 questions from each part. Questions involve calculating values, proving statements, finding probabilities, and more.
This document contains a mathematics exam with questions about complex numbers. It has four sections - straight objective type questions with single correct answers, multiple correct answer type questions, matrix matching questions, and a linked comprehension passage with follow up questions. The document tests knowledge of topics like complex number operations, properties of complex functions, geometry of complex numbers on the Argand plane, and relationships between complex number representations of geometric shapes.
The document is illegible and contains no discernible information. It appears to be random symbols and characters with no coherent words, sentences, or meaning.
This document contains questions from a Material Science and Metallurgy exam. It covers various topics:
- Crystal structures of BCC, FCC and HCP lattices and their properties. Diffusion of iron atoms in BCC lattice.
- Mechanical properties in the plastic region from stress-strain diagrams. True and conventional strain expressions. Twinning mechanism of plastic deformation.
- Fracture mechanisms based on Griffith's theory of brittle fracture. Factors affecting creep. Fatigue testing and S-N curves for materials.
- Solidification process and expression for critical nucleus radius. Cast metal structures. Solid solutions and Hume-Rothery rules. Phase diagrams and Gibbs phase rule.
The document discusses solving various differential equations using different numerical methods. It contains 6 questions related to numerical methods for solving differential equations. Specifically, it involves:
1) Using Taylor's series, Euler's method, and Adams-Bashforth method to solve differential equations.
2) Employing Picard's method and Runge-Kutta method to obtain approximate solutions of differential equations.
3) Using Milne's method to obtain an approximate solution of a differential equation.
4) Defining an analytic function and obtaining Cauchy-Riemann equations in polar form.
The questions cover a wide range of numerical methods for solving differential equations including Taylor series, Euler's method, Picard
This document appears to be an exam paper for an 8th semester software testing course. It contains 6 questions with subparts related to software testing topics. Question 1 asks about the definitions of error, fault, and failure and separation of actual vs observed behavior. Question 2 covers defect management, software vs hardware testing, and static testing. Question 3 is about cause-effect graphing and the BOR algorithm. Question 4 addresses infeasibility problems and structural testing criteria. Question 5 covers control and data dependence graphs, reaching definitions, and data flow analysis terms. Question 6 asks about test scaffolding, test oracles, and testing strategies like integration testing.
This document appears to be an exam for a Concrete Technology course, with questions covering various topics related to concrete materials and design. It includes two parts (A and B) with multiple choice questions. Part A questions cover topics like cement manufacturing processes, aggregate properties and testing, workability of concrete, and the role of chemical and mineral admixtures. Part B questions address factors influencing concrete strength, testing methods, elastic properties of concrete, durability, shrinkage and creep, and concrete mix design procedures. Students are instructed to answer any five full questions, selecting at least two from each part, and references are made to relevant Indian Standards for concrete.
Successful people replace words like "wish", "try", and "should" with "I will" in their speech and thinking. Ineffective people do not make this replacement and continue using weaker language. The document provides mathematical formulas and properties involving complex numbers, including formulas for roots of unity, sums of trigonometric series, representations of lines and circles using complex numbers, and other identities.
This document discusses fractal geometry and fractals. It provides a brief history of fractals, from their theoretical mathematical foundations to their modern discovery. Key properties of fractals are described, including self-similarity, iteration, and fractal dimension. Famous fractals like the Koch curve and Julia sets are examined. The Julia set is defined as the set of points that do not tend toward an attracting fixed point or infinity under iteration of a complex polynomial function. Overall, the document provides an introduction to fractal geometry and some of its most important concepts and examples.
1. Assume that an algorithm to solve a problem takes f(n) microse.docxSONU61709
1. Assume that an algorithm to solve a problem takes f(n) microseconds for some function f of the input size n. For each time t labeled across the top, determine the exact largest value of n which can be solved in time f(n) where f(n) ≤ t. Use a calculator! You will find it helpful to convert the t values to microseconds, and you may find it helpful to insert a row for n. Note that “lg n” is the log2 n. Note that the only row you can’t write out the values for fully is the “lg n” row—only there may you write 2x for the appropriate value of x. Use the Windows built-in scientific calculator (under Accessories menu) as necessary. A couple values are filled in to get you started. Important: “exact values” means precisely that. Check your answers with values above and below!
Time t =
f(n) =
1 second
1 hour
1 day
1 month
=30 days
n2
1,609,968
lg n
n3
2n
n lg n
2,755,147,513
2. Use loop counting to give a O( ) characterization of each of the following loops basing each upon the size of its input:
a. Algorithm Loop1(n):
s ← 0
for i ← 1 to n do
s ← s + i
b. Algorithm Loop2(p):
p ← 1
for i ← 1 to 2n do
p ← p * i
c. Algorithm Loop3(n):
p ← 1
for j ← 1 to n2 do
p ← p * i
d. Algorithm Loop4(n):
s ← 0
for j ← 1 to 2n do
for k ← 1 to j do
s ← s + j
e. Algorithm Loop5(n):
k ← 0
for r ← 1 to n2 do
for s ← 1 to r do
k ← k + r
3. Order the following functions from smallest to largest by their big-O notation—you can use the letters in your answer rather than copying each formula. Be clear which is smallest and which is largest, and which functions are asymptotically equivalent. For example, if g, h, and m are all O(n lg n), you would write g = h = m = O(n lg n).
a. 562 log3 108
b. n3
c. 2n lg n
d. lg nn
e. n3 lg n
f. (n3 lg n3)/2
g. nn
h. 56n
i. log5 (n!)
j. ncos n
k. n / lg n
l. lg* n
m.
4. a. Which of these equations is true, and why?
b. Which of these is smaller for very large n?
Trisecting the Circle: A Case for Euclidean Geometry
Author(s): Alfred S. Posamentier
Source: The Mathematics Teacher, Vol. 99, No. 6 (FEBRUARY 2006), pp. 414-418
Published by: National Council of Teachers of Mathematics
Stable URL: http://www.jstor.org/stable/27972006
Accessed: 09-02-2018 18:19 UTC
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This document provides a summary of a master's thesis in mathematics. The thesis studies generalisations of the fact that every Riesz homomorphism between spaces of continuous functions on compact Hausdorff spaces can be written as a composition multiplication operator. Specifically, the thesis examines Riesz homomorphisms between spaces of extended continuous functions on extremally disconnected spaces, known as Maeda-Ogasawara spaces. It proves that every Riesz homomorphism on such a space has a generalized composition multiplication form involving a continuous map between the underlying spaces. The thesis also applies these results to spaces of measurable functions and explores Riesz homomorphisms between spaces of continuous functions with values in a Banach lattice.
This document discusses research on identifying the densest arrangements, or close packings, of identical hard spheres confined within cylinders of varying diameters. The researchers adapted an adaptive-shrinking-cell and sequential-linear-programming technique to find close packings up to a cylinder diameter of 4.00 times the sphere diameter. They identified 17 new close packing structures, most of which have chiral, or handed, geometries. Beyond a cylinder diameter of around 2.85 times the sphere diameter, most structures consist of an outer shell and inner core competing to be closely packed. In some cases, the shell or core adopts its own maximum density arrangement, and in other cases the packing involves an interplay between the shell and core.
In this work we discuss how to compute KLE with complexity O(k n log n), how to approximate large covariance matrices (in H-matrix format), how to use the Lanczos method.
We solve elliptic PDE with uncertain coefficients. We apply Karhunen-Loeve expansion to separate stochastic part from spatial part. The corresponding eigenvalue problem with covariance function is solved via the Hierarchical Matrix technique. We also demonstrate how low-rank tensor method can be applied for high-dimensional problems (e.g., to compute higher order statistical moments) . We provide explicit formulas to compute statistical moments of order k with linear complexity.
Establishment of New Special Deductions from Gauss Divergence Theorem in a Ve...inventionjournals
International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
This document introduces complex integration and provides examples of evaluating integrals along paths in the complex plane. It expresses integrals in terms of real and imaginary parts involving line integrals of functions. Key points made include:
- Complex integrals can be interpreted as line integrals over paths in the complex plane.
- Integrals of analytic functions over closed paths, like the unit circle, may yield simple results like 2πi or 0.
- Blasius' theorem relates forces and moments on a cylinder in fluid flow to complex integrals around the cylinder boundary.
Computer Science
Active and Programmable Networks
Active safety systems
Ad Hoc & Sensor Network
Ad hoc networks for pervasive communications
Adaptive, autonomic and context-aware computing
Advance Computing technology and their application
Advanced Computing Architectures and New Programming Models
Advanced control and measurement
Aeronautical Engineering,
Agent-based middleware
Alert applications
Automotive, marine and aero-space control and all other control applications
Autonomic and self-managing middleware
Autonomous vehicle
Biochemistry
Bioinformatics
BioTechnology(Chemistry, Mathematics, Statistics, Geology)
Broadband and intelligent networks
Broadband wireless technologies
CAD/CAM/CAT/CIM
Call admission and flow/congestion control
Capacity planning and dimensioning
Changing Access to Patient Information
Channel capacity modelling and analysis
Civil Engineering,
Cloud Computing and Applications
Collaborative applications
Communication application
Communication architectures for pervasive computing
Communication systems
Computational intelligence
Computer and microprocessor-based control
Computer Architecture and Embedded Systems
Computer Business
Computer Sciences and Applications
Computer Vision
Computer-based information systems in health care
Computing Ethics
Computing Practices & Applications
Congestion and/or Flow Control
Content Distribution
Context-awareness and middleware
Creativity in Internet management and retailing
Cross-layer design and Physical layer based issue
Cryptography
Data Base Management
Data fusion
Data Mining
Data retrieval
Data Storage Management
Decision analysis methods
Decision making
Digital Economy and Digital Divide
Digital signal processing theory
Distributed Sensor Networks
Drives automation
Drug Design,
Drug Development
DSP implementation
E-Business
E-Commerce
E-Government
Electronic transceiver device for Retail Marketing Industries
Electronics Engineering,
Embeded Computer System
Emerging advances in business and its applications
Emerging signal processing areas
Enabling technologies for pervasive systems
Energy-efficient and green pervasive computing
Environmental Engineering,
Estimation and identification techniques
Evaluation techniques for middleware solutions
Event-based, publish/subscribe, and message-oriented middleware
Evolutionary computing and intelligent systems
Expert approaches
Facilities planning and management
Flexible manufacturing systems
Formal methods and tools for designing
Fuzzy algorithms
Fuzzy logics
GPS and location-based app
1. Motivation: why do we need low-rank tensors
2. Tensors of the second order (matrices)
3. CP, Tucker and tensor train tensor formats
4. Many classical kernels have (or can be approximated in ) low-rank tensor format
5. Post processing: Computation of mean, variance, level sets, frequency
This document summarizes Michael Kreisel's dissertation on the connection between Gabor frames for quasicrystals, the topology of the hull of a quasicrystal, and K-theory of an associated twisted groupoid algebra. The author constructs a finitely generated projective module over this algebra, where any multiwindow Gabor frame for the quasicrystal can be used to construct a projection representing this module in K-theory. As an application, results are obtained on the twisted version of Bellissard's gap labeling conjecture for quasicrystals.
ER Publication,
IJETR, IJMCTR,
Journals,
International Journals,
High Impact Journals,
Monthly Journal,
Good quality Journals,
Research,
Research Papers,
Research Article,
Free Journals, Open access Journals,
erpublication.org,
Engineering Journal,
Science Journals,
Classically, the point particle and the string exhibit the same kind of motion. For instance in flat space both of them move in straight lines albeit for string oscillations which occur because it has to obey the wave equation.
When we put it in AdS3 space both the point particle and the string move as if they are in a potential well. However, coordinate singularities arise in the numerical computation of the string so motion beyond ρ = 0 becomes computationally inac- cessible. Physically the string should still move beyond this point in empty AdS3 spacetime. This singularity is an artefact because coordinate systems in general are not physical. The behaviour of the string in the vicinity of a black hole background in AdS3 spacetime is well defined a fair bit away from the horizon. It moves in the same manner as in the AdS3 spacetime in the absence of the background. Un- fortunately, when the string approaches the horizon part of the string overshoots into the horizon. The solutions become divergent and the numerical solution fails before we can observe anything interesting.
1) The document provides an overview of the contents of Part II of a slideshow on modern physics, which covers topics such as charge and current densities, electromagnetic induction, Maxwell's equations, special relativity, tensors, blackbody radiation, photons, electrons, scattering problems, and waves.
2) It aims to provide a brief yet modern review of foundational concepts in electromagnetism and set the stage for introducing special relativity, quantum mechanics, and matter waves for undergraduate students.
3) The overview highlights that succeeding chapters will develop tensor formulations of electromagnetism and special relativity from first principles before discussing applications like blackbody radiation and early quantum models.
This summary provides the key details from the document in 3 sentences:
The document investigates the structure of unital 3-fields, which are fields where addition requires 3 summands rather than the usual 2. It is shown that unital 3-fields are isomorphic to the set of invertible elements in a local ring R with Z2Z as the residual field. Pairs of elements in the 3-field are used to define binary operations that allow reducing the arity and connecting the 3-field to binary algebra. The structure of finite 3-fields is examined, proving properties like the number of elements being a power of 2.
Or: From ice to R-matrices
This talk is a summary of the history of quantum groups, describe how they arose from questions in statistical mechanics. The keyword is the Yang-Baxter equation, which was crucial for the development of the field.
Disclaimer: No one is perfect, so please mind that there might be mistakes and typos.
dtubbenhauer@gmail.com
Corrected slides: dtubbenhauer.com/talks.html
On the Covering Radius of Codes Over Z6 ijitjournal
In this correspondence, we give lower and upper bounds on the covering radius of codes over the finite
ring Z6 with respect to different distances such as Hamming, Lee, Euclidean and Chinese Euclidean. We
also determine the covering radius of various Block Repetition Codes over Z6
This document introduces differential forms as an alternative approach to vector calculus. It provides a brief overview of 1-forms and 2-forms, including how to calculate line integrals and surface integrals of differential forms. The author explains that differential forms are similar to vector fields but written in a "funny notation" that is ultimately quite powerful. The document is intended as a supplement for teaching multivariable calculus using differential forms at the undergraduate level in a informal way without advanced linear algebra or manifolds.
Similar to Space fullerenes: A computer search of new Frank-Kasper structures (20)
The computation of automorphic forms for a group Gamma is
a major problem in number theory. The only known way to approach the higher rank cases is by computing the action of Hecke operators on the cohomology.
Henceforth, we consider the explicit computation of the cohomology by using cellular complexes. We then explain how the rational elements can be made to act on the complex when it originate from perfect forms. We illustrate the results obtained for the symplectic Sp4(Z) group.
A crystallographic group is a group acting on R^n that contains a translation subgroup Z^n as a finite index subgroup. Here we consider which Coxeter groups are crystallographic groups. We also expose the enumeration in dimension 2 and 3. Then we shortly give the principle under which the enumeration of N dimensional crystallographic groups is done.
A strictly face regular map is a k-valent plane graph on the sphere or the entire plane with faces of size a and b such that any a-gonal face is adjacent to exactly p a-gonal face and exactly q b-gonal faces. If only one of such rule is respected then we get a weak face-regular map.
We present here enumeration technique for the face regular maps that rely on polycycle and other techniques.
Fullerenes are 3-valent plane graph that have have faces of size 5 or 6. This class of graph can be parametrized with 10 complex eisenstein numbers by Thurston theory. Here we consider spheric analogs of this theory and found 8 different classes. For each we consider following notions: (a) possible groups (b) Goldberg Coxeter construction (c) zigzags and central circuits (d) parameterization by complex integers.
Besides that we consider generalization to icosahedrites, space fullerenes and d-dimensional fullerenes.
A polycycle is a 2-connected plane locally finite graph G with faces partitioned
in two faces F1 and F2. The faces in F1 are combinatorial i-gons.
The faces in F2 are called holes and are pair-wise disjoint.
All vertices have degree {2,...,q} with interior vertices of degree q.
Polycycles can be decomposed into elementary polycycles. For some parameters (i,q) the elementary polycycles can be classified and this allows to solve many different combinatorial problems.
The document discusses Wythoff constructions and l1-embeddings. It begins by introducing Wythoff constructions, which generate new complexes from an original complex based on subsets of face dimensions. It then discusses l1-embeddings, which isometrically embed graphs into l1 metric spaces. Regular polytopes and tilings are provided as examples that embed into hypercubes or half-cubes through l1-embeddings. Embeddability can be tested using properties like being bipartite and satisfying hypermetric inequalities.
Implicit schemes are needed in order to have fast runtime in wave models. Parallelization using the Message Passing Interface are needed in order to run on computers with thousands of processors. Implicit schemes rely on preconditioner in order for the iterative schemes to converge fast. Thus we need fast preconditioners and we present those here.
This document discusses methods for summarizing Lego-like sphere and torus maps. It begins by introducing the concept of ({a,b},k)-maps, which are k-valent maps with faces of size a or b. It then discusses several challenges in enumerating and drawing such maps, including enumerating all possible Lego decompositions. Specific enumeration methods are described, such as using exact covering problems or satisfiability problems. The document also discusses challenges in graph drawing representations, and suggests using primal-dual circle packings as a promising approach.
The accurate modeling of the ocean requires the computation of the horizontal pressure gradient. This gradient depends on the difference of two large quantities and thus is susceptible to numerical errors. A way to avoid this to smooth the bathymetry. However, we do not want to perturb too much the bathymetry. Here we proposed to use linear programming in order to get a good representation of the bathymetry.
The Goldberg-Coxeter construction takes two integers (k,l) a 3-or 4-valent plane graph and returns a 3- or 4-valent plane graph. This construction is useful in virus study, numerical analysis, architecture, chemistry and of course mathematics.
Here we consider the zigzags and central circuits of 3- or 4-valent plane graph. It turns out that we can define an algebraic construction of (k,l)-product that allows to find the length of the zigzags and central circuits in a compact way. All possible lengths of zigzags are determined by this (k,l)-product and the normal structure of the automorphism group allows to find them for some congruence conditions.
This document discusses fullerenes, which are spherical or cylindrical carbon molecules composed of hexagons and pentagons. It provides definitions and examples of different types of fullerenes, including icosahedral fullerenes like C60 and C80. The document also summarizes potential applications of fullerenes in chemistry, biology, nanotechnology, and other fields, such as uses in drug design, superconductors, and nanotubes. However, it notes that nanotubes remain too expensive for many applications currently.
Enumeration methods are very important in a variety of settings, both mathematical and applications. For many problems there is actually no real hope to do the enumeration in reasonable time since the number of solutions is so big. This talk is about how to compute at the limit.
The talk is decomposed into:
(a) Regular enumeration procedure where one uses computerized case distinction.
(b) Use of symmetry groups for isomorphism checks.
(c) The augmentation scheme that allows to enumerate object up to isomorphism without keeping the full list in memory.
(d) The homomorphism principle that allows to map a complex problem to a simpler one.
Lattice coverings are a simple case of covering problems. We will first expose methods for finding the covering density of a given lattice. If one considers a space of possible lattices, then one gets the theory of L-type. We will explain how this theory works out and how the over a given L-type the problem is a
semidefinite programming problem. Finally, we will explore the covering maxima of lattices, i.e. local behavior of the covering density function.
The technology uses reclaimed CO₂ as the dyeing medium in a closed loop process. When pressurized, CO₂ becomes supercritical (SC-CO₂). In this state CO₂ has a very high solvent power, allowing the dye to dissolve easily.
Travis Hills' Endeavors in Minnesota: Fostering Environmental and Economic Pr...Travis Hills MN
Travis Hills of Minnesota developed a method to convert waste into high-value dry fertilizer, significantly enriching soil quality. By providing farmers with a valuable resource derived from waste, Travis Hills helps enhance farm profitability while promoting environmental stewardship. Travis Hills' sustainable practices lead to cost savings and increased revenue for farmers by improving resource efficiency and reducing waste.
Phenomics assisted breeding in crop improvementIshaGoswami9
As the population is increasing and will reach about 9 billion upto 2050. Also due to climate change, it is difficult to meet the food requirement of such a large population. Facing the challenges presented by resource shortages, climate
change, and increasing global population, crop yield and quality need to be improved in a sustainable way over the coming decades. Genetic improvement by breeding is the best way to increase crop productivity. With the rapid progression of functional
genomics, an increasing number of crop genomes have been sequenced and dozens of genes influencing key agronomic traits have been identified. However, current genome sequence information has not been adequately exploited for understanding
the complex characteristics of multiple gene, owing to a lack of crop phenotypic data. Efficient, automatic, and accurate technologies and platforms that can capture phenotypic data that can
be linked to genomics information for crop improvement at all growth stages have become as important as genotyping. Thus,
high-throughput phenotyping has become the major bottleneck restricting crop breeding. Plant phenomics has been defined as the high-throughput, accurate acquisition and analysis of multi-dimensional phenotypes
during crop growing stages at the organism level, including the cell, tissue, organ, individual plant, plot, and field levels. With the rapid development of novel sensors, imaging technology,
and analysis methods, numerous infrastructure platforms have been developed for phenotyping.
ESR spectroscopy in liquid food and beverages.pptxPRIYANKA PATEL
With increasing population, people need to rely on packaged food stuffs. Packaging of food materials requires the preservation of food. There are various methods for the treatment of food to preserve them and irradiation treatment of food is one of them. It is the most common and the most harmless method for the food preservation as it does not alter the necessary micronutrients of food materials. Although irradiated food doesn’t cause any harm to the human health but still the quality assessment of food is required to provide consumers with necessary information about the food. ESR spectroscopy is the most sophisticated way to investigate the quality of the food and the free radicals induced during the processing of the food. ESR spin trapping technique is useful for the detection of highly unstable radicals in the food. The antioxidant capability of liquid food and beverages in mainly performed by spin trapping technique.
When I was asked to give a companion lecture in support of ‘The Philosophy of Science’ (https://shorturl.at/4pUXz) I decided not to walk through the detail of the many methodologies in order of use. Instead, I chose to employ a long standing, and ongoing, scientific development as an exemplar. And so, I chose the ever evolving story of Thermodynamics as a scientific investigation at its best.
Conducted over a period of >200 years, Thermodynamics R&D, and application, benefitted from the highest levels of professionalism, collaboration, and technical thoroughness. New layers of application, methodology, and practice were made possible by the progressive advance of technology. In turn, this has seen measurement and modelling accuracy continually improved at a micro and macro level.
Perhaps most importantly, Thermodynamics rapidly became a primary tool in the advance of applied science/engineering/technology, spanning micro-tech, to aerospace and cosmology. I can think of no better a story to illustrate the breadth of scientific methodologies and applications at their best.
The binding of cosmological structures by massless topological defectsSérgio Sacani
Assuming spherical symmetry and weak field, it is shown that if one solves the Poisson equation or the Einstein field
equations sourced by a topological defect, i.e. a singularity of a very specific form, the result is a localized gravitational
field capable of driving flat rotation (i.e. Keplerian circular orbits at a constant speed for all radii) of test masses on a thin
spherical shell without any underlying mass. Moreover, a large-scale structure which exploits this solution by assembling
concentrically a number of such topological defects can establish a flat stellar or galactic rotation curve, and can also deflect
light in the same manner as an equipotential (isothermal) sphere. Thus, the need for dark matter or modified gravity theory is
mitigated, at least in part.
The debris of the ‘last major merger’ is dynamically youngSérgio Sacani
The Milky Way’s (MW) inner stellar halo contains an [Fe/H]-rich component with highly eccentric orbits, often referred to as the
‘last major merger.’ Hypotheses for the origin of this component include Gaia-Sausage/Enceladus (GSE), where the progenitor
collided with the MW proto-disc 8–11 Gyr ago, and the Virgo Radial Merger (VRM), where the progenitor collided with the
MW disc within the last 3 Gyr. These two scenarios make different predictions about observable structure in local phase space,
because the morphology of debris depends on how long it has had to phase mix. The recently identified phase-space folds in Gaia
DR3 have positive caustic velocities, making them fundamentally different than the phase-mixed chevrons found in simulations
at late times. Roughly 20 per cent of the stars in the prograde local stellar halo are associated with the observed caustics. Based
on a simple phase-mixing model, the observed number of caustics are consistent with a merger that occurred 1–2 Gyr ago.
We also compare the observed phase-space distribution to FIRE-2 Latte simulations of GSE-like mergers, using a quantitative
measurement of phase mixing (2D causticality). The observed local phase-space distribution best matches the simulated data
1–2 Gyr after collision, and certainly not later than 3 Gyr. This is further evidence that the progenitor of the ‘last major merger’
did not collide with the MW proto-disc at early times, as is thought for the GSE, but instead collided with the MW disc within
the last few Gyr, consistent with the body of work surrounding the VRM.
Unlocking the mysteries of reproduction: Exploring fecundity and gonadosomati...AbdullaAlAsif1
The pygmy halfbeak Dermogenys colletei, is known for its viviparous nature, this presents an intriguing case of relatively low fecundity, raising questions about potential compensatory reproductive strategies employed by this species. Our study delves into the examination of fecundity and the Gonadosomatic Index (GSI) in the Pygmy Halfbeak, D. colletei (Meisner, 2001), an intriguing viviparous fish indigenous to Sarawak, Borneo. We hypothesize that the Pygmy halfbeak, D. colletei, may exhibit unique reproductive adaptations to offset its low fecundity, thus enhancing its survival and fitness. To address this, we conducted a comprehensive study utilizing 28 mature female specimens of D. colletei, carefully measuring fecundity and GSI to shed light on the reproductive adaptations of this species. Our findings reveal that D. colletei indeed exhibits low fecundity, with a mean of 16.76 ± 2.01, and a mean GSI of 12.83 ± 1.27, providing crucial insights into the reproductive mechanisms at play in this species. These results underscore the existence of unique reproductive strategies in D. colletei, enabling its adaptation and persistence in Borneo's diverse aquatic ecosystems, and call for further ecological research to elucidate these mechanisms. This study lends to a better understanding of viviparous fish in Borneo and contributes to the broader field of aquatic ecology, enhancing our knowledge of species adaptations to unique ecological challenges.
The ability to recreate computational results with minimal effort and actionable metrics provides a solid foundation for scientific research and software development. When people can replicate an analysis at the touch of a button using open-source software, open data, and methods to assess and compare proposals, it significantly eases verification of results, engagement with a diverse range of contributors, and progress. However, we have yet to fully achieve this; there are still many sociotechnical frictions.
Inspired by David Donoho's vision, this talk aims to revisit the three crucial pillars of frictionless reproducibility (data sharing, code sharing, and competitive challenges) with the perspective of deep software variability.
Our observation is that multiple layers — hardware, operating systems, third-party libraries, software versions, input data, compile-time options, and parameters — are subject to variability that exacerbates frictions but is also essential for achieving robust, generalizable results and fostering innovation. I will first review the literature, providing evidence of how the complex variability interactions across these layers affect qualitative and quantitative software properties, thereby complicating the reproduction and replication of scientific studies in various fields.
I will then present some software engineering and AI techniques that can support the strategic exploration of variability spaces. These include the use of abstractions and models (e.g., feature models), sampling strategies (e.g., uniform, random), cost-effective measurements (e.g., incremental build of software configurations), and dimensionality reduction methods (e.g., transfer learning, feature selection, software debloating).
I will finally argue that deep variability is both the problem and solution of frictionless reproducibility, calling the software science community to develop new methods and tools to manage variability and foster reproducibility in software systems.
Exposé invité Journées Nationales du GDR GPL 2024
Deep Software Variability and Frictionless Reproducibility
Space fullerenes: A computer search of new Frank-Kasper structures
1. Space fullerenes: A computer search of new
Frank-Kasper structures
Olaf Delgado Friedrichs
The Australian National
University, Canberra
Michel Deza
´Ecole Normale Sup´erieure, Paris
Mathieu Dutour Sikiri´c
Institut Rudjer Bo˘skovi´c, Zagreb
June 8, 2018
3. Fullerenes
A fullerene is a 3-valent plane graph, whose faces are 5 or
6-gonal.
They exist for any even n ≥ 20, n = 22.
There exist extremely efficient programs to enumerate them
(FullGen by G. Brinkman, CPF by T. Harmuth)
Fullerenes with isolated pentagons have n ≥ 60. The smallest
one:
Truncated icosahedron,
soccer ball,
Buckminsterfullerene
4. Frank Kasper structures
There are exactly 4 fullerenes with isolated hexagons:
20, Ih 24 D6d 26, D3h 28, Td
A Space-fullerene structure is a 4-valent 3-periodic tiling of
R3 by those 4 fullerenes.
They were introduced by Frank & Kasper in two papers in
1958, 1959 in order to explain a variety of crystallographic
structures in a unified way.
The basic problems are:
Find the possible structures, they are very rare.
Find some general constructions.
Find structural properties.
5. Known Physical phases I
group is the space group according to the crystallographic
tables
fund. dom. is the number of cells in a fundamental domain.
fraction (x20, x24, x26, x28) is the relative number of 20-, 24-,
26- and 28-cells in
phase rep. alloy group fund. dom. fraction
C14 MgZn2 P63/mmc 12 (2, 0, 0, 1)
C15 MgCu2 Fd3m 24 (2, 0, 0, 1)
C36 MgNi2 P63/mmc 24 (2, 0, 0, 1)
6-layers MgCuNi P63/mmc 36 (2, 0, 0, 1)
8-layers MgZn2 + 0.03MgAg2 P63/mmc 48 (2, 0, 0, 1)
9-layers MgZn2 + 0.07MgAg2 R3m 54 (2, 0, 0, 1)
10-layers MgZn2 + 0.1MgAg2 P63/mmc 60 (2, 0, 0, 1)
− Mg4Zn7 C2/m 110 (35, 2, 2, 16)
X Mn45Co40Si15 Pnnm 74 (23, 2, 2, 10)
T Mg32(Zn, Al)49 Im3 162 (49, 6, 6, 20)
C V2(Co, Si)3 C2/m 50 (15, 2, 2, 6)
7. The Laves phases
Laves phases are structures defined by stacking different layers
of F28 together with two choices at every step. Thus a symbol
(xi )−∞≤i≤∞ with xi = ±1 describes them.
All structures with x26 = x24 = 0 are Laves phases and a great
many compounds are of this type.
Frank & Kasper, 1959 generalize the construction to sequence
with xi = 0, ±1.
P63/mmc, 12 Fd3m, 24 P63/mmc, 24
P63/mmc, 36 P63/mmc, 48 R3m, 54
8. Some other structures
Also in some mixed clathrate “ice-like” hydrates:
t.c.p. alloys exp. clathrate # 20 # 24 # 26 # 28
A15 Cr3Si I:4Cl2.7H2O 1 3 0 0
C15 MgCu2 II:CHCl3.17H2O 2 0 0 1
Z Zr4Al3 III:Br2.86H2O 3 2 2 0
vertices are H2O, hydrogen bonds, cells are sites of solutes
(Cl, Br, . . . ).
At the olympic games:
9. Kelvin problem I
The general Kelvin problem is to partition the Euclidean space
En by some cells of equal volume and to minimize the surface
between cells.
In dimension 2 the solution is known to be the hexagonal
structure:
T. Hales, The honeycomb conjecture. Discrete Comput.
Geom. 25 (2001) 1–22.
The solution in dimension 3 is not known but Kelvin proposed
a structure, which was the example to beat.
F. Almgren proposed to try to beat it by doing variational
optimization over periodic structures
10. Kelvin problem II
Kelvin’s partition Weaire, Phelan’s partition
Weaire-Phelan partition (A15) is 0.3% better than Kelvin’s
(Voronoi polytope of A∗
3)
Best is unknown
12. Flags and flag operators
A cell complex C is a family of cells with inclusion relations
such that the intersection of any two cells is either empty or a
single cell.
We also assume it to be pure of dimension d, i.e. all inclusion
maximal cell have dimension d.
It is closed (or has no boundary) if any d − 1 dimensional cell
is contained in two d-dimensional cells.
A flag is an increasing sequence Fn0 ⊂ Fn1 ⊂ · · · ⊂ Fnr of cells
of dimension n0, . . . , nr . (n0, . . . , nr ) is the type of the flag.
A flag is complete if its type is (0, . . . , d).
Denote by F(C) the set of complete flags of C.
If f = (F0, . . . , Fd ) is a complete flag and 0 ≤ i ≤ d then the
flag σi (f ) is the one differing from f only in the dimension i.
A cell complex C is completely described by the action of σi
on F(C).
The problem is that F(C) may well be infinite or very large to
be workable with.
13. Illustration with the cube
A flag and its images
F
v
e
Flag f
F
e
v’
Flag σ0(f )
F
v
e’
Flag σ1(f )
v
e
F’
Flag σ2(f )
The vertices, edges and faces correspond to orbits of σ1, σ2 ,
σ0, σ2 and σ0, σ1 .
14. Delaney symbol
Suppose C is a cell complex, with a group G acting on it. The
Delaney symbol of C with respect to G is a combinatorial
object containing:
The orbits Ok of complete flags under G
The action of σi on those orbits for 0 ≤ i ≤ d.
For every orbit Ok , take f ∈ Ok , the smallest m such that
(σi σj )m
(f ) = f is independent of f and denoted mi,j (k).
C quotiented by G is an orbifold.
If G = Aut(C) we speak simply of Delaney symbol of C
Theorem: If C is a simply connected manifold, then it is
entirely described by its Delaney symbol.
A.W.M. Dress, Presentations of discrete groups, acting on
simply connected manifolds, in terms of parametrized systems
of Coxeter matrices—a systematic approach, Advances in
Mathematics 63-2 (1987) 196–212.
This is actually a reminiscence of Poincar´e polyhedron
theorem.
15. The inverse recognition problem
Suppose we have a Delaney symbol D, i.e. the data of
permutations (σi )0≤i≤d and the matrices mij (k).
We want to know what is the universal cover man-
ifold C (and if it is Euclidean space).
Some cases:
If we have only 1 orbit of flag then the Delaney symbol is
simply a Coxeter Dynkin diagram and the decision problem is
related to the eigenvalues of the Coxeter matrix.
If d = 2 then we can associate a curvature c(D) to the
Delaney symbol and the sign determines whether C is a sphere,
euclidean plane or hyperbolic plane.
If d = 3 then the problem is related to hard questions in
3-dimensional topology. But the software Gavrog/3dt by O.
Delgado Friedrichs can actually decide those questions.
16. Functionalities of Gavrog/3dt
It can
Test for euclidicity of Delaney symbols, that is recognize when
C is Euclidean space.
Find the minimal Delaney symbol, i.e. the representation with
smallest fundamental domain and maximal group of symmetry.
Compute the space group of the crystallographic structure.
Test for isomorphism amongst minimal Delaney symbols.
Create pictures, i.e. metric informations from Delaney symbols.
All this depends on difficult questions of 3-dimensional
topology, now mostly solved. This means that in theory the
program does not always works, but in practice it does.
« O. Delgado Friedrichs, 3dt - Systre,
http://gavrog.sourceforge.net
« O. Delgado Friedrichs, Euclidicity criteria, PhD thesis.
18. Proposed enumeration method
All periodic tilings can be described combinatorially by
Delaney symbol.
But is it good for enumeration? No, because the number of
flags may be too large.
So, we choose not to use it for the generation of the tilings.
We are enumerating closed orientable 3-dimensional manifolds
with N maximal cells, i.e. with an additional requirement:
Every maximal cell C is adjacent only to maximal cells C with
C = C.
The crystallographic structure is obtained as universal cover.
A partial tiling is an agglomeration of tiles, possibly with some
holes.
The method is thus to add tiles in all possibilities and to
consider adding tiles in all possible ways.
19. Tree search
When we are computing all possibilities, we are adding
possible tiles one by one. All options are considered
sequentially.
This means that we need to store in memory only the previous
choices, i.e. if a structure is made of N maximal cells
C1, . . . , CN, then we simply have to store:
{C1}
{C1, C2}
{C1, C2, C3}
...
{C1, C2, . . . , CN}
This is memory efficient.
There are two basic movement in the tree: go deeper or go to
the next choice (at the same or lower depth).
20. Computer Science & Orderly enumeration
The tree search needs very little memory, if N cells are put,
we need to store only N + 1 levels.
It is entirely processors limited, with the program going over
all possibilities.
Since it is a tree search, we can parallelize very easily: Each
process runs its own part of the tree.
The problem is that a same structure can be obtained in
many equivalent ways.
With orderly enumeration one can enumerate lexicographically
minimal representations among all equivalent representations
See for more details
M. Dutour Sikiri´c, O. Delgado-Friedrichs, M. Deza, Space
fullerenes: computer search for new Frank-Kasper structures,
Acta crystallographica A 66 (2010) 602–615
M. Dutour Sikiri, M. Deza, Space fullerenes: computer search
for new Frank-Kasper structures II, Structural Chemistry 23-4
(2012) 1103–1114
22. Enumeration results
We enumerate periodic structures having a fundamental
domain containing at most N maximal cells.
Note that the cells are not all congruent, Dodecahedron is not
necessarily regular and the faces of “polytopes” can be curved.
For every structure, we have a fractional formula
(x20, x24, x26, x28).
For N = 20, we get 84 structures in 1 month of computations
on about 200 processors. Going from N to N + 1,
computation time multiply by around 2.3.
(1, 3, 0, 0) 1 (2, 0, 0, 1) 5 (3, 2, 2, 0) 4
(3, 3, 0, 1) 3 (3, 3, 2, 0) 1 (3, 4, 2, 0) 3
(4, 5, 2, 0) 1 (5, 2, 2, 1) 20 (5, 3, 0, 2) 3
(5, 8, 2, 0) 2 (6, 5, 2, 1) 6 (6, 11, 2, 0) 1
(7, 2, 2, 2) 5 (7, 4, 2, 2) 1 (7, 7, 4, 0) 1
(7, 8, 2, 1) 1 (8, 4, 4, 1) 2 (8, 5, 2, 2) 2
(9, 2, 2, 3) 1 (10, 3, 6, 1) 3 (10, 5, 2, 3) 6
(11, 1, 4, 3) 1 (11, 2, 2, 4) 11
23. The A15 structure (1, 3, 0, 0)
Uniquely determined by fractional composition.
37. Tiling by buckminsterfullerene
Does there exist space-fullerenes with maximal cells being
soccer balls (i.e. buckminsterfullerenes)?
Given a type T of flag and a closed cell complex C it is
possible to build a cell complex C(T), named Wythoff
construction, Shadow geometry, Grassmann geometry,
Kaleidoscope construction, etc.
Examples:
If T = {0}, then C(T) = C (identity)
If T = {d}, then C(T) = C∗
(i.e. the dualof C)
If T = {0, . . . , d}, then C(T) is the order complex.
The answer is that such space fullerenes are obtained by
applying T = {0, 1} to the Coxeter geometry of diagram
(5, 3, 5), which is hyperbolic. So, no such object exist as a
polytope or as a space-fullerene
A. Pasini, Four-dimensional football, fullerenes and diagram
geometry, Discrete Math 238 (2001) 115–130.
38. A special tiling by fullerenes
Deza and Shtogrin: There exist tilings by fullerenes different from
F20, F24, F26 and F28(Td ). By F20, F24 and its elongation
F36(D6h) in ratio 7 : 2 : 1;
Delgado Friedrichs, O’Keeffe: All tiling by fullerenes with at
most 7 kinds of flags: A15, C15, Z, σ and this one.
39. Yarmolyuk Kripyakevich conjecture
They conjectured that for a space fullerene to exist, we should
have
−x20 +
x24
3
+
7
6
x26 + 2x28 = 0
But some counterexamples were found:
Some other conjecture are broken.
40. The Sadoc-Mosseri inflation I
Call snubPrism5 the Dodecahedron and snubPrism6 the
fullerene F24.
Given a space fullerene T by cells P, we define the inflation
IFM(T ) to be the simple tiling such that
Every cell P contains a shrunken copy P of P in its interior.
On every vertices of P a F28 has been put.
On every face of P with m edges, a snub Prismm is put which
is contained in P.
Thus for individual cells F20, F24, F26, F28 the operations goes
as follows:
F20 → F20 + 12F20 + 20
4 F28
F24 → F24 + {12F20 + 2F24} + 24
4 F28
F26 → F26 + {12F20 + 3F24} + 26
4 F28
F28 → F28 + {12F20 + 4F24} + 28
4 F28
41. The Sadoc-Mosseri inflation II
The inflation on the A15 structure: the shrunken cells of A15
and the generated F28
42. The Frank Kasper Sullivan construction I
The construction is first described in Frank & Kasper, 1959
but a better reference is:
J.M. Sullivan, New tetrahedrally closed-packed structures.
We take a tiling of the plane by regular triangle and regular
squares and define from it a space fullerene with x28 = 0.
Every edge of the graph is assigned a color (red or blue) such
that
Triangles are monochromatic
colors alternate around a square.
Local structure is
F 26
F24
F24
F20
43. The Frank Kasper Sullivan construction II
The construction explains a number of structures:
F−phase
sigma
H−phase
Z
A15
J−phase
K−phase
Actually a structure with x28 = 0 is physically realized if and
only if it is obtained by this construction.
Another name is Hexagonal t.c.p. since there are infinite
columns of F24 on each vertex of the tesselation by triangle
and squares.
44. Pentagonal t.c.p. I. general
Those structures are described in
Shoemaker C.B. and Shoemaker D., Concerning systems for
the generation and coding of layered, tetrahedrally
closed-packed structures of intermetallic compounds, Acta
Crystallographica (1972) B28 2957–2965.
They generalize Laves phases, generalized Laves phases (by
Frank and Kasper) and various constructions by Pearson
Shoemaker and Kripyakevich.
The input of the construction is a plane tiling by, not
necessarily regular, quadrangles and triangles with vertex
configuration (36), (33, 42), (44), (35), (34, 4) and (35, 4)
being allowed. Some of the edges are doubled and the
non-doubled edges are colored in red and blue so that:
Every square contains exactly two doubled edges on opposite
sides.
Every triangle contains exactly one double edge.
For every face the non-doubled edges are of the same color.
If two faces share a black edge then their color (red or blue) is
the same if and only if their size are different.
45. Pentagonal t.c.p. II. general
The result is a FK space fullerene with x24 = x26.
The structure is organized in layers with alternating structures.
We have:
chains of Dodecahedron on each vertex (hence the name
Pentagonal t.c.p.).
Dodecahedron on doubled edges
24-cells and 26-cells inside squres.
28-cells near the triangles.