Wiles defect for Hecke algebras that are not complete intersections

In his work on modularity theorems, Wiles proved a numerical criterion for a map of rings $R\to T$ to be an isomorphism of complete intersections. He used this to show that certain deformation rings and Hecke algebras associated to a mod $p$ Galois representation at non-minimal level are isomorphic and complete intersections, provided the same is true at minimal level. In this paper we study Hecke algebras acting on cohomology of Shimura curves arising from maximal orders in indefinite quaternion algebras over the rationals localized at a semistable irreducible mod $p$ Galois representation $\bar {\rho }$ . If $\bar {\rho }$ is scalar at some primes dividing the discriminant of the quaternion algebra, then the Hecke algebra is still isomorphic to the deformation ring, but is not a complete intersection, or even Gorenstein, so the Wiles numerical criterion cannot apply. We consider a weight-2 newform $f$ which contributes to the cohomology of the Shimura curve and gives rise to an augmentation $\lambda _f$ of the Hecke algebra. We quantify the failure of the Wiles numerical criterion at $\lambda _f$ by computing the associated Wiles defect purely in terms of the local behavior at primes dividing the discriminant of the global Galois representation $\rho _f$ which $f$ gives rise to by the Eichler–Shimura construction. One of the main tools used in the proof is Taylor–Wiles–Kisin patching.

Le gouvernement de l’homme royal dans le Politique : une utopie assumée

The object of this article, which analyses Statesman 291a1-303d3, is to show how the good, the object of politics qua knowledge, makes the regime with which it is associated a utopia. The good cannot be actualized anywhere in the sensible realm, because no city can be governed without laws, and the laws define what is good most often for the greatest number. A government of the good, without laws, is a utopia, but the laws, to the extent that they aim at the common interest, are in themselves the imprint of the good. I defend the thesis, at first sight paradoxical, that the true politicians are not those who know the good and imitate it, but those who recognise the utility of laws and agree to submit themselves to their authority. Thus Plato dismisses all existing regimes as sophistic regimes where power is exercised in the interest of the rulers and not of the ruled. Not every regime, however, bears the imprint of the good equally. By making politics a branch of knowledge, Plato imposes a numerical criterion on the classification of regimes; he crosses the criterion of number with that of law. I therefore argue for another paradox: the best possible regime is a monarchy in which the rulers obey the laws, the best existing regime is none other than democracy.

Technical note: A two-sided affine power scaling relationship to represent the concentration–discharge relationship

This technical note deals with the mathematical representation of concentration–discharge relationships. We propose a two-sided affine power scaling relationship (2S-APS) as an alternative to the classic one-sided power scaling relationship (commonly known as “power law”). We also discuss the identification of the parameters of the proposed relationship, using an appropriate numerical criterion. The application of 2S-APS to the high-frequency chemical time series of the Orgeval-ORACLE observatory is presented here (in calibration and validation mode): it yields better results for several solutes and for electrical conductivity in comparison with the power law relationship.

Technical Note: A two-sided affine power scaling relationship to represent the concentration–discharge relationship

This technical note deals with the mathematical representation of concentration–discharge relationships. We propose a two-sided affine power scaling relationship (2S-APS) as an alternative to the classic one-sided power scaling relationship (commonly known as power law). We also discuss the identification of the parameters of the proposed relationship, using an appropriate numerical criterion. The application of 2S-APS to the high-frequency chemical time series of the Orgeval–Oracle observatory is presented (in calibration and validation mode): It yields better results for several solutes and for electrical conductivity in comparison with the power law relationship.

The Af condition and relative conormal spaces for functions with nonvanishing derivative

We introduce a join construction, as a way of completing the description of the relative conormal space of an analytic function on a complex analytic space that has a non-vanishing derivative at the origin. Then we show how to obtain a numerical criterion for Thom’s [Formula: see text] condition.

Party Switching in Israel: Understanding the Split of the Labor Party in 2011

In January 2011, former Israeli Prime Minister Ehud Barak issued a surprising announcement to take four other members of his Labor Party’s Knesset faction with himself to set up a new political party, Haatzmaut (Independence). The conditions under which this split took place illustrate the ways in which the Israeli anti-defection law, passed in the 12th Knesset, incentivizes the behavior of elected legislators who seek to exit from the party that they were elected to represent. This article shows that the anti-defection law cannot keep a legislative party together that suffers from weak internal cohesion. In fact, by imposing numerical criterion (1/3) on prospective party switchers, the anti-defection law prolongs internal disunity, thereby further weakening an already low level of cohesion.

Multipoint Numerical Criterion for Manifolds to Guarantee Attractor Properties in the Problem of Synergetic Control Design

The paper presents the results of our research on the application of symbolic regression methods for the numerical solution of the problem of synthesis of synergetic control. Synergetic control is characterized by the presence of manifolds in the state space that must have the properties of an attractor, in particular terminal manifolds, as well as the existence of regions of state space through which system solutions do not pass, for example, phase constraints, that must have the properties of repellers. In the present paper we formulate a numerical criterion for ensuring the property of an attractor for a terminal manifold of arbitrary dimension, taking into account the dynamic behavior of the system in a neighborhood of a given manifold. The results of computational experiments on the effective application of the proposed criterion are presented. It is shown that the proposed approach makes it possible to automate the synthesis of synergistic control using the methods of symbolic regression.