The harmonic numbers are the sequence 1, 1 + 1/2, 1 + 1/2 + 1/3, • • • . Their asymptotic differe... more The harmonic numbers are the sequence 1, 1 + 1/2, 1 + 1/2 + 1/3, • • • . Their asymptotic difference from the sequence of the natural logarithm of the positive integers is Euler's constant gamma. We define a family of natural generalizations of the harmonic numbers. The jth iterated harmonic numbers are a sequence of rational numbers that nests the previous sequences and relates in a similar way to the sequence of the jth iterate of the natural logarithm of positive integers. The analogues of several well-known properties of the harmonic numbers also hold for the iterated harmonic numbers, including a generalization of Euler's constant. We reproduce the proof that only the first harmonic number is an integer and, providing some numeric evidence for the cases j = 2 and j = 3, conjecture that the same result holds for all iterated harmonic numbers. We also review another proposed generalization of harmonic numbers.
We show that for any two continuous real valued functions f, g on [0, 1], the problem Z 1 0 f(x) ... more We show that for any two continuous real valued functions f, g on [0, 1], the problem Z 1 0 f(x) dx · Z c
Please submit solutions and problem proposals to Dr. Harris Kwong, Department of Mathematical Sci... more Please submit solutions and problem proposals to Dr. Harris Kwong, Department of Mathematical Sciences, SUNY Fredonia, Fredonia, NY, 14063, or by email at [email protected]. If you wish to have receipt of your submission acknowledged by mail, please include a selfaddressed, stamped envelope. Each problem or solution should be typed on separate sheets. Solutions to problems in this issue must be received by May 15, 2018. If a problem is not original, the proposer should inform the Problem Editor of the history of the problem. A problem should not be submitted elsewhere while it is under consideration for publication in this Journal. Solvers are asked to include references rather than quoting “well-known results.” The content of the problem sections of The Fibonacci Quarterly are all available on the web free of charge at www.fq.math.ca/.
Visual proof that the limit of the recursive arithmetic mean sequence defined by is , where a1 an... more Visual proof that the limit of the recursive arithmetic mean sequence defined by is , where a1 and a2 are the initial values of the sequence.
The k-Fibonacci polynomials are the natural extension of the k-Fibonacci numbers and many of thei... more The k-Fibonacci polynomials are the natural extension of the k-Fibonacci numbers and many of their properties admit a straightforward proof. Here in particular, we present the derivatives of these polynomials in the form of convolution of k-Fibonacci polynomials. This fact allows us to present in an easy form a family of integer sequences in a new and direct way. Many relations for the derivatives of Fibonacci polynomials are proven.
In this paper we present a new version of the derefinement algorithm developed by Plaza et al. (i... more In this paper we present a new version of the derefinement algorithm developed by Plaza et al. (in Numerical Methodr in Engineering, Elsevier Science,
In this note we introduce the conforming longest-edge n-section algorithm and show that for n ≥ 4... more In this note we introduce the conforming longest-edge n-section algorithm and show that for n ≥ 4 it produces a family of triangulations which does not satisfy the maximum angle condition.
Summary In this paper we present a geometric explanation and construct a 2-D shape measure for tr... more Summary In this paper we present a geometric explanation and construct a 2-D shape measure for triangles. This then leads to a generic n-dimensional simplex shape measure, based on an existing tetahedral shape measure.6
Publications in the domain of three-dimensional unstructured tetrahedral mesh generation, adaptat... more Publications in the domain of three-dimensional unstructured tetrahedral mesh generation, adaptation and optimization abound, [2]. Different statistics on the quality of the mesh are used, and tetrahedron shape measures play a significant role in the design and development of 3D meshes and for a variety of applications, [3, 4]. In this paper we study the degeneracy of tetrahedral meshes which are obtained by the 8-Tetrahedra Longest eEdge partition, [1]. To test a wide variety of tetrahedra in the computations, we apply rotation and translation to vertices and this lead to a list of tetrahedra ordered from the most regular case to the more degenerated case. We compare some of the most interesting shape measures and give numerical evidence showing that the measure of degeneracy is bounded and converges asymptotically to a fixed value. The result also confirms that the property of auto-improvement of the 8-Tetrahedra Longest Edge partition is dependent of the initial geometry of the tetrahedra.
We prove that the longest-edge n-section of triangles for n 4 produces a sequence of triangle mes... more We prove that the longest-edge n-section of triangles for n 4 produces a sequence of triangle meshes with minimum interior angle converging to zero. The so called degeneracy property of LE for n 4 is proved.
En este trabajo se presenta un diagrama geométrico para el estudio de la calidad geométrica de lo... more En este trabajo se presenta un diagrama geométrico para el estudio de la calidad geométrica de los triángulos generados por la aplicación reiterada de la partición en cuatro triángulos por el lado mayor (4TLE). El diagrama proporciona una herramienta gráfica de utilidad para visualizar la evolución de los triángulos generados en mallas triangulares. Se usan funciones de variable compleja para hallar las curvas frontera de las diferentes clases de triángulos generadas en el diagrama. Asimismo, se presenta un método de subdivisión para mallas de triángulos que combina la partición 4TLE y la partición semejante (4TSS) en una nueva partición híbrida denominada 4TLE-SS. Se resaltan los beneficios de este nuevo método, al comprobar que el número de triángulos de 'buena' calidad generados con el algoritmo es mayor que con el método 4TLE. El trabajo que se aborda es de interés en el campo de generación y refinamiento de mallas triangulares.
The Longest-Edge (LE) trisection of a triangle is obtained by joining the two points which divide... more The Longest-Edge (LE) trisection of a triangle is obtained by joining the two points which divide the longest edge in three with the opposite vertex. If LE-trisection is iteratively applied to an initial triangle, then the maximum diameter of the resulting triangles is between two sharpened decreasing functions. This paper mathematically answers the question of how fast the diameters of a triangle mesh tend to zero as repeated trisection is performed, and completes the previous empirical studies presented in the MASCOT 2010 Meeting (Perdomo et al., 2010).
A tetrahedron is called regular if its six edges are of equal length. It is clear that, for an in... more A tetrahedron is called regular if its six edges are of equal length. It is clear that, for an initial regular tetrahedron R0, the iterative eight-tetrahedron longest-edge partition (8T-LE) of R0 produces an infinity sequence of tetrahedral meshes, τ0={R0}, τ1={Ri1}, τ2={Ri2},…, τn={Rin},…. In this paper, it is proven that, in the iterative process just mentioned, only two distinct similarity classes are generated. Therefore, the stability and the non-degeneracy of the generated meshes, as well as the minimum and maximum angle condition straightforwardly follow. Additionally, for a standard-shape tetrahedron quality measure (η) and any tetrahedron Rin∈τn, n>0, then ηRin≥23ηR0. The non-degeneracy constant is c=23 in the case of the iterative 8T-LE partition of a regular tetrahedron.
The harmonic numbers are the sequence 1, 1 + 1/2, 1 + 1/2 + 1/3, • • • . Their asymptotic differe... more The harmonic numbers are the sequence 1, 1 + 1/2, 1 + 1/2 + 1/3, • • • . Their asymptotic difference from the sequence of the natural logarithm of the positive integers is Euler's constant gamma. We define a family of natural generalizations of the harmonic numbers. The jth iterated harmonic numbers are a sequence of rational numbers that nests the previous sequences and relates in a similar way to the sequence of the jth iterate of the natural logarithm of positive integers. The analogues of several well-known properties of the harmonic numbers also hold for the iterated harmonic numbers, including a generalization of Euler's constant. We reproduce the proof that only the first harmonic number is an integer and, providing some numeric evidence for the cases j = 2 and j = 3, conjecture that the same result holds for all iterated harmonic numbers. We also review another proposed generalization of harmonic numbers.
We show that for any two continuous real valued functions f, g on [0, 1], the problem Z 1 0 f(x) ... more We show that for any two continuous real valued functions f, g on [0, 1], the problem Z 1 0 f(x) dx · Z c
Please submit solutions and problem proposals to Dr. Harris Kwong, Department of Mathematical Sci... more Please submit solutions and problem proposals to Dr. Harris Kwong, Department of Mathematical Sciences, SUNY Fredonia, Fredonia, NY, 14063, or by email at [email protected]. If you wish to have receipt of your submission acknowledged by mail, please include a selfaddressed, stamped envelope. Each problem or solution should be typed on separate sheets. Solutions to problems in this issue must be received by May 15, 2018. If a problem is not original, the proposer should inform the Problem Editor of the history of the problem. A problem should not be submitted elsewhere while it is under consideration for publication in this Journal. Solvers are asked to include references rather than quoting “well-known results.” The content of the problem sections of The Fibonacci Quarterly are all available on the web free of charge at www.fq.math.ca/.
Visual proof that the limit of the recursive arithmetic mean sequence defined by is , where a1 an... more Visual proof that the limit of the recursive arithmetic mean sequence defined by is , where a1 and a2 are the initial values of the sequence.
The k-Fibonacci polynomials are the natural extension of the k-Fibonacci numbers and many of thei... more The k-Fibonacci polynomials are the natural extension of the k-Fibonacci numbers and many of their properties admit a straightforward proof. Here in particular, we present the derivatives of these polynomials in the form of convolution of k-Fibonacci polynomials. This fact allows us to present in an easy form a family of integer sequences in a new and direct way. Many relations for the derivatives of Fibonacci polynomials are proven.
In this paper we present a new version of the derefinement algorithm developed by Plaza et al. (i... more In this paper we present a new version of the derefinement algorithm developed by Plaza et al. (in Numerical Methodr in Engineering, Elsevier Science,
In this note we introduce the conforming longest-edge n-section algorithm and show that for n ≥ 4... more In this note we introduce the conforming longest-edge n-section algorithm and show that for n ≥ 4 it produces a family of triangulations which does not satisfy the maximum angle condition.
Summary In this paper we present a geometric explanation and construct a 2-D shape measure for tr... more Summary In this paper we present a geometric explanation and construct a 2-D shape measure for triangles. This then leads to a generic n-dimensional simplex shape measure, based on an existing tetahedral shape measure.6
Publications in the domain of three-dimensional unstructured tetrahedral mesh generation, adaptat... more Publications in the domain of three-dimensional unstructured tetrahedral mesh generation, adaptation and optimization abound, [2]. Different statistics on the quality of the mesh are used, and tetrahedron shape measures play a significant role in the design and development of 3D meshes and for a variety of applications, [3, 4]. In this paper we study the degeneracy of tetrahedral meshes which are obtained by the 8-Tetrahedra Longest eEdge partition, [1]. To test a wide variety of tetrahedra in the computations, we apply rotation and translation to vertices and this lead to a list of tetrahedra ordered from the most regular case to the more degenerated case. We compare some of the most interesting shape measures and give numerical evidence showing that the measure of degeneracy is bounded and converges asymptotically to a fixed value. The result also confirms that the property of auto-improvement of the 8-Tetrahedra Longest Edge partition is dependent of the initial geometry of the tetrahedra.
We prove that the longest-edge n-section of triangles for n 4 produces a sequence of triangle mes... more We prove that the longest-edge n-section of triangles for n 4 produces a sequence of triangle meshes with minimum interior angle converging to zero. The so called degeneracy property of LE for n 4 is proved.
En este trabajo se presenta un diagrama geométrico para el estudio de la calidad geométrica de lo... more En este trabajo se presenta un diagrama geométrico para el estudio de la calidad geométrica de los triángulos generados por la aplicación reiterada de la partición en cuatro triángulos por el lado mayor (4TLE). El diagrama proporciona una herramienta gráfica de utilidad para visualizar la evolución de los triángulos generados en mallas triangulares. Se usan funciones de variable compleja para hallar las curvas frontera de las diferentes clases de triángulos generadas en el diagrama. Asimismo, se presenta un método de subdivisión para mallas de triángulos que combina la partición 4TLE y la partición semejante (4TSS) en una nueva partición híbrida denominada 4TLE-SS. Se resaltan los beneficios de este nuevo método, al comprobar que el número de triángulos de 'buena' calidad generados con el algoritmo es mayor que con el método 4TLE. El trabajo que se aborda es de interés en el campo de generación y refinamiento de mallas triangulares.
The Longest-Edge (LE) trisection of a triangle is obtained by joining the two points which divide... more The Longest-Edge (LE) trisection of a triangle is obtained by joining the two points which divide the longest edge in three with the opposite vertex. If LE-trisection is iteratively applied to an initial triangle, then the maximum diameter of the resulting triangles is between two sharpened decreasing functions. This paper mathematically answers the question of how fast the diameters of a triangle mesh tend to zero as repeated trisection is performed, and completes the previous empirical studies presented in the MASCOT 2010 Meeting (Perdomo et al., 2010).
A tetrahedron is called regular if its six edges are of equal length. It is clear that, for an in... more A tetrahedron is called regular if its six edges are of equal length. It is clear that, for an initial regular tetrahedron R0, the iterative eight-tetrahedron longest-edge partition (8T-LE) of R0 produces an infinity sequence of tetrahedral meshes, τ0={R0}, τ1={Ri1}, τ2={Ri2},…, τn={Rin},…. In this paper, it is proven that, in the iterative process just mentioned, only two distinct similarity classes are generated. Therefore, the stability and the non-degeneracy of the generated meshes, as well as the minimum and maximum angle condition straightforwardly follow. Additionally, for a standard-shape tetrahedron quality measure (η) and any tetrahedron Rin∈τn, n>0, then ηRin≥23ηR0. The non-degeneracy constant is c=23 in the case of the iterative 8T-LE partition of a regular tetrahedron.
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