Enrique Antoniano - Profile on Academia.edu (original) (raw)
Papers by Enrique Antoniano
On the Tammes Problem for 60 Points, 2025
In an attempt to solve the Tammes problem for 60 points, we analyzed the positioning obtained by ... more In an attempt to solve the Tammes problem for 60 points, we analyzed the positioning obtained by Laszlo Hars [1]. Although we did not achieve the goal, we found a couple of configurations in the induced triangulation that might shed some light on the way to the solution. As a result of this analysis, we achieved a slightly better position where the length of the minimum edge is increased by around 6 • 10-9 units.
Journal de Ciencia e Ingenieria, Dec 31, 2022
The real multiplication map ∅m,m : R m × R m → R 2m−1 induces a symetric immersion ∅m : S m−1 → R... more The real multiplication map ∅m,m : R m × R m → R 2m−1 induces a symetric immersion ∅m : S m−1 → RP m−1 → R 2m−2 which by a theorem of E.H.Brown has mod two Whitney invariant 1 if and only if m = 2 p for some p ≥ 1. As an explanation of this fact we provide an explicit regular homotopy from the immersion ∅m to another map essentially given by a polynomial self map of S m−1 whose degree equals the Whitney invariant of ∅m mod 2. Another choice of a polynomial self-map of S m−1 yields an immersion in the regular homotopy class of ∅m whose Whitney invariant is visible from its double point set.
The 𝑘-theory of projective Stiefel manifolds
Contemporary mathematics, 1987
Sections for Bundles over Projective Spaces
Contemporary mathematics, 1982
Journal de Ciencia e Ingenieria, Aug 31, 2019
We submit an outline for the solved cases of Tammes problem. We give an elementary proof that the... more We submit an outline for the solved cases of Tammes problem. We give an elementary proof that the Platonic polihedra; tetrahedom, octaedronm and icosahedrom are the solutions for 4, 5 and 12 respectively, based on the determination of the triangles with minmun area. We higlight the concept of star of a vertex, and how the determination of the minimun area stars lead to the solution for the cases 8 and 24.
![It is extremely interesting to note that according to Robinson [R], Tarsky’s general method [T], theoretically provides a solution to the problem of Tammes. In fact, given n, we can obtain an algebraic equation in a finite number of steps that is satisfied by a(n). Therefore, the solution to Tammes’s problem is an algebraic number. The solutions are known only for the values of n < 14 and n = 24. We will give some values of a, a and A, as well as a geometric description of the positioning in each case.](https://figures.academia-assets.com/107690759/table_001.jpg)
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Journal de Ciencia e Ingenieria, Dec 31, 2022
The real multiplication map ∅m,m : R m × R m → R 2m−1 induces a symetric immersion ∅m : S m−1 → R... more The real multiplication map ∅m,m : R m × R m → R 2m−1 induces a symetric immersion ∅m : S m−1 → RP m−1 → R 2m−2 which by a theorem of E.H.Brown has mod two Whitney invariant 1 if and only if m = 2 p for some p ≥ 1. As an explanation of this fact we provide an explicit regular homotopy from the immersion ∅m to another map essentially given by a polynomial self map of S m−1 whose degree equals the Whitney invariant of ∅m mod 2. Another choice of a polynomial self-map of S m−1 yields an immersion in the regular homotopy class of ∅m whose Whitney invariant is visible from its double point set.
Application of Hodgkin’s spectral sequence to the calculation of the K- theory of projective Stiefel manifolds
Once in class with Sam
Contemporary Mathematics, 1995
Delicate and novel analysis of a modified Postnikov tower is used to show that the geometric dime... more Delicate and novel analysis of a modified Postnikov tower is used to show that the geometric dimension of 36 times the Hopf bundle over plO is greater than 6.
Transactions of the American Mathematical Society
Using the Hodgkin spectral sequence we calculate K * (X m,k), the complex K-theory of the project... more Using the Hodgkin spectral sequence we calculate K * (X m,k), the complex K-theory of the projective Stiefel manifold X m,k , for mk even. For mk odd, we are only able to calculate K 0 (X m,k), but this is sufficient to determine the order of the complexified Hopf bundle over X m,k .
Boletin de la Sociedad Matemática Mexicana, 1984
The folwwing conditions are equivalent: i) There is an axial map of type (m, n, k). ii) The bundl... more The folwwing conditions are equivalent: i) There is an axial map of type (m, n, k). ii) The bundle (k + 1)~ over P" has m + 1 linearly independent cross-sections. And then it is also true that if there is a nonsingular skew-skew map R'"+1 x n•+ 1-+ n>+•, then there is one which is linear-skew. This result answers affirmatively a question raised by Daniel B. Shapiro in (10], relating skew-skew maps with linear-skew maps. §2. Proof of Theorem 1.6. Remember that O :s m :s n :s k, and write m = 2' + a, n = 2' + b, k = 2' + c, where O :s a < 2', 0 < b < 2' and O :s c < 2'. If n 2:: 2(kn) then 2n a:: 2k-n a:: k or 2•+ 1 + 2a 2:: 2' + c, sos+ 1 2:: r 2::
Boletin de la Sociedad Matematica Mexicana, 1986
Let v •. , be the Stiefel manifold of orthonormal s-frames in R" and let X •. , be the projective... more Let v •. , be the Stiefel manifold of orthonormal s-frames in R" and let X •. , be the projective Stiefel manifold obtained by identifying each s-frame in V •. , with its negative. The double covering v •. ,-+ X., , determines a line bundle over X •. , that we will call the Hopf bundle. In this paper we study the question of the parallelizability of X.,, and obtain the following results: Parallelizable Undecided Nonstably parallelizable X n,n-1, n 1 X12.s
Boletín de la Sociedad Matemática Mexicana, 1977
CONTEMPORARY MATHEMATICS, 1987
Here I want to sta~e some results I recently obtained In collaboration with Samuel Gltler and Jac... more Here I want to sta~e some results I recently obtained In collaboration with Samuel Gltler and Jack Ucci (theorem 6), about the K-theory of projective Stiefel manifolds. I will begin by explaining what these manifolds are and the reason of our interest in all this. If k-. n, the Stiefel manifold vn,k' consist of all orthonormal k-frames in Rn : vn,k•{(vl' .•. ,vk)lvieR and vi,vJ•6iJ} These manifo 1 ds are homogeneous spaces of the orthogona 1 group: Vn ,k • {left cosets of O(n-k)c O(n)} • O(n)/O(n-k) A ,. [~ ~) The projective Stiefel manifold xn,k is obtained by ident1fying each frame (v 1 , ... ,vk)' with its negative (-vl'•••••\l and it is also an homogeneous space of the orthogonal group: Xn,k • O(n)/O(n-k) x z 2 where Zz • {.tl}CO(n) Now, we have a double covering vn,k+\,k and this double covering is classified by a map: f: X k+ p"' n, Consider the question suggested by the following diagram: .,X k S? , n • ,' H pm'~p"' Given n and k, what is the maximal value of m for which there exists a function s making the diagram commutative? This question is related to other known problems as we can see from the next two propositions; see [l], [4]: PROPOSITION 1: There is a map s making the preceeding diagram commutative if and only if there is a skew-linear and non singular map
SECTIONS FOR BUNDLES OVER PROJECTIVE SPACES, 1982
Contemporary Mathematics, 1995
Delicate and novel analysis of a modified Postnikov tower is used to show that the geometric dime... more Delicate and novel analysis of a modified Postnikov tower is used to show that the geometric dimension of 36 times the Hopf bundle over plO is greater than 6.
Journal de Ciencia e Ingeniería, 2019
We submit an outline for the solved cases of Tammes problem. We give an elementary proof that the... more We submit an outline for the solved cases of Tammes problem. We give an elementary proof that the Platonic polihedra; tetrahedom, octaedronm and icosahedrom are the solutions for 4, 5 and 12 respectively, based on the determination of the triangles with minImun area. We higlight the concept of star of a vertex, and how the determination of the minimun area stars lead to the solution for the cases 8 and 24.
![It is extremely interesting to note that according to Robinson [R], Tarsky’s general method [T], theoretically provides a solution to the problem of Tammes. In fact, given n, we can obtain an algebraic equation in a finite number of steps that is satisfied by a(n). Therefore, the solution to Tammes’s problem is an algebraic number. The solutions are known only for the values of n < 14 and n = 24. We will give some values of a, a and A, as well as a geometric description of the positioning in each case.](https://figures.academia-assets.com/79386983/table_001.jpg)
](
On the Tammes Problem for 60 Points, 2025
In an attempt to solve the Tammes problem for 60 points, we analyzed the positioning obtained by ... more In an attempt to solve the Tammes problem for 60 points, we analyzed the positioning obtained by Laszlo Hars [1]. Although we did not achieve the goal, we found a couple of configurations in the induced triangulation that might shed some light on the way to the solution. As a result of this analysis, we achieved a slightly better position where the length of the minimum edge is increased by around 6 • 10-9 units.
Journal de Ciencia e Ingenieria, Dec 31, 2022
The real multiplication map ∅m,m : R m × R m → R 2m−1 induces a symetric immersion ∅m : S m−1 → R... more The real multiplication map ∅m,m : R m × R m → R 2m−1 induces a symetric immersion ∅m : S m−1 → RP m−1 → R 2m−2 which by a theorem of E.H.Brown has mod two Whitney invariant 1 if and only if m = 2 p for some p ≥ 1. As an explanation of this fact we provide an explicit regular homotopy from the immersion ∅m to another map essentially given by a polynomial self map of S m−1 whose degree equals the Whitney invariant of ∅m mod 2. Another choice of a polynomial self-map of S m−1 yields an immersion in the regular homotopy class of ∅m whose Whitney invariant is visible from its double point set.
The 𝑘-theory of projective Stiefel manifolds
Contemporary mathematics, 1987
Sections for Bundles over Projective Spaces
Contemporary mathematics, 1982
Journal de Ciencia e Ingenieria, Aug 31, 2019
We submit an outline for the solved cases of Tammes problem. We give an elementary proof that the... more We submit an outline for the solved cases of Tammes problem. We give an elementary proof that the Platonic polihedra; tetrahedom, octaedronm and icosahedrom are the solutions for 4, 5 and 12 respectively, based on the determination of the triangles with minmun area. We higlight the concept of star of a vertex, and how the determination of the minimun area stars lead to the solution for the cases 8 and 24.
Journal de Ciencia e Ingenieria, Dec 31, 2022
The real multiplication map ∅m,m : R m × R m → R 2m−1 induces a symetric immersion ∅m : S m−1 → R... more The real multiplication map ∅m,m : R m × R m → R 2m−1 induces a symetric immersion ∅m : S m−1 → RP m−1 → R 2m−2 which by a theorem of E.H.Brown has mod two Whitney invariant 1 if and only if m = 2 p for some p ≥ 1. As an explanation of this fact we provide an explicit regular homotopy from the immersion ∅m to another map essentially given by a polynomial self map of S m−1 whose degree equals the Whitney invariant of ∅m mod 2. Another choice of a polynomial self-map of S m−1 yields an immersion in the regular homotopy class of ∅m whose Whitney invariant is visible from its double point set.
Application of Hodgkin’s spectral sequence to the calculation of the K- theory of projective Stiefel manifolds
Once in class with Sam
Contemporary Mathematics, 1995
Delicate and novel analysis of a modified Postnikov tower is used to show that the geometric dime... more Delicate and novel analysis of a modified Postnikov tower is used to show that the geometric dimension of 36 times the Hopf bundle over plO is greater than 6.
Transactions of the American Mathematical Society
Using the Hodgkin spectral sequence we calculate K * (X m,k), the complex K-theory of the project... more Using the Hodgkin spectral sequence we calculate K * (X m,k), the complex K-theory of the projective Stiefel manifold X m,k , for mk even. For mk odd, we are only able to calculate K 0 (X m,k), but this is sufficient to determine the order of the complexified Hopf bundle over X m,k .
Boletin de la Sociedad Matemática Mexicana, 1984
The folwwing conditions are equivalent: i) There is an axial map of type (m, n, k). ii) The bundl... more The folwwing conditions are equivalent: i) There is an axial map of type (m, n, k). ii) The bundle (k + 1)~ over P" has m + 1 linearly independent cross-sections. And then it is also true that if there is a nonsingular skew-skew map R'"+1 x n•+ 1-+ n>+•, then there is one which is linear-skew. This result answers affirmatively a question raised by Daniel B. Shapiro in (10], relating skew-skew maps with linear-skew maps. §2. Proof of Theorem 1.6. Remember that O :s m :s n :s k, and write m = 2' + a, n = 2' + b, k = 2' + c, where O :s a < 2', 0 < b < 2' and O :s c < 2'. If n 2:: 2(kn) then 2n a:: 2k-n a:: k or 2•+ 1 + 2a 2:: 2' + c, sos+ 1 2:: r 2::
Boletin de la Sociedad Matematica Mexicana, 1986
Let v •. , be the Stiefel manifold of orthonormal s-frames in R" and let X •. , be the projective... more Let v •. , be the Stiefel manifold of orthonormal s-frames in R" and let X •. , be the projective Stiefel manifold obtained by identifying each s-frame in V •. , with its negative. The double covering v •. ,-+ X., , determines a line bundle over X •. , that we will call the Hopf bundle. In this paper we study the question of the parallelizability of X.,, and obtain the following results: Parallelizable Undecided Nonstably parallelizable X n,n-1, n 1 X12.s
Boletín de la Sociedad Matemática Mexicana, 1977
CONTEMPORARY MATHEMATICS, 1987
Here I want to sta~e some results I recently obtained In collaboration with Samuel Gltler and Jac... more Here I want to sta~e some results I recently obtained In collaboration with Samuel Gltler and Jack Ucci (theorem 6), about the K-theory of projective Stiefel manifolds. I will begin by explaining what these manifolds are and the reason of our interest in all this. If k-. n, the Stiefel manifold vn,k' consist of all orthonormal k-frames in Rn : vn,k•{(vl' .•. ,vk)lvieR and vi,vJ•6iJ} These manifo 1 ds are homogeneous spaces of the orthogona 1 group: Vn ,k • {left cosets of O(n-k)c O(n)} • O(n)/O(n-k) A ,. [~ ~) The projective Stiefel manifold xn,k is obtained by ident1fying each frame (v 1 , ... ,vk)' with its negative (-vl'•••••\l and it is also an homogeneous space of the orthogonal group: Xn,k • O(n)/O(n-k) x z 2 where Zz • {.tl}CO(n) Now, we have a double covering vn,k+\,k and this double covering is classified by a map: f: X k+ p"' n, Consider the question suggested by the following diagram: .,X k S? , n • ,' H pm'~p"' Given n and k, what is the maximal value of m for which there exists a function s making the diagram commutative? This question is related to other known problems as we can see from the next two propositions; see [l], [4]: PROPOSITION 1: There is a map s making the preceeding diagram commutative if and only if there is a skew-linear and non singular map
SECTIONS FOR BUNDLES OVER PROJECTIVE SPACES, 1982
Contemporary Mathematics, 1995
Delicate and novel analysis of a modified Postnikov tower is used to show that the geometric dime... more Delicate and novel analysis of a modified Postnikov tower is used to show that the geometric dimension of 36 times the Hopf bundle over plO is greater than 6.
Journal de Ciencia e Ingeniería, 2019
We submit an outline for the solved cases of Tammes problem. We give an elementary proof that the... more We submit an outline for the solved cases of Tammes problem. We give an elementary proof that the Platonic polihedra; tetrahedom, octaedronm and icosahedrom are the solutions for 4, 5 and 12 respectively, based on the determination of the triangles with minImun area. We higlight the concept of star of a vertex, and how the determination of the minimun area stars lead to the solution for the cases 8 and 24.