Tunisian Journal of Mathematics Articles (Project Euclid)
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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
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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 ESTLooijenga line bundles in complex analytic elliptic cohomologyhttps://projecteuclid.org/euclid.tunis/1554170461<strong>Charles Rezk</strong>. <p><strong>Source: </strong>Tunisian Journal of Mathematics, Volume 2, Number 1, 142.</p><p><strong>Abstract:</strong><br/>
We present a calculation that shows how the moduli of complex analytic elliptic curves arises naturally from the Borel cohomology of an extended moduli space of [math] bundles on a torus. Furthermore, we show how the analogous calculation, applied to a moduli space of principal bundles for a [math] central extension of [math] , gives rise to Looijenga line bundles. We then speculate on the relation of these calculations to the construction of complex analytic equivariant elliptic cohomology.
</p>projecteuclid.org/euclid.tunis/1554170461_20190401220107Mon, 01 Apr 2019 22:01 EDTFronts d'onde des représentations tempérées et de réduction unipotente pour $\mathrm{SO}(2n+1)$https://projecteuclid.org/euclid.tunis/1554170462<strong>JeanLoup Waldspurger</strong>. <p><strong>Source: </strong>Tunisian Journal of Mathematics, Volume 2, Number 1, 4395.</p><p><strong>Abstract:</strong><br/>
Let [math] be a special orthogonal group [math] defined over a [math] adic field [math] . Let [math] be an admissible irreducible representation of [math] which is tempered and of unipotent reduction. We prove that [math] has a wave front set. In some particular cases, we give a method to compute this wave front set.
</p>projecteuclid.org/euclid.tunis/1554170462_20190401220107Mon, 01 Apr 2019 22:01 EDTSpectral Mackey functors and equivariant algebraic $K$theory, IIhttps://projecteuclid.org/euclid.tunis/1554170463<strong>Clark Barwick</strong>, <strong>Saul Glasman</strong>, <strong>Jay Shah</strong>. <p><strong>Source: </strong>Tunisian Journal of Mathematics, Volume 2, Number 1, 97146.</p><p><strong>Abstract:</strong><br/>
We study the “higher algebra” of spectral Mackey functors, which the first named author introduced in Part I of this paper. In particular, armed with our new theory of symmetric promonoidal [math] categories and a suitable generalization of the second named author’s Day convolution, we endow the [math] category of Mackey functors with a wellbehaved symmetric monoidal structure. This makes it possible to speak of spectral Green functors for any operad [math] . We also answer a question of Mathew, proving that the algebraic [math] theory of group actions is lax symmetric monoidal. We also show that the algebraic [math] theory of derived stacks provides an example. Finally, we give a very short, new proof of the equivariant Barratt–Priddy–Quillen theorem, which states that the algebraic [math] theory of the category of finite [math] sets is simply the [math] equivariant sphere spectrum.
</p>projecteuclid.org/euclid.tunis/1554170463_20190401220107Mon, 01 Apr 2019 22:01 EDTTwisted Calabi–Yau ring spectra, string topology, and gauge symmetryhttps://projecteuclid.org/euclid.tunis/1554170464<strong>Ralph L. Cohen</strong>, <strong>Inbar Klang</strong>. <p><strong>Source: </strong>Tunisian Journal of Mathematics, Volume 2, Number 1, 147196.</p><p><strong>Abstract:</strong><br/>
In this paper we import the theory of “Calabi–Yau” algebras and categories from symplectic topology and topological field theories, to the setting of spectra in stable homotopy theory. Twistings in this theory will be particularly important. There will be two types of Calabi–Yau structures in the setting of ring spectra: one that applies to compact algebras and one that applies to smooth algebras. The main application of twisted compact Calabi–Yau ring spectra that we will study is to describe, prove, and explain a certain duality phenomenon in string topology. This is a duality between the manifold string topology of Chas and Sullivan (1999) and the Lie group string topology of Chataur and Menichi (2012). This will extend and generalize work of Gruher (2007). Then, generalizing work of Cohen and Jones (2017), we show how the gauge group of the principal bundle acts on this compact Calabi–Yau structure, and we compute some explicit examples. We then extend the notion of the Calabi–Yau structure to smooth ring spectra, and prove that Thom ring spectra of (virtual) bundles over the loop space, [math] , have this structure. In the case when [math] is a sphere, we will use these twisted smooth Calabi–Yau ring spectra to study Lagrangian immersions of the sphere into its cotangent bundle. We recast the work of Abouzaid and Kragh (2016) to show that the topological Hochschild homology of the Thom ring spectrum induced by the [math] principle classifying map of the Lagrangian immersion detects whether that immersion can be Lagrangian isotopic to an embedding. We then compute some examples. Finally, we interpret these Calabi–Yau structures directly in terms of topological Hochschild homology and cohomology.
</p>projecteuclid.org/euclid.tunis/1554170464_20190401220107Mon, 01 Apr 2019 22:01 EDTSemiclassical approximation of the magnetic Schrödinger operator on a strip: dynamics and spectrumhttps://projecteuclid.org/euclid.tunis/1554170465<strong>Mouez Dimassi</strong>. <p><strong>Source: </strong>Tunisian Journal of Mathematics, Volume 2, Number 1, 197215.</p><p><strong>Abstract:</strong><br/>
In the semiclassical regime (i.e., [math] ), we study the effect of a slowly varying potential [math] on the magnetic Schrödinger operator [math] on a strip [math] . The potential [math] is assumed to be smooth. We derive the semiclassical dynamics and we describe the asymptotic structure of the spectrum and the resonances of the operator [math] for [math] small enough. All our results depend on the eigenvalues corresponding to [math] on [math] with Dirichlet boundary condition.
</p>projecteuclid.org/euclid.tunis/1554170465_20190401220107Mon, 01 Apr 2019 22:01 EDTDuality relations among multiple series with three parametershttps://projecteuclid.org/euclid.tunis/1554170466<strong>Masahiro Igarashi</strong>. <p><strong>Source: </strong>Tunisian Journal of Mathematics, Volume 2, Number 1, 217236.</p><p><strong>Abstract:</strong><br/>
We prove a duality relation among multiple series with three parameters. As special case of it, we obtain some new identities for multiple Hurwitz zeta values, which are relations among extensions of the multiple Hurwitz zeta value.
</p>projecteuclid.org/euclid.tunis/1554170466_20190401220107Mon, 01 Apr 2019 22:01 EDT$G$symmetric monoidal categories of modules over equivariant commutative ring spectrahttps://projecteuclid.org/euclid.tunis/1565661715<strong>Andrew J. Blumberg</strong>, <strong>Michael A. Hill</strong>. <p><strong>Source: </strong>Tunisian Journal of Mathematics, Volume 2, Number 2, 237286.</p><p><strong>Abstract:</strong><br/>
We describe the multiplicative structures that arise on categories of equivariant modules over certain equivariant commutative ring spectra. Building on our previous work on [math] ring spectra, we construct categories of equivariant operadic modules over [math] rings that are structured by equivariant linear isometries operads. These categories of modules are endowed with equivariant symmetric monoidal structures, which amounts to the structure of an “incomplete Mackey functor in homotopical categories”. In particular, we construct internal norms which satisfy the double coset formula. One application of the work of this paper is to provide a context in which to describe the behavior of Bousfield localization of equivariant commutative rings. We regard the work of this paper as a first step towards equivariant derived algebraic geometry.
</p>projecteuclid.org/euclid.tunis/1565661715_20190812220215Mon, 12 Aug 2019 22:02 EDTStatistics of $K$groups modulo $p$ for the ring of integers of a varying quadratic number fieldhttps://projecteuclid.org/euclid.tunis/1565661717<strong>Bruce W. Jordan</strong>, <strong>Zev Klagsbrun</strong>, <strong>Bjorn Poonen</strong>, <strong>Christopher Skinner</strong>, <strong>Yevgeny Zaytman</strong>. <p><strong>Source: </strong>Tunisian Journal of Mathematics, Volume 2, Number 2, 287307.</p><p><strong>Abstract:</strong><br/>
For each odd prime [math] , we conjecture the distribution of the [math] torsion subgroup of [math] as [math] ranges over real quadratic fields, or over imaginary quadratic fields. We then prove that the average size of the [math] torsion subgroup of [math] is as predicted by this conjecture.
</p>projecteuclid.org/euclid.tunis/1565661717_20190812220215Mon, 12 Aug 2019 22:02 EDTOn $p$adic vanishing cycles of log smooth familieshttps://projecteuclid.org/euclid.tunis/1565661718<strong>Shuji Saito</strong>, <strong>Kanetomo Sato</strong>. <p><strong>Source: </strong>Tunisian Journal of Mathematics, Volume 2, Number 2, 309335.</p><p><strong>Abstract:</strong><br/>
In this paper, we will prove that the sheaf of [math] adic vanishing cycles on a regular log smooth family is generated by Milnor symbols, assuming that the base dvr contains a primitive [math] th root of unity. Our result generalizes the surjectivity results of Bloch and Kato ( Inst. Hautes É tudes Sci. Publ. Math. 63 (1986), 107–152) and Hyodo ( Invent. Math. 91 :3 (1988), 543–557) to a regular log smooth case.
</p>projecteuclid.org/euclid.tunis/1565661718_20190812220215Mon, 12 Aug 2019 22:02 EDTTame multiplicity and conductor for local Galois representationshttps://projecteuclid.org/euclid.tunis/1565661719<strong>Colin J. Bushnell</strong>, <strong>Guy Henniart</strong>. <p><strong>Source: </strong>Tunisian Journal of Mathematics, Volume 2, Number 2, 337357.</p><p><strong>Abstract:</strong><br/>
Let [math] be a nonArchimedean locally compact field of residual characteristic [math] . Let [math] be an irreducible smooth representation of the absolute Weil group [math] of [math] and [math] the Swan exponent of [math] . Assume [math] . Let [math] be the inertia subgroup of [math] and [math] the wild inertia subgroup. There is an essentially unique, finite, cyclic group [math] , of order prime to [math] , such that [math] . In response to a query of Mark Reeder, we show that the multiplicity in [math] of any character of [math] is bounded by [math] .
</p>projecteuclid.org/euclid.tunis/1565661719_20190812220215Mon, 12 Aug 2019 22:02 EDTNilpotence theorems via homological residue fieldshttps://projecteuclid.org/euclid.tunis/1565661721<strong>Paul Balmer</strong>. <p><strong>Source: </strong>Tunisian Journal of Mathematics, Volume 2, Number 2, 359378.</p><p><strong>Abstract:</strong><br/>
We prove nilpotence theorems in tensortriangulated categories using suitable Gabriel quotients of the module category, and discuss examples.
</p>projecteuclid.org/euclid.tunis/1565661721_20190812220215Mon, 12 Aug 2019 22:02 EDTFinitedimensional reduction of a supercritical exponent equationhttps://projecteuclid.org/euclid.tunis/1565661722<strong>Mohamed Ben Ayed</strong>. <p><strong>Source: </strong>Tunisian Journal of Mathematics, Volume 2, Number 2, 379397.</p><p><strong>Abstract:</strong><br/>
We present a finitedimensional reduction for a supercritical exponent PDE. We reduce the existence of a solution of the problem
−
Δ
u
=
K

u

4
∕
(
n
−
2
)
+
ε
u
in
Ω
(with
ε
>
0
)
,
u
=
0
on
∂
Ω
,
to finding a critical point of a function defined in some set [math] .
</p>projecteuclid.org/euclid.tunis/1565661722_20190812220215Mon, 12 Aug 2019 22:02 EDTPotentially good reduction loci of Shimura varietieshttps://projecteuclid.org/euclid.tunis/1565661723<strong>Naoki Imai</strong>, <strong>Yoichi Mieda</strong>. <p><strong>Source: </strong>Tunisian Journal of Mathematics, Volume 2, Number 2, 399454.</p><p><strong>Abstract:</strong><br/>
We give a notion of the potentially good reduction locus of a Shimura variety. It consists of the points which should be related with motives having potentially good reductions in some sense. We show the existence of such locus for a Shimura variety of preabelian type. Further, we construct a partition of the adic space associated to a Shimura variety of preabelian type, which is expected to describe degenerations of motives. Using this partition, we prove that the cohomology of the potentially good reduction locus is isomorphic to the cohomology of a Shimura variety up to nonsupercuspidal parts.
</p>projecteuclid.org/euclid.tunis/1565661723_20190812220215Mon, 12 Aug 2019 22:02 EDTMonodromy and log geometryhttps://projecteuclid.org/euclid.tunis/1576206288<strong>Piotr Achinger</strong>, <strong>Arthur Ogus</strong>. <p><strong>Source: </strong>Tunisian Journal of Mathematics, Volume 2, Number 3, 455534.</p><p><strong>Abstract:</strong><br/>
A now classical construction due to Kato and Nakayama attaches a topological space (the “Betti realization”) to a log scheme over [math] . We show that in the case of a log smooth degeneration over the standard log disc, this construction allows one to recover the topology of the germ of the family from the log special fiber alone. We go on to give combinatorial formulas for the monodromy and the [math] differentials acting on the nearby cycle complex in terms of the log structures. We also provide variants of these results for the Kummer étale topology. In the case of curves, these data are essentially equivalent to those encoded by the dual graph of a semistable degeneration, including the monodromy pairing and the Picard–Lefschetz formula.
</p>projecteuclid.org/euclid.tunis/1576206288_20191212220459Thu, 12 Dec 2019 22:04 ESTThe Markov sequence problem for the Jacobi polynomials and on the simplexhttps://projecteuclid.org/euclid.tunis/1576206289<strong>Dominique Bakry</strong>, <strong>Lamine Mbarki</strong>. <p><strong>Source: </strong>Tunisian Journal of Mathematics, Volume 2, Number 3, 535566.</p><p><strong>Abstract:</strong><br/>
The Markov sequence problem aims at the description of possible eigenvalues of symmetric Markov operators with some given orthonormal basis as eigenvector decomposition. A fundamental tool for their description is the hypergroup property. We first present the general Markov sequence problem and provide the classical examples, most of them associated with the classical families of orthogonal polynomials. We then concentrate on the hypergroup property, and provide a general method to obtain it, inspired by a fundamental work of Carlen, Geronimo and Loss. Using this technique and a few properties of diffusion operators having polynomial eigenvectors, we then provide a simplified proof of the hypergroup property for the Jacobi polynomials (Gasper’s theorem) on the unit interval. We finally investigate various generalizations of this property for the family of Dirichlet laws on the simplex.
</p>projecteuclid.org/euclid.tunis/1576206289_20191212220459Thu, 12 Dec 2019 22:04 ESTThe cohomology of $C_2$equivariant $\mathcal{A}(1)$ and the homotopy of $ko_{C_2}$https://projecteuclid.org/euclid.tunis/1576206290<strong>Bertrand J. Guillou</strong>, <strong>Michael A. Hill</strong>, <strong>Daniel C. Isaksen</strong>, <strong>Douglas Conner Ravenel</strong>. <p><strong>Source: </strong>Tunisian Journal of Mathematics, Volume 2, Number 3, 567632.</p><p><strong>Abstract:</strong><br/>
We compute the cohomology of the subalgebra [math] of the [math] equivariant Steenrod algebra [math] . This serves as the input to the [math] equivariant Adams spectral sequence converging to the completed [math] graded homotopy groups of an equivariant spectrum [math] . Our approach is to use simpler [math] motivic and [math] motivic calculations as stepping stones.
</p>projecteuclid.org/euclid.tunis/1576206290_20191212220459Thu, 12 Dec 2019 22:04 ESTDegeneracy loci, virtual cycles and nested Hilbert schemes, Ihttps://projecteuclid.org/euclid.tunis/1576206291<strong>Amin Gholampour</strong>, <strong>Richard P. Thomas</strong>. <p><strong>Source: </strong>Tunisian Journal of Mathematics, Volume 2, Number 3, 633665.</p><p><strong>Abstract:</strong><br/>
Given a map of vector bundles on a smooth variety, consider the deepest degeneracy locus where its rank is smallest. We show it carries a natural perfect obstruction theory whose virtual cycle can be calculated by the Thom–Porteous formula.
We show nested Hilbert schemes of points on surfaces can be expressed as degeneracy loci. We show how to modify the resulting obstruction theories to recover the virtual cycles of Vafa–Witten and reduced local DT theories. The result computes some Vafa–Witten invariants in terms of Carlsson–Okounkov operators. This proves and extends a conjecture of Gholampour, Sheshmani, and Yau and generalises a vanishing result of Carlsson and Okounkov.
</p>projecteuclid.org/euclid.tunis/1576206291_20191212220459Thu, 12 Dec 2019 22:04 ESTAlmost ${\mathbb C}_p$ Galois representations and vector bundleshttps://projecteuclid.org/euclid.tunis/1576206292<strong>JeanMarc Fontaine</strong>. <p><strong>Source: </strong>Tunisian Journal of Mathematics, Volume 2, Number 3, 667732.</p><p><strong>Abstract:</strong><br/>
Let [math] be a finite extension of [math] and [math] the absolute Galois group. Then [math] acts on the fundamental curve [math] of [math] adic Hodge theory and we may consider the abelian category [math] of coherent [math] modules equipped with a continuous and semilinear action of [math] .
An almost [math] representation of [math] is a [math] adic Banach space [math] equipped with a linear and continuous action of [math] such that there exists [math] , two [math] stable finite dimensional sub [math] vector spaces [math] of [math] , [math] of [math] , and a [math] equivariant isomorphism
V
∕
U
+
→
ℂ
p
d
∕
U
−
.
These representations form an abelian category [math] . The main purpose of this paper is to prove that [math] can be recovered from [math] by a simple construction (and viceversa) inducing, in particular, an equivalence of triangulated categories
D
b
(
ℳ
(
G
K
)
)
→
D
b
(
C
(
G
K
)
)
.
</p>projecteuclid.org/euclid.tunis/1576206292_20191212220459Thu, 12 Dec 2019 22:04 ESTOn log motiveshttps://projecteuclid.org/euclid.tunis/1584669649<strong>Tetsushi Ito</strong>, <strong>Kazuya Kato</strong>, <strong>Chikara Nakayama</strong>, <strong>Sampei Usui</strong>. <p><strong>Source: </strong>Tunisian Journal of Mathematics, Volume 2, Number 4, 733789.</p><p><strong>Abstract:</strong><br/>
We define the categories of log motives and log mixed motives. The latter gives a new formulation for the category of mixed motives. We prove that the former is a semisimple abelian category if and only if the numerical equivalence and homological equivalence coincide, and that it is also equivalent to the latter being a Tannakian category. We discuss various realizations, formulate Tate and Hodge conjectures, and verify them in the curve case.
</p>projecteuclid.org/euclid.tunis/1584669649_20200319220051Thu, 19 Mar 2020 22:00 EDTEquidistribution and counting of orbit points for discrete rank one isometry groups of Hadamard spaceshttps://projecteuclid.org/euclid.tunis/1584669650<strong>Gabriele Link</strong>. <p><strong>Source: </strong>Tunisian Journal of Mathematics, Volume 2, Number 4, 791839.</p><p><strong>Abstract:</strong><br/>
Let [math] be a proper, geodesically complete Hadamard space, and [math] a discrete subgroup of isometries of [math] with the fixed point of a rank one isometry of [math] in its infinite limit set. In this paper we prove that if [math] has nonarithmetic length spectrum, then the Ricks–Bowen–Margulis measure — which generalizes the wellknown Bowen–Margulis measure in the CAT [math] setting — is mixing. If in addition the Ricks–Bowen–Margulis measure is finite, then we also have equidistribution of [math] orbit points in [math] , which in particular yields an asymptotic estimate for the orbit counting function of [math] . This generalizes wellknown facts for nonelementary discrete isometry groups of Hadamard manifolds with pinched negative curvature and proper CAT [math] spaces.
</p>projecteuclid.org/euclid.tunis/1584669650_20200319220051Thu, 19 Mar 2020 22:00 EDTA generalization of a powerconjugacy problem in torsionfree negatively curved groupshttps://projecteuclid.org/euclid.tunis/1584669651<strong>Rita Gitik</strong>. <p><strong>Source: </strong>Tunisian Journal of Mathematics, Volume 2, Number 4, 841849.</p><p><strong>Abstract:</strong><br/>
Let [math] and [math] be quasiconvex subgroups of a negatively curved torsionfree group [math] . We give an algorithm which decides whether an element of [math] is conjugate in [math] to an element of [math] .
</p>projecteuclid.org/euclid.tunis/1584669651_20200319220051Thu, 19 Mar 2020 22:00 EDTA simple proof of the Hardy inequality on Carnot groups and for some hypoelliptic families of vector fieldshttps://projecteuclid.org/euclid.tunis/1584669652<strong>François Vigneron</strong>. <p><strong>Source: </strong>Tunisian Journal of Mathematics, Volume 2, Number 4, 851880.</p><p><strong>Abstract:</strong><br/>
We give an elementary proof of the classical Hardy inequality on any Carnot group, using only integration by parts and a fine analysis of the commutator structure, which was not deemed possible until now. We also discuss the conditions under which this technique can be generalized to deal with hypoelliptic families of vector fields, which, in this case, leads to an open problem regarding the symbol properties of the gauge norm.
</p>projecteuclid.org/euclid.tunis/1584669652_20200319220051Thu, 19 Mar 2020 22:00 EDTTrigonometric series with a given spectrumhttps://projecteuclid.org/euclid.tunis/1584669653<strong>Yves Meyer</strong>. <p><strong>Source: </strong>Tunisian Journal of Mathematics, Volume 2, Number 4, 881906.</p><p><strong>Abstract:</strong><br/>
Let [math] be a closed and discrete set. The vector space consisting of all trigonometric sums whose frequencies belong to [math] is denoted by [math] . Given an exponent [math] we say that [math] is [math] coherent if there exist a compact set [math] and a continuous function [math] defined on [math] with values in [math] such that for every [math] and every [math] one has [math] . Several properties of [math] coherent sets are proved in this essay.
</p>projecteuclid.org/euclid.tunis/1584669653_20200319220051Thu, 19 Mar 2020 22:00 EDT