A classical probability question asks for the expected waiting time for flipping a coin (fair or not) until a series of consecutive k heads occur. Now instead of k heads, …
A classical probability question asks for the expected waiting time for flipping a coin (fair or not) until a series of consecutive k heads occur. Now instead of k heads, we can ask for the expected waiting time for a prescribed string such as HTHHTT (H for ‘head’ and T for ‘tail’), and furthermore, the following more general setting: replacing coin flipping by taking a letter, one at a time, what is the expected waiting time until a prescribed string (a series of letters) is reached? Here we allow different probabilities for the occurrence of different letters. We give an exposition to this problem by offering an elementary algorithm and implementing it to compute the corresponding probability generating function: we show that there exists a universal program taking as inputs the choice of letters with given probabilities and the prescribed string, and as output, returning the probability generating function for the waiting time. The same method is applied to solve the problem of several competing strings, which asks for the probability (or more generally the probability generating function) of one of the given strings occurring before the remaining strings. In particular, this solves the problem of finding the expectation and variance for the waiting time random variable of the first problem.
Motivated by the result of Fibonacci numbers for which the ratio of successive terms tends to a limit, which is commonly known as the Golden Ratio, we prove an immediate …
Motivated by the result of Fibonacci numbers for which the ratio of successive terms tends to a limit, which is commonly known as the Golden Ratio, we prove an immediate generalization for a wider class of recurrence sequences.We note that such limiting behavior for ratio of successive terms of general linear recurrence sequences has been well discussed, but still they need to satisfy specific conditions for the limit to exist.Our contribution is that we show that such conditions are indeed satisfied for the cases we are considering.For an application of our main result, we find a natural way to approximate an algebraic number, which is a zero for some class of polynomial equations, by rational numbers.As recently there seem to be renewed interests on Fibonacci numbers and related recurrence sequences, we hope that our elementary methods and results may shed some light for solving the related problems.
In a paper by J. Deutsch [1], a quaternionic proof of the universality of seven quaternary quadratic forms was given. The proof relies on a construction very similar to that …
In a paper by J. Deutsch [1], a quaternionic proof of the universality of seven quaternary quadratic forms was given. The proof relies on a construction very similar to that of Hurwitz quaternions, and its associated division algorithm. Of course, these results are evident, if one uses the Conway-Schneeberger Fifteen Theorem [2], as the author also mentioned, however it is interesting to give a direct proof for some specific quadratic forms based on simple argument. It is the purpose of this short note to prove five of the seven quadratic forms mentioned and proven by Deutsch, using the universality of the classical quadratic form associated to the celebrated Lagrange’s Theorem of Four Squares and Euler’s trick. Mathematics Subject Classification: 11A67, 11R52, 11E25
In a recent preprint, Gullerud and Walker [2] proved a theorem and made a conjecture about the correctness of efficiently generating Bezout trees for Pythagorean pairs. In this note, we …
In a recent preprint, Gullerud and Walker [2] proved a theorem and made a conjecture about the correctness of efficiently generating Bezout trees for Pythagorean pairs. In this note, we give a simple proof of their theorem, confirm that their conjecture is true, and furthermore we give a generalization.
In the plane, three distinct lines in general position (in which no lines are parallel and not all three lines intersect at a point) characterize a triangle. For simplicity, if …
In the plane, three distinct lines in general position (in which no lines are parallel and not all three lines intersect at a point) characterize a triangle. For simplicity, if the three lines are ...
"A Variant of the Proof of the Pythagorean Theorem by Huygens." Mathematics Magazine, 92(1), pp. 70–71
"A Variant of the Proof of the Pythagorean Theorem by Huygens." Mathematics Magazine, 92(1), pp. 70–71
In a recent preprint, Gullerud and Walker proved a theorem and made a conjecture about thecorrectness of efficiently generating Bézout trees for Pythagorean pairs. In this note, we give a …
In a recent preprint, Gullerud and Walker proved a theorem and made a conjecture about thecorrectness of efficiently generating Bézout trees for Pythagorean pairs. In this note, we give a simple proof of their theorem, conrm that their conjecture is true, and furthermore we give a generalization.
In 2014, Paul Yiu constructed two equilateral triangles inscribed in a Kiepert hyperbola associated with a reference triangle. It was asserted that each of the equilateral triangles is triply perspective …
In 2014, Paul Yiu constructed two equilateral triangles inscribed in a Kiepert hyperbola associated with a reference triangle. It was asserted that each of the equilateral triangles is triply perspective to the reference triangle, and in each case, the corresponding three perspectors are collinear. In this note, we give a proof of his assertions. Furthermore as an analogue of Lemoine's problem, we formulated and answered the question about how to recover the reference triangle given a Kiepert hyperbola, one of the two Fermat points and one vertex of the reference triangle.
In a recent preprint, Gullerud and Walker [2] proved a theorem and made a conjecture about the correctness of efficiently generating B\'ezout trees for Pythagorean pairs. In this note, we …
In a recent preprint, Gullerud and Walker [2] proved a theorem and made a conjecture about the correctness of efficiently generating B\'ezout trees for Pythagorean pairs. In this note, we give a simple proof of their theorem, confirm that their conjecture is true, and furthermore we give a generalization.
In a recent preprint, Gullerud and Walker proved a theorem and made a conjecture about thecorrectness of efficiently generating Bézout trees for Pythagorean pairs. In this note, we give a …
In a recent preprint, Gullerud and Walker proved a theorem and made a conjecture about thecorrectness of efficiently generating Bézout trees for Pythagorean pairs. In this note, we give a simple proof of their theorem, conrm that their conjecture is true, and furthermore we give a generalization.
"A Variant of the Proof of the Pythagorean Theorem by Huygens." Mathematics Magazine, 92(1), pp. 70–71
"A Variant of the Proof of the Pythagorean Theorem by Huygens." Mathematics Magazine, 92(1), pp. 70–71
In 2014, Paul Yiu constructed two equilateral triangles inscribed in a Kiepert hyperbola associated with a reference triangle. It was asserted that each of the equilateral triangles is triply perspective …
In 2014, Paul Yiu constructed two equilateral triangles inscribed in a Kiepert hyperbola associated with a reference triangle. It was asserted that each of the equilateral triangles is triply perspective to the reference triangle, and in each case, the corresponding three perspectors are collinear. In this note, we give a proof of his assertions. Furthermore as an analogue of Lemoine's problem, we formulated and answered the question about how to recover the reference triangle given a Kiepert hyperbola, one of the two Fermat points and one vertex of the reference triangle.
In the plane, three distinct lines in general position (in which no lines are parallel and not all three lines intersect at a point) characterize a triangle. For simplicity, if …
In the plane, three distinct lines in general position (in which no lines are parallel and not all three lines intersect at a point) characterize a triangle. For simplicity, if the three lines are ...
In a recent preprint, Gullerud and Walker [2] proved a theorem and made a conjecture about the correctness of efficiently generating Bezout trees for Pythagorean pairs. In this note, we …
In a recent preprint, Gullerud and Walker [2] proved a theorem and made a conjecture about the correctness of efficiently generating Bezout trees for Pythagorean pairs. In this note, we give a simple proof of their theorem, confirm that their conjecture is true, and furthermore we give a generalization.
In a recent preprint, Gullerud and Walker [2] proved a theorem and made a conjecture about the correctness of efficiently generating B\'ezout trees for Pythagorean pairs. In this note, we …
In a recent preprint, Gullerud and Walker [2] proved a theorem and made a conjecture about the correctness of efficiently generating B\'ezout trees for Pythagorean pairs. In this note, we give a simple proof of their theorem, confirm that their conjecture is true, and furthermore we give a generalization.
Motivated by the result of Fibonacci numbers for which the ratio of successive terms tends to a limit, which is commonly known as the Golden Ratio, we prove an immediate …
Motivated by the result of Fibonacci numbers for which the ratio of successive terms tends to a limit, which is commonly known as the Golden Ratio, we prove an immediate generalization for a wider class of recurrence sequences.We note that such limiting behavior for ratio of successive terms of general linear recurrence sequences has been well discussed, but still they need to satisfy specific conditions for the limit to exist.Our contribution is that we show that such conditions are indeed satisfied for the cases we are considering.For an application of our main result, we find a natural way to approximate an algebraic number, which is a zero for some class of polynomial equations, by rational numbers.As recently there seem to be renewed interests on Fibonacci numbers and related recurrence sequences, we hope that our elementary methods and results may shed some light for solving the related problems.
A classical probability question asks for the expected waiting time for flipping a coin (fair or not) until a series of consecutive k heads occur. Now instead of k heads, …
A classical probability question asks for the expected waiting time for flipping a coin (fair or not) until a series of consecutive k heads occur. Now instead of k heads, we can ask for the expected waiting time for a prescribed string such as HTHHTT (H for ‘head’ and T for ‘tail’), and furthermore, the following more general setting: replacing coin flipping by taking a letter, one at a time, what is the expected waiting time until a prescribed string (a series of letters) is reached? Here we allow different probabilities for the occurrence of different letters. We give an exposition to this problem by offering an elementary algorithm and implementing it to compute the corresponding probability generating function: we show that there exists a universal program taking as inputs the choice of letters with given probabilities and the prescribed string, and as output, returning the probability generating function for the waiting time. The same method is applied to solve the problem of several competing strings, which asks for the probability (or more generally the probability generating function) of one of the given strings occurring before the remaining strings. In particular, this solves the problem of finding the expectation and variance for the waiting time random variable of the first problem.
In a paper by J. Deutsch [1], a quaternionic proof of the universality of seven quaternary quadratic forms was given. The proof relies on a construction very similar to that …
In a paper by J. Deutsch [1], a quaternionic proof of the universality of seven quaternary quadratic forms was given. The proof relies on a construction very similar to that of Hurwitz quaternions, and its associated division algorithm. Of course, these results are evident, if one uses the Conway-Schneeberger Fifteen Theorem [2], as the author also mentioned, however it is interesting to give a direct proof for some specific quadratic forms based on simple argument. It is the purpose of this short note to prove five of the seven quadratic forms mentioned and proven by Deutsch, using the universality of the classical quadratic form associated to the celebrated Lagrange’s Theorem of Four Squares and Euler’s trick. Mathematics Subject Classification: 11A67, 11R52, 11E25