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MATHPAGES
Areas of Conic Quadrilaterals . A quadrilateral is said to be conic for a given eccentricity if all four vertices lie on a conic locus with that eccentricity.COPRIME PARTITIONS
Coprime Partitions Let F(N,k) denote the number of distinct partitions of N into exactly k mutually coprime parts. Obviously F(N,1)=1 for allN.
GRAVITATIONAL ACCELERATION IN SCHWARZSCHILD COORDINATES 6.7 Gravitational Acceleration in Schwarzschild Coordinates . If bodies, moved in any manner among themselves, are urged in the direction of parallel lines by equal accelerative forces, they will all continue to move among themselves, after the same manner as if they had not been urged by those forces. POLYOMINO ENUMERATIONS Polyomino Enumerations A popular computer game called "Tetris" involves manipulations of various clusters of four squares. The seven distinct types of clusters are illustrated below Clusters of connected squares like this were named "polyominoes" by Solomon Golomb. In particular, if four squares are joined together the result is called atetromino.
ANALYSIS OF RELATIVISTIC ORBITS Analysis of Relativistic Orbits . Two quite distinct methods have historically been used to derive the precession rate of relativistic orbits for a test particle moving in the gravitational field of a stationary spherical body of mass m.ROEMER'S HYPOTHESIS
Roemer's Hypothesis . Soon after the invention of the telescope, Galileo discovered the four largest moons of Jupiter. Subsequently many astronomers made careful observations of those moons, and already by the 1660's detailed tables of their movements PARTITIONS INTO DISTINCT PARTS Partitions Into Distinct Parts . For any positive integers n and k, let p k (n) denote the number of ways in which the integer n can be expressed as a sum of exactly k distinct positive integers, without regard to order. For example, the integer n = 12 can be expressed as a sum of three distinct positive integers in the following seven ways: THE PRISMOIDAL FORMULA The Prismoidal Formula: One of the oldest documents in existence is a papyrus roll written in Egypt around 1890 BC. This is sometimes called the Golenischev Papyrus, after the Russian who purchased it in Egypt in 1893 and brought it to Moscow, where it remains today. RATIONAL SINES OF RATIONAL MULTIPLES OF P Rational Sines of Rational Multiples of π . For which rational multiples of π is the sine rational? We have the three trivial cases . and we wish to show that these are essentially the only distinct rational sines of rational multiples of π. MATHPAGESREFLECTIONS ON RELATIVITYALGEBRAQUOTATIONSANIMATED ILLUSTRATIONSSET THEORY AND FOUNDATIONS Number Theory Combinatorics Geometry Algebra Calculus & Diff Eqs Probability & Statistics Set Theory & Foundations: Reflections on Relativity History PhysicsMATHPAGES
Areas of Conic Quadrilaterals . A quadrilateral is said to be conic for a given eccentricity if all four vertices lie on a conic locus with that eccentricity.COPRIME PARTITIONS
Coprime Partitions Let F(N,k) denote the number of distinct partitions of N into exactly k mutually coprime parts. Obviously F(N,1)=1 for allN.
GRAVITATIONAL ACCELERATION IN SCHWARZSCHILD COORDINATES 6.7 Gravitational Acceleration in Schwarzschild Coordinates . If bodies, moved in any manner among themselves, are urged in the direction of parallel lines by equal accelerative forces, they will all continue to move among themselves, after the same manner as if they had not been urged by those forces. POLYOMINO ENUMERATIONS Polyomino Enumerations A popular computer game called "Tetris" involves manipulations of various clusters of four squares. The seven distinct types of clusters are illustrated below Clusters of connected squares like this were named "polyominoes" by Solomon Golomb. In particular, if four squares are joined together the result is called atetromino.
ANALYSIS OF RELATIVISTIC ORBITS Analysis of Relativistic Orbits . Two quite distinct methods have historically been used to derive the precession rate of relativistic orbits for a test particle moving in the gravitational field of a stationary spherical body of mass m.ROEMER'S HYPOTHESIS
Roemer's Hypothesis . Soon after the invention of the telescope, Galileo discovered the four largest moons of Jupiter. Subsequently many astronomers made careful observations of those moons, and already by the 1660's detailed tables of their movements PARTITIONS INTO DISTINCT PARTS Partitions Into Distinct Parts . For any positive integers n and k, let p k (n) denote the number of ways in which the integer n can be expressed as a sum of exactly k distinct positive integers, without regard to order. For example, the integer n = 12 can be expressed as a sum of three distinct positive integers in the following seven ways: THE PRISMOIDAL FORMULA The Prismoidal Formula: One of the oldest documents in existence is a papyrus roll written in Egypt around 1890 BC. This is sometimes called the Golenischev Papyrus, after the Russian who purchased it in Egypt in 1893 and brought it to Moscow, where it remains today. RATIONAL SINES OF RATIONAL MULTIPLES OF P Rational Sines of Rational Multiples of π . For which rational multiples of π is the sine rational? We have the three trivial cases . and we wish to show that these are essentially the only distinct rational sines of rational multiples of π.MATHPAGES
Number Theory Combinatorics Geometry Algebra Calculus & Diff Eqs Probability & Statistics Set Theory & Foundations: Reflections on Relativity History PhysicsMATHPAGES
Timed Markov Models . One of the defining attributes of ordinary Markov models is that the state transitions are exponentially distributed, meaning that the transition rates are constant and therefore the transitions are continuous.MATHPAGES
From Schwarzschild to Droste . Love's not Time's fool, though rosy lips and cheeks. Within his bending sickle's compass come: Love alters not with his brief hours and weeks,MATHPAGES
Round Robin Ties . Intreat me not to leave thee, or to return from following after thee, for whither thou goest, I will go, and where thou lodgest, I will lodge.MATHPAGES
Areas of Conic Quadrilaterals . A quadrilateral is said to be conic for a given eccentricity if all four vertices lie on a conic locus with that eccentricity.MATHPAGES
2.4 Doppler Shift for Sound and Light. I was much further out than you thought. And not waving but drowning. Stevie Smith, 1957COPRIME PARTITIONS
Coprime Partitions Let F(N,k) denote the number of distinct partitions of N into exactly k mutually coprime parts. Obviously F(N,1)=1 for allN.
PARTITIONS INTO DISTINCT PARTS Partitions Into Distinct Parts . For any positive integers n and k, let p k (n) denote the number of ways in which the integer n can be expressed as a sum of exactly k distinct positive integers, without regard to order. For example, the integer n = 12 can be expressed as a sum of three distinct positive integers in the following seven ways:ANGULAR ANGST
Angular Angst . What is the angle denoted by α in the figure below? It isn't too difficult to determine with simple trigonometry that α is very nearly equal to 30 degrees, but to rigorously prove that it is exactly 30 degrees is not as trivial as one might think. THE FUNDAMENTAL ANAGRAM OF CALCULUS The Fundamental Anagram of Calculus . Here's an interesting quote from the correspondence of Isaac Newton: This is from the 2nd letter that Newton wrote to Leibniz (via Oldenburg) in 1677. MATHPAGESREFLECTIONS ON RELATIVITYALGEBRAQUOTATIONSANIMATED ILLUSTRATIONSSET THEORY AND FOUNDATIONS Number Theory Combinatorics Geometry Algebra Calculus & Diff Eqs Probability & Statistics Set Theory & Foundations: Reflections on Relativity History PhysicsMATHPAGES
Areas of Conic Quadrilaterals . A quadrilateral is said to be conic for a given eccentricity if all four vertices lie on a conic locus with that eccentricity. PARTITIONS INTO DISTINCT PARTS Partitions Into Distinct Parts . For any positive integers n and k, let p k (n) denote the number of ways in which the integer n can be expressed as a sum of exactly k distinct positive integers, without regard to order. For example, the integer n = 12 can be expressed as a sum of three distinct positive integers in the following seven ways:COPRIME PARTITIONS
Coprime Partitions. Let F (N,k) denote the number of distinct partitions of N into exactly k mutually coprime parts. Obviously F (N,1)=1 for all N. Also, we have F (N,2) = phi (N)/2 for all N, where phi () is the Euler totient function. The sequence of values F(N-1+j,j)
ROEMER'S HYPOTHESIS
Roemer's Hypothesis . Soon after the invention of the telescope, Galileo discovered the four largest moons of Jupiter. Subsequently many astronomers made careful observations of those moons, and already by the 1660's detailed tables of their movements DIFFERENCES BETWEEN POWERS Differences Between Powers . Now my charms are all o’erthrown. And what strength I have’s mine own, Which is most faint. Catalan conjectured that the equation THE PRISMOIDAL FORMULA The Prismoidal Formula: One of the oldest documents in existence is a papyrus roll written in Egypt around 1890 BC. This is sometimes called the Golenischev Papyrus, after the Russian who purchased it in Egypt in 1893 and brought it to Moscow, where it remains today. ANALYSIS OF RELATIVISTIC ORBITS Analysis of Relativistic Orbits . Two quite distinct methods have historically been used to derive the precession rate of relativistic orbits for a test particle moving in the gravitational field of a stationary spherical body of mass m. GRAVITATIONAL ACCELERATION IN SCHWARZSCHILD COORDINATES 6.7 Gravitational Acceleration in Schwarzschild Coordinates . If bodies, moved in any manner among themselves, are urged in the direction of parallel lines by equal accelerative forces, they will all continue to move among themselves, after the same manner as if they had not been urged by those forces. GAMALIEL'S PRINCIPLE Gamaliel's Principle . The new testament tells of how, after the apostles were imprisoned in Jerusalem, they were brought before the council of high priests, who questioned them about why they had disobeyed the order to stop preaching the word of Jesus. MATHPAGESREFLECTIONS ON RELATIVITYALGEBRAQUOTATIONSANIMATED ILLUSTRATIONSSET THEORY AND FOUNDATIONS Number Theory Combinatorics Geometry Algebra Calculus & Diff Eqs Probability & Statistics Set Theory & Foundations: Reflections on Relativity History PhysicsMATHPAGES
Areas of Conic Quadrilaterals . A quadrilateral is said to be conic for a given eccentricity if all four vertices lie on a conic locus with that eccentricity. PARTITIONS INTO DISTINCT PARTS Partitions Into Distinct Parts . For any positive integers n and k, let p k (n) denote the number of ways in which the integer n can be expressed as a sum of exactly k distinct positive integers, without regard to order. For example, the integer n = 12 can be expressed as a sum of three distinct positive integers in the following seven ways:COPRIME PARTITIONS
Coprime Partitions. Let F (N,k) denote the number of distinct partitions of N into exactly k mutually coprime parts. Obviously F (N,1)=1 for all N. Also, we have F (N,2) = phi (N)/2 for all N, where phi () is the Euler totient function. The sequence of values F(N-1+j,j)
ROEMER'S HYPOTHESIS
Roemer's Hypothesis . Soon after the invention of the telescope, Galileo discovered the four largest moons of Jupiter. Subsequently many astronomers made careful observations of those moons, and already by the 1660's detailed tables of their movements DIFFERENCES BETWEEN POWERS Differences Between Powers . Now my charms are all o’erthrown. And what strength I have’s mine own, Which is most faint. Catalan conjectured that the equation THE PRISMOIDAL FORMULA The Prismoidal Formula: One of the oldest documents in existence is a papyrus roll written in Egypt around 1890 BC. This is sometimes called the Golenischev Papyrus, after the Russian who purchased it in Egypt in 1893 and brought it to Moscow, where it remains today. ANALYSIS OF RELATIVISTIC ORBITS Analysis of Relativistic Orbits . Two quite distinct methods have historically been used to derive the precession rate of relativistic orbits for a test particle moving in the gravitational field of a stationary spherical body of mass m. GRAVITATIONAL ACCELERATION IN SCHWARZSCHILD COORDINATES 6.7 Gravitational Acceleration in Schwarzschild Coordinates . If bodies, moved in any manner among themselves, are urged in the direction of parallel lines by equal accelerative forces, they will all continue to move among themselves, after the same manner as if they had not been urged by those forces. GAMALIEL'S PRINCIPLE Gamaliel's Principle . The new testament tells of how, after the apostles were imprisoned in Jerusalem, they were brought before the council of high priests, who questioned them about why they had disobeyed the order to stop preaching the word of Jesus.MATHPAGES
Number Theory Combinatorics Geometry Algebra Calculus & Diff Eqs Probability & Statistics Set Theory & Foundations: Reflections on Relativity History PhysicsMATHPAGES
Timed Markov Models . One of the defining attributes of ordinary Markov models is that the state transitions are exponentially distributed, meaning that the transition rates are constant and therefore the transitions are continuous.COPRIME PARTITIONS
Coprime Partitions. Let F (N,k) denote the number of distinct partitions of N into exactly k mutually coprime parts. Obviously F (N,1)=1 for all N. Also, we have F (N,2) = phi (N)/2 for all N, where phi () is the Euler totient function. The sequence of values F(N-1+j,j)
MATHPAGES
Round Robin Ties . Intreat me not to leave thee, or to return from following after thee, for whither thou goest, I will go, and where thou lodgest, I will lodge.MATHPAGES
2.4 Doppler Shift for Sound and Light. I was much further out than you thought. And not waving but drowning. Stevie Smith, 1957 RATIONAL SINES OF RATIONAL MULTIPLES OF P Rational Sines of Rational Multiples of π . For which rational multiples of π is the sine rational? We have the three trivial cases . and we wish to show that these are essentially the only distinct rational sines of rational multiples of π.PRISCA SAPIENTIA
Prisca Sapientia . It's ironic that most of the men who participated in the "scientific revolution", whose contributions seem (to us) so original and innovative, were themselves convinced that they were merely re- discovering the vast body of pristine knowledge (prisca sapientia) that had been possessed by the ancients, but somehow lost and forgotten during the centuries that came to be calledMARKOV MODELING
Markov Modeling for Reliability – Part 4: Examples . 4.1 Primary/Backup System with Internal/External Fault Monitoring . The system shown schematically in the figure below consists of a primary unit (Unit 1) with continuous internal fault monitoring, a backup unit (Unit 2) with no self-monitoring, and an external monitoring unit (Unit 3) whose function is to monitor the health of the backup INTRODUCTION TO MARKOV MODELING FOR RELIABILITY Introduction to Markov Modeling for Reliability Here are sample chapters (early drafts) from the book “Markov Models andReliability”:
ON GAUSS' MOUNTAINS
8.6 On Gauss's Mountains . Grossmann is getting his doctorate on a topic that is connected with fiddling around and non-Euclideangeometry.
NUMBER THEORY
COMBINATORICS
GEOMETRY
ALGEBRA
CALCULUS & DIFF EQS
PROBABILITY & STATISTICS SET THEORY & FOUNDATIONS REFLECTIONS ON RELATIVITYHISTORY
PHYSICS
ANIMATED ILLUSTRATIONS COMBINED LIST OF ARTICLESQUOTATIONS
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