1. The triangle PQR is equilateral if the lines l1 and l2 intersecting at K satisfy KP = KQ. This is proved by showing that ∆KPO1O2 and ∆PQR are isosceles, with angles of 30 degrees, making ∆PQR equilateral.
2. The only positive integer solutions to m(4m^2 + m + 12) = 3(pn - 1) are m = 12, n = 4, p = 7.
3. The polynomial x^4 - ax^3 - bx^2 - cx - d cannot have an integer solution because its roots must be either integers or irrational in pairs, but
The document contains a 10 question diagnostic math test involving proportional relationships between variables. The questions test concepts such as direct and inverse variation, using tables of values to determine relationships, and setting up and solving equations involving proportional variables.
1) The document contains examples of direct, inverse, and joint variations. It provides the definitions and formulas for each type of variation.
2) Examples are given for expressing variables in terms of other variables for different situations involving direct, inverse, and joint variations. The values of constants are calculated.
3) Tables are included that require calculating missing values based on the given variations and values.
A note on arithmetic progressions in sets of integersLukas Nabergall
This document presents a new upper bound on r3(n), the maximum size of a set of integers between 1 and n that contains no three elements in arithmetic progression. The author proves that r3(n) = O(n/log^h n) for any arbitrarily large h, improving on previous bounds. The proof uses the fundamental theorem of discrete calculus and the pigeonhole principle to show that any sufficiently dense set of integers must contain arbitrarily long arithmetic progressions. This verifies a 1936 conjecture of Erdos and improves understanding of a major problem in combinatorics.
The document summarizes research on finding polychromatic solutions to linear equations in an r-bounded coloring of the natural numbers. The research builds on previous work exploring rainbow analogues of classical Ramsey theory results. Key findings include:
1) Theorems were proven showing that for equations of the form ax-by=c with integers a,b,c and gcd(a,b)|c, there exists a polychromatic solution in any r-bounded coloring.
2) Explicit recurrence relations and formulas were defined to generate chains of solutions to equations with two variables.
3) The results for two variables were used to show polychromatic solutions exist for equations with three variables of the form a1
Construction of BIBD’s Using Quadratic Residuesiosrjce
IOSR Journal of Mathematics(IOSR-JM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
This document contains a summary of 8 mathematics questions on the topic of sets. Each question contains 1-4 parts asking students to shade regions on Venn diagrams, list set elements, or calculate set properties like union and intersection. The document also provides the answers to each question in point form for easy reference.
The document contains a 10 question diagnostic math test involving proportional relationships between variables. The questions test concepts such as direct and inverse variation, using tables of values to determine relationships, and setting up and solving equations involving proportional variables.
1) The document contains examples of direct, inverse, and joint variations. It provides the definitions and formulas for each type of variation.
2) Examples are given for expressing variables in terms of other variables for different situations involving direct, inverse, and joint variations. The values of constants are calculated.
3) Tables are included that require calculating missing values based on the given variations and values.
A note on arithmetic progressions in sets of integersLukas Nabergall
This document presents a new upper bound on r3(n), the maximum size of a set of integers between 1 and n that contains no three elements in arithmetic progression. The author proves that r3(n) = O(n/log^h n) for any arbitrarily large h, improving on previous bounds. The proof uses the fundamental theorem of discrete calculus and the pigeonhole principle to show that any sufficiently dense set of integers must contain arbitrarily long arithmetic progressions. This verifies a 1936 conjecture of Erdos and improves understanding of a major problem in combinatorics.
The document summarizes research on finding polychromatic solutions to linear equations in an r-bounded coloring of the natural numbers. The research builds on previous work exploring rainbow analogues of classical Ramsey theory results. Key findings include:
1) Theorems were proven showing that for equations of the form ax-by=c with integers a,b,c and gcd(a,b)|c, there exists a polychromatic solution in any r-bounded coloring.
2) Explicit recurrence relations and formulas were defined to generate chains of solutions to equations with two variables.
3) The results for two variables were used to show polychromatic solutions exist for equations with three variables of the form a1
Construction of BIBD’s Using Quadratic Residuesiosrjce
IOSR Journal of Mathematics(IOSR-JM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
This document contains a summary of 8 mathematics questions on the topic of sets. Each question contains 1-4 parts asking students to shade regions on Venn diagrams, list set elements, or calculate set properties like union and intersection. The document also provides the answers to each question in point form for easy reference.
This document contains lecture notes from a Calculus I class discussing optimization problems. It begins with announcements about upcoming exams and courses the professor is teaching. It then presents an example problem about finding the rectangle of a fixed perimeter with the maximum area. The solution uses calculus techniques like taking the derivative to find the critical points and determine that the optimal rectangle is a square. The notes discuss strategies for solving optimization problems and summarize the key steps to take.
Birkhoff coordinates for the Toda Lattice in the limit of infinitely many par...Alberto Maspero
We study the Birkhoff coordinates (Cartesian action angle coordinates) of the Toda lattice with periodic boundary condition in the limit where the number N of the particles tends to infinity. We prove that the transformation introducing such coordinates maps analytically a complex ball of radius R/Nα (in discrete Sobolev-analytic norms) into a ball of radius R′/Nα (with R,R′>0 independent of N) if and only if α≥2. Then we consider the problem of equipartition of energy in the spirit of Fermi-Pasta-Ulam. We deduce that corresponding to initial data of size R/N2, 0<R≪1, and with only the first Fourier mode excited, the energy remains forever in a packet of Fourier modes exponentially decreasing with the wave number. Finally we consider the original FPU model and prove that energy remains localized in a similar packet of Fourier modes for times one order of magnitude longer than those covered by previous results which is the time of formation of the packet. The proof of the theorem on Birkhoff coordinates is based on a new quantitative version of a Vey type theorem by Kuksin and Perelman which could be interesting in itself.
Talk given at the Twelfth Workshop on Non-Perurbative Quantum Chromodynamics ...Marco Frasca
This document summarizes the key steps in deriving an effective Nambu–Jona-Lasinio (NJL) model from QCD in the infrared limit. It shows that QCD can be written as a Gaussian theory for gluon fields, with a trivial infrared fixed point. This leads to a Yukawa interaction between quarks and an effective scalar field, along with a nonlocal four-quark interaction. Truncating to the lowest scalar excitation reproduces the NJL model, with couplings determined from the gluon propagator.
The document presents a decomposition method for solving indefinite quadratic programming problems with n variables and m linear constraints. The method decomposes the original problem into at most m subproblems, each with dimension n-1 and m linear constraints. All global minima, isolated local minima, and some non-isolated local minima of the original problem can be obtained by combining the solutions of the subproblems. The subproblems can then be further decomposed into smaller subproblems until 1-dimensional subproblems are reached, which can be solved directly.
This document provides examples and exercises related to deriving rate laws from reaction mechanisms using steady-state and rate-determining step approximations. It includes examples of applying these approximations to mechanisms with elementary steps and intermediates to obtain overall rate expressions in terms of the initial reactants and rate constants. It also asks the reader to derive rate laws and relaxation times for several reaction mechanisms following these same approaches.
1. The document contains 10 mathematics word problems involving matrix operations and inverses.
2. The problems require finding inverse matrices, solving systems of equations using matrices, and calculating values that satisfy matrix equations.
3. Detailed step-by-step solutions are provided for each problem.
This document contains a past paper for the IITJEE physics exam from 2009. It includes 3 sections - multiple choice questions with single answers, multiple choice questions with multiple answers, and questions requiring integer answers. The document provides the questions and options/choices for each question but does not include the solutions. It addresses topics in physics including photoelectric effect, simple harmonic motion, centripetal force, gas laws, electromagnetic induction, sound waves, and fluid mechanics.
Correspondence analysis is a technique for approximating a contingency table with lower rank tables to analyze the relationship between two categorical variables. It works by finding pairs of correspondence factors that have unit variance with respect to the marginal distributions and are maximally correlated. The correspondence factors and their correlations are obtained from the singular value decomposition of a normalized contingency table. Hypothesis tests can then be conducted to test the independence of the categorical variables and how well a lower rank approximation fits the data. The analysis also provides a spatial representation of the row and column categories in lower dimensions.
This document summarizes Dorian Ehrlich's thesis investigating the quantum case of Horn's question. Horn's question asks for which sequences of real numbers γ can there exist Hermitian matrices A, B, and C=A+B with eigenvalues α, β, and γ. Knutson and Tao provided a solution using intersections of Schubert varieties. Ehrlich seeks to generalize this to intersections linked by smooth curves, known as the quantum case. The document provides background on Hermitian matrices, Schubert varieties, and Knutson and Tao's solution. It then outlines Ehrlich's plan to prove a "quantum analogue" of Knutson and Tao's saturation conjecture to address the quantum case of Horn's question.
The document discusses recursive algorithms and recurrence relations. It provides examples of solving recurrence relations for different algorithms like Towers of Hanoi, selection sort, and merge sort. Recurrence relations define algorithms recursively in terms of smaller inputs. They are solved to find closed-form formulas for the running time of algorithms.
This presentation provides an introduction to Galois fields, which are finite fields with a prime number of elements. The objectives are to discuss preliminaries like sets and groups, introduce Galois fields and provide examples, discuss related theorems, and describe the computational approach. A sample computation in FORTRAN verifies the theorem that any element in a Galois field can be expressed as the sum of two squares.
This document summarizes Markowitz's mean-variance portfolio theory and the two-fund theorem.
[1] Markowitz formulated the mean-variance model, which minimizes portfolio variance subject to a target expected return. The optimal weights are a function of the covariance matrix and target mean.
[2] The two-fund theorem states that any efficient portfolio can be replicated as a combination of two "fundamental" portfolios. Investors only need to invest in these two funds.
[3] The minimum variance set forms the left boundary of the feasible region in mean-variance space. Portfolios on this boundary are efficient funds.
This document provides the answer key to homework #7 for CHEM 444. It includes 3 chemistry problems dealing with equations of state for gases and thermodynamic derivatives. The solutions show the steps to derive the requested equations, citing relevant equations from the textbook. A point value is given for each part of each problem. Notes are included to explain aspects of the solutions and emphasize conceptual understanding over just citing equations.
The document discusses recursive definitions of sequences, functions, sets, and strings. It provides examples of recursively defining the Fibonacci sequence, factorial function, set of prices using quarters and dimes, and set of binary numbers. It also discusses recursively defining the length, empty string, concatenation, and reversal of strings.
This document discusses the divide and conquer algorithm called merge sort. It begins by explaining the general divide and conquer approach of dividing a problem into subproblems, solving the subproblems recursively, and then combining the solutions. It then provides an example of how merge sort uses this approach to sort a sequence. It walks through the recursive merge sort algorithm on a sample input. The document explains the merge procedure used to combine the sorted subproblems and proves its correctness. It analyzes the running time of merge sort using recursion trees and determines it is O(n log n). Finally, it introduces recurrence relations and methods like substitution, recursion trees, and the master theorem for solving recurrences.
This document summarizes the solutions to seven graph equations involving line graphs L(G), complements G, and n-th power graphs Gn. The authors solve equations of the form L(G) = H, G = H, and L(G)n = H for various graphs G and H. They prove theorems characterizing the graphs that satisfy equations like G = L(G)n for n ≥ 2. The solutions generalize previous results on related equations.
This document discusses sequences and their properties. It defines sequences and provides examples of different types of sequences including arithmetic, geometric, harmonic, and Fibonacci sequences. It discusses finding formulas for the nth term of sequences and calculating sums of sequence terms. Examples are provided to demonstrate finding sequence terms, recognizing sequence types based on patterns of terms, and using recurrence relations to define sequences.
This document provides instructions for the 28th Indian National Mathematical Olympiad exam to be held on February 03, 2013. It states that calculators and protractors are not allowed, but rulers and compasses are. It includes 6 multi-part math problems to be solved on separate pages with clear numbering. The problems cover topics like properties of circles touching externally, positive integer solutions to equations, properties of polynomial equations, subsets with integer mean averages, relationships between areas of triangles formed by triangle centers, and inequalities relating positive real numbers.
This document discusses different coordinate systems used to describe points in two-dimensional and three-dimensional spaces, including polar, cylindrical, and spherical coordinates. It provides the key formulas for converting between Cartesian and these other coordinate systems, and gives examples of performing these conversions as well as writing equations of basic geometric shapes in different coordinate systems.
This document discusses different coordinate systems used to describe points in 2D and 3D space, including polar, cylindrical, and spherical coordinates. It provides the key formulas for converting between Cartesian and these other coordinate systems. Examples are given of converting points and equations between the different coordinate systems. The key points are that polar coordinates use an angle and distance to specify a 2D point, cylindrical coordinates extend this to 3D using a z-value, and spherical coordinates specify a 3D point using a distance from the origin, an angle, and an azimuthal angle.
The shortest distance between skew linesTarun Gehlot
The document discusses finding the angle and distance between two skew lines. It provides solutions for when a point and direction are given on each line, and for when the edges of a tetrahedron are known. The angle can be found using the dot product of the lines' direction vectors. The distance is the projection of the vector between points onto the cross product of the direction vectors. For a tetrahedron, the angle and distance are related to the lengths of opposite edges and the tetrahedron's volume.
This document contains lecture notes from a Calculus I class discussing optimization problems. It begins with announcements about upcoming exams and courses the professor is teaching. It then presents an example problem about finding the rectangle of a fixed perimeter with the maximum area. The solution uses calculus techniques like taking the derivative to find the critical points and determine that the optimal rectangle is a square. The notes discuss strategies for solving optimization problems and summarize the key steps to take.
Birkhoff coordinates for the Toda Lattice in the limit of infinitely many par...Alberto Maspero
We study the Birkhoff coordinates (Cartesian action angle coordinates) of the Toda lattice with periodic boundary condition in the limit where the number N of the particles tends to infinity. We prove that the transformation introducing such coordinates maps analytically a complex ball of radius R/Nα (in discrete Sobolev-analytic norms) into a ball of radius R′/Nα (with R,R′>0 independent of N) if and only if α≥2. Then we consider the problem of equipartition of energy in the spirit of Fermi-Pasta-Ulam. We deduce that corresponding to initial data of size R/N2, 0<R≪1, and with only the first Fourier mode excited, the energy remains forever in a packet of Fourier modes exponentially decreasing with the wave number. Finally we consider the original FPU model and prove that energy remains localized in a similar packet of Fourier modes for times one order of magnitude longer than those covered by previous results which is the time of formation of the packet. The proof of the theorem on Birkhoff coordinates is based on a new quantitative version of a Vey type theorem by Kuksin and Perelman which could be interesting in itself.
Talk given at the Twelfth Workshop on Non-Perurbative Quantum Chromodynamics ...Marco Frasca
This document summarizes the key steps in deriving an effective Nambu–Jona-Lasinio (NJL) model from QCD in the infrared limit. It shows that QCD can be written as a Gaussian theory for gluon fields, with a trivial infrared fixed point. This leads to a Yukawa interaction between quarks and an effective scalar field, along with a nonlocal four-quark interaction. Truncating to the lowest scalar excitation reproduces the NJL model, with couplings determined from the gluon propagator.
The document presents a decomposition method for solving indefinite quadratic programming problems with n variables and m linear constraints. The method decomposes the original problem into at most m subproblems, each with dimension n-1 and m linear constraints. All global minima, isolated local minima, and some non-isolated local minima of the original problem can be obtained by combining the solutions of the subproblems. The subproblems can then be further decomposed into smaller subproblems until 1-dimensional subproblems are reached, which can be solved directly.
This document provides examples and exercises related to deriving rate laws from reaction mechanisms using steady-state and rate-determining step approximations. It includes examples of applying these approximations to mechanisms with elementary steps and intermediates to obtain overall rate expressions in terms of the initial reactants and rate constants. It also asks the reader to derive rate laws and relaxation times for several reaction mechanisms following these same approaches.
1. The document contains 10 mathematics word problems involving matrix operations and inverses.
2. The problems require finding inverse matrices, solving systems of equations using matrices, and calculating values that satisfy matrix equations.
3. Detailed step-by-step solutions are provided for each problem.
This document contains a past paper for the IITJEE physics exam from 2009. It includes 3 sections - multiple choice questions with single answers, multiple choice questions with multiple answers, and questions requiring integer answers. The document provides the questions and options/choices for each question but does not include the solutions. It addresses topics in physics including photoelectric effect, simple harmonic motion, centripetal force, gas laws, electromagnetic induction, sound waves, and fluid mechanics.
Correspondence analysis is a technique for approximating a contingency table with lower rank tables to analyze the relationship between two categorical variables. It works by finding pairs of correspondence factors that have unit variance with respect to the marginal distributions and are maximally correlated. The correspondence factors and their correlations are obtained from the singular value decomposition of a normalized contingency table. Hypothesis tests can then be conducted to test the independence of the categorical variables and how well a lower rank approximation fits the data. The analysis also provides a spatial representation of the row and column categories in lower dimensions.
This document summarizes Dorian Ehrlich's thesis investigating the quantum case of Horn's question. Horn's question asks for which sequences of real numbers γ can there exist Hermitian matrices A, B, and C=A+B with eigenvalues α, β, and γ. Knutson and Tao provided a solution using intersections of Schubert varieties. Ehrlich seeks to generalize this to intersections linked by smooth curves, known as the quantum case. The document provides background on Hermitian matrices, Schubert varieties, and Knutson and Tao's solution. It then outlines Ehrlich's plan to prove a "quantum analogue" of Knutson and Tao's saturation conjecture to address the quantum case of Horn's question.
The document discusses recursive algorithms and recurrence relations. It provides examples of solving recurrence relations for different algorithms like Towers of Hanoi, selection sort, and merge sort. Recurrence relations define algorithms recursively in terms of smaller inputs. They are solved to find closed-form formulas for the running time of algorithms.
This presentation provides an introduction to Galois fields, which are finite fields with a prime number of elements. The objectives are to discuss preliminaries like sets and groups, introduce Galois fields and provide examples, discuss related theorems, and describe the computational approach. A sample computation in FORTRAN verifies the theorem that any element in a Galois field can be expressed as the sum of two squares.
This document summarizes Markowitz's mean-variance portfolio theory and the two-fund theorem.
[1] Markowitz formulated the mean-variance model, which minimizes portfolio variance subject to a target expected return. The optimal weights are a function of the covariance matrix and target mean.
[2] The two-fund theorem states that any efficient portfolio can be replicated as a combination of two "fundamental" portfolios. Investors only need to invest in these two funds.
[3] The minimum variance set forms the left boundary of the feasible region in mean-variance space. Portfolios on this boundary are efficient funds.
This document provides the answer key to homework #7 for CHEM 444. It includes 3 chemistry problems dealing with equations of state for gases and thermodynamic derivatives. The solutions show the steps to derive the requested equations, citing relevant equations from the textbook. A point value is given for each part of each problem. Notes are included to explain aspects of the solutions and emphasize conceptual understanding over just citing equations.
The document discusses recursive definitions of sequences, functions, sets, and strings. It provides examples of recursively defining the Fibonacci sequence, factorial function, set of prices using quarters and dimes, and set of binary numbers. It also discusses recursively defining the length, empty string, concatenation, and reversal of strings.
This document discusses the divide and conquer algorithm called merge sort. It begins by explaining the general divide and conquer approach of dividing a problem into subproblems, solving the subproblems recursively, and then combining the solutions. It then provides an example of how merge sort uses this approach to sort a sequence. It walks through the recursive merge sort algorithm on a sample input. The document explains the merge procedure used to combine the sorted subproblems and proves its correctness. It analyzes the running time of merge sort using recursion trees and determines it is O(n log n). Finally, it introduces recurrence relations and methods like substitution, recursion trees, and the master theorem for solving recurrences.
This document summarizes the solutions to seven graph equations involving line graphs L(G), complements G, and n-th power graphs Gn. The authors solve equations of the form L(G) = H, G = H, and L(G)n = H for various graphs G and H. They prove theorems characterizing the graphs that satisfy equations like G = L(G)n for n ≥ 2. The solutions generalize previous results on related equations.
This document discusses sequences and their properties. It defines sequences and provides examples of different types of sequences including arithmetic, geometric, harmonic, and Fibonacci sequences. It discusses finding formulas for the nth term of sequences and calculating sums of sequence terms. Examples are provided to demonstrate finding sequence terms, recognizing sequence types based on patterns of terms, and using recurrence relations to define sequences.
This document provides instructions for the 28th Indian National Mathematical Olympiad exam to be held on February 03, 2013. It states that calculators and protractors are not allowed, but rulers and compasses are. It includes 6 multi-part math problems to be solved on separate pages with clear numbering. The problems cover topics like properties of circles touching externally, positive integer solutions to equations, properties of polynomial equations, subsets with integer mean averages, relationships between areas of triangles formed by triangle centers, and inequalities relating positive real numbers.
This document discusses different coordinate systems used to describe points in two-dimensional and three-dimensional spaces, including polar, cylindrical, and spherical coordinates. It provides the key formulas for converting between Cartesian and these other coordinate systems, and gives examples of performing these conversions as well as writing equations of basic geometric shapes in different coordinate systems.
This document discusses different coordinate systems used to describe points in 2D and 3D space, including polar, cylindrical, and spherical coordinates. It provides the key formulas for converting between Cartesian and these other coordinate systems. Examples are given of converting points and equations between the different coordinate systems. The key points are that polar coordinates use an angle and distance to specify a 2D point, cylindrical coordinates extend this to 3D using a z-value, and spherical coordinates specify a 3D point using a distance from the origin, an angle, and an azimuthal angle.
The shortest distance between skew linesTarun Gehlot
The document discusses finding the angle and distance between two skew lines. It provides solutions for when a point and direction are given on each line, and for when the edges of a tetrahedron are known. The angle can be found using the dot product of the lines' direction vectors. The distance is the projection of the vector between points onto the cross product of the direction vectors. For a tetrahedron, the angle and distance are related to the lengths of opposite edges and the tetrahedron's volume.
This document contains the solutions to problems from the 2018 Canadian Mathematical Olympiad. The first summary discusses a problem about arranging tokens on a plane and moving them to the same point via midpoint moves. The solution proves that every arrangement is collapsible if and only if the number of tokens is a power of 2. The second summary is about points on a circle where two lengths are equal, and proving a line is perpendicular to another line. The third summary asks for all positive integers with at least three divisors that can be arranged in a circle such that adjacent divisors are prime-related, and the solution shows these are integers that are neither a perfect square nor a power of a prime.
(1) The document is the front cover and instructions for a mathematics preliminary examination. It provides instructions such as writing one's name and index number, answering all questions, showing working, and bundling all work together at the end.
(2) The examination contains 14 pages with 80 total marks across multiple choice and written answer questions involving topics like algebra, trigonometry, calculus, statistics, and geometry.
(3) Several mathematical formulas are provided for reference, including formulas for compound interest, mensuration, trigonometry, and statistics. Candidates are advised to use these formulas where appropriate.
This document provides information on the electron configuration and arrangement of electrons in atoms. It shows the proton and neutron makeup of elements from helium to aluminum. It then illustrates the electron configuration using shorthand notation for elements oxygen through sodium. The rest of the document defines quantum numbers such as principal energy level, azimuthal, magnetic, electron spin, and how they relate to the shape and orientation of atomic orbitals in an atom.
This document contains sample problems and questions related to thermodynamic processes and the first law of thermodynamics. It defines key terms like work (w), heat (q), internal energy change (ΔU), and enthalpy change (ΔH) for various thermodynamic processes including isobaric, isochoric, isothermal, reversible adiabatic, and irreversible processes. It then provides examples of calculating w, q, ΔU, and ΔH for gas expansion/compression processes under different conditions. Finally, it includes some multiple choice questions testing understanding of concepts like signs of w and q and properties of closed, open, and isolated systems.
The document discusses specific energy, which is the total energy of a channel flow referenced to the channel bed. Specific energy is constant for uniform flow but can increase or decrease for varied flow. Critical flow occurs when specific energy is at a minimum, corresponding to a Froude number of 1. For a rectangular channel, the critical depth formula and specific energy at critical depth are derived. The analysis is also extended to a triangular channel.
This document contains instructions and questions for a mathematics exam. It provides information about the exam such as the date, time allowed, materials permitted, and instructions for completing and submitting the exam. The exam contains 7 multi-part questions testing a variety of mathematics concepts including algebra, geometry, trigonometry, statistics, and matrix operations.
This document provides information about algebraic formulae including variables, constants, writing formulae based on situations, finding the subject of a formula, and solving for variable values. It includes examples and practice problems with solutions related to these concepts. The document is divided into sections covering variables and constants, formulae, the subject of a formula, and finding the value of a variable. Practice questions with answers are provided throughout for additional examples.
Let D be the given determinant. Then,
D = |1 x x^2|
|x 1 x|
|x^2 x 1|
Using C1 → C1,
D = |1-(x^3) x x^2|
|x 1 x|
|x^2 x 1|
Using C2 → C2 - xC1,
D = |1-(x^3) x x^2|
|0 1-x|
|x^2 x 1|
Using C3 → C3 - x^2C1,
D = |1-(x^3) x x^2|
|0 1
Ordinary abelian varieties having small embedding degreePaula Valenca
International Workshop on Pairings in Cryptography 12-15 June 2005, Dublin, Ireland and
`Mathematical Problems and Techniques in Cryptology' workshop, Barcelona, June 2005
Slides for the 2005 paper: S. D. Galbraith, J. McKee and P. Valenca, "Ordinary abelian varieties having small embedding degree"
1) The document contains solutions to 4 problems involving sequences, triangles, and number theory.
2) In problem 1, it is shown that if P and Q are points of intersection of lines drawn from a point M inside a triangle ABC to its circumcircle, then PQ is parallel to one of the sides of ABC.
3) Problem 2 finds all natural numbers n such that n^2 does not divide (n-2)!. The solutions are primes, twice a prime, and 8 and 9.
4) Problem 3 solves a system of equations and finds the only real solution is x=y=z=1/3.
The document discusses key concepts in seismology including:
1. Snell's law is generalized for spherical earth models using ray parameters.
2. The ray equation relates the change in ray geometry to variations in seismic velocity.
3. Radius of curvature is determined by velocity gradients and ray parameters.
4. Amplitude is affected by geometrical spreading and focusing/defocusing of rays due to velocity variations.
5. Tau-p analysis represents seismic travel times through intercept time curves as a function of ray parameter.
1) The Klein-Gordon equation describes spin-0 particles and satisfies relativistic energy-momentum relationships. It reduces to the Schrodinger equation in the non-relativistic limit.
2) The Klein-Gordon equation has issues with negative probability densities that are resolved by defining a conserved 4-current. This leads to a correct definition of particle density.
3) Scattering processes involving Klein-Gordon particles can be described using Feynman rules with additional factors for vertices and internal photon lines. This allows calculating scattering amplitudes relativistically.
1. The document discusses using the binomial expansion and Stirling's formula to estimate the value of r that maximizes a binomial coefficient expression as n becomes large. Taking the limit as n approaches infinity, the optimal value of r is shown to be nq.
2. A example is given of estimating the probability of winning $40 or more by betting $1 on number 8 in roulette 500 times. Using the normal approximation, this probability is estimated to be about 25.8%.
1. The document discusses linear elastic springs and examples of springs connected in series and in parallel. It introduces concepts of force-deformation relationships, compatibility, and using these together with equilibrium equations to solve statically indeterminate problems.
2. For springs in series, the total displacement is the sum of the individual spring displacements. The effective spring constant is calculated from the individual spring constants.
3. Solving examples involves setting up free body diagrams, writing equilibrium equations, using force-deformation relationships, and adding compatibility equations to solve for unknown forces and displacements.
This document discusses using complex numbers to solve combinatorial problems involving integer divisibility in an elegant way. It shows that the sum of the kth powers of the roots of unity is 0 if k is not divisible by the prime p, allowing one to determine divisibility of k by p. As an example, it uses this to count the number of subsets of an n-element set with cardinalities divisible by 3 by substituting roots of unity into the binomial theorem. Complex numbers allow expressing this concisely and performing algebraic manipulations.
A Visual Guide to 1 Samuel | A Tale of Two HeartsSteve Thomason
These slides walk through the story of 1 Samuel. Samuel is the last judge of Israel. The people reject God and want a king. Saul is anointed as the first king, but he is not a good king. David, the shepherd boy is anointed and Saul is envious of him. David shows honor while Saul continues to self destruct.
Gender and Mental Health - Counselling and Family Therapy Applications and In...PsychoTech Services
A proprietary approach developed by bringing together the best of learning theories from Psychology, design principles from the world of visualization, and pedagogical methods from over a decade of training experience, that enables you to: Learn better, faster!
How to Make a Field Mandatory in Odoo 17Celine George
In Odoo, making a field required can be done through both Python code and XML views. When you set the required attribute to True in Python code, it makes the field required across all views where it's used. Conversely, when you set the required attribute in XML views, it makes the field required only in the context of that particular view.
Level 3 NCEA - NZ: A Nation In the Making 1872 - 1900 SML.pptHenry Hollis
The History of NZ 1870-1900.
Making of a Nation.
From the NZ Wars to Liberals,
Richard Seddon, George Grey,
Social Laboratory, New Zealand,
Confiscations, Kotahitanga, Kingitanga, Parliament, Suffrage, Repudiation, Economic Change, Agriculture, Gold Mining, Timber, Flax, Sheep, Dairying,
How to Setup Warehouse & Location in Odoo 17 InventoryCeline George
In this slide, we'll explore how to set up warehouses and locations in Odoo 17 Inventory. This will help us manage our stock effectively, track inventory levels, and streamline warehouse operations.
This presentation was provided by Rebecca Benner, Ph.D., of the American Society of Anesthesiologists, for the second session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session Two: 'Expanding Pathways to Publishing Careers,' was held June 13, 2024.
1. 28th Indian National Mathematical Olympiad-2013
Time : 4 hours Februray 03, 2013
Instructions :
• Calculators (in any form) and protractors are not allowed.
• Rulers and compasses are allowed.
• Answer all the questions. All questions carry equal marks.
• Answer to each question should start on a new page. Clearly indicate the questions number.
Let and Γ be two circles touching each other externally at R. Let be a line which is tangent to Γ
1. Γ l at
1 2 2
P and passing through the centre of Γ . Similarly, be a line which is tangent to Γ at Q and
O let l passing
2
1 1
through the centre of Γ . Suppose l are not parallel and intersect at K. If KP = KQ, prove that
O and l the
2 1
triangle PQR is equilateral.
Q P
K
O
1 R O2
Sol
.
KP = KQ, So K lies on the common tangent
(Radical Axis)
Now ∆ KPQ ~ ∆KO1O2 & PQK is isoceles
∠KQP = ∠KO1O2
PQO1O2 is cyclic
∠KPQ = ∠KO2O1
So, ∆KO1O2 is also Isosceles So, KO1 = KO2 &
O1R = O2R, clearly in ∆O1PO2 ,
O1
PO2 O2
=
2
∠PO1O2 = 30º & similarly
so, ∠QO2O1 = 30º
∠O1KR = ∠O2KR = 60º & ∆PQR is
so, equilateral.
2. Page # 1
Find all positive integers m, n and primes p ≥ 5 such
2. that
m(4m2 + m + 12) = 3(pn – 1).
4
Sol m
. 3
+ m2 + 12m + 3 = 3pn
+
m 4m + 1 =
2 3 3pn
(
⇒ ) ; p ≥ 5 & prime
2
{so, m + 3 must be odd so m is even let m = 2a
4
a + 8a + 1 =
2
3 3pn
⇒ )
(
{Now a must be 3b or 3b + 1 because 3 is a factor so,
Case-1 - Let a = 3b
(36b2 + 3)(24b + 1) = 3pn
(12b2 + 1)(24b + 1) = pn
Now 24b + 1 must divide 12b2 + 1 & hence it must divide b -
2, so the only possibility is b = 2 & hence m = 12 & p = 7, n =
4 Case-2 : if a = 3b + 1
(36b2 + 24b + 7)(24b + 9) = 3pn
(36b2 + 24b + 7)(8b + 3) = pn
2
+ 24b +
so, (8b + 3) must divide 36b 7
Hence divides 49 which is not possible for b ∈ Ι
m,n =
so, 12,4
( ) ( )
3. Let a,b,c,d be positive integers such that a ≥ b ≥ c ≥ d. Prove that the equation x4 – ax3 – bx2 – cx – d =
0 has no integer solution.
2
−
Sol a − bx − cx − d = 0 & a ≥ b ≥ c ≥
. x x d
a,b,c,d ∈ N
p
is
Let α be a factor of d because other roots can’t be of the
q as coefficient of
1
form x .
so, roots are either integers or unreal or irrational in pairs. Now there may be atleast one more
root (say β )which is integer & it is also a factor of d.
3. −d ≤ α,β
So, ≤d
}
Now, f 0 = −d < 0 & f −1 = 1+ a − b + c − d > 0
) ( )
f(x) < 0 for x ∈ , So there is no positive integral
also 0,d root.
[
Also. for x ∈ −d,−1 ; f(x) > 0 so, no integral root in [-d, -1].
[ ]
Hence there is no integral root. {Though roots are in (-1, 0)}.
4. Let n be a positive integer. Call a nonempty subset S of {1,2,3,.....,n} good if the arithemtic mean of the
elements of S is also an integer. Further let to denote the number of good subsets of {1,2,3,.....,n}.
Prove that tn and n are both odd or both even.
Let A = {x1, x2 , x3 ,...xr } be a good subset, then there must be
Sol
. a
B n − n −
n+1 + x + − x ,... n
= x, 1 , 1 +1 x
s { r } which is also good. So, good subsets occur in
e )
(
2
t ( ) (
) 3 ( ) a
pair.
However, there are few cases when A = B, which means if x ∈
A n + 1 − x ∈ A . To count the
⇒
i ( ) i
number of these subsets.
Case-1 : If n is odd.
a. If the middle element is excluded, the no. of elements in such subsets is 2k.
(k before middle, & k elements after). So sum of hte elements will be k(n + 1), Apparently these sets
n −1
are good. So no. of these subsets is 2 2 1(i.e. odd)
−
Page # 2
4. b. Similarly if mid term
these subsets will
good.
So number os sub
22
so toal number of s
2
5 S
. o
,
if
n
i
s
o
d
d
.
R
e
s
t
o
S f
t
o h
e
s
u
b
s
e
t
s
a
r
e
o
c
c
u
r
i
n
g
i
n
p
a
i
r
a
n
d
t
h
e
c
o
m
p
l
e
t
5. e
C
a3cos A =
I 2
f si
n
A B
g si
s n
o C
I =
n 3
A −
E c
o
s
F (
H B
O +
C
G )
⇒ 3c
B Now,
ta
B
S
o 2s
1 C
⇒
(
Page
#3
Equ. 1 -
⇒
⇒
Equ. 2 -
so,
Let
a,b,c,x
,y,z be
positiv
e real
numb
ers
such