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Michael Shub

Publications and source records attributed to Michael Shub.

18 recordsLinked to original sources
arXiv2024

An average intersection estimate for families of diffeomorphisms

We show that for any sufficiently rich compact family $\mathcal{H}$ of $C^1$ diffeomorphisms of a closed Riemannanian manifold $M$, the average geometric intersection number over $h \in \mathcal{H}$ between $h(V)$ and $W$, for $V, W$ any complementary dimensional submanifolds of $M$, is approximately (i.e. up to a uniform multiplicative error depending only on $\mathcal{H}$) the product of their volumes. We also give a construction showing that such families always exist.

math.DG↗
arXiv2022

Random and mean Lyapunov exponents for $\mathrm{GL}_n(\mathbb{R})$

We consider orthogonally invariant probability measures on $\mathrm{GL}_n(\mathbb{R})$ and compare the mean of the logs of the moduli of eigenvalues of the matrices to the Lyapunov exponents of random matrix products independently drawn with respect to the measure. We give a lower bound for the former in terms of the latter. The results are motivated by Dedieu-Shub\cite{DS}. A novel feature of our treatment is the use of the theory of spherical polynomials in the proof of our main result.

math.DS↗
arXiv2022

Cusp Bifurcation in Metastatic Breast Cancer Cells

Ordinary differential equations (ODEs) can model the transition of cell states over time. Bifurcation theory is a branch of dynamical systems which studies changes in the behavior of an ODE system while one or more parameters are varied. We have found that concepts in bifurcation theory may be applied to model metastatic cell behavior. Our results show how a specific phenomenon called a cusp bifurcation describes metastatic cell state transitions, separating two qualitatively different transition modalities. Moreover, we show how the cusp bifurcation models other genetic networks, and we relate the dynamics after the bifurcation to observed phenomena in commitment to enter the cell cycle.

q-bio.CB↗
arXiv2021

Disease Prediction with a Maximum Entropy Method

In this paper, we propose a maximum entropy method for predicting disease risks. It is based on a patient's medical history with diseases coded in ICD-10 which can be used in various cases. The complete algorithm with strict mathematical derivation is given. We also present experimental results on a medical dataset, demonstrating that our method performs well in predicting future disease risks and achieves an accuracy rate twice that of the traditional method. We also perform a comorbidity analysis to reveal the intrinsic relation of diseases.

cs.LG↗
arXiv2018

Stable and non-symmetric pitchfork bifurcations

In this paper, we present a criterion for pitchfork bifurcation of smooth vector fields based on a topological argument. Our result expands Rajapakse and Smale's result \cite{RS2} significantly. Based on our criterion, we present a class of families of non-symmetric vector fields undergoing a pitchfork bifurcation.

math.DS↗
arXiv2016

Computing the Homology of Real Projective Sets

We describe and analyze a numerical algorithm for computing the homology (Betti numbers and torsion coefficients) of real projective varieties. Here numerical means that the algorithm is numerically stable (in a sense to be made precise). Its cost depends on the condition of the input as well as on its size and is singly exponential in the number of variables (the dimension of the ambient space) and polynomial in the condition and the degrees of the defining polynomials. In addition, we show that outside of an exceptional set of measure exponentially small in the size of the data, the algorithm takes exponential time.

math.AG↗
arXiv2015

Condition length and complexity for the solution of polynomial systems

Smale's 17th problem asks for an algorithm which finds an approximate zero of polynomial systems in average polynomial time (see Smale 2000). The main progress on Smale's problem is Beltr\'an-Pardo (2011) and B\"urgisser-Cucker (2010). In this paper we will improve on both approaches and we prove an important intermediate result. Our main results are Theorem 1 on the complexity of a randomized algorithm which improves the result of Beltr\'an-Pardo (2011), Theorem 2 on the average of the condition number of polynomial systems which improves the estimate found in B\"urgisser-Cucker (2010), and Theorem 3 on the complexity of finding a single zero of polynomial systems. This last Theorem is the main result of B\"urgisser-Cucker (2010). We give a proof of it relying only on homotopy methods, thus removing the need for the elimination theory methods used in B\"urgisser-Cucker (2010). We build on methods developed in Armentano et al. (2015).

math.NA↗
arXiv2015

A stable, polynomial-time algorithm for the eigenpair problem

We describe algorithms for computing eigenpairs (eigenvalue-eigenvector pairs) of a complex $n\times n$ matrix $A$. These algorithms are numerically stable, strongly accurate, and theoretically efficient (i.e., polynomial-time). We do not believe they outperform in practice the algorithms currently used for this computational problem. The merit of our paper is to give a positive answer to a long-standing open problem in numerical linear algebra.

math.NA↗
arXiv2014

Amino acid metabolism conflicts with protein diversity

The twenty protein coding amino acids are found in proteomes with different relative abundances. The most abundant amino acid, leucine, is nearly an order of magnitude more prevalent than the least abundant amino acid, cysteine. Amino acid metabolic costs differ similarly, constraining their incorporation into proteins. On the other hand, sequence diversity is necessary for protein folding, function and evolution. Here we present a simple model for a cost-diversity trade-off postulating that natural proteomes minimize amino acid metabolic flux while maximizing sequence entropy. The model explains the relative abundances of amino acids across a diverse set of proteomes. We found that the data is remarkably well explained when the cost function accounts for amino acid chemical decay. More than one hundred proteomes reach comparable solutions to the trade-off by different combinations of cost and diversity. Quantifying the interplay between proteome size and entropy shows that proteomes can get optimally large and diverse.

q-bio.PE↗
arXiv2012

The complexity and geometry of numerically solving polynomial systems

These pages contain a short overview on the state of the art of efficient numerical analysis methods that solve systems of multivariate polynomial equations. We focus on the work of Steve Smale who initiated this research framework, and on the collaboration between Stephen Smale and Michael Shub, which set the foundations of this approach to polynomial system--solving, culminating in the more recent advances of Carlos Beltran, Luis Miguel Pardo, Peter Buergisser and Felipe Cucker.

math.NA↗
arXiv2012

Periodic Points on the 2-sphere

For a $C^{1}$ degree two latitude preserving endomorphism $f$ of the 2-sphere, we show that $f$ has $2^{n}$ periodic points.

math.DS↗
arXiv2012

Hölder foliations, revisited

We investigate transverse Hölder regularity of some canonical leaf conjugacies in partially hyperbolic dynamical systems and transverse Hölder regularity of some invariant foliations. Our results validate claims made elsewhere in the literature.

math.DS↗
arXiv2012

Smale's Fundamental Theorem of Algebra reconsidered

In his 1981 Fundamental Theorem of Algebra paper Steve Smale initiated the complexity theory of finding a solution of polynomial equations of one complex variable by a variant of Newton's method. In this paper we reconsider his algorithm in the light of work done in the intervening years. Smale's upper bound estimate was infinite average cost. Our's is polynomial in the B\'ezout number and the dimension of the input. Hence polynomial for any range of dimensions where the B\'ezout number is polynomial in the input size. In particular not just for the case that Smale considered but for a range of dimensions as considered by B\"urgisser-Cucker where the max of the degrees is greater than or equal to $n^{1+\epsilon}$ for some fixed $\epsilon$. It is possible that Smale's algorithm is polynomial cost in all dimensions and our main theorem raises some problems that might lead to a proof of such a theorem.

math.NA↗
arXiv2011

Adaptative Step Size Selection for Homotopy Methods to Solve Polynomial Equations

Given a C^1 path of systems of homogeneous polynomial equations f_t, t in [a,b] and an approximation x_a to a zero zeta_a of the initial system f_a, we show how to adaptively choose the step size for a Newton based homotopy method so that we approximate the lifted path (f_t,zeta_t) in the space of (problems, solutions) pairs. The total number of Newton iterations is bounded in terms of the length of the lifted path in the condition metric.

math.NA↗
arXiv2007

Complexity of Bezout's Theorem VI: Geodesics in the Condition (Number) Metric

We introduce a new complexity measure of a path of (problems, solutions) pairs in terms of the length of the path in the condition metric which we define in the article. The measure gives an upper bound for the number of Newton steps sufficient to approximate the path discretely starting from one end and thus produce an approximate zero for the endpoint. This motivates the study of short paths or geodesics in the condition metric.

math.NA↗
arXiv2002

Partial differentiability of invariant splittings

A key feature of a general nonlinear partially hyperbolic dynamical system is the absence of differentiability of its invariant splitting. In this paper, we show that often partial derivatives of the splitting exist and the splitting depends smoothly on the dynamical system itself.

math.DS↗