Infinite Matrices and Their Recent Applications

Shivakumar, P. N.; Sivakumar, K. C.; Zhang, Yang

doi:10.1007/978-3-319-30180-8

Cited by 18 publications

(7 citation statements)

References 104 publications

(188 reference statements)

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“…where e i indicates the i th column unit vector. Importantly, for any φ ≥ 0, M + φI is a "weakly chained diagonally dominant" matrix with negative off-diagonals, which means that it is invertible and all entries of the inverse (M + φI) −1 are non-negative [73]. This allows us to express the steadystate intermediate concentrations as…”

Section: Discussion and Future Workmentioning

confidence: 99%

A thermodynamic threshold for Darwinian evolution

Kolchinsky¹

2021

Preprint

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Understanding the thermodynamics of Darwinian evolution has important implications for biophysics, evolutionary biology, and the study of the origin of life. We show that in a population of nonequilibrium autocatalytic replicators, the critical selection coefficient (minimal fitness difference visible to selection) is lower bounded by the Gibbs free energy dissipated per replication event. This bound presents a fundamental thermodynamic threshold for Darwinian evolution, analogous to thresholds that arise from finite population sizes or large mutation rates. Our results apply to a large class of molecular replicators, including many types of multistep autocatalytic mechanisms and autocatalytic sets. We illustrate our approach on a model of simple replicators in a chemostat.

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Section: Discussion and Future Workmentioning

confidence: 99%

A thermodynamic threshold for Darwinian evolution

Kolchinsky¹

2021

Preprint

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“…This theorem can be extended to operators that act on an infinite dimensional space, particularly to infinite matrices. There is a great quantity of literature on the subject (see for example [40] and references therein) and indeed the Gershgorin theorem gives rise to an entire theory, called the Gershgorin theory. Lemmas 4.5 and 4.6 are examples of the achievements of this theory.…”

Section: Gershgorin Theorymentioning

confidence: 99%

Non‐harmonic Gohberg's lemma, Gershgorin theory and heat equation on manifolds with boundary

Ruzhansky

Velasquez-Rodriguez

2021

Mathematische Nachrichten

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We use Operator Ideals Theory and Gershgorin theory to obtain explicit information in terms of the symbol concerning the spectrum of pseudo‐differential operators, on a smooth manifold Ω with boundary ∂Ω, in the context of the non‐harmonic analysis of boundary value problems introduced in [29] in terms of a model operator frakturL. For symbols in the Hörmander class S1,00true(Ω¯×scriptItrue), we provide a non‐harmonic version of Gohberg's lemma, and a sufficient and necessary condition to ensure that the corresponding pseudo‐differential operator is a compact operator in L2false(normalΩfalse), or a Riesz operator in Lpfalse(normalΩfalse) in the case of Riemannian manifolds with smooth boundary. We extend to the context of the non‐harmonic analysis of boundary value problems the well known theorems about the exact domain of elliptic operators, and discuss some applications of the obtained results to evolution equations.

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“…This in fact makes little difference as the rise is dominated by the C i factors in the M matrix elements, which grow asymptotically very rapidly. The size of the eigenvalues can be estimated by taking finite 'sections' (submatrices) [15], [16] in conjunction with the theory of Geršgorin disks [14]. Listing the eigenvalues λ (k) in increasing order, one has the estimate λ…”

Section: Noting That Xmentioning

confidence: 99%

Multigravity and the cosmological constant

Tomboulis¹

2021

Preprint

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The possibility of flat metric solutions in the presence of arbitrary, in particular, Planck mass sized cosmological constants is investigated within multigravity theory, the ghost-free theory of (N + 1) rank-2 tensor fields comprising one massless and N massive spin-2 fields. It is found that, indeed, for any N , no matter how large, no flat solutions exist without finetuning parameters in the potential. For an infinite number of fields, however, flat solutions exist within an infinite-dimensional parameter space. Such a ground state may be viewed as a collective state that cannot be approximated by any finite N .

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Infinite Matrices and Their Recent Applications

Cited by 18 publications

References 104 publications

A thermodynamic threshold for Darwinian evolution

A thermodynamic threshold for Darwinian evolution

Non‐harmonic Gohberg's lemma, Gershgorin theory and heat equation on manifolds with boundary

Multigravity and the cosmological constant

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