Tunisian Journal of Mathematics Articles (Project Euclid)
https://projecteuclid.org/euclid.tunis
The latest articles from Tunisian Journal of Mathematics on Project Euclid, a site for mathematics and statistics resources.enusCopyright 2018 Cornell University LibraryEuclidL@cornell.edu (Project Euclid Team)Mon, 03 Dec 2018 11:31 ESTMon, 03 Dec 2018 11:31 ESThttps://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
https://projecteuclid.org/
Nonlocal selfimproving properties: a functional analytic approach
https://projecteuclid.org/euclid.tunis/1543854680
<strong>Pascal Auscher</strong>, <strong>Simon Bortz</strong>, <strong>Moritz Egert</strong>, <strong>Olli Saari</strong>. <p><strong>Source: </strong>Tunisian Journal of Mathematics, Volume 1, Number 2, 151183.</p><p><strong>Abstract:</strong><br/>
A functional analytic approach to obtaining selfimproving properties of solutions to linear nonlocal elliptic equations is presented. It yields conceptually simple and very short proofs of some previous results due to Kuusi–Mingione–Sire and Bass–Ren. Its flexibility is demonstrated by new applications to nonautonomous parabolic equations with nonlocal elliptic part and questions related to maximal regularity.
</p>projecteuclid.org/euclid.tunis/1543854680_20181203113124Mon, 03 Dec 2018 11:31 ESTSaturated morphisms of logarithmic schemes
https://projecteuclid.org/euclid.tunis/1543854681
<strong>Takeshi Tsuji</strong>. <p><strong>Source: </strong>Tunisian Journal of Mathematics, Volume 1, Number 2, 185220.</p><p><strong>Abstract:</strong><br/>
The notion of universally saturated morphisms between saturated log schemes was introduced by Kazuya Kato. In this paper, we study universally saturated morphisms systematically by introducing the notion of saturated morphisms between integral log schemes as a relative analogue of saturated log structures. We eventually show that a morphism of saturated log schemes is universally saturated if and only if it is saturated. We prove some fundamental properties and characterizations of universally saturated morphisms via this interpretation.
</p>projecteuclid.org/euclid.tunis/1543854681_20181203113124Mon, 03 Dec 2018 11:31 ESTQuantum meanfield asymptotics and multiscale analysis
https://projecteuclid.org/euclid.tunis/1543854682
<strong>Zied Ammari</strong>, <strong>Sébastien Breteaux</strong>, <strong>Francis Nier</strong>. <p><strong>Source: </strong>Tunisian Journal of Mathematics, Volume 1, Number 2, 221272.</p><p><strong>Abstract:</strong><br/>
We study, via multiscale analysis, a defectofcompactness phenomenon which occurs in bosonic and fermionic quantum meanfield problems. The approach relies on a combination of meanfield asymptotics and second microlocalized semiclassical measures. The phase space geometric description is illustrated by various examples.
</p>projecteuclid.org/euclid.tunis/1543854682_20181203113124Mon, 03 Dec 2018 11:31 ESTA nonlinear estimate of the life span of solutions of the three dimensional Navier–Stokes equations
https://projecteuclid.org/euclid.tunis/1543854683
<strong>JeanYves Chemin</strong>, <strong>Isabelle Gallagher</strong>. <p><strong>Source: </strong>Tunisian Journal of Mathematics, Volume 1, Number 2, 273293.</p><p><strong>Abstract:</strong><br/>
The purpose of this article is to establish bounds from below for the life span of regular solutions to the incompressible Navier–Stokes system, which involve norms not only of the initial data, but also of nonlinear functions of the initial data. We provide examples showing that those bounds are significant improvements to the one provided by the classical fixed point argument. One of the important ingredients is the use of a scaleinvariant energy estimate.
</p>projecteuclid.org/euclid.tunis/1543854683_20181203113124Mon, 03 Dec 2018 11:31 ESTRigid local systems and alternating groupshttps://projecteuclid.org/euclid.tunis/1544842815<strong>Robert M. Guralnick</strong>, <strong>Nicholas M. Katz</strong>, <strong>Pham Huu Tiep</strong>. <p><strong>Source: </strong>Tunisian Journal of Mathematics, Volume 1, Number 3, 295320.</p><p><strong>Abstract:</strong><br/>
We show that some very simple to write one parameter families of exponential sums on the affine line in characteristic [math] have alternating groups as their geometric monodromy groups.
</p>projecteuclid.org/euclid.tunis/1544842815_20181214220036Fri, 14 Dec 2018 22:00 ESTLocal estimates for Hörmander's operators with Gevrey coefficients and application to the regularity of their Gevrey vectorshttps://projecteuclid.org/euclid.tunis/1544842816<strong>Makhlouf Derridj</strong>. <p><strong>Source: </strong>Tunisian Journal of Mathematics, Volume 1, Number 3, 321345.</p><p><strong>Abstract:</strong><br/>
Given a general Hörmander’s operator [math] in an open set [math] , where [math] are smooth real vector fields in [math] , [math] , and given also an open, relatively compact set [math] with [math] , and [math] , [math] , such that the coefficients of [math] are in [math] and [math] satisfies a [math] Sobolev estimate in [math] , our aim is to establish local estimates reflecting local domination of ordinary derivatives by powers of [math] , in [math] . As an application, we give a direct proof of the [math] regularity of any [math] vector of [math] .
</p>projecteuclid.org/euclid.tunis/1544842816_20181214220036Fri, 14 Dec 2018 22:00 ESTGeneric colourful tori and inverse spectral transform for Hankel operatorshttps://projecteuclid.org/euclid.tunis/1544842819<strong>Patrick Gérard</strong>, <strong>Sandrine Grellier</strong>. <p><strong>Source: </strong>Tunisian Journal of Mathematics, Volume 1, Number 3, 347372.</p><p><strong>Abstract:</strong><br/>
This paper explores the regularity properties of an inverse spectral transform for Hilbert–Schmidt Hankel operators on the unit disc. This spectral transform plays the role of actionangle variables for an integrable infinite dimensional Hamiltonian system: the cubic Szegő equation. We investigate the regularity of functions on the tori supporting the dynamics of this system, in connection with some wave turbulence phenomenon, discovered in a previous work and due to relative small gaps between the actions. We revisit this phenomenon by proving that generic smooth functions and a [math] dense set of irregular functions do coexist on the same torus. On the other hand, we establish some uniform analytic regularity for tori corresponding to rapidly decreasing actions which satisfy some specific property ruling out the phenomenon of small gaps.
</p>projecteuclid.org/euclid.tunis/1544842819_20181214220036Fri, 14 Dec 2018 22:00 ESTRamification groups of coverings and valuationshttps://projecteuclid.org/euclid.tunis/1544842820<strong>Takeshi Saito</strong>. <p><strong>Source: </strong>Tunisian Journal of Mathematics, Volume 1, Number 3, 373426.</p><p><strong>Abstract:</strong><br/>
We give a purely scheme theoretic construction of the filtration by ramification groups of the Galois group of a covering. The valuation need not be discrete but the normalizations are required to be locally of complete intersection.
</p>projecteuclid.org/euclid.tunis/1544842820_20181214220036Fri, 14 Dec 2018 22:00 ESTAlmost sure local wellposedness for the supercritical quintic NLShttps://projecteuclid.org/euclid.tunis/1544842823<strong>Justin T. Brereton</strong>. <p><strong>Source: </strong>Tunisian Journal of Mathematics, Volume 1, Number 3, 427453.</p><p><strong>Abstract:</strong><br/>
This paper studies the quintic nonlinear Schrödinger equation on [math] with randomized initial data below the critical regularity [math] for [math] . The main result is a proof of almost sure local wellposedness given a Wiener randomization of the data in [math] for [math] . The argument further develops the techniques introduced in the work of Á. Bényi, T. Oh and O. Pocovnicu on the cubic problem. The paper concludes with a condition for almost sure global wellposedness.
</p>projecteuclid.org/euclid.tunis/1544842823_20181214220036Fri, 14 Dec 2018 22:00 ESTGrothendieck–Messing deformation theory for varieties of K3 typehttps://projecteuclid.org/euclid.tunis/1545102020<strong>Andreas Langer</strong>, <strong>Thomas Zink</strong>. <p><strong>Source: </strong>Tunisian Journal of Mathematics, Volume 1, Number 4, 455517.</p><p><strong>Abstract:</strong><br/>
Let [math] be an artinian local ring with perfect residue class field [math] . We associate to certain [math] displays over the small ring of Witt vectors [math] a crystal on [math] .
Let [math] be a scheme of K3 type over [math] . We define a perfect bilinear form on the second crystalline cohomology group [math] which generalizes the Beauville–Bogomolov form for hyperKähler varieties over [math] . We use this form to prove a lifting criterion of Grothendieck–Messing type for schemes of K3 type. The crystalline cohomology [math] is endowed with the structure of a [math] display such that the Beauville–Bogomolov form becomes a bilinear form in the sense of displays. If [math] is ordinary, the infinitesimal deformations of [math] correspond bijectively to infinitesimal deformations of the [math] display of [math] with its Beauville–Bogomolov form. For ordinary K3 surfaces [math] we prove that the slope spectral sequence of the de Rham–Witt complex degenerates and that [math] has a canonical Hodge–Witt decomposition.
</p>projecteuclid.org/euclid.tunis/1545102020_20181217220032Mon, 17 Dec 2018 22:00 ESTPurity of crystalline stratahttps://projecteuclid.org/euclid.tunis/1545102021<strong>Jinghao Li</strong>, <strong>Adrian Vasiu</strong>. <p><strong>Source: </strong>Tunisian Journal of Mathematics, Volume 1, Number 4, 519538.</p><p><strong>Abstract:</strong><br/>
Let [math] be a prime. Let [math] . Let [math] be an [math] crystal over a locally noetherian [math] scheme [math] . Let [math] . We show that the reduced locally closed subscheme of [math] whose points are exactly those [math] such that [math] is a break point of the Newton polygon of the fiber [math] of [math] at [math] is pure in [math] , i.e., it is an affine [math] scheme. This result refines and reobtains previous results of de Jong and Oort, of Vasiu, and of Yang. As an application, we show that for all [math] the reduced locally closed subscheme of [math] whose points are exactly those [math] for which the [math] rank of [math] is [math] is pure in [math] ; the case [math] was previously obtained by Deligne (unpublished) and the general case [math] refines and reobtains a result of Zink.
</p>projecteuclid.org/euclid.tunis/1545102021_20181217220032Mon, 17 Dec 2018 22:00 ESTOn the mod$2$ cohomology of $\operatorname{SL}_3\bigl(\mathbb Z\bigl[\frac{1}{2},i\bigr]\bigr)$https://projecteuclid.org/euclid.tunis/1545102022<strong>HansWerner Henn</strong>. <p><strong>Source: </strong>Tunisian Journal of Mathematics, Volume 1, Number 4, 539560.</p><p><strong>Abstract:</strong><br/>
Let [math] , let [math] be any mod [math] acyclic [math] CW complex on which [math] acts with finite stabilizers and let [math] be the [math] singular locus of [math] . We calculate the mod [math] cohomology of the Borel construction of [math] with respect to the action of [math] . This cohomology coincides with the mod [math] cohomology of [math] in cohomological degrees bigger than [math] and the result is compatible with a conjecture of Quillen which predicts the structure of the cohomology ring [math] .
</p>projecteuclid.org/euclid.tunis/1545102022_20181217220032Mon, 17 Dec 2018 22:00 ESTGeometric origin and some properties of the arctangential heat equationhttps://projecteuclid.org/euclid.tunis/1545102023<strong>Yann Brenier</strong>. <p><strong>Source: </strong>Tunisian Journal of Mathematics, Volume 1, Number 4, 561584.</p><p><strong>Abstract:</strong><br/>
We establish the geometric origin of the nonlinear heat equation with arctangential nonlinearity: [math] by deriving it, together and in duality with the mean curvature flow equation, from the minimal surface equation in Minkowski spacetime, through a suitable quadratic change of time. After examining various properties of the arctangential heat equation (in particular through its optimal transport interpretation à la Otto and its relationship with the Born–Infeld theory of electromagnetism), we briefly discuss its possible use for image processing, once written in nonconservative form and properly discretized.
</p>projecteuclid.org/euclid.tunis/1545102023_20181217220032Mon, 17 Dec 2018 22:00 ESTHorn's problem and Fourier analysishttps://projecteuclid.org/euclid.tunis/1545102024<strong>Jacques Faraut</strong>. <p><strong>Source: </strong>Tunisian Journal of Mathematics, Volume 1, Number 4, 585606.</p><p><strong>Abstract:</strong><br/>
Let [math] and [math] be two [math] Hermitian matrices. Assume that the eigenvalues [math] of [math] are known, as well as the eigenvalues [math] of [math] . What can be said about the eigenvalues of the sum [math] ? This is Horn’s problem. We revisit this question from a probabilistic viewpoint. The set of Hermitian matrices with spectrum [math] is an orbit [math] for the natural action of the unitary group [math] on the space of [math] Hermitian matrices. Assume that the random Hermitian matrix [math] is uniformly distributed on the orbit [math] and, independently, the random Hermitian matrix [math] is uniformly distributed on [math] . We establish a formula for the joint distribution of the eigenvalues of the sum [math] . The proof involves orbital measures with their Fourier transforms, and Heckman’s measures.
</p>projecteuclid.org/euclid.tunis/1545102024_20181217220032Mon, 17 Dec 2018 22:00 ESTPartial resolution by toroidal blowupshttps://projecteuclid.org/euclid.tunis/1551495678<strong>János Kollár</strong>. <p><strong>Source: </strong>Tunisian Journal of Mathematics, Volume 1, Number 1, 312.</p><p><strong>Abstract:</strong><br/>
We give an alternate proof of a theorem of Tevelev about improving a nontoroidal ideal sheaf by a sequence of toroidal blowups.
</p>projecteuclid.org/euclid.tunis/1551495678_20190301220136Fri, 01 Mar 2019 22:01 ESTConstruction of a stable blowup solution with a prescribed behavior for a nonscalinginvariant semilinear heat equationhttps://projecteuclid.org/euclid.tunis/1551495679<strong>Giao Ky Duong</strong>, <strong>Van Tien Nguyen</strong>, <strong>Hatem Zaag</strong>. <p><strong>Source: </strong>Tunisian Journal of Mathematics, Volume 1, Number 1, 1345.</p><p><strong>Abstract:</strong><br/>
We consider the semilinear heat equation
∂
t
u
=
Δ
u
+

u

p
−
1
u
ln
α
(
u
2
+
2
)
in the whole space [math] , where [math] and [math] . Unlike the standard case [math] , this equation is not scaling invariant. We construct for this equation a solution which blows up in finite time [math] only at one blowup point [math] , according to the asymptotic dynamic
u
(
x
,
t
)
∼
ψ
(
t
)
(
1
+
(
p
−
1
)

x
−
a

2
4
p
(
T
−
t
)

ln
(
T
−
t
)

)
−
1
∕
(
p
−
1
)
as
t
→
T
,
where [math] is the unique positive solution of the ODE
ψ
′
=
ψ
p
ln
α
(
ψ
2
+
2
)
,
lim
t
→
T
ψ
(
t
)
=
+
∞
.
The construction relies on the reduction of the problem to a finitedimensional one and a topological argument based on the index theory to get the conclusion. By the interpretation of the parameters of the finitedimensional problem in terms of the blowup time and the blowup point, we show the stability of the constructed solution with respect to perturbations in initial data. To our knowledge, this is the first successful construction for a genuinely nonscaleinvariant PDE of a stable blowup solution with the derivation of the blowup profile. From this point of view, we consider our result as a breakthrough.
</p>projecteuclid.org/euclid.tunis/1551495679_20190301220136Fri, 01 Mar 2019 22:01 ESTTroisième groupe de cohomologie non ramifiée des hypersurfaces de Fanohttps://projecteuclid.org/euclid.tunis/1551495680<strong>JeanLouis ColliotThélène</strong>. <p><strong>Source: </strong>Tunisian Journal of Mathematics, Volume 1, Number 1, 4757.</p><p><strong>Abstract:</strong><br/>
We establish the vanishing of degree three unramified cohomology for several new types of Fano hypersurfaces when the ground field is either finite or algebraically closed of arbitrary characteristic.
</p>projecteuclid.org/euclid.tunis/1551495680_20190301220136Fri, 01 Mar 2019 22:01 ESTOn the ultimate energy bound of solutions to some forced secondorder evolution equations with a general nonlinear damping operatorhttps://projecteuclid.org/euclid.tunis/1551495681<strong>Alain Haraux</strong>. <p><strong>Source: </strong>Tunisian Journal of Mathematics, Volume 1, Number 1, 5972.</p><p><strong>Abstract:</strong><br/>
Under suitable growth and coercivity conditions on the nonlinear damping operator [math] which ensure nonresonance, we estimate the ultimate bound of the energy of the general solution to the equation [math] , [math] , where [math] is a positive selfadjoint operator on a Hilbert space [math] and [math] is a bounded forcing term with values in [math] . In general the bound is of the form [math] , where [math] stands for the [math] norm of [math] with values in [math] and the growth of [math] does not seem to play any role. If [math] behaves like a power for large values of the velocity, the ultimate bound has quadratic growth with respect to [math] and this result is optimal. If [math] is antiperiodic, we obtain a much lower growth bound and again the result is shown to be optimal even for scalar ODEs.
</p>projecteuclid.org/euclid.tunis/1551495681_20190301220136Fri, 01 Mar 2019 22:01 ESTOn the irreducibility of some induced representations of real reductive Lie groupshttps://projecteuclid.org/euclid.tunis/1551495682<strong>Wee Teck Gan</strong>, <strong>Atsushi Ichino</strong>. <p><strong>Source: </strong>Tunisian Journal of Mathematics, Volume 1, Number 1, 73107.</p><p><strong>Abstract:</strong><br/>
We prove the irreducibility of some standard modules of the metaplectic group [math] and some nonstandard modules of the split odd special orthogonal group [math] .
</p>projecteuclid.org/euclid.tunis/1551495682_20190301220136Fri, 01 Mar 2019 22:01 ESTTruncated operads and simplicial spaceshttps://projecteuclid.org/euclid.tunis/1551495683<strong>Michael S. Weiss</strong>. <p><strong>Source: </strong>Tunisian Journal of Mathematics, Volume 1, Number 1, 109126.</p><p><strong>Abstract:</strong><br/>
It was shown by Boavida de Brito and Weiss ( J. Topol. 11 :1 (2018), 65–143) that a wellknown construction which to a (monochromatic, symmetric) topological operad associates a topological category and a functor from it to the category of finite sets is homotopically fully faithful, under mild conditions on the operads. The main result here is a generalization of that statement to [math] truncated topological operads. A [math] truncated operad is a weaker version of operad where all operations have arity [math] .
</p>projecteuclid.org/euclid.tunis/1551495683_20190301220136Fri, 01 Mar 2019 22:01 ESTFrom compressible to incompressible inhomogeneous flows in the case of large datahttps://projecteuclid.org/euclid.tunis/1551495684<strong>Raphaël Danchin</strong>, <strong>Piotr Bogusław Mucha</strong>. <p><strong>Source: </strong>Tunisian Journal of Mathematics, Volume 1, Number 1, 127149.</p><p><strong>Abstract:</strong><br/>
We are concerned with the mathematical derivation of the inhomogeneous incompressible Navier–Stokes equations (INS) from the compressible Navier–Stokes equations (CNS) in the large volume viscosity limit. We first prove a result of largetime existence of regular solutions for (CNS). Next, as a consequence, we establish that the solutions of (CNS) converge to those of (INS) when the volume viscosity tends to infinity. Analysis is performed in the twodimensional torus [math] for general initial data. Compared to prior works, the main breakthrough is that we are able to handle large variations of density.
</p>projecteuclid.org/euclid.tunis/1551495684_20190301220136Fri, 01 Mar 2019 22:01 EST