On residuals of finite groups

Theorem C in [S. Dolfi, M. Herzog, G. Kaplan and A. Lev, The size of the solvable residual in finite groups, Groups Geom. Dyn. 1 (2007), 4, 401–407] asserts that, in a finite group with trivial Fitting subgroup, the size of the soluble residual of the group is bounded from below by a certain power of the group order and that the inequality is sharp. Inspired by this result and some of the arguments in the above article, we establish the following generalisation: if 𝔛 is a subgroup-closed Fitting formation of full characteristic which does not contain all finite groups and X ¯ \overline{\mathfrak{X}} is the extension-closure of 𝔛, then there exists an (explicitly known and optimal) constant 𝛾 depending only on 𝔛 such that, for all non-trivial finite groups 𝐺 with trivial 𝔛-radical, | G X ¯ | > | G | γ \lvert G^{\overline{\mathfrak{X}}}\rvert>\lvert G\rvert^{\gamma} , where G X ¯ G^{\overline{\mathfrak{X}}} is the X ¯ \overline{\mathfrak{X}} -residual of 𝐺. When X = N \mathfrak{X}=\mathfrak{N} , the class of finite nilpotent groups, it follows that X ¯ = S \overline{\mathfrak{X}}=\mathfrak{S} , the class of finite soluble groups; thus we recover the original theorem of Dolfi, Herzog, Kaplan, and Lev. In the last section of our paper, building on J. G. Thompson’s classification of minimal simple groups, we exhibit a family of subgroup-closed Fitting formations 𝔛 of full characteristic such that S ⊂ X ¯ ⊂ E \mathfrak{S}\subset\overline{\mathfrak{X}}\subset\mathfrak{E} , where 𝔈 denotes the class of all finite groups, thus providing applications of our main result beyond the reach of the above theorem.

Partition-theoretic formulas for arithmetic densities, II

International audience In earlier work generalizing a 1977 theorem of Alladi, the authors proved a partition-theoretic formula to compute arithmetic densities of certain subsets of the positive integers N as limiting values of q-series as q → ζ a root of unity (instead of using the usual Dirichlet series to compute densities), replacing multiplicative structures of N by analogous structures in the integer partitions P. In recent work, Wang obtains a wide generalization of Alladi's original theorem, in which arithmetic densities of subsets of prime numbers are computed as values of Dirichlet series arising from Dirichlet convolutions. Here the authors prove that Wang's extension has a partition-theoretic analogue as well, yielding new q-series density formulas for any subset of N. To do so, we outline a theory of q-series density calculations from first principles, based on a statistic we call the "q-density" of a given subset. This theory in turn yields infinite families of further formulas for arithmetic densities.

Quantum Ostrogradsky theorem

The Ostrogradsky theorem states that any classical Lagrangian that contains time derivatives higher than the first order and is nondegenerate with respect to the highest-order derivatives leads to an unbounded Hamiltonian which linearly depends on the canonical momenta. Recently, the original theorem has been generalized to nondegeneracy with respect to non-highest-order derivatives. These theorems have been playing a central role in construction of sensible higher-derivative theories. We explore quantization of such non-degenerate theories, and prove that Hamiltonian is still unbounded at the level of quantum field theory.

Aging Population, Retirement, and Risk Taking: Reply

In theorem 1 given in my paper, “Aging Population, Retirement, and Risk Taking” [Levy H (2016a) Aging population, retirement, and risk taking. Management Sci. 62(5):1415–1430.], there is indeed a technical error. Yet, adding one condition to the theorem (which can be added in two alternate ways) is sufficient to ensure the dominance of stocks over bonds in the very long run. For the commonly employed preferences, the empirical evidence conforms with the claim given in my original theorem 1, asserting that the portfolio with the higher geometric mean (stocks) dominates the other portfolio under consideration (bonds) as the investment horizon increases indefinitely. Thus, as advocated in my paper, stocks dominate bonds for investors with typical preferences who save for retirement. This paper was accepted by Karl Deither, finance.

The unreasonable effectiveness of Nonstandard Analysis

As suggested by the title, the aim of this paper is to uncover the vast computational content of classical Nonstandard Analysis. To this end, we formulate a template ${\mathfrak{C}\mathfrak{I}}$ which converts a theorem of ‘pure’ Nonstandard Analysis, i.e. formulated solely with the nonstandard definitions (of continuity, integration, differentiability, convergence, compactness, etc.), into the associated effective theorem. The latter constitutes a theorem of computable mathematics no longer involving Nonstandard Analysis. To establish the huge scope of ${\mathfrak{C}\mathfrak{I}}$, we apply this template to representative theorems from the Big Five categories from Reverse Mathematics. The latter foundational program provides a classification of the majority of theorems from ‘ordinary’, i.e. non-set theoretical, mathematics into the aforementioned five categories. The Reverse Mathematics zoo gathers exceptions to this classification, and is studied in [ 74, 77] using ${\mathfrak{C}\mathfrak{I}}$. Hence, the template ${\mathfrak{C}\mathfrak{I}}$ is seen to apply to essentially all of ordinary mathematics, thanks to the Big Five classification (and associated zoo) from Reverse Mathematics. Finally, we establish that certain ‘highly constructive’ theorems, called Herbrandizations, also imply the original theorem of Nonstandard Analysis from which they were obtained via ${\mathfrak{C}\mathfrak{I}}$.

Algebraic Values of Entire Functions with Extremal Growth Orders: An Extension of a Theorem of Boxall and Jones

Given an entire function $f$ of finite order $\unicode[STIX]{x1D70C}$ and positive lower order $\unicode[STIX]{x1D706}$, Boxall and Jones proved a bound of the form $C(\log H)^{\unicode[STIX]{x1D702}(\unicode[STIX]{x1D706},\unicode[STIX]{x1D70C})}$ for the density of algebraic points of bounded degree and height at most $H$ on the restrictions to compact sets of the graph of $f$. The constant $C$ and exponent $\unicode[STIX]{x1D702}$ are effectively computable from certain data associated with the function. In this followup note, using different measures of the growth of entire functions, we obtain similar bounds for other classes of functions to which the original theorem does not apply.

Algorithms in A ∞-algebras

Based on Kadeishvili’s original theorem inducing{A_{\infty}}-algebra structures on the homology of dg-algebras, several directions of algorithmic research in{A_{\infty}}-algebras have been pursued. In this paper, we survey the work done on calculating explicit{A_{\infty}}-algebra structures from homotopy retractions, in group cohomology and in persistent homology.

An Epistemic Theory of Democracy

One attractive feature of democracy is its ability to track the truth by information aggregation. The formal support for this claim goes back to Condorcet’s famous jury theorem. However, the theorem has often been dismissed as a mathematical curiosity because the assumptions on which the theorem is based are demanding. Such quick dismissals tend to misunderstand the original theorem. They also fail to appreciate how Condorcet’s assumptions can be weakened to obtain jury theorems that are readily applicable in the real world. The first part of the book explains the original theorem and its various extensions and introduces results to deal with the challenge of voter dependence. Part II considers opportunities to make democracies perform better in epistemic terms by improving voter competence and diversity, by dividing epistemic labour, and by preceding voting with deliberation. In the third part, political practices are looked at through an epistemic lens, focusing on the influence of tradition, following opinion leaders or cues, and on settings in which the electorate falls into diverging factions. Part IV analyses the implications for the structures of government. While arguing against the case for epistocracy, the use of deliberation and expert advice in representative democracy can lead to improved truth-tracking, provided epistemic bottlenecks are avoided. The final part summarizes the results and explores how epistemic democracy might be undermined, using as case studies the Trump and Brexit campaigns.

Global Existence for a System of Quasi-Linear Wave Equations in 3D Satisfying the Weak Null Condition

We show global existence of small solutions to the Cauchy problem for a system of quasi-linear wave equations in three space dimensions. The feature of the system lies in that it satisfies the weak null condition, though we permit the presence of some quadratic nonlinear terms which do not satisfy the null condition. Due to the presence of such quadratic terms, the standard argument no longer works for the proof of global existence. To get over this difficulty, we extend the ghost weight method of Alinhac so that it works for the system under consideration. The original theorem of Alinhac for the scalar unknowns is also refined.

Walker's Cancellation Theorem

Walker's cancellation theorem says that if B + Z isisomorphic to C + Z in the category of abeliangroups, then B is isomorphic to C. We construct an example ina diagram category of abelian groups where the theorem fails. As aconsequence, the original theorem does not have a constructiveproof. In fact, in our example B and C are subgroups ofZ<sup>2</sup>. Both of these results contrast with a group whoseendomorphism ring has stable range one, which allows aconstructive proof of cancellation and also a proof in any diagramcategory.