arXiv subjects

Oscar Perdomo

Publications and source records attributed to Oscar Perdomo.

At least 19 recordsLinked to original sources

arXiv2026

New Explicit Eigenfunctions of the stability operator on some minimal hypersurfaces

For any $n$-dimensional compact minimal hypersurface of the $(n+1)$-dimensional sphere, we have that the coordinate functions of the Gauss map are eigenfunctions of the stability operator associated with the eigenvalue $-n$. In this paper we consider minimal immersions from $\mbS^{k}\times\mbS^{\ell}\times\mbS^{1}$ to $\mbS^{k+\ell+2}$ of the form $\phi(y,z,t)=\left(f(t) y, f_2(t) z, f_1(t)\right)$ and we explicitly show new eigenfunctions for the stability operator associated with the same eigenvalue. We also show that the stability index of these minimal immersions is at least $k\ell+3k+3\ell+8$.

math.DG↗

arXiv2026

The stability index and Yau's conjecture for Carlotto-Schulz minimal hypertori, part II

For any closed minimal hypersurface $M$ in the $N+1$-dimensional Euclidean sphere $S^{N+1}$, $-N$ is an eigenvalue of the stability operator. In this paper we show that the multiplicity of this eigenvalue for the Carlotto and Schulz minimal embedding $X_{CS}^{n}:S^{n-1}\times S^{n-1}\times S^{1}\to S^{2n}$ is at least $2n+1+n^2$. We conjecture that if $n>2$, then the stability index of $X_{CS}^{n}$ is $\frac{1}{3} \left(n^3+9 n^2+11 n+3\right)$ and for the hypertorus in $S^4$ (case $n=2$) the stability index is $27$. We numerically verify the conjecture for the first 100 values of $n$. We also numerically verify that Yau's conjecture on the first eigenvalue of the Laplacian holds when $2\le n\le 260$.

math.DG↗

arXiv2026

Families of periodic solutions of the 4- and 6-body problem using a gradient-free continuation method

In this paper, we describe a gradient-free method to solve a system of equations, and we use it to construct two families of pseudo-periodic planar solutions of the 4- and 6-body problem. The method is a stochastic black-box procedure that uses only function evaluations. For the 4-body problem, bodies 1 and 2 have mass 1 and move opposite to each other, and bodies 3 and 4 have mass $m_2$ and also move opposite to each other. For the 6-body problem, bodies 1, 2, and 3 have mass 1 and move on the vertices of an equilateral triangle centered at the origin, and bodies 4, 5, and 6 have mass $m_2$ and also move on the vertices of an equilateral triangle. In both cases, we compute families of periodic solutions by imposing return conditions up to rotation and relabeling.

math.DS↗

arXiv2025

Spectrum of the Laplacian and the Jacobi operator on Generalized rotational minimal hypersurfaces of spheres

Let $M\subset S^{n+1}$ be the hypersurface generated by rotating a hypersurface $M_0$ contained in the interior of the unit ball of $\mathbb{R}^{n-k+1}$. More precisely, $M=\{(\sqrt{1-|m|^2}\, y, m):y\in S^k, m\in M_0\}$. We derive the equation for the mean curvature of $M$ in terms of the principal curvatures of $M_0$. For the particular case when $M_0$ is a surface of revolution in $\mathbb{R}^3$, we provide a method for finding the eigenvalues of the Laplace and stability operators. To illustrate this method, we consider an example of a minimal embedded hypersurface in $S^6$ and numerically compute all the eigenvalues of the Laplace operator less than 12, as well as all non-positive eigenvalues of the stability operators. For this example, we show that the stability index (the number of negative eigenvalues of the stability operator, counted with multiplicity) is 77, and the nullity (the multiplicity of the eigenvalue $λ=0$ of the stability operator) is 14. Similar results are found in the case where $M_0$ is a hypersurface in $\mathbb{R}^{l+2}$ of the form $(f_2(u)z, f_1(u))$, with $z$ in the $l$-dimensional unit sphere $S^l$. Carlotto and Schulz have found examples of embedded minimal hypersurfaces in the case where $M_0=S^k\times S^1$.

math.DG↗

arXiv2025

The stability index and Yau's conjecture for Carlotto-Schulz minimal hypertori

Recently, for any n>1, Carlotto and Schulz showed the existence of a minimal embedding in the 2n-dimensional unit sphere. In this paper, we show that the stability index of these embedded minimal hypersurfaces is at least n^2+4n+3. We also show that Yau's conjecture holds for these examples if and only if the solution of the differential equation z''(t)+a_n(t)z'(t)+(2n-1)z(t)=0 with z(0)=1 and z'(0)=0 satisfies z'(T)>0. Here,T and the T-periodic function a_n(t) are determined in terms of the functions defining the minimal immersion.

math.DG↗

arXiv2025

Existence of a constant-mean-curvature hypertorus in $S^4$ via computer assistance

The round Taylor method uses rational arithmetic, allowing control of both round-off and truncation errors in approximating solutions of differential equations. In this paper, we employ this method together with the Poincare-Miranda theorem to prove the existence of a new embedded constant mean curvature (CMC) hypertorus in the unit four dimensional sphere

math.DG↗

arXiv2025

Navigating the Space of Compact CMC Hypersurfaces in Spheres, Part II

In R^3, let M be the infinite union of unit spheres whose centers lie at even integers on the x-axis; every pair of consecutive spheres touches at (2m+1, 0, 0). Desingularizing these point contacts yields Delaunay's classical constant mean curvature (CMC) surfaces, including unduloids and nodoids. Motivated by this picture, we construct an analogue in the unit sphere S^4. We begin with the piecewise-smooth hypersurface M contained in S^4, obtained by gluing two carefully chosen totally umbilical 3-spheres to two specific Clifford hypersurfaces, all four components sharing the same constant mean curvature and meeting along four disjoint circles. We provide numerical evidence that these circles can be desingularized: there exists a smooth one-parameter family Sigma_b, each lying in S^4, of CMC hypersurfaces such that Sigma_b approaches M as b tends to 0. The mean curvature H(b) varies smoothly along the family and vanishes at a single non-embedded minimal member. Moreover, there is a threshold B_1 in (0, B) such that when b < B_1 the hypersurface Sigma_b is embedded ("unduloid type"), whereas for b >= B_1 it is non-embedded ("nodoid type"). As b increases toward B, the hypersurfaces converge to a minimal hypersurface with two singular points.

math.DG↗

arXiv2025

Navigating the Space of Compact CMC Hypersurfaces in Spheres, Part I

In this paper, we describe a family of embedded hypersurfaces with constant mean curvature (CMC) in the $(n+1)$-dimensional unit sphere. In the process, we provide evidence for new CMC embedded examples. In particular, for some examples with $H=0$, we verify Yau's conjecture stating that among the embedded, non-totally umbilical minimal hypersurfaces in spheres, the Clifford hypersurfaces have the least area.

math.DG↗

arXiv2023

Lagrange points of Euler's solutions of the 3-body problem

In this paper we classify the central configurations of the circular restricted 4-body problem with three primaries at the collinear configuration of the 3-body problem and an infinitesimal mass. The case where the three primaries have the same mass, with one of the bodies staying motionless at the center of mass, was considered in 2021 by Llibre. The video \url{https://youtu.be/PWFtqxd4RUA} goes over some of the results in this paper.

math.DS↗

arXiv2023

Five-qubit states generated by Clifford gates

The Clifford group is the set of gates generated by controlled-Z gates, the phase gate and the Hadamard gate. We will say that a n-qubit state is a Clifford state if it can be prepared using Clifford gates. These states are known as the stabilizer states and they arise in quantum error correction. In this paper we study the set of all 5-qubit Clifford states. By using an exhaustive method we start by confirming that there are 19388160 states. The main goal of the paper is to understand the action of the controlled-Z gates action on the 5-qubit states. With this goal in mind, we partition the Clifford states into orbits using the equivalence relation: two states are equivalent if they differ by a local Clifford gate. We show that there are 93 orbits, and we label each orbit in such a way that it is easy to see the effect of the controlled-Z gates. Diagrams and tables explaining the action of the CZ gates on all the orbits are presented in the paper. A similar work is done for the real Clifford 5-qubits states, this is, for states that can be prepared with Controlled-Z gates, the Z gate and the Hadamard gate.

quant-ph↗

arXiv2022

Periodic oscillations in the restricted Hip-Hop 2N+1 body problem

We prove the existence of periodic solutions of the restricted $(2N+1)$-body problem when the $2N$-primaries move on a periodic Hip-Hop solution and the massless body moves on the line that contains the center of mass and is perpendicular to the base of the antiprism formed by the $2N$-primaries.

math.DS↗

arXiv2022

Periodic oscillations in a 2N-body problem

Hip-Hop solutions of the $2N$-body problem are solutions that satisfy at every instance of time, that the $2N$ bodies with the same mass $m$, are at the vertices of two regular $N$-gons, each one of these $N$-gons are at planes that are equidistant from a fixed plane $Π_0$ forming an antiprism. In this paper, we first prove that for every $N$ and every $m$ there exists a family of periodic hip-hop solutions. For every solution in these families the oriented distance to the plane $Π_0$, which we call $d(t)$, is an odd function that is also even with respect to $t=T$ for some $T>0.$ For this reason we call solutions in these families, double symmetric solutions. By exploring more carefully our initial set of periodic solutions, we numerically show that some of the branches stablished in our existence theorem have bifurcations that produce branches of solutions with the property that the oriented distance function $d(t)$ is not even with respect to any $T>0$, we call these solutions single symmetry solutions. We prove that no single symmetry solution is a choreography. We also display explicit double symmetric solutions that are choreographies.

math.DS↗

arXiv2022

Preparation of 3-qubit states

We will call a pure qubit state real if all its amplitudes are real numbers. We show that any real 3-qubit state can be prepared using $R_y(θ)$ gates and at most four controlled-$Z$ gates, and we conjecture that four is optimal. We also present an algorithm -- different from the 2008 algorithm given by Znidaric, Giraud and Georgeot -- that prepares any 3-qubit state using local gates and at most three controlled-$Z$ gates. Videos showing how our method works for two- and three-qubit states can be found at https://youtu.be/LIdYSs-rE-o and https://youtu.be/Kne0Vq7gyzQ

quant-ph↗

arXiv2021

CMC hypersurfaces with two principal curvatures

This paper explains the construction of all hypersurfaces with constant mean curvature -- cmc -- and exactly two principal curvatures on any space form endowed with a semi-riemannian metric. Here we will consider riemannian hypersurfaces as well as hypersurfaces with semi-riemannian metrics induced by the ambient space. We show explicit immersions for all these hypersurfaces. We will see that for any given ambient space, the family of cmc hypersurfaces with two principal curvatures depends on two parameters, $H$ and $C$. We describe the range for all possible $(H,C)$ associated with complete cmc hypersurfaces. We end the paper by finding optimal bounds for the traceless second fundamental form in terms of $H$ and $n$.

math.DG↗

arXiv2021

A Small Variation of the Circular Hodograph Theorem and the Best Elliptical Trajectory of the Planets

A small variation of the circular shape of the hodograph theorem states that for every elliptical solution of the two-body problem, it is possible to find an appropriate inertial frame such that the speed of the bodies is constant. We use this result and data from the NASA JPL Horizon Web Interface to find the best fitting ellipse for the trajectory of Mercury, Venus, Earth, Mars, and Jupiter. The process requires us to find procedures to obtain the plane and ellipse that best fit a collection of points in space. We show that if we aim for the plane that minimizes the sum of the square distances from the given points to the unknown plane, we obtain three planes that appear to divide the set of points equally into octants, one of these being our desired plane of best fit. We provide a detailed proof of the hodograph theorem.

astro-ph.EP↗

arXiv2020

Schmidt representation of 3-qubits with real amplitudes

From the Schmidt representation we have that, up to local gates, every 2-qubit state can be written as $[ϕ\rangle=λ_1 [00\rangle+λ_2 [11\rangle$ with $λ_1$ and $λ_2$ real numbers. For 3-qubits states, it is known, PRL 2000 85, that up to local gates every 3-qubit state can be written as $[ϕ\rangle=λ_1 [000\rangle+λ_2 e^{i θ}[100\rangle+λ_3[101\rangle+λ_4[110\rangle+λ_5[111\rangle$ with $λ_i\ge0$ and $0\le θ\le π$. In this paper, we show that no every 3-qubit state with real amplitudes can be transform in the form $[ϕ\rangle=λ_1 [000\rangle+λ_2 [100\rangle+λ_3[101\rangle+λ_4[110\rangle+λ_5[111\rangle$ by using local gates in the orthogonal group (the group generated by $R_y(θ)$ and $X$ gates). We also show that, up to local gates in the orthogonal group, every 3-qubit with real amplitudes can be written as $[ϕ\rangle=λ_1 [000\rangle+λ_2 [011\rangle+λ_3[101\rangle+λ_4[110\rangle+λ_5[111\rangle$ with the $λ_i$ real numbers. An explanation of this result can be found in the youtube video \url{https://youtu.be/gDN20QHzsoQ}

quant-ph↗

arXiv2020

Bifurcation of periodic orbits for the $N$-body problem, from a non geometrical family of solutions

Given two positive real numbers $M$ and $m$ and an integer $n>2$, it is well known that we can find a family of solutions of the $(n+1)$-body problem where the body with mass $M$ stays put at the origin and the other $n$ bodies, all with the same mass $m$, move on the $x$-$y$ plane following ellipses with eccentri\-city $e$. It is expected that this geometrical family that depends on $e$, has some bifurcations that produce solutions where the body in the center moves on the $z$-axis instead of staying put in the origin. By doing an analytic continuation of a periodic numerical solution of the $4$-body problem --the one displayed on the video http://youtu.be/2Wpv6vpOxXk --we surprisingly discovered that the origin of this periodic solution is not part of the geometrical family of elliptical solutions parametrized by the eccentricity $e$. It comes from a not so geometrical but easier to describe family. Having noticed this new family, the authors find an exact formula for the bifurcation point in this new family and use it to show the existence of a non-planar periodic solution for any pair of masses $M$, $m$, and any integer $n$. As a particular example, we find a solution where three bodies with mass $3$ move around a body with mass $7$ that moves up and down.

math.DS↗

arXiv2020

Controlled not connectivity in the Clifford group

The Clifford group is the set of gates generated by the CZ gate, and the two local gates: the Hadamard and the Pi/2 phase shift gate. It is known that, for a two qubit system, the Clifford group C2 is a subgroup of order 92160 of the group of 4 by 4 unitary matrices. It is also known that the local Clifford gates LC2 is a subgroup of order 4608 of the group C2. In order to better understand the set C2, we make two matrices U1 and U2 in C2 equivalent if U_1=VU_2 for some V in LC2. We show that this equivalence relation splits C2 into 20 orbits, O1, ..., O20, each with 4608 elements. Moreover, for each orbit Oi, CZOi intersects 9 different orbits Oi1, ...,Oi9. Moreover, the intersection of Oij and CZOi has 512 matrices for each j=1,2, ..., ,9. The link https://www.youtube.com/watch?v=lcYtB2tnXFw&t=685s leads you to a YouTube video that explains the most important results in this paper.

quant-ph↗