On the minimum value of the condition number of polynomials

The condition number of a polynomial is a natural measure of the sensitivity of the roots under small perturbations of the polynomial coefficients. In 1993 Shub and Smale posed the problem of finding a sequence of univariate polynomials of degree N with condition number bounded above by N⁠. In Beltrán et al. (2021, A sequence of polynomials with optimal condition number. J. Amer. Math. Soc., 34, 219–244) it was proved that the optimal value of the condition number is of the form O(N−−√)⁠, and the sequence demanded by Shub and Smale was described by a closed formula for large enough N⩾N0 with N0 unknown, and by a search algorithm for the rest of the cases. In this paper we find concrete estimates for the constant hidden in the O(N−−√) term and we describe a simple formula for a sequence of polynomials whose condition number is at most N⁠, valid for all N=4M2⁠, with M a positive integer

A generalization of the spherical ensemble to even-dimensional spheres

In a recent article [1], Alishahi and Zamani discuss the spherical ensemble, a rotationally invariant determinantal point process on S2. In this paper we extend this process in a natural way to the 2d-dimensional sphere S2d. We prove that the expected value of the Riesz s-energy associated to this determinantal point process has a reasonably low value compared to the known asymptotic expansion of the minimal Riesz s-energy

Sobre el problema número 7 de Smale

En este artículo describimos la situación de la investigación sobre el conocido como Problema número 7 de Smale, propuesto originalmente por Mike Shub y Steve Smale en 1993, y posteriormente añadido a la lista de Smale de Problemas para el Siglo XXI. Tratamos de describir las diferentes facetas del problema, así como muchos de los resultados que se conocen, evitando tecnicismos innecesarios.El autor está financiado por los proyectos MTM2017-83816-P y MTM2017-90682-REDT del Ministerio de Ciencia, Innovación y Universidades, y por el proyecto 21.SI01.64658 del Banco Santander en colaboración con la Universidad de Cantabria

The Average Condition Number of Most Tensor Rank Decomposition Problems is Infinite

The tensor rank decomposition, or canonical polyadic decomposition, is the decomposition of a tensor into a sum of rank-1 tensors. The condition number of the tensor rank decomposition measures the sensitivity of the rank-1 summands with respect to structured perturbations. Those are perturbations preserving the rank of the tensor that is decomposed. On the other hand, the angular condition number measures the perturbations of the rank-1 summands up to scaling. We show for random rank-2 tensors that the expected value of the condition number is infinite for a wide range of choices of the density. Under a mild additional assumption, we show that the same is true for most higher ranks r?3r?3 as well. In fact, as the dimensions of the tensor tend to infinity, asymptotically all ranks are covered by our analysis. On the contrary, we show that rank-2 tensors have finite expected angular condition number. Based on numerical experiments, we conjecture that this could also be true for higher ranks. Our results underline the high computational complexity of computing tensor rank decompositions. We discuss consequences of our results for algorithm design and for testing algorithms computing tensor rank decompositions.We thank the reviewers for helpful suggestions. Part of this work was made while the second and third author were visiting the Universidad de Cantabria, supported by the funds of Grant 21.SI01.64658 (Banco Santander and Universidad de Cantabria), Grant MTM2017-83816-P from the Spanish Ministry of Science. The third author was additionally supported by the FWO Grant for a long stay abroad V401518N. We thank these institutions for their support

How well-conditioned can the eigenvector problem be?

The condition number for eigenvector computations is a well– studied quantity. But how small can it possibly be?: Specifically, what matrices are perfectly conditioned with respect to eigenvector computations? In this note we answer this question for n × n matrices, giving a solution that is exact to first-order as n → ∞

On the number of interference alignment solutions for the K-user MIMO channel with constant coefficients

In this paper, we study the number of different interference alignment (IA) solutions in a K-user multiple-input multiple-output (MIMO) interference channel, when the alignment is performed via beamforming and no symbol extensions are allowed. We focus on the case where the number of IA equations matches the number of variables. In this situation, the number of IA solutions is finite and constant for any channel realization out of a zero-measure set and, as we prove in this paper, it is given by an integral formula that can be numerically approximated using Monte Carlo integration methods. More precisely, the number of alignment solutions is the scaled average of the determinant of a certain Hermitian matrix related to the geometry of the problem. Interestingly, while the value of this determinant at an arbitrary point can be used to check the feasibility of the IA problem, its average (properly scaled) gives the number of solutions. For single-beam systems, the asymptotic growth rate of the number of solutions is analyzed and some connections with classical combinatorial problems are presented. Nonetheless, our results can be applied to arbitrary interference MIMO networks, with any number of users, antennas, and streams per user.The work of Ó. González and I. Santamaría was supported by MICINN (Spanish Ministry for Science and Innovation) under grants TEC2013-47141-C4-3-R (RACHEL), TEC2010-19545-C04-03 (COSIMA), CONSOLIDER-INGENIO 2010 CSD2008-00010 (COMONSENS) and FPU grant AP2009-1105. Carlos Beltrán was partially supported by MICINN grant MTM2010-16051

TEMPLUM: A Process Adapted Numerical Simulation Code for The 3D Predictive Assessment of Laser Surface Heat Treatments in Planar Geometry

A process adapted numerical simulation code for the 3D predictive assessment of laser heat treatment of materials has been developed. Primarily intended for the analysis of the laser transformation hardening of steels, the code has been successfully applied for the predictive characterization of other metallic and non metallic materials posing specific difficulties from the numerical point of view according to their extreme thermal and absorption properties. Initially based on a conventional FEM calculational structure, the developed code (TEMPLUM) reveals itself as an extremely useful prediction tool with specific process adapted features (not usually available in FEM heat transfer codes) in the field of laser heat treatment applications

A sequence of polynomials with optimal condition number

We find an explicit sequence of univariate polynomials of arbitrary degree with optimal condition number. This solves a problem posed by Michael Shub and Stephen Smale in 1993.The first and second authors were partially supported by Ministerio de Econom´ıa y Competitividad, Gobierno de Espa˜na, through grants MTM2017-83816-P and MTM2017-90682-REDT, and by the Banco de Santander and Universidad de Cantabria grant 21.SI01.64658. The second author was also supported by the Austrian Science Fund FWF project F5503 (part of the Special Research Program (SFB) Quasi-Monte Carlo Methods: Theory and Applications). The third and fourth authors have been partially supported by grant MTM2017-83499-P by the Ministerio de Econom´ıa y Competitividad, Gobierno de Espa˜na and by the Generalitat de Catalunya (project 2017 SGR 358)

Union bound minimization approach for designing grassmannian constellations

In this paper, we propose an algorithm for designing unstructured Grassmannian constellations for noncoherent multiple-input multiple-output (MIMO) communications over Rayleigh block-fading channels. Unlike the majority of existing unitary space-time or Grassmannian constellations, which are typically designed to maximize the minimum distance between codewords, in this work we employ the asymptotic pairwise error probability (PEP) union bound (UB) of the constellation as the design criterion. In addition, the proposed criterion allows the design of MIMO Grassmannian constellations specifically optimized for a given number of receiving antennas. A rigorous derivation of the gradient of the asymptotic UB on a Cartesian product of Grassmann manifolds, is the main technical ingredient of the proposed gradient descent algorithm. A simple modification of the proposed cost function, which weighs each pairwise error term in the UB according to the Hamming distance between the binary labels assigned to the respective codewords, allows us to jointly solve the constellation design and the bit labeling problem. Our simulation results show that the constellations designed with the proposed method outperform other structured and unstructured Grassmannian designs in terms of symbol error rate (SER) and bit error rate (BER), for a wide range of scenarios.This work was supported by Huawei Technologies, Sweden under the project GRASSCOM. The work of D. Cuevas was also partly supported under grant FPU20/03563 funded by Ministerio de Universidades (MIU), Spain. The work of Carlos Beltr´an was also partly supported under grant PID2020-113887GB-I00 funded by MCIN/ AEI /10.13039/501100011033. The work of I. Santamaria was also partly supported under grant PID2019-104958RB-C43 (ADELE) funded by MCIN/ AEI /10.13039/501100011033

A measure preserving mapping for structured Grassmannian constellations in SIMO channels

In this paper, we propose a new structured Grassmannian constellation for noncoherent communications over single-input multiple-output (SIMO) Rayleigh block-fading channels. The constellation, which we call Grass-Lattice, is based on a measure preserving mapping from the unit hypercube to the Grassmannian of lines. The constellation structure allows for on-the-fly symbol generation, low-complexity decoding, and simple bit-to-symbol Gray coding. Simulation results show that Grass-Lattice has symbol error rate performance close to that of a numerically optimized unstructured constellation, and is more power efficient than other structured constellations proposed in the literature.This work was supported by Huawei Technologies Sweden, under the project GRASSCOM. The work of D. Cuevas was also partly supported under grant FPU20/03563 funded by Ministerio de Universidades (MIU), Spain. The work of Carlos Beltr´an was also partly supported under grant PID2020-113887GB-I00 funded by MCIN/AEI /10.13039/501100011033. The work ofI. Santamaria was also partly supported under grant PID2019-104958RB-C43(ADELE) funded by MCIN/ AEI /10.13039/501100011033