The catenary is one of the most common curves; it is, in fact, the shape assumed by a homogeneous... more The catenary is one of the most common curves; it is, in fact, the shape assumed by a homogeneous and inextensible chain, fixed to the extremities, which is subject only to its own weight. The catenary has always fascinated not only mathematicians but also architects and engineers, who have often used it in their works due to its remarkable properties. In this note, the catenary is introduced by determining its equation and considering its history from Galileo Galilei to the present day, pointing out a historical mistake. Then, some of its applications to architecture and engineering are shown and the catenary is compared to the parabola, a curve that at first sight can look similar to catenary but must not be confused with it. Finally, some variants of the catenary are considered: the weighted catenary and the catenary of equal resistance, highlighting their properties and their practical applications.
In this work we will deal with ellipses and ovals, comparing them both from the geometric point o... more In this work we will deal with ellipses and ovals, comparing them both from the geometric point of view and from the one of applications. There is a notable similarity between these curves so often it’s not possible to recognize which of the two figures is, unless we consider other elements to distinguish them. We will show the presence of both curves in architectural works and in treatises, motivating their use, when it’s possible, with geometric and technological considerations.
The Reuleaux triangle is a figure with the remarkable property of having constant width, a typica... more The Reuleaux triangle is a figure with the remarkable property of having constant width, a typical property of the circle. It takes its name from Franz Reuleaux, a 19th century German engineer, who studied its properties, in particular the ones related to applications to mechanics. However, this figure was previously known: actually, we find it in the shape of the windows and in the ornaments of some Gothic architecture. Furthermore, Leonardo da Vinci, to represent the terrestrial globe, used eight Reuleaux triangles, each one corresponding to an octant of the spherical surface. Even the mathematician Euler encountered this figure in his study of geometric forms with constant width.
with n ≥ 2, d ≥ 3 does not satisfy property Np (according to Green and Lazarsfeld) if p ≥ 3d − 2.... more with n ≥ 2, d ≥ 3 does not satisfy property Np (according to Green and Lazarsfeld) if p ≥ 3d − 2. We make the conjecture that also the converse holds. This is true for n = 2 and for n = d = 3. Abstract- We prove that the Veronese embedding ϕ OIP n(d) : IP n ֒ → IP N
This is an introductory survey, from a geometric perspective, on the Singular Value Decomposition... more This is an introductory survey, from a geometric perspective, on the Singular Value Decomposition (SVD) for real matrices, focusing on the role of the Terracini Lemma. We extend this point of view to tensors, we define the singular space of a tensor as the space spanned by singular vector tuples and we study some of its basic properties.
The catenary is one of the most common curves; it is, in fact, the shape assumed by a homogeneous... more The catenary is one of the most common curves; it is, in fact, the shape assumed by a homogeneous and inextensible chain, fixed to the extremities, which is subject only to its own weight. The catenary has always fascinated not only mathematicians but also architects and engineers, who have often used it in their works due to its remarkable properties. In this note, the catenary is introduced by determining its equation and considering its history from Galileo Galilei to the present day, pointing out a historical mistake. Then, some of its applications to architecture and engineering are shown and the catenary is compared to the parabola, a curve that at first sight can look similar to catenary but must not be confused with it. Finally, some variants of the catenary are considered: the weighted catenary and the catenary of equal resistance, highlighting their properties and their practical applications.
In this work we will deal with ellipses and ovals, comparing them both from the geometric point o... more In this work we will deal with ellipses and ovals, comparing them both from the geometric point of view and from the one of applications. There is a notable similarity between these curves so often it’s not possible to recognize which of the two figures is, unless we consider other elements to distinguish them. We will show the presence of both curves in architectural works and in treatises, motivating their use, when it’s possible, with geometric and technological considerations.
The Reuleaux triangle is a figure with the remarkable property of having constant width, a typica... more The Reuleaux triangle is a figure with the remarkable property of having constant width, a typical property of the circle. It takes its name from Franz Reuleaux, a 19th century German engineer, who studied its properties, in particular the ones related to applications to mechanics. However, this figure was previously known: actually, we find it in the shape of the windows and in the ornaments of some Gothic architecture. Furthermore, Leonardo da Vinci, to represent the terrestrial globe, used eight Reuleaux triangles, each one corresponding to an octant of the spherical surface. Even the mathematician Euler encountered this figure in his study of geometric forms with constant width.
with n ≥ 2, d ≥ 3 does not satisfy property Np (according to Green and Lazarsfeld) if p ≥ 3d − 2.... more with n ≥ 2, d ≥ 3 does not satisfy property Np (according to Green and Lazarsfeld) if p ≥ 3d − 2. We make the conjecture that also the converse holds. This is true for n = 2 and for n = d = 3. Abstract- We prove that the Veronese embedding ϕ OIP n(d) : IP n ֒ → IP N
This is an introductory survey, from a geometric perspective, on the Singular Value Decomposition... more This is an introductory survey, from a geometric perspective, on the Singular Value Decomposition (SVD) for real matrices, focusing on the role of the Terracini Lemma. We extend this point of view to tensors, we define the singular space of a tensor as the space spanned by singular vector tuples and we study some of its basic properties.
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