1 - Oberwolfach Preprints (OWP)
http://publications.mfo.de/handle/mfo/19
The Oberwolfach Preprints (OWP) mainly contain research results related to a longer stay in Oberwolfach. In particular, this concerns the Research in Pairs program and the Oberwolfach Leibniz Fellows, but this can also include an Oberwolfach Lecture, for example.Thu, 16 Jul 2020 00:26:29 GMT2020-07-16T00:26:29ZSpace-Time Euler Discretization Schemes for the Stochastic 2D Navier-Stokes Equations
http://publications.mfo.de/handle/mfo/3744
Space-Time Euler Discretization Schemes for the Stochastic 2D Navier-Stokes Equations
Bessaih, Hakima; Millet, Annie
We prove that the implicit time Euler scheme coupled with finite elements space discretization for the 2D Navier-Stokes equations on the torus subject to a random perturbation converges in $L^2(\Omega)$, and describe the rate of convergence for an $H^1$-valued initial condition. This refines previous results which only established the convergence in probability of these numerical approximations. Using exponential moment estimates of the solution of the stochastic Navier-Stokes equations and convergence of a localized scheme, we can prove strong convergence of this space-time approximation. The speed of the $L^2(\Omega)$-convergence depends on the diffusion coefficient and on the viscosity parameter. In case of Scott-Vogelius mixed elements and for an additive noise, the convergence is polynomial.
Wed, 06 May 2020 00:00:00 GMThttp://publications.mfo.de/handle/mfo/37442020-05-06T00:00:00ZBessaih, HakimaMillet, AnnieWe prove that the implicit time Euler scheme coupled with finite elements space discretization for the 2D Navier-Stokes equations on the torus subject to a random perturbation converges in $L^2(\Omega)$, and describe the rate of convergence for an $H^1$-valued initial condition. This refines previous results which only established the convergence in probability of these numerical approximations. Using exponential moment estimates of the solution of the stochastic Navier-Stokes equations and convergence of a localized scheme, we can prove strong convergence of this space-time approximation. The speed of the $L^2(\Omega)$-convergence depends on the diffusion coefficient and on the viscosity parameter. In case of Scott-Vogelius mixed elements and for an additive noise, the convergence is polynomial.l-Torsion Bounds for the Class Group of Number Fields with an l -Group as Galois Group
http://publications.mfo.de/handle/mfo/3742
l-Torsion Bounds for the Class Group of Number Fields with an l -Group as Galois Group
Klüners, Jürgen; Wang, Jiuya
We describe the relations among the $\ell$-torsion conjecture for $\ell$-extensions, the discriminant multiplicity conjecture for nilpotent extensions and a conjecture of Malle giving an upper bound for the number of nilpotent extensions. We then prove all of these conjectures in these cases.
Mon, 04 May 2020 00:00:00 GMThttp://publications.mfo.de/handle/mfo/37422020-05-04T00:00:00ZKlüners, JürgenWang, JiuyaWe describe the relations among the $\ell$-torsion conjecture for $\ell$-extensions, the discriminant multiplicity conjecture for nilpotent extensions and a conjecture of Malle giving an upper bound for the number of nilpotent extensions. We then prove all of these conjectures in these cases.On Weakly Complete Universal Enveloping Algebras of pro-Lie Algebras
http://publications.mfo.de/handle/mfo/3740
On Weakly Complete Universal Enveloping Algebras of pro-Lie Algebras
Hofmann, Karl Heinrich; Kramer, Linus
Mon, 27 Apr 2020 00:00:00 GMThttp://publications.mfo.de/handle/mfo/37402020-04-27T00:00:00ZHofmann, Karl HeinrichKramer, LinusTheoretical Analysis and Simulation Methods for Hawkes Processes and their Diffusion Approximation
http://publications.mfo.de/handle/mfo/3719
Theoretical Analysis and Simulation Methods for Hawkes Processes and their Diffusion Approximation
Chevallier, Julien; Melnykova, Anna; Tubikanec, Irene
Oscillatory systems of interacting Hawkes processes with Erlang memory kernels were introduced in Ditlevsen and Löcherbach (2017). They are piecewise deterministic Markov processes (PDMP) and can be approximated by a stochastic diffusion. First, a strong error bound between the PDMP and the diffusion is proved. Second, moment bounds for the resulting diffusion are derived. Third, approximation schemes for the diffusion, based on the numerical splitting approach, are proposed. These schemes are proved to converge with meansquare order 1 and to preserve the properties of the diffusion, in particular the hypoellipticity, the ergodicity and the moment bounds. Finally, the PDMP and the diffusion are compared through numerical experiments, where the PDMP is simulated with an adapted thinning procedure.
Mon, 30 Mar 2020 00:00:00 GMThttp://publications.mfo.de/handle/mfo/37192020-03-30T00:00:00ZChevallier, JulienMelnykova, AnnaTubikanec, IreneOscillatory systems of interacting Hawkes processes with Erlang memory kernels were introduced in Ditlevsen and Löcherbach (2017). They are piecewise deterministic Markov processes (PDMP) and can be approximated by a stochastic diffusion. First, a strong error bound between the PDMP and the diffusion is proved. Second, moment bounds for the resulting diffusion are derived. Third, approximation schemes for the diffusion, based on the numerical splitting approach, are proposed. These schemes are proved to converge with meansquare order 1 and to preserve the properties of the diffusion, in particular the hypoellipticity, the ergodicity and the moment bounds. Finally, the PDMP and the diffusion are compared through numerical experiments, where the PDMP is simulated with an adapted thinning procedure.Singularities and Bifurcations of Pseudospherical Surfaces
http://publications.mfo.de/handle/mfo/3710
Singularities and Bifurcations of Pseudospherical Surfaces
Brander, David; Tari, Farid
We study singularities and bifurcations of constant negative curvature surfaces in Euclidean 3-space via their association with Lorentzian harmonic maps. This preprint presents the basic results on this, the full proofs of which will appear in an article under preparation. We show that the generic bifurcations in 1-parameter families of such surfaces are the Cuspidal Butterfly, Cuspidal Lips, Cuspidal Beaks, 2/5 Cuspidal edge and Shcherbak bifurcations.
Tue, 17 Mar 2020 00:00:00 GMThttp://publications.mfo.de/handle/mfo/37102020-03-17T00:00:00ZBrander, DavidTari, FaridWe study singularities and bifurcations of constant negative curvature surfaces in Euclidean 3-space via their association with Lorentzian harmonic maps. This preprint presents the basic results on this, the full proofs of which will appear in an article under preparation. We show that the generic bifurcations in 1-parameter families of such surfaces are the Cuspidal Butterfly, Cuspidal Lips, Cuspidal Beaks, 2/5 Cuspidal edge and Shcherbak bifurcations.Generating Finite Coxeter Groups with Elements of the Same Order
http://publications.mfo.de/handle/mfo/3709
Generating Finite Coxeter Groups with Elements of the Same Order
Hart, Sarah; Kelsey, Veronica; Rowley, Peter
Supposing $G$ is a group and $k$ a natural number, $d_k(G)$ is defined to be the minimal number of elements of $G$ of order $k$ which generate $G$ (setting $d_k(G) = 0$ if $G$ has no such generating sets). This paper investigates $d_k(G)$ when $G$ is a finite Coxeter group either of type $B_n$ or $D_n$ or of exceptional type. Together with Garzoni [3] and Yu [10], this determines $d_k(G)$ for all finite irreducible Coxeter groups $G$ when 2$ \leq k \leq$rank$(G)$ (rank$(G) + 1$ when $G$ is of type $A_n$).
Mon, 16 Mar 2020 00:00:00 GMThttp://publications.mfo.de/handle/mfo/37092020-03-16T00:00:00ZHart, SarahKelsey, VeronicaRowley, PeterSupposing $G$ is a group and $k$ a natural number, $d_k(G)$ is defined to be the minimal number of elements of $G$ of order $k$ which generate $G$ (setting $d_k(G) = 0$ if $G$ has no such generating sets). This paper investigates $d_k(G)$ when $G$ is a finite Coxeter group either of type $B_n$ or $D_n$ or of exceptional type. Together with Garzoni [3] and Yu [10], this determines $d_k(G)$ for all finite irreducible Coxeter groups $G$ when 2$ \leq k \leq$rank$(G)$ (rank$(G) + 1$ when $G$ is of type $A_n$).Nonexistence of Subcritical Solitary Waves
http://publications.mfo.de/handle/mfo/3708
Nonexistence of Subcritical Solitary Waves
Kozlov, Vladimir; Lokharu, Evgeniy; Wheeler, Miles H.
We prove the nonexistence of two-dimensional solitary gravity water waves with subcritical wave speeds and an arbitrary distribution of vorticity. This is a longstanding open problem, and even in the irrotational case there are only partial results relying on sign conditions or smallness assumptions. As a corollary, we obtain a relatively complete classification of solitary waves: they must be supercritical, symmetric, and monotonically decreasing on either side of a central crest. The proof introduces a new function which is related to the so-called flow force and has several surprising properties. In addition to solitary waves, our nonexistence result applies to "half-solitary" waves (e.g. bores) which decay in only one direction.
Sun, 15 Mar 2020 00:00:00 GMThttp://publications.mfo.de/handle/mfo/37082020-03-15T00:00:00ZKozlov, VladimirLokharu, EvgeniyWheeler, Miles H.We prove the nonexistence of two-dimensional solitary gravity water waves with subcritical wave speeds and an arbitrary distribution of vorticity. This is a longstanding open problem, and even in the irrotational case there are only partial results relying on sign conditions or smallness assumptions. As a corollary, we obtain a relatively complete classification of solitary waves: they must be supercritical, symmetric, and monotonically decreasing on either side of a central crest. The proof introduces a new function which is related to the so-called flow force and has several surprising properties. In addition to solitary waves, our nonexistence result applies to "half-solitary" waves (e.g. bores) which decay in only one direction.Rational Functions with Small Value Set
http://publications.mfo.de/handle/mfo/3707
Rational Functions with Small Value Set
Bartoli, Daniele; Borges, Herivelto; Quoos, Luciane
In connection with Galois Theory and Algebraic Curves, this paper investigates rational
functions $h(x) = f(x)/g(x) \in \mathbb{F}_q(x)$ for which the value set $V_h = {\{h(α) | α \in \mathbb{F}_q \cup\{\infty\}}\}$ is relatively small. In particular, under certain circumstances, it proves that $h(x)$ having a small value set is equivalent to the field extension $\mathbb{F}_q(x)/\mathbb{F}_q(h(x))$ being Galois.
Sat, 14 Mar 2020 00:00:00 GMThttp://publications.mfo.de/handle/mfo/37072020-03-14T00:00:00ZBartoli, DanieleBorges, HeriveltoQuoos, LucianeIn connection with Galois Theory and Algebraic Curves, this paper investigates rational
functions $h(x) = f(x)/g(x) \in \mathbb{F}_q(x)$ for which the value set $V_h = {\{h(α) | α \in \mathbb{F}_q \cup\{\infty\}}\}$ is relatively small. In particular, under certain circumstances, it proves that $h(x)$ having a small value set is equivalent to the field extension $\mathbb{F}_q(x)/\mathbb{F}_q(h(x))$ being Galois.Multivariate Hybrid Orthogonal Functions
http://publications.mfo.de/handle/mfo/3706
Multivariate Hybrid Orthogonal Functions
Bracciali, Cleonice F.; Pérez, Teresa E.
We consider multivariate orthogonal functions satisfying hybrid orthogonality conditions
with respect to a moment functional. This kind of orthogonality means that the product of
functions of different parity order is computed by means of the moment functional, and the product of elements of the same parity order is computed by a modification of the original moment functional.
Results about existence conditions, three term relations with matrix coefficients, a Favard type theorem for this kind of hybrid orthogonal functions are proved. In addition, a method to construct bivariate hybrid
orthogonal functions from univariate orthogonal polynomials and univariate orthogonal
functions is presented. Finally, we give a complete description of a sequence of hybrid orthogonal
functions on the unit disk on $\mathbb{R}^2$, that includes, as particular case, the classical orthogonal polynomials on the disk.
Thu, 12 Mar 2020 00:00:00 GMThttp://publications.mfo.de/handle/mfo/37062020-03-12T00:00:00ZBracciali, Cleonice F.Pérez, Teresa E.We consider multivariate orthogonal functions satisfying hybrid orthogonality conditions
with respect to a moment functional. This kind of orthogonality means that the product of
functions of different parity order is computed by means of the moment functional, and the product of elements of the same parity order is computed by a modification of the original moment functional.
Results about existence conditions, three term relations with matrix coefficients, a Favard type theorem for this kind of hybrid orthogonal functions are proved. In addition, a method to construct bivariate hybrid
orthogonal functions from univariate orthogonal polynomials and univariate orthogonal
functions is presented. Finally, we give a complete description of a sequence of hybrid orthogonal
functions on the unit disk on $\mathbb{R}^2$, that includes, as particular case, the classical orthogonal polynomials on the disk.Splitting Necklaces, with Constraints
http://publications.mfo.de/handle/mfo/3698
Splitting Necklaces, with Constraints
Jojic, Dusko; Panina, Gaiane; Zivaljevic, Rade
We prove several versions of Alon's "necklace-splitting theorem", subject to additional constraints, as illustrated by the following results.
(1) The "almost equicardinal necklace-splitting theorem" claims that, without increasing the number of cuts, one guarantees the existence of a fair splitting such that each thief is allocated (approximately) one and the same number of pieces of the necklace, provided the number of thieves $r=p^\nu$ is a prime power.
(2) The "binary splitting theorem" claims that if $r=2^d$ and the thieves are associated with the vertices of a d-cube then, without increasing the number of cuts, one can guarantee the existence of a fair splitting such that
adjacent pieces are allocated to thieves that share an edge of the cube. This result provides a positive answer to the "binary splitting necklace conjecture" of Asada at al. (Conjecture 2.11 in [5]) in the case $r=2^d$.
Tue, 11 Feb 2020 00:00:00 GMThttp://publications.mfo.de/handle/mfo/36982020-02-11T00:00:00ZJojic, DuskoPanina, GaianeZivaljevic, RadeWe prove several versions of Alon's "necklace-splitting theorem", subject to additional constraints, as illustrated by the following results.
(1) The "almost equicardinal necklace-splitting theorem" claims that, without increasing the number of cuts, one guarantees the existence of a fair splitting such that each thief is allocated (approximately) one and the same number of pieces of the necklace, provided the number of thieves $r=p^\nu$ is a prime power.
(2) The "binary splitting theorem" claims that if $r=2^d$ and the thieves are associated with the vertices of a d-cube then, without increasing the number of cuts, one can guarantee the existence of a fair splitting such that
adjacent pieces are allocated to thieves that share an edge of the cube. This result provides a positive answer to the "binary splitting necklace conjecture" of Asada at al. (Conjecture 2.11 in [5]) in the case $r=2^d$.